We provide the existence of a solution for quasilinear elliptic equation -div(a∞(x)|∇u|p-2∇u+ã(x,|∇u|)∇u)=λm(x)|u|p-2u+f(x,u)+h(x) in Ω under the Neumann boundary condition. Here, we consider the condition that ã(x,t)=o(tp-2) as t→+∞ and f(x,u)=o(|u|p-1) as |u|→∞. As a special case, our result implies that the following p-Laplace equation has at least one solution: -Δpu=λm(x)|u|p-2u+μ|u|r-2u+h(x) in Ω,∂u/∂ν=0 on ∂Ω for every 1<r<p<∞, λ∈ℝ, μ≠0 and m,h∈L∞(Ω) with ∫Ωmdx≠0. Moreover, in the nonresonant case, that is, λ is not an eigenvalue of the p-Laplacian with weight m, we present the existence of a solution of the above p-Laplace equation for every 1<r<p<∞, μ∈ℝ and m,h∈L∞(Ω).
1. Introduction
In this paper, we consider the existence of a solution for the following quasilinear elliptic equation:
(P;λ,m,h)-divA(x,∇u)=λm(x)|u|p-2u+f(x,u)+h(x)inΩ,∂u∂ν=0on∂Ω,
where Ω⊂ℝN is a bounded domain with C2 boundary ∂Ω, ν denotes the outward unit normal vector on ∂Ω, λ∈ℝ, 1<p<∞ and m,h∈L∞(Ω). We assume that f is a Carathéodory function on Ω×ℝ satisfying
(1.1)lim|t|→∞f(x,t)|t|p-2t=0uniformlyinx∈Ω,
and that f(x,t) is bounded on a bounded set (admitting f≡0 in the nonresonant case). Here, A:Ω-×ℝN→ℝN is a map which is strictly monotone in the second variable and satisfies certain regularity conditions (see the following assumption (A)). The equation (P;λ,m,h) contains the corresponding p-Laplacian problem as a special case. Although the operator A is nonhomogeneous in the second variable in general, we assume that A(x,y) is asymptotically (p-1)-homogeneous at infinity in the following sense (AH).
Throughout this paper, we assume that the map A satisfies the following assumptions (AH) and (A):
there exist a positive function a∞∈C1(Ω-,ℝ) and a continuous function a~(x,t) on Ω-×ℝ such that
(1.2)A(x,y)=a∞(x)|y|p-2y+a~(x,|y|)yforeveryx∈Ω,y∈RN,limt→+∞a~(x,t)tp-2=0uniformlyinx∈Ω-.
A(x,y)=a(x,|y|)y, where a(x,t)>0 for all (x,t)∈Ω-×(0,+∞) and
A∈C0(Ω-×ℝN,ℝN)∩C1(Ω-×(ℝN∖{0}),ℝN);
there exists C1>0 such that
(1.3)|DyA(x,y)|≤C1|y|p-2foreveryx∈Ω-,y∈RN∖{0};
there exists C0>0 such that
(1.4)DyA(x,y)ξ⋅ξ≥C0|y|p-2|ξ|2foreveryx∈Ω-,y∈RN∖{0},ξ∈RN;
there exists C2>0 such that
(1.5)|DxA(x,y)|≤C2(1+|y|p-1)foreveryx∈Ω-,y∈RN∖{0}.
A similar hypothesis to (A) is considered in the study of quasilinear elliptic problems (cf. [1, Example 2.2], [2–6]). It is easily seen that many examples as in the above references satisfy the condition (AH). Also, the following example satisfies our hypotheses:
(1.6)div((|∇u|p-2+|∇u|q-2)(1+|∇u|q)(p-q)/q∇u)for1<p≤q<∞.
In particular, for A(x,y)=|y|p-2y, that is, divA(x,∇u) stands for the usual p-Laplacian Δpu, we can take C0=C1=p-1 in (A). Conversely, in the case where C0=C1=p-1 holds in (A), by the inequalities in Remark 1.4 (ii) and (iii), we see a(x,t)=|t|p-2 whence A(x,y)=|y|p-2y.
Concerning the weight m, throughout this paper, we assume that
(1.7)|{m>0}|:=|{x∈Ω;m(x)>0}|>0
holds, where |X| denotes the Lebesgue measure of a measurable set X.
Because A(x,y) is asymptotically (p-1)-homogeneous at infinity, the solvability of our equation is related to the following homogeneous equation (see Theorem 1.1):
(EV;m)-div(a∞(x)|∇u|p-2∇u)=λm(x)|u|p-2uinΩ,∂u∂ν=0on∂Ω,
where a∞ is the positive function as in (AH). We say that λ∈ℝ is an eigenvalue of (EV;m) if the equation (EV;m) has a nontrivial solution.
There are few existence results of a solution to our equation (and also the p-Laplace equation). For example, if λ<0 and m≡1 hold, then the standard argument guarantees the existence of a solution. For the p-Laplacian as a special case of our problem, it is shown in [7] that the equation
(1.8)-Δpu=λm|u|p-2u+hinΩ∂u∂ν=0on∂Ω
has a unique positive solution provided 0<λ<λ*(m), ∫Ωmdx<0 and 0≢h∈L∞(Ω)+, where λ*(m) is the principal eigenvalue defined in Section 2.1 with a∞≡1. In [8], although the resonant case where λ=λ1(m) or λ=λ2(m) is considered under the assumptions to f(x,u)=f(u), its result does not cover the case of f(u)=|u|r-2u with 1<r<p, where λi(m) (i=1, 2) is ith eigenvalue of the p-Laplacian with weight m. For the Laplace problem under the Neumann boundary condition, we can refer to [9, 10]. Under the Dirichlet boundary condition, the existence results for the Laplace problem are well known when m≡1 and λ is not an eigenvalue of the Laplacian (cf. [11]). Moreover, under the Dirichlet (or blow-up) boundary condition, many authors study various equations involving the p-Laplace (Laplace) operator with (indefinite) weight. For example, we refer to [12] for boundary blow-up problems with Laplacian, [13] for periodic reaction-diffusion problems and [14, 15] for singular quasilinear elliptic problems.
Recently, the present author shows the existence of a solution for our problem in the case where λ is between the principal eigenvalue and the second eigenvalue in [6] (for f≡0). In addition, a similar situation is treated in [5]. However, existence results are not seen in the case when λ is greater than the second eigenvalue for our problem. Therefore, the first purpose of this paper is to present an existence result of a solution in the nonresonant case where λ is not an eigenvalue of (EV;m). Then, it studied the existence of at least one solution in the resonant case under assumptions that cover the case f(u)=μ|u|r-2u with 1<r<p and μ≠0.
For the proof of our result, it is necessary to study the weighted eigenvalue problem (EV;m). Thus, in Section 2, we introduce two sequences {λn(m)}n and {μn(m)}n of an eigenvalue of (EV;m) defined by Ljusternik-Schnirelman theory or Drábek-Robinson's method (cf. [16]), respectively. Then, we show several properties of above eigenvalues. In Section 3, we give the proof in the nonresonant case by using {μn(m)}n. In Sections 4 and 5, we handle the resonant case.
1.1. Statements of Our Existence Results
First, we state the existence result of a solution in the nonresonant case.
Theorem 1.1.
Assume that λ∈ℝ is not an eigenvalue of (EV;m). Then, (P;λ,m,h) has at least one solution.
To state our existence result in the resonant case, we introduce some conditions. Set
(1.9)F(x,u):=∫0uf(x,s)ds,G~(x,y):=∫0|y|a~(x,t)tdt,
where a~ is the function as in (AH).
there exist 0≤q≤p-1 and H0>0 such that
(1.10)lim|y|→∞pG~(x,y)-a~(x,|y|)|y|2|y|1+q=+∞uniformlyina.e.x∈Ω,f(x,t)t-pF(x,t)≥-H0(1+|t|1+q)fora.e.x∈Ω,everyt∈R;
there exist 0≤q≤p-1 and H0>0 such that
(1.11)lim|y|→∞pG~(x,y)-a~(x,|y|)|y|2|y|1+q=-∞uniformlyina.e.x∈Ω,f(x,t)t-pF(x,t)≤H0(|t|1+q+1)fora.e.x∈Ω,everyt∈R;
there exist 0≤q≤p-1 and H0>0 such that
(1.12)pG~(x,y)-a~(x,|y|)|y|2≥-H0(1+|y|1+q)foreveryx∈Ω,y∈RN,lim|t|→∞f(x,t)t-pF(x,t)|t|1+q=+∞uniformlyina.e.x∈Ω;
there exist 0≤q≤p-1 and H0>0 such that
(1.13)pG~(x,y)-a~(x,|y|)|y|2≤H0(1+|y|1+q)foreveryx∈Ω,y∈RN,lim|t|→∞f(x,t)t-pF(x,t)|t|1+q=-∞uniformlyina.e.x∈Ω.
Theorem 1.2.
Assume one of the following conditions:
λ=0 and (HF+) or (HF-) hold;
λ≠0, ∫Ωmdx≠0 and one of (H+), (H-), (HF+) and (HF-) hold;
λ≠0, ∫Ωmdx=0 and (H+) or (HF+) hold;
Then, (P;λ,m,h) has at least one solution.
In the special case where a~(x,t)≡0 and f(x,u)=μ|u|r-2u for 1<r<p, we easily see that (HF+) or (HF-) holds with 0≤q<r-1 provided μ<0 or μ>0, respectively. Therefore, the following result is proved according to Theorem 1.2.
Corollary 1.3.
Let 1<r<p<∞, μ≠0 and ∫Ωmdx≠0. Then, the following equation has at least one solution:
(1.14)-div(a∞(x)|∇u|p-2∇u)=λm(x)|u|p-2u+μ|u|r-2u+h(x)
in
Ω,∂u∂ν=0on∂Ω.
1.2. Properties of the Map A
In what follows, the norm on W1,p(Ω) is given by ∥u∥p:=∥∇u∥pp+∥u∥pp, where ∥u∥q denotes the norm of Lq(Ω) for u∈Lq(Ω) (1≤q≤∞). Setting G(x,y):=∫0|y|a(x,t)tdt, then we can easily see that
(1.15)∇yG(x,y)=A(x,y),G(x,0)=0
for every x∈Ω-.
Remark 1.4.
It is easily seen that the following assertions hold under condition (A):
for all x∈Ω-, A(x,y) is maximal monotone and strictly monotone in y;
|A(x,y)|≤(C1/(p-1))|y|p-1 for every (x,y)∈Ω-×ℝN;
A(x,y)y≥(C0/(p-1))|y|p for every (x,y)∈Ω-×ℝN;
G(x,y) is convex in y for all x and satisfies the following inequalities:
(1.16)A(x,y)y≥G(x,y)≥C0p(p-1)|y|p,G(x,y)≤C1p(p-1)|y|p,
for every (x,y)∈Ω-×ℝN, where C0 and C1 are the positive constants in (A).
The following result is proved in [3]. It plays an important role for our poof.
Proposition 1.5 (see [3, Proposition 1]).
Let A:W1,p(Ω)→W1,p(Ω)* be the map defined by
(1.17)〈A(u),v〉=∫ΩA(x,∇u)∇vdx,
for u, v∈W1,p(Ω). Then, A has the (S)+ property, that is, any sequence {un} weakly convergent to u with limsupn→∞〈A(un),un-u〉≤0 strongly converges to u.
2. The Weighted Eigenvalue Problems2.1. Preliminaries
The following lemmas can be easily shown by way of contradiction because ∫Ωa∞|∇u|pdx is equivalent to ∥∇u∥pp (note that a∞ is positive). Here, we omit the proofs (refer to [7]).
Lemma 2.1.
Assume ∫Ωmdx<0. Then, there exists a constant c>0 such that ∫Ωa∞|∇u|pdx≥c∥u∥pp for every u∈W1,p(Ω) with ∫Ωm|u|pdx>0.
Lemma 2.2.
Assume that ∫Ωmdx≠0 and ξ>0. Then, there exists a constant b(m,ξ)>0 such that
(2.1)∫Ωa∞|∇u|pdx-ξ∫Ωm|u|pdx≥b(m,ξ)∫Ω|u|pdx
for every u∈B(m):={u∈W1,p(Ω);∫Ωm|u|pdx≤0}.
Lemma 2.3.
Assume that m≥0 in Ω. Then, for every ξ>0 there existed d(m,ξ)>0 such that
(2.2)∫Ωa∞|∇u|pdx-ξ∫Ωm|u|pdx≥d(m,ξ)∫Ω|u|pdx
for every u∈W1,p(Ω).
First, we recall the following principle eigenvalue λ*(m):
(2.3)λ*(m)≔inf{∫Ωa∞|∇u|pdx;u∈W1,p(Ω),∫Ωm|u|pdx=1}.
Because of ∞>supx∈Ωa∞(x)≥infx∈Ωa∞(x)>0, we have the following result as the same argument as in the case of the p-Laplacian.
Proposition 2.4 (see [7, Proposition 2.2]).
The following assertions hold:
If ∫Ωmdx≥0 holds, then λ*(m)=0;
If ∫Ωmdx<0 holds, then λ*(m)>0 is a simple eigenvalue and it admits a positive eigenfunction. In addition, the open interval (0,λ*(m)) contains no eigenvalues of (EV;m).
Lemma 2.5.
Assume ∫Ωmdx<0. Then, one has λ*(m+ɛ)<λ*(m)<λ*(m-ɛ′) for every ɛ>0 and ɛ′>0 with |{m-ɛ′>0}|>0.
Proof.
We choose a minimizer u for λ*(m) because Proposition 2.4 guarantees the existence of it. Then, for every ɛ>0, we have
(2.4)λ*(m+ɛ)≤∫Ωa∞|∇u|pdx∫Ω(m+ɛ)|u|pdx<∫Ωa∞|∇u|pdx∫Ωm|u|pdx=∫Ωa∞|∇u|pdx=λ*(m)
by the definition of λ*(m+ɛ). By applying the same argument to a minimizer for λ*(m-ɛ), we obtain λ*(m)<λ*(m-ɛ′) for ɛ′>0 with |{m-ɛ′>0}|>0.
2.2. Other Eigenvalues
Here, we introduce two unbounded sequences {λn(m)}n and {μn(m)}n as follows:
(2.5)J(u):=∫Ωa∞|∇u|pdxforu∈W1,p(Ω),J~:=J∣S(m),S(m):={u∈W1,p(Ω);∫Ωm|u|pdx=1},Sn(m):={X⊂S(m);compact,symmetricandγ(X)≥n},Fn(m):={g∈C(Sn-1,S(m));gisodd},λn(m):=infX∈Sn(m)maxu∈XJ~(u),μn(m):=infg∈Fn(m)maxz∈Sn-1J~(g(z)),
where γ(X) denotes the Krasnoselskii genus of X (see [17, Definition 5.1] for the definition) and Sn-1 denotes the usual unit sphere in ℝn. We see that λn(m) is defined by Ljusternik-Schnirelman theory and it is known that the definition of μn(m) is introduced by Drábek and Robinson ([16]) under the p-Laplace Dirichlet problem with m≡1.
Remark 2.6.
The following assertions can be shown easily:
λ1(m)=μ1(m)=λ*(m);
𝒮n(m)≠∅ and ℱn(m)≠∅ for every n∈ℕ;
g(Sn-1)⊂𝒮n(m) for every g∈ℱn(m);
μn(m)≥λn(m) for every n∈ℕ;
λn+1(m)≥λn(m) and μn+1(m)≥μn(m) for every n∈ℕ,
see [18] for the proof of (ii).
Define a C1 functional Φm on W1,p(Ω) by Φm(u):=∫Ωm|u|pdx for u∈W1,p(Ω). Because 1∈ℝ is a regular value of Φm, it is well known that the norm of the derivative at u∈S(m) of the restriction of J to S(m) is defined as follows:
(2.6)‖J~′(u)‖*:=min{‖J′(u)-tΦm′(u)‖W1,p(Ω)*;t∈R}=sup{〈J′(u),v〉;v∈Tu(S(m)),‖v‖=1},
where Tu(S(m)) denotes the tangent space of S(m) at u, that is, Tu(S(m))={v∈W1,p(Ω);∫Ωm|u|p-2uvdx=0}. Here, we recall the definition of the Palais-Smale condition for J~.
Definition 2.7.
J~ is said to satisfy the bounded Palais-Smale condition if any bounded sequence un∈S(m) such that ∥J~′(un)∥*→0 has a convergent subsequence. Moreover, we say that J~ satisfies the Palais-Smale condition at level c∈ℝ if any sequence un∈S(m) such that J~(un)→c and ∥J~′(un)∥*→0 as n→∞ has a convergent subsequence. In addition, we say that J~ satisfies the Palais-Smale condition if J~ satisfies the Palais-Smale condition for every c∈ℝ.
The following result can be proved by the same argument as in [19, Proposition 3.3] (which treats the case of the p-Laplacian, i.e., a∞≡1) because of ∞>supx∈Ωa∞(x)≥infx∈Ωa∞(x)>0. Here, we omit the proof.
Lemma 2.8.
The following assertions hold:
J~ satisfies the bounded Palais-Smale condition;
J~ satisfies the Palais-Smale condition provided ∫Ωmdx≠0.
Proposition 2.9.
λn(m) and μn(m) are eigenvalues of (EV;m) such that
(2.7)limn→∞λn(m)=limn→∞μn(m)=+∞.
Proof.
In the case of ∫Ωmdx≠0, since J~ satisfies the Palais-Smale condition, we can apply the first deformation lemma on C1 manifold (refer to [20]). Thus, by the standard argument, we can prove that λn(m) and μn(m) are critical values of J~. This means that λn(m) and μn(m) are eigenvalues of (EV;m) by the Lagrange multiplier rule. In addition, we can easily show limn→∞λn(m)=+∞ by the standard argument via the first deformation lemma on C1 manifold (refer to [21, Proposition 3.14.7], [22] or [17] in the case of a Banach space). Hence, limn→∞μn(m)=+∞ holds because of μn(m)≥λn(m) for every n∈ℕ.
In the case of ∫Ωmdx=0, by the same argument as in [18], our conclusion can be proved. For readers' convenience, we give a sketch of the proof. For ɛ>0, we define Jɛ(u):=J(u)+ɛ∥u∥pp and J~ɛ:=Jɛ∣S(m). Moreover, we set minimax values λnɛ(m) and μnɛ(m) of J~ɛ by
(2.8)λnɛ(m):=infX∈Sn(m)maxu∈XJ~ɛ(u),μnɛ(m):=infg∈Fn(m)maxz∈Sn-1J~ɛ(g(z)).
Because any Palais-Smale sequence of J~ɛ is bounded, it is easily shown that J~ɛ satisfies the Palais-Smale condition (refer to [19, Proposition 3.3]) Hence, it can be proved that λnɛ(m) and μnɛ(m) are critical values of J~ɛ. Furthermore, it follows from the argument as in [18, Lemma 3.5] that λnɛ(m)→λn(m) and μnɛ(m)→μn(m) as ɛ→0+. Therefore, by noting that Jɛ is p-homogeneous, we can obtain a solution uɛ with ∥uɛ∥=1 for -div(a∞|∇u|p-2∇u)=cɛm|u|p-2u in Ω, ∂u/∂ν=0 on ∂Ω, where cɛ=λnɛ(m) or μnɛ(m). Because of ∥uɛ∥=1, it follows from the standard argument that uɛ has a subsequence strongly convergent to a solution u for
(2.9)-div(a∞|∇u|p-2∇u)=cm|u|p-2uinΩ,∂u∂ν=0on∂Ω,
where c=limɛ→0+cɛ. Thus, λn(m) and μn(m) are eigenvalues of (EV;m). To prove limn→∞λn(m)=+∞, by considering a function mδ(x):=max{m(x),δ} for δ>0, we have λn(mδ)≤λn(m) (refer to Proposition 2.10). Because we can apply our fist assertion to mδ (note ∫Ωmδdx>0), we obtain limn→∞μn(m)≥limn→∞λn(m)≥limn→∞λn(mδ)=+∞.
Proposition 2.10.
Let 1<r<∞ if N≤p and p*/(p*-p)≤r<∞ if N>p. Then, the following assertions hold:
if m′≥m in Ω, then μk(m′)≤μk(m);
if limn→∞mn=m in Lr(Ω), then limsupn→∞μk(mn)≤μk(m);
if ∫Ωmdx≠0 and limn→∞mn=m in Lr(Ω), then limn→∞μk(mn)=μk(m).
Moreover, the same conclusion holds for λk(m).
Proof.
We only treat μk(m) because we can give the proof for λk(m) similarly.
Let m′≥m in Ω. Fix an arbitrary ɛ>0. Then, by the definition of μk(m), there exists a g∈ℱk(m) such that maxz∈Sk-1J(g(z))<μk(m)+ɛ. Set g~(z):=g(z)/(∫Ωm′|g(z)|pdx)1/p for z∈Sk-1 (note ∫Ωm′|g(z)|pdx≥∫Ωm|g(z)|pdx=1), then g~∈ℱk(m′) holds. Therefore, by the definition of μk(m′), we have
(2.10)μk(m′)≤maxz∈Sk-1J(g~(z))=maxz∈Sk-1J(g(z))∫Ωm′|g(z)|pdx≤maxz∈Sk-1J(g(z))<μk(m)+ɛ.
because of ∫Ωm′|g(z)|pdx≥∫Ωm|g(z)|pdx=1 for every z∈Sk-1. Since ɛ>0 is arbitrary, we obtain μk(m′)≤μk(m).
Let limn→∞mn=m in Lr(Ω) and fix an arbitrary ɛ>0. By the definition of μk(m), there exists a g∈ℱk(m) such that maxz∈Sk-1J(g(z))<μk(m)+ɛ/2. Since g(Sk-1) is compact and pr′:=pr/(r-1)≤p*, we set M:=maxu∈g(Sk-1)∥u∥pr′. Then, due to Hölder's inequality and mn→m in Lr(Ω), there exists an n0∈ℕ such that
(2.11)∫Ωmn|u|pdx=1+∫Ω(mn-m)|u|pdx≥1-‖mn-m‖rMp>0
for every u∈g(Sk-1) and n≥n0. Therefore, by a similar argument to (i), we obtain
(2.12)μk(mn)≤maxz∈Sk-1J(g(z))∫Ωmn|g(z)|pdx≤μk(m)+ɛ/21-‖mn-m‖rMp<μk(m)+ɛ
for sufficiently large n. Hence, limsupn→∞μk(mn)≤μk(m)+ɛ follows. Since ɛ>0 is arbitrary, our conclusion is proved.
Let limn→∞mn=m in Lr(Ω) and ∫Ωmdx≠0. We fix an arbitrary ɛ>0. Due to our assertion (ii), there exists an n1∈ℕ such that μk(mn)≤μk(m)+ɛ/2. For every n≥n1, by the definition of μk(mn), we can take gn∈ℱk(mn) satisfying maxz∈Sk-1J(gn(z))<μk(mn)+ɛ/2.
Here, we will prove
(2.13)supn≥n1max{‖u‖p;u∈gn(Sk-1)}<∞.
If u∈gn(Sk-1) satisfies ∫Ωm|u|pdx≤0, then we obtain
(2.14)b(m,1)‖u‖pp≤J(u)-∫Ωm|u|pdx=J(u)-∫Ωmn|u|pdx+∫Ω(mn-m)|u|pdx≤μk(mn)+ɛ2-1+‖mn-m‖r‖u‖pr′p≤μk(m)+ɛ+C‖mn-m‖r‖u‖pp+CJ(u)‖mn-m‖rinfΩa∞≤(1+C‖mn-m‖rinfΩa∞)(μk(m)+ɛ)+C‖mn-m‖r‖u‖pp
by Lemma 2.2 and Hölder's inequality (note ∥∇u∥pp≤J(u)/infΩa∞ and μk(mn)≤μk(m)+ɛ/2), where C>0 is a constant (independent of n and u) obtained by the continuity of W1,p(Ω) into Lpr′(Ω). Therefore, if we take an n2≥n1 satisfying C∥mn-m∥r≤b(m,1)/2 for every n≥n2, then we obtain
(2.15)‖u‖pp≤2b(m,1)(1+b(m,1)2infΩa∞)(μk(m)+ɛ)
for every u∈gn(Sk-1) provided ∫Ωm|u|pdx≤0 and n≥n2. Similarly, in the case where m changes sign, for every u∈gn(Sk-1) satisfying ∫Ωm|u|pdx>0, we have
(2.16)b(-m,1)‖u‖pp≤J(u)-∫Ω(-m)|u|pdx≤(1+C‖mn-m‖rinfΩa∞)(μk(m)+ɛ)+1+C‖mn-m‖r‖u‖pp.
Hence, by taking a sufficiently large n3≥n2, we get the inequality
(2.17)‖u‖pp≤2b(-m,1)(1+b(-m,1)2infΩa∞)(μk(m)+ɛ+1),
for every u∈gn(Sk-1) with ∫Ωm|u|pdx>0 and n≥n3. In the case of m≥0 in Ω, by using Lemma 2.3 instead of Lemma 2.2, we have a similar inequality
(2.18)‖u‖pp≤2d(m,1)(1+d(m,1)2infΩa∞)(μk(m)+ɛ+1),
for every u∈gn(Sk-1) provided n≥n4 (some sufficiently large n4≥n3). Consequently, our claim follows from (2.15), (2.17), and (2.18).
Let us return to the proof of (iii). Because
(2.19)sup{‖u‖pr′;u∈gn(Sk-1),n≥n1}=:R<+∞
holds by (2.13), J(u)≤μk(m)+ɛ/2 and the continuity of W1,p(Ω) into Lpr′(Ω), we see the inequality
(2.20)∫Ωm|u|pdx=1-∫Ω(mn-m)|u|pdx>1-‖mn-m‖rRp>0,
for every u∈gn(Sk-1) and n≥n5 (some sufficiently large n5≥n4). By considering g~n(·):=gn(·)/(∫Ωm|gn(·)|pdx)1/p∈ℱk(m), we obtain
(2.21)μk(m)≤maxz∈Sk-1J(g~n(z))≤maxz∈Sk-1J(gn(z))1-‖mn-m‖rRp≤μk(mn)+ɛ/21-‖mn-m‖rRp.
Because of ∥mn-m∥r→0, we get μk(mn)≥μk(m)-ɛ for sufficiently large n, and hence our conclusion holds.
Finally, we recall the second eigenvalue of (EV;m) obtained by the mountain pass theorem.
(2.22)Σ(m):={η∈C([0,1],S(m));η(0)∈P,η(1)∈(-P)},c(m):=infη∈Σ(m)maxt∈[0,1]J~(η(t)),
where P:={u∈W1,p(Ω);u(x)≥0fora.e.x∈Ω}.
Since ∞>supx∈Ωa∞(x)≥infx∈Ωa∞(x)>0 holds, the following result can be shown by the same argument as in [19] (although they handle the asymmetry case, it is sufficient to consider the case of m≡n in this paper). See [19, Theorem 3.2] for the proof.
Theorem 2.11.
c(m) is an eigenvalue of (EV;m) which satisfies λ*(m)<c(m). Moreover, there is no eigenvalues of (EV;m) between λ*(m) and c(m).
Now, we have the following result.
Proposition 2.12.
(2.23)λ2(m)=μ2(m)=c(m)
holds, where c(m) is a minimax value defined by (2.22).
Proof.
First, we prove the inequality c(m)≥μ2(m). Because c(m) is an eigenvalue (note that the following equation is homogeneous), we can choose a solution u∈W1,p(Ω) with ∫Ωm|u|pdx=1 for
(2.24)-div(a∞(x)|∇u|p-2∇u)=c(m)m(x)|u|p-2uinΩ,∂u∂ν=0on∂Ω.
Note that u is a sign-changing function because any eigenfunction associated with any eigenvalue greater than the principal eigenvalue changes sign (refer to [18, Proposition 4.3]). Thus, we have
(2.25)0<∫Ωa∞|∇u±|pdx=c(m)∫Ωmu±pdx
by taking ±u± as test function (recall that u±:=max{±u,0}). Hence, we may assume that ∫Ωmu±pdx=1 by the normalization. Set X:={su+-tu-;|s|p+|t|p=1}⊂S(m). Then, because X is homeomorphic to S1, there exists g∈ℱ2(m) such that g(S1)=X. Since the value of J is equal to c(m) on X, we obtain
(2.26)μ2(m)≤maxz∈S1J~(g(z))=c(m)
by the definition of μ2(m) and X.
Next, we will prove the inequality c(m)≤λ2(m) by dividing into two cases: ∫Ωmdx≠0 and ∫Ωmdx=0.
Case of ∫Ωmdx≠0: by way of contradiction, we assume that λ2(m)<c(m). Then, λ*(m)=λ1(m)=λ2(m) follows from Theorem 2.11. Note that J~ satisfies the Palais-Smale condition in this case (see Lemma 2.8), and hence we can apply the first deformation lemma to J~. Therefore, by the standard argument (cf. [22], [17, Lemma 5.6]), we see that γ(K)≥2, where K:={u∈S(m);J~′(u)=0,J~(u)=λ*(m)}. This means that K is an infinite set, that is, the following equation has infinite many solutions:
(2.27)-div(a∞(x)|∇u|p-2∇u)=λ*(m)m(x)|u|p-2uinΩ,∂u∂ν=0on∂Ω
due to the Lagrange multiplier's rule. This contradicts to the fact described as in Proposition 2.4 that λ*(m) is simple. As a result, we have shown that c(m)=λ2(m)=μ2(m) holds in the case of ∫Ωmdx≠0 (note λn(m)≤μn(m)).
Case of ∫Ωmdx=0: According to Proposition 2.10 (i) for λ2(m), we have λ2(m)≥λ2(m+ɛ)=c(m+ɛ) for every ɛ>0 since we can apply the first result to m+ɛ. Because we prove limɛ→0+c(m+ɛ)=c(m) by the same argument as in [6, Lemma 2.9] (for the case a∞≡1), our conclusion is proved by taking ɛ↓0 in the inequality λ2(m)≥c(m+ɛ).
3. Proof of Theorem 1.1
We define a functional Iλ,m on W1,p(Ω) as follows:
(3.1)Iλ,m(u)=∫ΩG(x,∇u)dx-λp∫Ωm|u|pdx-∫ΩF(x,u)dx-∫Ωhudx=1p∫Ωa∞|∇u|pdx+∫ΩG~(x,∇u)dx-λp∫Ωm|u|pdx-∫ΩF(x,u)dx-∫Ωhudx
for u∈W1,p(Ω) ((1.15) or (1.9) for the definition of G, G~, and F). It is easily seen that Iλ,m is well defined and class of C1 on W1,p(Ω) by (1.1), (1.16) and the continuity of W1,p(Ω)↪Lp(Ω).
Remark 3.1.
Let u∈W1,p(Ω) be a critical point of Iλ,m, namely, u satisfies the equality
(3.2)∫ΩA(x,∇u)∇φdx=λ∫Ωm|u|p-2uφdx+∫Ωf(x,u)φdx+∫Ωhφdx
for every φ∈W1,p(Ω). Then, u∈L∞(Ω) by the Moser iteration process (refer to Theorem C in [4]). Therefore, u∈C1,α(Ω-) (0<α<1) follows from the regularity result in [23]. Furthermore, due to [24, Theorem 3], u satisfies (P;λ,m,h) in the distribution sense and the boundary condition
(3.3)0=∂u∂νA=A(⋅,∇u)ν=a(⋅,|∇u|)∂u∂νinW-1/q,q(∂Ω)
for every 1<q<∞ (see [24] for the definition of W-1/q,q(∂Ω)). Since u∈C1,α(Ω-) and a(x,t)>0 for every t≠0, u satisfies the Neumann boundary condition, that is, (∂u/∂ν)(x)=0 for every x∈∂Ω.
3.1. The Palais-Smale Condition in the Nonresonant Case
First, we recall the definition of the Palais-Smale condition.
Definition 3.2.
A C1 functional Ψ on a Banach space X is said to satisfy the Palais-Smale condition at c∈ℝ if a Palais-Smale sequence {un}⊂X at level c, namely,
(3.4)Ψ(un)⟶c,‖Ψ′(un)‖X*⟶0asn⟶∞
has a convergent subsequence. We say that Ψ satisfies the Palais-Smale condition if Ψ satisfies the Palais-Smale condition at any c∈ℝ. Moreover, we say that Ψ satisfies the bounded Palais-Smale condition if any bounded sequence {un} such that {Ψ(un)} is bounded and ∥Ψ′(un)∥X*→0 as n→∞ has a convergent subsequence.
Concerning the Palais-Smale condition, we state the following result developed from [6, Proposition 7].
Proposition 3.3.
If λ is not an eigenvalue of (EV;m), then Iλ,m satisfies the Palais-Smale condition.
Proof.
Let {un} be a Palais-Smale sequence of Iλ,m, namely,
(3.5)Iλ,m(un)⟶c,‖Iλ,m′(un)‖W1,p(Ω)*⟶0asn⟶∞
for some c∈ℝ. It is sufficient to prove only the boundedness of ∥un∥ because the operator A:W1,p(Ω)→W1,p(Ω)* described in Proposition 1.5 has the (S)+ property.
To prove the boundedness of ∥un∥, it suffices to show that ∥un∥p is bounded because of the inequality |f(x,u)|≤C(|u|p-1+1) (obtained by (1.1)) and the following inequality:
(3.6)〈Iλ,m′(un),un〉+λ∫Ωm|un|pdx+∫Ωf(x,un)undx+∫Ωhundx,=∫ΩA(x,∇un)∇undx≥C0p-1‖∇un‖pp,
where we use Remark 1.4 (iii) in the last inequality. By way of contradiction, we may assume that ∥un∥p→∞ as n→∞ by choosing a subsequence if necessary. Set vn:=un/∥un∥p. Then, since the inequality (3.6) guarantees that {vn} is bounded in W1,p(Ω), we may suppose, by choosing a subsequence, that vn⇀v0 in W1,p(Ω) and vn→v0 in Lp(Ω) for some v0.
Here, we will prove that
(3.7)limn→∞‖f(⋅,un)‖p′‖un‖pp-1=0,
where p′=p/(p-1). Fix an arbitrary ɛ>0. It follows from (1.1) that there exists a Cɛ>0 such that
(3.8)|f(x,u)|≤ɛ|u|p-1+Cɛforeveryu∈R,
a.e.x∈Ω.
Then, we obtain
(3.9)∫Ω|f(x,un)|p′dx≤2p′∫Ω(ɛp′|un|p+Cɛp′)dx≤2p′ɛp′‖un‖pp+2p′Cɛp′|Ω|.
Since we are assuming that ∥un∥p→∞ as n→∞, there exists n0∈ℕ such that for every n≥n0(3.10)‖f(⋅,un)‖p′‖un‖pp-1≤4ɛ
holds. This shows that limn→∞∥f(·,un)∥p′/∥un∥pp-1=0 because ɛ>0 is arbitrary.
Here, we recall the following result proved in [6]:
(3.11)limn→∞∫Ωa~(x,|∇un|)∇un‖un‖pp-1∇(vn-v0)dx=limn→∞∫Ωa~(x,|∇un|)∇un‖un‖pp-1∇φdx=0,
for every φ∈W1,p(Ω). Thus, by considering
(3.12)o(1)=〈Iλ,m′(un),vn-v0〉‖un‖pp-1=∫Ωa∞|∇vn|p-2∇vn∇(vn-v0)dx+o(1),
we see that vn strongly converges to v0 in W1,p(Ω) (note that p-Laplacian has the (S)+ property). Therefore, by taking a limit in o(1)=〈Iλ,m′(un),φ〉/∥un∥pp-1 for any φ∈W1,p(Ω) and by noting (3.7) and (3.11), we know that v0 is a nontrivial solution (note ∥v0∥p=1) of
(3.13)-div(a∞|∇u|p-2∇u)=λm|u|p-2uinΩ,∂u∂ν=0on∂Ω.
This means that λ is an eigenvalue of (EV;m). This is a contradiction. Hence, ∥un∥p is bounded.
3.2. Key Lemmas
To show the linking lemma, we define
(3.14)Y(μ,m):={u∈W1,p(Ω);∫Ωa∞|∇u|pdx≥μ∫Ωm|u|pdx}forμ∈ℝ.
Lemma 3.4.
Let g0∈C(Sk-1,W1,p(Ω)∖{0}) be odd and 0<μ≤μk+1(m). Then, g(S+k)∩Y(μ,m)≠∅ for every g∈C(S+k,W1,p(Ω)) with g∣Sk-1=g0, where Y(μ,m) is the set introduced in (3.14) and S+k is the upper hemisphere in ℝk+1 with boundary Sk-1.
Proof.
Fix any g∈C(S+k,W1,p(Ω)) such that g∣Sk-1=g0. If u∈g(S+k) satisfies ∫Ωm|u|pdx≤0, then u∈Y(μ,m) holds. So, we may assume that ∫Ωm|u|pdx>0 for every u∈g(S+k). Define g~∈ℱk+1(m) as follows:
(3.15)g~(z):={g(z)(∫Ωm|g(z)|pdx)1/pifz∈S+k,-g(-z)(∫Ωm|g(-z)|pdx)1/pifz∈S-k.
By the definition of μk+1(m), there exists z0∈Sk such that J~(g~(z0))≥μk+1(m). Since g~ is odd and J is even, we may suppose z0∈S+k. So, this yields the inequality J(g(z0))≥μk+1(m)∫Ωm|g(z0)|pdx≥μ∫Ωm|g(z0)|pdx, whence g(z0)∈Y(μ,m) holds.
Lemma 3.5.
Let μk(m)<λ. Then, there exists g0∈ℱk(m) such that
(3.16)maxz∈Sk-1J(g0(z))<λ,maxz∈Sk-1Iλ,m(Tg0(z))⟶-∞
as
|T|⟶∞,
where μk(m) is defined by (2.5).
Proof.
Choose ɛ0>0 such that μk(m)+ɛ0<λ. By the definition of μk(m), there exists g0∈ℱk(m) such that
(3.17)maxz∈Sk-1J(g0(z))<μk(m)+ɛ0.
Due to the compactness of g0(Sk-1), we put M:=maxz∈Sk-1∥g0(z)∥p. By the property of the function a~ as in (AH) and Young's inequality, for every ɛ>0 there exist constants Cɛ>0 and Cɛ′>0 such that
(3.18)|G~(x,y)|≤ɛ2|y|p+Cɛ|y|≤ɛ|y|p+Cɛ′≤ɛinfΩa∞a∞(x)|y|p+Cɛ′
for every x∈Ω and y∈ℝN. Moreover, the hypothesis (1.1) ensures that for every ɛ′>0 there exist constants Dɛ′>0 satisfying
(3.19)|F(x,u)|≤ɛ′2|u|p+Dɛ′|u|≤ɛ′|u|p+Dɛ′′
for every u∈ℝ and a.e. x∈Ω. Hence, we have
(3.20)Iλ,m(Tu)≤Tpp(1+pɛa_)∫Ωa∞|∇u|pdx-Tp(λ-pɛ′Mp)p+T‖h‖∞‖u‖1+C≤Tpp{(1+pɛa_)(μk(m)+ɛ0)-λ+pMpɛ′}+TM‖h‖∞|Ω|(p-1)/p+C
for every T>0, u∈g0(Sk-1), ɛ>0 and ɛ′>0 since g0(Sk-1)⊂S(m), (3.17), (3.18) and (3.19), where C=(Cɛ′+Dɛ′′)|Ω| and a_=infx∈Ωa∞(x)>0. By taking ɛ>0 and ɛ′>0 satisfying (1+pɛ/a_)(μk(m)+ɛ0)-λ+pMpɛ′<0, we show that maxz∈Sk-1Iλ,m(Tg0(z))→-∞ as T→+∞. Thus, our conclusion follows because g0(Sk-1) is symmetric.
3.3. The Case ∫Ωmdx≠0Lemma 3.6.
Let ∫Ωmdx<0 and 0<λ<λ*(m). Then, Iλ,m is bounded from below, coercive and weakly lower semicontinuous (w.l.s.c.) on W1,p(Ω).
Proof.
Φ(u):=∫ΩG(x,∇u)dx is w.l.s.c. on W1,p(Ω) because Φ is convex and continuous on W1,p(Ω) (cf. [25, Theorem 1.2]). Thus, Iλ,m is also w.l.s.c. on W1,p(Ω) since the inclusion from W1,p(Ω) to Lp(Ω) is compact.
Choose ɛ>0 such that pɛ<a_(1-λ/λ*(m)), where a_:=infΩa∞. By an easy estimation, (3.18) and (3.19) as in Lemma 3.5, we have
(3.21)Iλ,m(u)≥a_-ɛppa_∫Ωa∞|∇u|pdx-λp∫Ωm|u|pdx-ɛ′‖u‖pp-‖h‖∞‖u‖p|Ω|(p-1)/p-(Cɛ′+Dɛ′′)|Ω|
for every u∈W1,p(Ω) and ɛ′>0.
Let u∈W1,p(Ω) satisfy ∫Ωm|u|pdx≤0. Then, the following inequality follows from Lemma 2.2:
(3.22)D0∫Ωa∞|∇u|pdx-λ∫Ωm|u|pdx≥D02∫Ωa∞|∇u|pdx+b(m,ξ)‖u‖pp,
where b(m,ξ) is a positive constant independent of u with ξ=2λ/D0 and D0=(a_-ɛp)/a_.
For every u∈W1,p(Ω) such that ∫Ωm|u|pdx>0, we obtain
(3.23)D0∫Ωa∞|∇u|pdx-λ∫Ωm|u|pdx≥(D0-λλ*(m))∫Ωa∞|∇u|pdx≥12(D0-λλ*(m))∫Ωa∞|∇u|pdx+c2(D0-λλ*(m))‖u‖pp
by the definition of λ*(m), Lemma 2.1 and D0-λ/λ*(m)>0, where c>0 is a constant obtained by Lemma 2.1.
Consequently, if we choose a ɛ′>0 satisfying ɛ′<min{b(m,ξ)/p,c(D0-λ/λ*(m))/(2p)}, then we obtain positive constants d1 and d2 (independent of u) such that
(3.24)Iλ,m(u)≥d1∫Ωa∞|∇u|pdx+d2‖u‖pp-‖h‖∞‖u‖p|Ω|(p-1)/p-(Cɛ′+Dɛ′′)|Ω|≥min{a_d1,d2}‖u‖p-‖h‖∞‖u‖|Ω|(p-1)/p-(Cɛ′+Dɛ′′)|Ω|
for every u∈W1,p(Ω) by (3.21), (3.22), and (3.23). Because of p>1, our conclusion is shown.
Lemma 3.7.
Let m≥0 in Ω and m≢0. If λ<0 holds, then Iλ,m is bounded from below, coercive and w.l.s.c. on W1,p(Ω).
Proof.
First, as the same reason in Lemma 3.6, it follows that Iλ,m is w.l.s.c. on W1,p(Ω). By a similar argument to Lemma 3.6, for every ɛ′>0 and 0<ɛ<a_/p where a_=infΩa∞, we obtain
(3.25)Iλ,m(u)≥a_-ɛppa_∫Ωa∞|∇u|pdx+|λ|p∫Ωm|u|pdx-ɛ′‖u‖pp-‖h‖∞‖u‖p|Ω|(p-1)/p-(Cɛ′+Dɛ′′)|Ω|
for every u∈W1,p(Ω) (note λ<0). Here, from Lemma 2.3,
(3.26)D0∫Ωa∞|∇u|pdx+|λ|∫Ωm|u|pdx≥D02∫Ωa∞|∇u|pdx+D02b(ξ,m)‖u‖pp
for every u∈W1,p(Ω) follows, where D0:=(a_-ɛp)/a_, ξ:=2|λ|/D0 and b(ξ,m) is a constant obtained in Lemma 2.3. Therefore, by choosing a ɛ′ such that 0<ɛ′<D0b(ξ,m)/2, we can prove our conclusion.
Lemma 3.8.
Let ∫Ωmdx≠0 and 0<λ<μ. Then, Iλ,m is bounded from below on Y(μ,m), where Y(μ,m) is the set introduced in (3.14).
Proof.
Due to the same inequalities concerning G and F as in Lemma 3.5, for every ɛ>0 and ɛ′>0, there exists C=C(ɛ,ɛ′)>0 such that
(3.27)Iλ,m(u)≥a_-pɛpa_∫Ωa∞|∇u|pdx-λp∫Ωm|u|pdx-ɛ′‖u‖pp-‖h‖∞‖u‖1-C|Ω|
for every u∈W1,p(Ω), where a_:=infx∈Ωa∞(x). Choose positive constants ɛ and δ such that D0:=1-pɛ/a_>δ>λ/μ (note λ/μ<1).
First, we consider the case of m≥0 in Ω. For every u∈Y(μ,m), we obtain
(3.28)D0∫Ωa∞|∇u|pdx-λ∫Ωm|u|pdx≥(D0-δ)∫Ωa∞|∇u|pdx+(δμ-λ)∫Ωm|u|pdx≥d(m,ξ1)(D0-δ)‖u‖pp
by Lemma 2.3 with ξ1=(δμ-λ)/(D0-δ) (note δμ-λ>0 and D0-δ>0).
Next, we handle with the case where m changes sign. Let u∈W1,p(Ω) satisfy ∫Ωm|u|pdx≤0. Then, we have for such u(3.29)D0∫Ωa∞|∇u|pdx-λ∫Ωm|u|pdx≥b(m,ξ2)D0‖u‖pp
by Lemma 2.2, where D0=1-pɛ/a_ and ξ2:=λ/D0.
On the other hand, for u∈Y(μ,m) with ∫Ωm|u|pdx>0, the following inequality follows from Lemma 2.2:
(3.30)D0∫Ωa∞|∇u|pdx-λ∫Ωm|u|pdx≥(D0-δ)∫Ωa∞|∇u|pdx-(δμ-λ)∫Ω(-m)|u|pdx≥b(-m,ξ1)(D0-δ)‖u‖pp.
Consequently, by (3.27), (3.29), (3.28), and (3.30), there exists d>0 independent of u such that
(3.31)Iλ,m(u)≥(d-ɛ′)‖u‖pp-‖h‖∞‖u‖p|Ω|(p-1)/p-C|Ω|
for every u∈Y(μ,m). Hence, our conclusion is shown by taking ɛ′>0 satisfying ɛ′<d.
Proof of Theorem 1.1 in the Case ∫Ωmdx≠0.
First, if either m≥0 on Ω and λ<0 or 0<λ<λ*(m)=μ1(m) (i.e., ∫Ωmdx<0) holds, then Lemma 3.7 or Lemma 3.6 guarantees the existence of a global minimizer of Iλ,m, respectively (cf. [25, Theorem 1.1]). Hence, (P;λ,m,h) has a solution.
Since λ is an eigenvalue of (EV;m) if and only if -λ is one of (EV;-m), it suffices to consider the case of λ>λ*(m)≥0. Furthermore, by Proposition 2.9, Remark 2.6 (i), and our hypothesis that λ is not an eigenvalue of (EV;m), we may assume that there exists a k∈ℕ such that μk(m)<λ<μk+1(m). By Lemmas 3.5 and 3.8, we can choose T>0 and g0∈ℱk(m) satisfying
(3.32)maxz∈Sk-1Iλ,m(Tg0(z))<inf{Iλ,m(u);u∈Y(μk+1(m),m)}=:α.
Put
(3.33)Σ≔{g∈C(S+k,W1,p(Ω));g∣Sk-1=Tg0},c≔infg∈Σmaxz∈S+kIλ,m(g(z)).
Then, it follows from Lemma 3.4 and (3.32) that c≥α>maxz∈Sk-1Iλ,m(Tg0(z)) holds. Since Iλ,m satisfies the Palais-Smale condition by Proposition 3.3, the minimax theorem guarantees (cf. [25, Theorem 4.6]) that c is a critical value of Iλ,m. Hence, (P;λ,m,h) has at least one solution.
3.4. The Case ∫Ωmdx=0
First, we introduce an approximate functional Iλ,m,n+ as follows:
(3.34)Iλ,m,n+(u):=Iλ,m(u)+1pn‖u‖pp=Iλ,m-1/(λn)(u)foru∈W1,p(Ω).
Lemma 3.9.
Let 0<λ<μ. Then, there exists an n0∈ℕ such that for each n≥n0, Iλ,m,n+ is bounded from below on Y(μ,m-1/λn), where Y(μ,m-1/λn) is the set introduced in (3.14).
Proof.
Choose n0∈ℕ such that 1/n0<λesssupx∈Ωm(x)/2. Then, for every n≥n0, Lemma 3.8 guarantees that Iλ,m,n+=Iλ,m-1/(λn) bounded from below on Y(μ,m-1/(λn)) because of ∫Ω(m-1/(λn))dx<0 and |{m-1/(λn)>0}|>0.
Proof of Theorem 1.1 in the Case ∫Ωmdx=0.
By noting that λm=(-λ)(-m) and μ1(m)=λ*(m)=0, we may assume that μk(m)<λ<μk+1(m) for some k∈ℕ. Let n0 be a natural number obtained by Lemma 3.9. Due to Proposition 2.10 (i) and (ii), there exists an n1≥n0 such that
(3.35)μk(m)≤μk(m-1nλ)≤μk(m-1n1λ)<λ<μk+1(m)≤μk+1(m-1nλ)
for every n≥n1. Thus, for every n≥n1, we can take Tn>0 and gn∈ℱk(m-1/(nλ)) satisfying
(3.36)maxz∈Sk-1Iλ,m,n+(Tngn(z))<inf{Iλ,m,n(u);u∈Y(μk+1(m-1(nλ)),m-1(nλ))}
by applying Lemmas 3.5 and 3.9 to Iλ,m,n+=Iλ,m-1/(nλ) (note (3.35)). Set
(3.37)Σn:={g∈C(S+k,W1,p(Ω));g∣Sk-1=Tngn},cn:=infg∈Σnmaxz∈S+kIλ,m,n+(g(z))
for each n≥n1. Then, for each n≥n1, we can obtain un satisfying
(3.38)|Iλ,m,n+(un)-cn|<1n,‖(Iλ,m,n+)′(un)‖W1,p(Ω)<1n
by applying Ekeland's variational principle to each Iλ,m,n+ (refer to [25, Theorem 4.3]). In addition, we can see that {un} is bounded in W1,p(Ω). Indeed, if there exists a subsequence {unl}l satisfying ∥unl∥p→∞ as l→∞, then we can show that λ is an eigenvalue of (EV;m) by the same argument as in Proposition 3.3. This contradicts to our assumption that λ is not an eigenvalue of (EV;m). Moreover, the boundedness of ∥∇un∥p follows from a similar inequality to (3.6) as in Proposition 3.3 under the boundedness of ∥un∥p.
Therefore, we may assume, by choosing a subsequence that {un} is a Palais-Smale sequence of Iλ,m since Iλ,m is bounded on a bounded set and according to the following inequality:
(3.39)‖Iλ,m′(un)‖(W1,p(Ω))*≤‖Iλ,m′(un)-(Iλ,m,n+)′(un)‖(W1,p(Ω))*+1n≤1n‖un‖pp-1+1n.
Therefore, because Iλ,m satisfies the Palais-Smale condition by Proposition 3.3, Iλ,m has a critical point, whence (P;λ,m,h) has at least one solution.
4. Proof of Theorem 1.2
First, we will prove the following result concerning the Palais-Smale condition under the additional hypothesis (H±) or (HF±).
Proposition 4.1.
Assume that one of the following conditions hold:
λ=0 and (HF+) or (HF-);
λ≠0 and one of (H+), (H-), (HF+) and (HF-).
Then, Iλ,m satisfies the Palais-Smale condition.
Proof.
As the same reason in Proposition 3.3, it suffices to prove the boundedness of a Palais-Smale sequence {un} such that Iλ,m(un)→c (for some c∈ℝ) and ∥Iλ,m′(un)∥W*→0 as n→∞. By way of contradiction, we may assume that ∥un∥p→∞ as n→∞ by choosing a subsequence. Set vn:=un/∥un∥p. Then, by the same argument as in Proposition 3.3, {vn} has a subsequence strongly convergent to v0 being a nontrivial solution of
(4.1)-div(a∞(x)|∇u|p-2∇u)=λm(x)|u|p-2uinΩ,∂u∂ν=0on∂Ω.
To simplify the notation, we denote the above subsequence strongly convergent to v0 by {vn}, again. Thus, |un(x)|→∞ as n→∞ for a.e. x∈Ω0:={x′∈Ω;v0(x′)≠0} (note ∥v0∥p=1).
Assume (HF+) or (HF-). Then, we can obtain
(4.2)(I):=∫Ωf(x,un)un-pF(x,un)‖un‖p1+qdx⟶±∞if(HF±),respectively.
Indeed, it follows from (HF+) that there exist R>0 and C>0 independent of n such that f(x,t)t-pF(x,t)≥0 if |t|≥R and a.e. x∈Ω, and |f(x,t)t-pF(x,t)|≤C for every |t|≤R and a.e. x∈Ω. Therefore, since |un(x)|→∞ a.e. x∈Ω0 and |Ω0|>0 (note ∥v0∥p=1), we have (4.2) if (HF+) holds, by applying Fatou's lemma to the following inequality:
(4.3)(I)≥∫Ω0f(x,un)un-pF(x,un)|un|1+q|vn|1+qdx-C|Ω∖Ω0|‖un‖p1+q.
In the case of (HF-), by considering -f instead of f as in the above argument, we can show our claim (4.2).
Furthermore, by Hölder's inequality, we have
(4.4)(II):=∫ΩpG~(x,∇un)-a~(x,|∇un|)|∇un|2‖un‖p1+qdx≤H0∫Ω(|∇vn|1+q+1‖un‖p1+q)dx≤H0‖∇vn‖p1+q|Ω|(p-1-q)/p+o(1)≤H0‖∇v0‖p1+q|Ω|(p-1-q)/p+o(1)
in the case of (HF-) because vn→v0 in W1,p(Ω), where q∈[0,p-1] and H0>0 are constants as in (HF-). Similarly, we obtain
(4.5)(II)≥-H0‖∇v0‖p1+q|Ω|(p-1-q)/p+o(1)
in the case of (HF+).
Hence, we have a contradiction because of (4.2), (4.4), or (4.5) by taking a limit inferior or superior in the following equality:
(4.6)o(1)=pIλ,m(un)-〈Iλ,m′(un),un〉‖un‖p1+q=(II)+(I)+(1-p)∫Ωhvn‖un‖pqdx,
where we use the fact that ∥un∥/∥un∥p1+q=∥vn∥/∥un∥pq is bounded because of q≥0.
Assume λ≠0 and (H+) or (H-): because v0 is a nontrivial solution of (4.1) with λ≠0, v0 is not a constant function, that is, ∥∇v0∥p>0. Therefore, we have |∇un(x)|→∞ as n→∞ for a.e. x∈Ω~0:={x′∈Ω;|∇v0(x′)|≠0}. Because of |Ω~0|>0, we can show
(4.7)∫ΩpG~(x,∇un)-a~(x,|∇un|)|∇un|2‖un‖p1+qdx⟶±∞if(H±),respectively,
by a similar argument to one for f in the above. In addition, we can easily obtain the following inequality:
(4.8)±∫Ωf(x,un)un-pF(x,un)‖un‖p1+qdx≥-H0‖vn‖1+q1+q+o(1)=-H0‖v0‖1+q1+q+o(1)
in the case of (H±), respectively. Hence, we have a contradiction by considering o(1)=(pIλ,m(un)-〈Iλ,m′(un),un〉)/∥un∥p1+q.
By a similar way to the case ∫Ωmdx=0, we introduce the following approximate functionals on W1,p(Ω):
(4.9)Iλ,m,n±(u):=Iλ,m(u)±1pn‖u‖ppforu∈W1,p(Ω).
Note Iλ,m,n±(u)=Iλ,m∓1/(λn)(u) on W1,p(Ω) provided λ≠0.
Proposition 4.2.
If either λ≠0 and (H+) or (HF+) (resp., either λ≠0 and (H-) or (HF-)) and {un} satisfies
(4.10)supn∈NIλ,m,n+(un)<+∞,limn→∞‖(Iλ,m,n+)′(un)‖W1,p(Ω)*=0,(4.11)(resp.infn∈NIλ,m,n-(un)>-∞,limn→∞‖(Iλ,m,n-)′(un)‖W1,p(Ω)*=0),
then {un} is bounded in W1,p(Ω).
Proof.
First, we note that the boundedness of ∥un∥p guarantees that ∥un∥ is bounded by limn→∞∥(Iλ,m,n±)′(un)∥W1,p(Ω)*=0 (refer to (3.6) as in the proof of Proposition 3.3). Moreover, because of the following equality:
(4.12)pIλ,m,n±(un)-〈(Iλ,m,n±)′(un),un〉‖un‖p1+q=(1-p)∫Ωhvn‖un‖pqdx,+∫ΩpG~(x,∇un)-a~(x,|∇un|)|∇un|2‖un‖p1+qdx+∫Ωf(x,un)un-pF(x,un)‖un‖p1+qdx,
we can prove the boundedness of ∥un∥p by the same argument as in Proposition 4.1.
Proof of Theorem 1.2.
Because of λm=(-λ)(-m), we may assume λ≥0. In the case where ∫Ωmdx≠0 and μk(m)<λ<μk+1(m) for some k∈ℕ, the proof of Theorem 1.1 implies the existence of a critical point of Iλ,m because Iλ,m satisfies the Palais-Smale condition by Proposition 4.1. Concerning other cases, in the next section, we will prove the existence of a bounded sequence {un} satisfying (Iλ,m,n+)′(un)→0 or (Iλ,m,n-)′(un)→0 in W1,p(Ω)* as n→∞. Because Iλ,m is bounded on a bounded set, we may assume that Iλ,m(un) converges to some c∈ℝ by choosing a subsequence. In addition, by noting the inequality ∥Iλ,m′(un)∥W1,p(Ω)*≤∥(Iλ,m,n±)′(un)∥W1,p(Ω)*+∥un∥pp-1/n, we easily see that {un} is a bounded Palais-Smale sequence of Iλ,m. Therefore, Iλ,m has a critical point since Iλ,m satisfies the Palais-Smale condition by Proposition 4.1.
5. Construction of a Bounded Palais-Smale Sequence
In this section, due to the reason stated in the proof of Theorem 1.2, we will construct a bounded sequence {un} satisfying (Iλ,m,n+)′(un)→0 or (Iλ,m,n-)′(un)→0 in W1,p(Ω)* as n→∞. It implies the existence of a bounded Palais-Smale sequence of Iλ,m.
5.1. The Case λ=0 Assume (HF+)
In this c ase, we can show that for each n∈ℕ, Iλ,m,n+ has a global minimizer un. Indeed, for 0<ɛ<1/(pn), there exists Cɛ>0 such that Iλ,m,n+(u)≥C0∥∇u∥pp/(p(p-1))+(1/(pn)-ɛ)∥u∥pp-∥h∥∞∥u∥1-Cɛ for every u∈W1,p(Ω) by (1.1), (1.16) and λ=0 (refer to the inequality as in the proof of Lemma 3.5). This means that Iλ,m,n+ is coercive and bounded from below on W1,p(Ω). Therefore, Iλ,m,n+ has a global minimizer un since Iλ,m,n+ is w.l.s.c. on W1,p(Ω) as the same reason in Lemma 3.6.
Furthermore, because of (Iλ,m,n+)′(un)=0 in W1,p(Ω)* and Iλ,m,n+(un)=minW1,p(Ω)Iλ,m,n+≤Iλ,m,n+(0)=0, it follows from Proposition 4.2 that {un} is bounded.
Assume (HF-)
Choose n0∈ℕ such that 1/n0<c(1)=μ2(1), where c(1) is the second eigenvalue of (EV;1) (so the weight function m≡1 and see (2.22) for the definition). Then, by noting that I0,m,n0-=I1/n0,1, we have
(5.1)α:=inf{I0,m,n0-(u);u∈Y(c(1),1)}>-∞
by Lemma 3.8, where Y(c(1),1) is a subset defined by (3.14) with the weight m≡1, that is,
(5.2)Y(c(1),1):={u∈W1,p(Ω);∫Ωa∞|∇u|pdx≥c(1)‖u‖pp}.
Moreover, inf{I0,m,n-(u);u∈Y(c(1),1)}≥α for every n≥n0 holds because I0,m,n-(u)≥I0,m,n0-(u) for every u∈W1,p(Ω). Since ∫ΩF(x,u)dx=o(1)∥u∥pp as ∥u∥p→∞ by (1.1), there exists Tn>0 such that I0,m,n-(±Tn)=-Tnp(|Ω|/(np)-o(1))<α-2.
Define
(5.3)Σn≔{g∈C([0,1],W1,p(Ω));g(0)=Tn,g(1)=-Tn},cn≔infg∈Σnmaxt∈[0,1]I0,m,n-(g(t))
for n≥n0. By the definition of c(1), we easily see that g([0,1])∩Y(c(1),1)≠∅ for every g∈Σn (refer to [6] or Lemma 3.4). Hence,
(5.4)cn≥inf{I0,m,n-(u);u∈Y(c(1),1)}≥α>I0,m,n(±Tn)
holds, whence cn is bounded from below. Moreover, by applying Ekeland's variational principle to each I0,m,n-, we can obtain a sequence {un} satisfying |I0,m,n-(un)-cn|<1/n and ∥(I0,m,n-)′(un)∥W1,p(Ω)*<1/n. Since cn is bounded from below, it follows from Proposition 4.2 that {un} is bounded. As a result, we can construct a bounded sequence {un} satisfying (I0,m,n-)′(un)→0 as n→∞ in W1,p(Ω)*.
5.2. The Case λ=λ*(m)=μ1(m) with ∫Ωmdx<0 Assume (H+) or (HF+)
Since we see that Iλ,m,n+=Iλ,m-1/(nλ) and λ*(m-1/(nλ))>λ*(m)=λ>0 (according to Lemma 2.5), Iλ,m,n+ is coercive, bounded from below and w.l.s.c. on W1,p(Ω) by Lemma 3.6. Thus, we obtain a global minimizer un of Iλ,m,n+ for sufficiently large n such that |{m-1/(nλ)>0}|>0. Because of Iλ,m,n+(un)≤Iλ,m,n+(0)=0 for every n, Proposition 4.2 guarantees that {un} is bounded.
Assume (H-) or (HF-)
First, we note that Iλ,m,n-=Iλ,m+1/(nλ) and 0<λ*(m+1/(nλ))<λ*(m)=λ by Lemma 2.5 for sufficiently large n such that ∫Ω(m+1/(nλ))dx<0. Moreover, it follows from Proposition 2.10 and μ1(m)<μ2(m) that there exists an n0∈ℕ satisfying ∫Ωm+1/(n0λ)dx<0 and
(5.5)λ*(m+1nλ)<λ=μ1(m)<μ2(m+1n0λ)≤μ2(m+1nλ)≤μ2(m)
for every n≥n0. By applying Theorem 1.1 to each case of a weight m+1/(nλ) (note that λ is not an eigenvalue of (EV;m+1/(nλ)) by (5.5), there exists un satisfying (Iλ,m,n-)′(un)=0 (note Iλ,m,n-=Iλ,m+1/(nλ)) and
(5.6)Iλ,m,n-(un)=cn≥inf{Iλ,m,n-(u);u∈Y(μ2(mn0),mn0)},
where the last inequality follows from Lemma 3.4 with mn0:=m+1/(n0λ). On the other hand, because Iλ,m,n-(u)≥Iλ,m,n0-(u)=Iλ,mn0(u) for every u∈W1,p(Ω) and n≥n0, we have
(5.7)cn≥inf{Iλ,mn0(u);u∈Y(μ2(mn0),mn0)}>-∞
for every n≥n0, where the last inequality follows from Lemma 3.8. Thus, cn is bounded from below. Hence, Proposition 4.2 guarantees the boundedness of {un}.
5.3. The Case λ=μk+1(m) with ∫Ωmdx≠0 Assume (H+) or (HF+)
We may assume μk(m)<μk+1(m)=λ by taking k anew if necessary (note that we have already proved the case of μk(m)<λ<μk+1(m) in Section 4). Here, we can choose an n0∈ℕ such that ∫Ω(m-1/(nλ))dx≠0, |{m-1/(nλ)>0}|>0 and
(5.8)μk(m-1nλ)≤μk(m-1n0λ)<λ-1n‖m‖∞<λ=μk+1(m)≤μk+1(m-1nλ)
for every n≥n0 by ∫Ωmdx≠0 and Proposition 2.10 (i), (iii). Note the following inequality:
(5.9)Iλ,m,n0+(u)≥Iλ,m,n+(u)≥Iλ-1/(n‖m‖∞),m(u)
for every u∈W1,p(Ω) and n≥n0, where the last inequality is obtained by ∥u∥pp≥∫Ωm|u|pdx/∥m∥∞. Let n≥n0. It follows from Lemma 3.8 and (5.8) that Iλ-1/(n∥m∥∞),m is bounded from below on Y(λ,m). Hence, (5.9) yields that Iλ,m,n+ is also bounded from below on Y(λ,m), namely,
(5.10)αn:=inf{Iλ,m,n+(u);u∈Y(λ,m)}>-∞.
On the other hand, because of μk(m-1/(n0λ))<λ (see (5.8)), Lemma 3.5 guarantees the existence of g0∈ℱk(m-1/(n0λ)) satisfying
(5.11)maxz∈Sk-1Iλ,m,n0+(Tg0(z))=maxz∈Sk-1Iλ,m-1/(n0λ)(Tg0(z))⟶-∞as|T|⟶∞.
Thus, for each n≥n0, we can take Tn>0 such that
(5.12)maxz∈Sk-1Iλ,m,n+(Tng0(z))≤maxz∈Sk-1Iλ,m,n0+(Tng0(z))≤αn-1,
(note (5.9) for the first inequality). Set
(5.13)Σn≔{g∈C(S+k,W1,p(Ω));g∣Sk-1=Tng0},cn+≔infg∈Σnmaxz∈S+kIλ,m,n+(g(z))
for n≥n0. Since g(S+k)∩Y(λ,m)≠∅ for every g∈Σn by Lemma 3.4 and λ=μk+1(m), we have cn+≥αn>maxz∈Sk-1Iλ,m,n+(Tng0(z)). Therefore, Ekeland's variational principle (refer to [25, Theorem 4.3]) guarantees the existence of un satisfying |Iλ,m,n+(un)-cn|<1/n and ∥(Iλ,m,n+)′(un)∥W1,p(Ω)*<1/n.
Finally, to show the boundedness of {un} due to Proposition 4.2, we will prove that cn+ is bounded from above. For each n≥n0, we define a continuous map gn from S+k to W1,p(Ω) by
(5.14)gn(z):={(1-zk+1)Tng0(z′1-zk+12)forz=(z′,zk+1)∈S+kwith0≤zk+1<1,0forz=(z′,zk+1)∈S+kwithzk+1=1.
Then, gn∈Σn holds. This leads to
(5.15)cn+≤supt≥0,z∈Sk-1Iλ,m,n+(tg0(z))≤supt≥0,z∈Sk-1Iλ,m,n0+(tg0(z))<+∞
because of (5.9), (5.11), and the compactness of g0(Sk-1).
Assume (H-) or (HF-)
Because the case of μ1(m)=λ*(m) is already shown (see Sections 5.1 and 5.2), We may assume (0<)μk(m)=λ<μk+1(m) for some k≥2 by taking k anew if necessary. Here, we can choose an n0∈ℕ such that ∫Ω(m+1/(nλ))dx≠0 and
(5.16)μk(m+1nλ)≤μk(m)=λ<μk+1(m+1n0λ)≤μk+1(m+1nλ)≤μk+1(m)
for every n≥n0 by ∫Ωmdx≠0 and Proposition 2.10 (i), (iii). Moreover, we note the following inequality:
(5.17)Iλ,m,n0-(u)≤Iλ,m,n-(u)=Iλ,m+1/(nλ)(u)≤Iλ+1/(n‖m‖∞),m(u)
for every u∈W1,p(Ω) and n≥n0. It follows from Lemma 3.8 and (5.16) (note (5.17) also) that Iλ,m,n0-=Iλ,m0 is bounded from below on Y(μk+1(m0),m0) with m0:=m+1/(n0λ). Hence, (5.17) implies
(5.18)inf{Iλ,m,n-(u);u∈Y(μk+1(m0),m0)}≥inf{Iλ,m,n0-(u);u∈Y(μk+1(m0),m0)}=:α0>-∞
for every n≥n0. Because of λ+1/(n∥m∥∞)>λ=μk(m), there exist gn∈ℱk(m) and Tn>0 such that
(5.19)maxz∈Sk-1Iλ,m,n-(Tngn(z))≤maxz∈Sk-1Iλ+1/(n‖m‖∞),m(Tngn(z))<α0-1
by Lemma 3.5. Define
(5.20)Σn≔{g∈C(S+k,W1,p(Ω));g∣Sk-1=Tngn},cn-≔infg∈Σnmaxz∈S+kIλ,m,n-(g(z))
for n≥n0. Then, cn-≥α0 occurs (see (5.18)) since g(S+k)∩Y(μk+1(m0),m0)≠∅ for every g∈Σn by Lemma 3.4. This means that cn- is bounded from below. Consequently, we can obtain a desired bounded sequence by the same argument as in Sections 5.1 and 5.2.
5.4. The Case (iii) as in Theorem 1.2
First, note that we are assuming the hypothesis (H+) or (HF+) in this case (iii). In addition, as the reason in the proof of Theorem 1.2, it suffices to handle with λ>0.
Let k∈ℕ satisfy μk(m)<λ≤μk+1(m). According to Proposition 2.10 (i) and (ii), we can take an n0∈ℕ such that |{m-1/(nλ)>0}|>0 and
(5.21)μk(m-12nλ)≤μk(m-1n0λ)<λ-12n‖m‖∞<λ≤μk+1(m)≤μk+1(m-12nλ)
for every n≥n0. The following inequality follows from the easy estimates:
(5.22)Iλ,m,n0+(u)≥Iλ,m,n+(u)=Iλ,m-1/(nλ)(u)≥Iλ-1/(2n‖m‖∞),m-1/(2nλ)(u)
for every u∈W1,p(Ω) and n≥n0. Let n≥n0 and set mn:=m-1/(2nλ). Because of (5.21), Lemma 3.8 implies that Iλ-1/(2n∥m∥∞),mn is bounded from below on Y(μk+1(mn),mn) with (note ∫Ωmndx≠0). Hence, (5.22) yields that
(5.23)αn:=inf{Iλ,m,n+(u);u∈Y(μk+1(mn),mn)}>-∞
for each n≥n0. On the other hand, because of μk(m-1/(n0λ))<λ (see (5.21)), Lemma 3.5 guarantees the existence of g0∈ℱk(m-1/(n0λ)) satisfying
(5.24)maxz∈Sk-1Iλ,m,n0+(Tg0(z))=maxz∈Sk-1Iλ,m-1/(n0λ)(Tg0(z))⟶-∞asT⟶∞.
Therefore, for each n≥n0, we can choose Tn>0 such that
(5.25)maxz∈Sk-1Iλ,m,n+(Tng0(z))≤maxz∈Sk-1Iλ,m,n0+(Tng0(z))≤αn-1,
(note (5.22) for the first inequality). Set
(5.26)Σn≔{g∈C(S+k,W1,p(Ω));g∣Sk-1=Tng0},cn+≔infg∈Σnmaxz∈S+kIλ,m,n+(g(z))
for n≥n0. Since g(S+k)∩Y(μk+1(mn),mn)≠∅ for every g∈Σn by Lemma 3.4, we have cn+≥αn>maxz∈Sk-1Iλ,m,n+(Tng0(z)). Moreover, by the same argument as in Section 5.3 (note (5.24)), we have
(5.27)cn+≤supt≥0,z∈Sk-1Iλ,m,n+(tg0(z))≤supt≥0,z∈Sk-1Iλ,m,n0+(tg0(z))<+∞,
and hence our conclusion is shown.
Remark 5.1.
If ∫Ωmdx=0 holds, then we can not show the continuity of μk(m) with respect to m (refer to Proposition 2.10). Hence, we are not able to construct a bounded Palais-Smale sequence under (H-) or (HF-). However, if we have the additional information about the existence of a suitable m′∈L∞(Ω) such that m′≥m in Ω, ∫Ωm′dx≠0 and μk(m)≤λ<μk+1(m′) when μk(m)≤λ<μk+1(m) occurs, then we can still easily prove that equation (P;λ,m,h) has a solution in the case also where λ≠0, ∫Ωmdx=0 and (H-) or (HF-). In fact, let 0<μk(m)≤λ<μk+1(m′) for some k≥2. Note the following inequality:
(5.28)Iλ+1/(n‖m‖∞),m(u)≥Iλ,m,n-(u)≥Iλ,m′(u)-1np‖u‖pp=Iλ,m′-1/(nλ)(u)
for every u∈W1,p(Ω) and n. Fix n0∈ℕ such that ∫Ωm′-1/(n0λ)dx>0 and |{m′-1/(n0λ)>0}|>0. Set m0′:=m′-1/(n0λ). Because of λ<μk+1(m′)≤μk+1(m0′) (the last inequality follows from Proposition 2.10 (i)), Lemma 3.8 implies that Iλ,m0′ is bounded from below on Y(μk+1(m0′),m0′) (note ∫Ωm0′dx>0). By combining this fact and (5.28), we have
(5.29)infn≥n0inf{Iλ,m,n-(u);u∈Y(μk+1(m0′),m0′)}≥inf{Iλ,m0′(u);u∈Y(μk+1(m0′),m0′)}>-∞.
Because of λ+1/(n∥m∥∞)>λ≥μk(m), for each n≥n0, we can take a gn∈ℱk(m) satisfying
(5.30)maxz∈Sk-1Iλ,m,n-(Tgn(z))≤maxz∈Sk-1Iλ+1/(n‖m‖∞),m(Tgn(z))⟶-∞
as T→∞ by Lemma 3.5.
Since any extension g∈C(S+k,W1,p(Ω)) of Tgn (T>0) links Y(μk+1(m0′),m0′) by Lemma 3.4, we can construct a desired sequence by the same argument as in Section 5.3 under (H-) or (HF-).
Acknowledgments
The author would like to express her sincere thanks to Professor Shizuo Miyajima for helpful comments and encouragement. The author thanks the referees for his helpful comments and suggestions.
MotreanuD.PapageorgiouN. S.Multiple solutions for nonlinear Neumann problems driven by a nonhomogeneous differential operator2011139103527353510.1090/S0002-9939-2011-10884-02813384ZBL1226.35021DamascelliL.Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results199815449351610.1016/S0294-1449(98)80032-21632933ZBL0911.35009MotreanuD.MotreanuV. V.PapageorgiouN. S.Papageorgiou, multiple constant sign and nodal solutions for nonlinear neumann eigenvalue problems2011105729755MiyajimaS.MotreanuD.TanakaM.Multiple existence results of solutions for the Neumann problems via super- and sub-solutions20122621921195310.1016/j.jfa.2011.11.028RobinsonS. T.On the second eigenvalue for nonhomogeneous quasi-linear operators20043551241124910.1137/S00361410034260082050200ZBL1061.35071TanakaM.The antimaximum principle and the existence of a solution
for the generalized p-Laplace equations with indefinite weight201244GodoyT.GossezJ.-P.PaczkaS.On the antimaximum principle for the p-Laplacian with indefinite weight200251344946710.1016/S0362-546X(01)00839-21942756AnaneA.DakkakA.Nonresonance conditions on the potential for a Neumann problem2002229New York, NY, USADekker85102Lecture Notes in Pure and Applied Mathematics1913322ZBL1142.35400GossezJ.-P.OmariP.A necessary and sufficient condition of nonresonance for a semilinear Neumann problem199211424334421091181ZBL0755.35041OmariP.ZanolinF.Nonresonance conditions on the potential for a second-order periodic boundary value problem19931171125135114302110.1090/S0002-9939-1993-1143021-2ZBL0766.34020AmbrosettiA.ProdiG.199534Cambridge, UKCambridge University Press171Cambridge Studies in Advanced Mathematics1336591ZBL0936.00033ChenY.WangM.Large solutions for quasilinear elliptic equation with nonlinear gradient term201112145546310.1016/j.nonrwa.2010.06.0312729034ZBL1205.35106PangP. Y. H.WangY.YinJ.Periodic solutions for a class of reaction-diffusion equations with P-Laplacian201011132333110.1016/j.nonrwa.2008.11.0062570552JiaG.ZhaoQ.DaiC.-Y.Singular quasilinear elliptic problems with indefinite weights and critical potential20122815716410.1007/s10255-012-0131-0ZhangG.ManS.ZhangW.On a class of critical singular quasilinear elliptic problem with indefinite weights201174144771478410.1016/j.na.2011.04.0462810716ZBL1218.35155DrábekP.RobinsonS. B.Resonance problems for the p-Laplacian19991691189200172675210.1006/jfan.1999.3501StruweM.1999New York, NY, USASpringerHabibS. E.TsouliN.On the spectrum of the p-Laplacian operator for Neumann eigenvalue problems with weights20052005181190AriasM.CamposJ.CuestaM.GossezJ.-P.An asymmetric Neumann problem with weights200825226728010.1016/j.anihpc.2006.07.0062396522ZBL1138.35074CorvellecJ.-N.A general approach to the min-max principle19971624054331459966ZBL0884.58025PereraK.AgarwalR. P.O'ReganD.2010161Providence, RI, USAAmerican Mathematical Society141Mathematical Surveys and Monographs2640827SzulkinA.Ljusternik-Schnirelmann theory on C1-manifolds198852119139954468LiebermanG. M.Boundary regularity for solutions of degenerate elliptic equations198812111203121910.1016/0362-546X(88)90053-3969499ZBL0675.35042CasasE.FernandezL. A.A Green's formula for quasilinear elliptic operators19891421627310.1016/0022-247X(89)90164-91011409ZBL0704.35047MawhinJ.WillemM.1989New York, NY, USASpringer277982267