This paper shows the existence and multiplicity of nontrivial solutions of the p-Laplacian problem -Δpu=1/σ(∂F(x,u)/∂u)+λa(x)|u|q-2u for x∈Ω with zero Dirichlet boundary conditions, where Ω is a bounded open set in ℝn, 1<q<p<σ<p*(p*=np/(n-p) if p<n, p*=∞ if p≥n), λ∈ℝ∖{0}, a is a smooth function which may change sign in Ω̅,, and F∈C1(Ω̅ × ℝ,ℝ). The method is based on Nehari results on three submanifolds of the space W01,p(Ω).

1. Introduction

In this paper, we are concerned with the multiplicity of nontrivial nonnegative solutions of the following elliptic equation:
(1)-Δpu=1σ∂F(x,u)∂u+λa(x)|u|q-2uinΩ,u=0,on∂Ω,
where Ω is a bounded domain of ℝn(n≥3), 1<q<p<σ<p*(p*=np/(n-p) if p<n, p*=∞ if p≥n), λ∈ℝ∖{0}, F∈C1(Ω¯×ℝ,ℝ) is positively homogeneous of degree σ; that is, F(x,tu)=tσF(x,u) holds for all (x,u)∈Ω¯×ℝ and the sign-changing weight function a satisfies the following condition:

(A) a∈C(Ω) with ∥a∥∞=1, a+≔max(+a,0)≢0, and a-≔max(-a,0)≢0.

In recent years, several authors have used the Nehari manifold and fibering maps (i.e., maps of the form t↦Jλ(tu), where Jλ is the Euler function associated with the equation) to solve semilinear and quasilinear problems. For instance, we cite papers [1–9] and references therein. More precisely, Brown and Zhang [10] studied the following subcritical semilinear elliptic equation with sign-changing weight function:
(2)-Δu=λa(x)u+b(x)|u|γ-2u(x),inΩ,u=0,on∂Ω,
where γ>2. Also, the authors in [10] by the same arguments considered the following semilinear elliptic problem:
(3)-Δu=λf(x)|u|q-2u(x)+g(x)|u|p-2u(x),inΩ,u=0,on∂Ω,
where 1<q<2<p. Exploiting the relationship between the Nehari manifold and fibering maps, they gave an interesting explanation of the well-known bifurcation result. In fact, the nature of the Nehari manifold changes as the parameter λ crosses the bifurcation value.

Inspired by the work of Brown and Zhang [10], Nyamouradi [11] treated the following problem:
(4)Δ(|Δu|p-2Δu)=1p*f(x,u)+λ|u|q-2u(x);inΩ,u=Δu=0,on∂Ω,
where f is positively homogeneous of degree p*-1.

In this work, motivated by the above works, we give a very simple variational method to prove the existence of at least two nontrivial solutions of problem (1). In fact, we use the decomposition of the Nehari manifold as λ vary to prove our main result.

Before stating our main result, we need the following assumptions:

F:Ω¯×ℝ→ℝ is a C1 function such that
(5)F(x,tu)=tσF(x,u)(t>0)∀x∈Ω¯,u∈ℝ.

F(x,0)=(∂F/∂u)(x,0)=0, F+(x,u)=max(+F(x,u),0)≢0, and F-(x,u)=max(-F(x,u),0)≢0 for all u≠0.

We remark that using assumption (H_{1}), for all x∈Ω¯, u∈ℝ, we have the so-called Euler identity:
(6)u∂F(x,u)∂u=σF(x,u),|F(x,u)|≤K|u|σforsomeconstantK>0.
Our main result is the following.
Theorem 1.

Under the assumptions (A), (H_{1}), and (H_{2}), there exists λ0>0 such that for all 0<|λ|<λ0, problem (1) has at least two nontrivial nonnegative solutions.

This paper is organized as follows. In Section 2, we give some notations and preliminaries and we present some technical lemmas which are crucial in the proof of Theorem 1. Theorem 1 is proved in Section 3.

2. Some Notations and Preliminaries

Throughout this paper, we denote by Sl the best Sobolev constant for the operators W01,p(Ω)↪Ll(Ω), given by
(7)Sl=infu∈W01,p(Ω)∖{0}∫Ω|∇u|pdx(∫Ω|u|l)p/ldx,
where 1<l≤p*. In particular, we have
(8)∫Ω|u|ldx≤Sl-l/p∥u∥l∀u∈W01,p(Ω),
with the standard norm
(9)∥u∥=(∫Ω|∇u|pdx)1/p.
Problem (1) is posed in the framework of the Sobolev space E=W01,p(Ω). Moreover, a function u in E is said to be a weak solution of problem (1) if
(10)∫Ω|∇u|p-2∇u∇φdx-1σ∫Ω∂F(x,u)∂uφdx-λ∫Ωa|u|q-2uφdx=0,∀φ∈E.
Thus, by (6) the corresponding energy functional of problem (1) is defined in E by
(11)Jλ(u)=1p∥u∥p-1σ∫ΩF(x,u)dx-λq∫Ωa(x)|u|qdx.
In order to verify Jλ∈C1(E,ℝ), we need the following lemmas.

Lemma 2.

Assume that F∈C1(Ω¯×ℝ,ℝ) is positively homogeneous of degree σ; then ∂F/∂u∈C(Ω¯×ℝ,ℝ) is positively homogeneous of degree σ-1.

Proof.

The proof is the same as that in Chu and Tang [4].

In addition, by Lemma 2, we get the existence of positive constant M such that
(12)|∂F(x,u)∂u|≤M|u|σ-1,∀x∈Ω¯,u∈ℝ.

Lemma 3 (see [<xref ref-type="bibr" rid="B11">12</xref>], Theorem A.2).

Let p,r∈[1,∞) and f∈C(Ω¯×ℝ,ℝ) such that
(13)|f(x,u)|≤c(1+|u|p/r),∀x∈Ω¯,∀u∈ℝ.
Then for every u∈Lp(Ω), one has f(·,u)∈Lr(Ω); moreover the operator A:Lp(Ω)→Lr(Ω) defined by A(u)(x)=f(x,u(x)) is continuous.

Lemma 4 (See Proposition 1 in [<xref ref-type="bibr" rid="B16">13</xref>]).

Suppose that ∂F(x,u)/∂u∈C(Ω¯×ℝ,ℝ) verifies condition (12). Then, the functional Jλ belongs to C1(E,ℝ), and
(14)〈Jλ′(u),u〉=∥u∥p-∫ΩF(x,u)dx-λ∫Ωa(x)|u|qdx,
where 〈·,·〉 denotes the usual duality between E and E*:=W-1,p′(Ω) (the dual space of the sobolev space E).

As the energy functional Jλ is not bounded below in E, it is useful to consider the functional on the Nehari manifold:
(15)Nλ={u∈E∖{0}:〈Jλ′(u),u〉=0}.
Thus, u∈Nλ if and only if
(16)∥u∥p-∫ΩF(x,u)dx-λ∫Ωa(x)|u|qdx=0.
Note that Nλ contains every nonzero solution of problem (1). Moreover, one has the following result.

Lemma 5.

The energy functional Jλ is coercive and bounded below on Nλ.

Proof.

If u∈Nλ, then by (16) and condition (A) we obtain
(17)Jλ(u)=σ-pσp∥u∥p-λσ-qσq∫Ωa(x)|u|qdx,≥σ-pσp∥u∥p-|λ|σ-qσq∫Ω|u|qdx.
So, it follows from (8) that
(18)Jλ(u)≥σ-pσp∥u∥p-|λ|Sqq/pσ-qσq∥u∥q.
Thus, Jλ is coercive and bounded below on Nλ.

Define
(19)ϕλ(u)=〈Jλ′(u),u〉.
Then, by (16) it is easy to see that for u∈Nλ,
(20)〈ϕλ′(u),u〉=p∥u∥p-σ∫ΩF(x,u)dx-λq∫Ωa(x)|u|qdx,(21)=λ(p-q)∫Ωa(x)|u|qdx-(σ-p)∫ΩF(x,u)dx,(22)=λ(σ-q)∫Ωa(x)|u|qdx-(σ-p)∥u∥p,(23)=(p-q)∥u∥p-(σ-q)∫ΩF(x,u)dx.
Now, we split Nλ into three parts
(24)Nλ+={u∈Nλ:〈ϕλ′(u),u〉>0},Nλ0={u∈Nλ:〈ϕλ′(u),u〉=0},Nλ-={u∈Nλ:〈ϕλ′(u),u〉<0}.

Lemma 6.

Assume that u0 is a local minimizer for Jλ on Nλ and that u0∉Nλ0. Then, Jλ′(u0)=0 in E-1 (the dual space of the Sobolev space E).

Proof.

Our proof is the same as that in Brown-Zhang [10, Theorem 2.3].

Lemma 7.

One has the following:

if u∈Nλ+, then λ∫Ωa(x)|u|qdx>0;

if u∈Nλ0, then λ∫Ωa(x)|u|qdx>0 and ∫ΩF(x,u)dx>0;

if u∈Nλ-, then ∫ΩF(x,u)dx>0.

Proof.

The proof is immediate from (21), (22), and (23).

From now on, we denote by λ0 the constant defined by
(25)λ0=q(σ-p)p(σ-q)Sqq/p(p-qK(σ-q)Sσσ/p)(p-q)(σ-p),
then we have the following.

Lemma 8.

If 0<|λ|<λ0, then Nλ0=⌀.

Proof.

Suppose otherwise, that 0<|λ|<λ0 such that Nλ0≠⌀. Then for u∈Nλ0, we have
(26)0=〈ϕλ′(u),u〉=λ(σ-q)∫Ωa(x)|u|qdx-(σ-p)∥u∥p(27)=(p-q)∥u∥p-(σ-q)∫ΩF(x,u)dx.
From the Hölder inequality, (6) and (8), it follows that
(28)∫ΩF(x,u)dx≤∫Ω|F(x,u)|dx≤K∫Ω|u|σdx≤KSσ-σ/p∥u∥σ.
Hence, it follows from (27) that
(29)∥u∥p=σ-qp-q∫ΩF(x,u)dx≤σ-qp-qKSσ-σ/p∥u∥σ,
then,
(30)∥u∥≥(p-qK(σ-q)Sσσ/p)1/(σ-p).
On the other hand, from condition (A), (8) and (26) we have(31)∥u∥p=λσ-qσ-p∫Ωa(x)|u|qdx≤|λ|σ-qσ-pSq-q/p∥u∥q.
So,
(32)∥u∥≤(|λ|σ-qσ-pSq-q/p)1/(p-q).
Combining (30) and (32), we obtain λ0≤|λ|, which is a contradiction.

By Lemma 8, for 0<|λ|<λ0, we write Nλ=Nλ+∪Nλ- and define
(33)θλ=infu∈NλJλ(u),θλ+=infu∈Nλ+Jλ(u),θλ-=infu∈Nλ-Jλ(u).
Then, we have the following.

Lemma 9.

If 0<|λ|<λ0, then
(34)θλ≤θλ+<0,θλ->d0
for some d0>0 depending on p,q,σ,K,λ,Sq, and Sσ.

Proof.

Let u∈Nλ+. Then, from (23) we have
(35)p-qσ-q∥u∥p>∫ΩF(x,u)dx.
So
(36)Jλ(u)=q-ppq∥u∥p+σ-qσq∫ΩF(x,u)dx<(q-ppq+σ-qσqp-qσ-q)∥u∥p=-(p-q)(σ-p)pqσ∥u∥p<0.
Thus, from the definition of θλ and θλ+, we can deduce that θλ≤θλ+<0.

Now, let u∈Nλ-. Then, using (6) and (8) we obtain
(37)p-qσ-q∥u∥p<∫ΩF(x,u)dx≤KSσ-σ/p∥u∥σ,
this implies that
(38)∥u∥>(p-qσ-qSσσ/pK)1/(σ-p),∀u∈Nλ-.
In addition, by (18) and (38)
(39)Jλ(u)≥σ-ppσ∥u∥p-|λ|Sq-q/pσ-pσq∥u∥q≥∥u∥q[σ-ppσ∥u∥p-q-|λ|Sq-q/pσ-qσq]>(p-qσ-qSσσ/pK)q/(σ-p)(σ-ppσ(p-qσ-qSσσ/pK)(p-q)/(σ-p)-|λ|Sq-q/pσ-qσqσ-ppσ(p-qσ-qSσσ/pK)(p-q)/(σ-p)).
Thus, since 0<|λ|<λ0, we conclude that Jλ>d0 for some d0>0. This completes the proof.

For u∈E with ∫ΩF(x,u)dx>0, set
(40)T=((p-q)∥u∥p(σ-q)∫ΩF(x,u)dx)1/(σ-p)>0.
Then, the following lemma holds.

Lemma 10.

For each u∈E with ∫ΩF(x,u)dx>0, one has the following:

if λ∫Ωa(x)|u|qdx≤0, then there exists unique t->T such that t-u∈Nλ- and
(41)Jλ(t-u)=supt≥0Jλ(tu);

if λ∫Ωa(x)|u|qdx>0, then there are unique 0<t+<T<t- such that (t-u,t+u)∈Nλ-×Nλ+ and
(42)Jλ(t-u)=supt≥0Jλ(tu),Jλ(t+u)=inf0≤t≤TJλ(tu).

Proof.

We fix u∈E with ∫ΩF(x,u)dx>0 and we let
(43)m(t)=tp-q∥u∥p-tσ-q∫ΩF(x,u)dxfort≥0.
Then, it is easy to check that m(t) achieves its maximum at T. Moreover,
(44)m(T)=∥u∥q[(p-qσ-q)(p-q)(σ-p)-(p-qσ-q)(σ-q)/(σ-p)]×(∥u∥σ∫ΩF(x,u)dx)(p-q)/(σ-p)≥∥u∥q(σ-pσ-q)((σ-q)Sσσ/pK(p-q))(p-q)/(σ-p).

(i) We suppose that λ∫Ωa(x)|u|qdx≤0. Since m(0)=0,m(t)→-∞ as t→∞, m′(t)>0 for t<T and m′(t)<0 for t>T. There is a unique t->T such that m(t-)=λ∫Ωa(x)|u|qdx≤0.

Now, it follows from (14) and (27) that
(45)ϕλ′(t-u)t-u=(t-)1+qm′(t-)<0,Jλ′(t-u)t-u=(t-)q(m(t-)-λ∫Ωa(x)|u|qdx)=0.
Hence, t-u∈Nλ-. On the other hand, it is easy to see that for all t>T(46)d2dt2Jλ(tu)<0,ddtJλ(tu)=0fort=t-.
Thus, Jλ(t-u)=supt≥0Jλ(tu).

(ii) We suppose that λ∫Ωa(x)|u|qdx>0. Then, by (A), (8) and the fact that |λ|<λ0 we obtain
(47)m(0)=0<λ∫Ωa(x)|u|qdx≤|λ|Sq-q/p∥u∥q<m(T).
Then, there are unique t+ and t- such that 0<t+<T<t-, m(t+)=λ∫Ωa(x)|u|qdx=m(t-), and m′(t-)<0<m′(t+). We have (t-u,t+u)∈Nλ-×Nλ+, and
(48)Jλ(t+u)≤Jλ(tu)≤Jλ(t-u)∀t∈[t+,t-],Jλ(t+u)≤Jλ(tu)∀0≤t≤t+.
Thus,
(49)Jλ(t-u)=supt≥0Jλ(tu),Jλ(t+u)=inf0≤t≤TJλ(tu).
This completes the proof.

For each u∈E with λ∫Ωa(x)|u|qdx>0, set
(50)T~=((σ-q)λ∫Ωa(x)|u|qdx(σ-p)∥u∥p)1/(p-q)>0.
Then we have the following.

Lemma 11.

For each u∈E with λ∫Ωa(x)|u|qdx>0, one has the following:

if ∫ΩF(x,u)dx≤0, then there exists a unique 0<t+<T~ such that t+u∈Nλ+ and
(51)Jλ(t+u)=inft≥0Jλ(tu);

if ∫ΩF(x,u)dx>0, then there are unique 0<t+<T~<t- such that (t-u,t+u)∈Nλ-×Nλ+ and
(52)Jλ(t-u)=supt≥0Jλ(tu),Jλ(t+u)=inf0≤t≤T~Jλ(tu).

Proof.

For u∈E with λ∫Ωa(x)|u|qdx>0, we can take
(53)m~(t)=tp-σ∥u∥p-λtq-σ∫Ωa(x)|u|qdxfort>0,
and similar to the argument in Lemma 9, we obtain the results of Lemma 10.

Proposition 12.

(i) There exist minimizing sequences {un+} in Nλ+ such that
(54)Jλ(un+)=θλ++∘(1),Jλ′(un+)=∘(1)inE-1.

(ii) There exist minimizing sequences {un-} in Nλ- such that
(55)Jλ(un-)=θλ-+∘(1),Jλ′(un-)=∘(1)inE-1.

Proof.

The proof is almost the same as that in Wu [14, Proposition 9] and is omitted here.

3. Proof of Our Result

Throughout this section, the norm Ls is denoted by ∥·∥s for 1≤s≤∞ and the parameter λ satisfies 0<|λ|<λ0.

Theorem 13.

If 0<|λ|<λ0, then, problem (1) has a positive solution u0+ in Nλ+ such that
(56)Jλ(u0+)=θλ=θλ+.

Proof.

By Proposition 12(i), there exists a minimizing sequence {un+} for Jλ on Nλ+ such that
(57)Jλ(un+)=θλ++o(1),Jλ′(un+)=o(1)inE-1.
Then by Lemma 5, there exists a subsequence {un} and u0+ in E such that
(58)un⇀u0+weaklyinE,un⟶u0+stronglyinLq(Ω)andinLσ(Ω).
This implies that ∫Ωa(x)|un|qdx→∫Ωa(x)|u0+|qdx as n→∞.

Next, we will show that
(59)∫ΩF(x,un)dx⟶∫ΩF(x,u0+)dxasn⟶∞.
By Lemma 3, we have
(60)∂F(x,un)∂u∈Lγ(Ω),∂F(x,un)∂u⟶∂F(x,u0+)∂uinLγ(Ω),
where γ=σ/(σ-1). On the other hand, it follows from the Hölder inequality that
(61)∫Ω|un∂F(x,un)∂u-u0+∂F(x,u0+)∂u|dx≤∫Ω|(un-u0+)∂F(x,un)∂u|dx+∫Ω|u0+||∂F(x,un)∂u-∂F(x,u0+)∂u|dx≤∥un-u0+∥σ∥∂F(x,un)∂u∥γ+∥u0+∥σ∥∂F(x,un)∂u-∂F(x,u0+)∂u∥γ⟶0asn⟶∞.
Hence, ∫ΩF(x,un)dx→∫ΩF(x,u0+)dx as n→∞.

By (57) and (58) it is easy to prove that u0+ is a weak solution of (1).

Since
(62)Jλ(un)=σ-ppσ∥un∥p-λσ-qqσ∫Ωa(x)|un|qdx≥-λσ-qqσ∫Ωa(x)|un|qdx
then by (57) and Lemma 9, we have Jλ(un)→θλ<0 as n→∞. Letting n→∞, we obtain
(63)λ∫Ωa(x)|u0+|qdx>0.
Now, we aim to prove that un→u0+ strongly in E and Jλ(u0+)=θλ.

Using the fact that u0+∈Nλ and by Fatou's lemma, we get
(64)θλ≤Jλ(u0+)=1p∥u0+∥p-1σ∫ΩF(x,u0+)dx-λq∫Ωa(x)|u0+|qdx≤liminfn→∞(1p∥un∥p-1σ∫ΩF(x,un)dx-λq∫Ωa(x)|un|qdx)≤liminfn→∞Jλ(un)=θλ.
This implies that
(65)Jλ(u0+)=θλ,limn→∞∥un∥p=∥u0+∥p.
Let u~n=un-u0+; then by Brézis-Lieb Lemma [3] we obtain
(66)limn→∞(∥un∥p-∥u~n∥p)=∥u0+∥p.
Therefore, un→u0+ strongly in E.

Moreover, we have u0+∈Nλ+. In fact, if u0+∈Nλ- then, there exist t0+,t0- such that t0-u0+∈Nλ- and t0+u0+∈Nλ+. In particular we have t0+<t0-=1. Since
(67)d2dt2Jλ(t0+u0+)>0,ddtJλ(t0+u0+)=0,
there exists t0+<t~<t0- such that Jλ(t0+u0+)<Jλ(t~u0+). By Lemma 10, we have
(68)Jλ(t0+u0+)<Jλ(t~u0+)≤Jλ(t0-u0+)=Jλ(u0+),
which is a contradiction.

Finally, by (63) we may assume that u0+ is a nontrivial nonnegative solution of problem (1).

Theorem 14.

If 0<|λ|<λ0, then, problem (1) has a positive solution u0- in Nλ- such that
(69)Jλ(u0-)=θλ-.

Proof.

By Proposition 12(ii), there exists a minimizing sequence {un} for Jλ on Nλ- such that
(70)Jλ(un)=θλ-+o(1),Jλ′(un)=o(1)inE-1,(71)un⇀u0-weaklyinE,un⟶u0-stronglyinLq(Ω)andinLσ(Ω).
Moreover, by (23) we obtain
(72)∫ΩF(x,un)dx>p-qσ-q∥un∥p.
So, by (38) and (72) there exists a positive constant C~ such that
(73)∫ΩF(x,un)>C~.
This implies that
(74)∫ΩF(x,u0-)≥C~.
By (70) and (71), we obtain clearly that u0- is a weak solution of (1).

Now, we aim to prove that un→u0- strongly in E. Supposing otherwise, then
(75)∥u0-∥<liminfn→∞∥un∥.
By Lemma 9, there is a unique t0- such that t0-u0-∈Nλ-. Since un∈Nλ-, Jλ(un)≥Jλ(tun) for all t≥0, we have
(76)Jλ(t0-u0-)<limn→∞Jλ(t0-un)≤limn→∞Jλ(un)=θλ-,
which is a contradiction. Hence un→u0- strongly in E.

This imply that
(77)Jλ(un)⟶Jλ(u0-)=θλ-asn⟶∞.
By Lemma 5 and (74) we may assume that u0- is a nontrivial solution of problem (1).

Now, we begin to show the proof of Theorem 1: by Theorem 13, we obtain that for all 0<λ<λ0, problem (1) has a nontrivial solution u0+∈Nλ+. On the other hand, from Theorem 14, we get the second solution u0-∈Nλ-. Since Nλ-∩Nλ+=∅, then u0- and u0+ are distinct.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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