Using variational arguments we prove some existence and nonexistence results for positive solutions of a class of elliptic boundary-value problems involving the p-Laplacian.

1. Introduction

In a recent paper, Rădulescu and Repovš [1] studied the existence and nonexistence of positive solutions of the nonlinear elliptic problem
(1.1)-Δu=λk(x)uq±h(x)upinΩ,u|∂Ω=0,u>0inΩ,
where Ω is a smooth bounded domain in ℝn, λ>0 is a parameter, 0<q<1<p, and h, k in L∞(Ω) such that
(1.2)essinfx∈Ωk(x)>0,essinfx∈Ωh(x)>0.
They showed using sub-supersolutions arguments and monotonicity methods that the problem (1.1)_{+} has a minimal solution, provided that λ>0 is small enough. The next result is concerned with problem (1.1)_{−} and asserts that there is some λ*>0 such that (1.1)_{−} has a nontrivial solution if λ>λ* and no solution exists provided that λ<λ*.

In the present paper we consider that the corresponding quasilinear problem
(1.3)-Δpu=λk(x)uq±h(x)urinΩ,u|∂Ω=0,u>0inΩ,
where Δpu=div(|∇u|p-2∇u), denotes the p-Laplacian operator, 1<p<∞, λ>0, 0≤q<p-1<r<p*-1, with p*=Np/(N-p) if p<N, and p*=+∞ otherwise, and h, k in L∞(Ω) such that
(1.4)essinfx∈Ωk(x)>0,essinfx∈Ωh(x)>0.
We are concerned with the existence of weak solutions of problems (1.3)_{+} and (1.3)_{−}, that is, for functions u∈W01,p(Ω) satisfying essinfKu>0 over every compact set K⊂Ω and
(1.5)∫Ω|∇u|p-2∇u⋅∇ϕdx=λ∫Ωk(x)uqϕdx±∫Ωh(x)urϕdx
for all ϕ∈Cc∞(Ω). As usual, Cc∞(Ω) denotes the space of all C∞ functions ϕ:Ω→ℝ with compact support. Using variational methods, we will prove the following theorems.

Theorem 1.1.

Assume 0≤q<p-1<r<p*-1. Then there exists a positive number Λ such that the following properties hold:

for all λ∈(0,Λ) problem (1.3)_{+} has a minimal solution uλ;

Problem (1.3)_{+} has a solution if λ=Λ;

Problem (1.3)_{+} does not have any solution if λ>Λ.

Theorem 1.2.

Assume 0≤q<p-1<r<p*-1. Then there exists a positive number Λ such that the following properties hold:

If λ>Λ, then problem (1.3)_{−} has at least one solution;

If λ<Λ, then problem (1.3)_{−} does not have any solution.

2. Proof of Theorem <xref ref-type="statement" rid="thm1.1">1.1</xref>

At first, we give the definition of weak supersolution and subsolution of (1.3)_{+}. By definition u∈W01,p(Ω) is a weak subsolution to (1.3)_{+} if u>0 in Ω and
(2.1)∫Ω|∇u|p-2∇u⋅∇ϕdx≤λ∫Ωk(x)uqϕdx±∫Ωh(x)urϕdx
for all ϕ∈Cc∞(Ω). Similarly u∈W01,p(Ω) is a weak supersolution to (1.3)_{+} if in the above the reverse inequalities hold.

Let us define
(2.2)Λ=defsup{λ>0:(1.3)+hasaweaksolution}
and the energy functional Eλ:W01,p(Ω)→ℝ defined by
(2.3)Eλ(u)=1p∫Ω|∇u|pdx-λq+1∫Ωk(x)uq+1dx-1r+1∫Ωh(x)uq+1dx
in the Sobolev space W01,p(Ω).

The proof of the theorem is organized in several steps.

Step 1 (existence of minimal solution for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M56"><mml:mn mathvariant="normal">0</mml:mn><mml:mo><</mml:mo><mml:mi>λ</mml:mi><mml:mo><</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></inline-formula>).

To show the existence of a solution to (1.3)_{+}, we construct a subsolution u_λ, and a supersolution u¯λ, such that u_λ≤u¯λ.

We introduce the following Dirichlet problem:
(2.4)-Δpu~=λk(x)u~qinΩ,u~|∂Ω=0,u~>0inΩ.
From [2] we know there exists a unique solution, say u~, satisfying the problem (2.4). Define u_λ=ϵu~. Then -Δp(u_λ)=λk(x)ϵp-1u~q and u_λ is a subsolution of the problem (1.3)_{+} if
(2.5)λk(x)ϵp-1u~q≤λk(x)ϵqu~q+h(x)ϵru~r.
Indeed, for ϵ small enough we get
(2.6)λk(x)ϵp-1u~q≤λk(x)ϵqu~q≤λk(x)ϵqu~q+h(x)ϵru~r.
(Since q<p-1 and for ϵ∈(0,1)). Then ϵu~ is a subsolution of the problem (1.3)_{+}.

On the other hand, let v the solution to the following problem be:
(2.7)-Δpv=λ+1inΩ,v|∂Ω=0,v>0inΩ.

Then 0<v<K in Ω. By simplicity of writing we call
(2.8)F(u)=λk(x)uq+h(x)ur.

Define u¯λ(x)=Tv(x) where T is a constant that will be chosen in such a way that
(2.9)-Δpu¯λ≥F(TM)≥F(u¯λ),
where M=max{1,∥v∥∞}. Now -Δpu¯λ=Tp-1(λ+1) and
(2.10)F(u¯λ)≡λk(x)Tqvq+Trvr≤λc1TqMq+c2TrMr,
where c1=∥k∥L∞ et c2=∥h∥L∞. Then, it is sufficient to find T such that
(2.11)(λ+1)≥λc1Tq+1-pMq+c2Tr+1-pMr.

We call
(2.12)φ(T)=λATq+1-p+BTr+1-p,
with A=c1Mq,B=c2Mr. Then
(2.13)limT→0+φ(T)=limT→∞φ(T)=∞,
because q+1-p<0<r+1-p; then φ attains a minimum in [0,∞). Elementary computations shows that this function attains its minimum for T0=Cλ1/(r-q) where C=[AB-1(r-p+1)(p-q-1)-1]1/(r-q). For the validity of (2.11) it suffices that
(2.14)φ(T0)≤λ+1,
that is,
(2.15)Dλ(r+1-p)/(r-q)<λ+1,
where D is a constant, depends on p,q, and M. Then there exists λ0 such that for 0<λ<λ0,u¯(x)=T0v is a supersolution of problem (1.3)_{+}. It remains to show that ϵu~≤T0v. In turn, fix the supersolution, that is, T, for ϵ small enough, we get
(2.16)-Δpu_λ=λk(x)ϵp-1u~q≤λϵp-1≤-Δp(u¯λ).
Consequently, we may apply the weak comparison principle (see Proposition 2.3 in [3]) in order to conclude that u_λ≤u¯λ. Thus, By the classical iteration method (1.3)_{+} has a solution between the subsolution and supersolution.

Let us now prove that uλ is a minimal solution of (1.3)_{+}. We use here the weak comparison principle (see Proposition 2.3 in Cuesta and Takác˘ [3]) and the following monotone iterative scheme:
(2.17)-Δpun=λk(x)un-1q+h(x)un-1rinΩ;un|∂Ω=0,
where u0=u_λ, the unique solution to (2.4). Note that u0 is a weak subsolution to (1.3)_{+}. and u0≤U where U is any weak solution to (1.3)_{+}. Then, from the weak comparison principle, we get easily that u0≤u1 and {un}n=1∞ is a nondecreasing sequence. Furthermore, un≤U and {un}n=1∞ is uniformly bounded in W01,p(Ω). Hence, it is easy to prove that {un} converges weakly in W01,p(Ω) and pointwise to u^λ, a weak solution to (1.3)_{+}. Let us show that u^λ is the minimal solution to (1.3)_{+} for any 0<λ<Λ. Let vλ a weak solution to (1.3)_{+} for any 0<λ<Λ. Then, u0=u_λ≤vλ. From the weak comparison principle, un≤vλ for any n≥0. Letting n→∞, we get u^λ≤vλ. This completes the proof of the Step 1.

Step 2 (there exists <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M130"><mml:mi mathvariant="normal">Λ</mml:mi><mml:mo>></mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math></inline-formula> such that (<xref ref-type="disp-formula" rid="EEq1.3">1.3</xref>)<sub>+</sub> has no positive solution for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M131"><mml:mi>λ</mml:mi><mml:mo>></mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></inline-formula>).

From the definition of Λ, problem (1.3)_{+} does not have any solution if λ>Λ. In what follows we claim that Λ<∞. We argue by contradiction: suppose there exists a sequence λn→∞ such that (1.3)_{+} admits a solution un. Denote
(2.18)m:=min{essinfx∈Ωk(x),essinfx∈Ωh(x)}>0.
There exists λ*>0 such that
(2.19)m(λtq+tr)≥(λ1+ϵ)tp-1∀t>0,ϵ∈(0,1),λ>λ*,
where λ1 is the first Dirichlet eigenvalue of -Δp is positive and is given by
(2.20)λ1=minu≠0∫Ω|∇u|p∫Ω|u|p
(see Lindqvist [4]). Choose λn>λ*. Clearly un is a supersolution of the problem
(2.21)-Δpu=(λ1+ϵ)up-1inΩ,u>0,u|∂Ω=0
for all ϵ∈(0,1). We now use the result in [2] to choose μ<λ1+ϵ small enough so that μϕ1(x)<un(x) and μϕ1 is a subsolution to problem (2.8). By a monotone interation procedure we obtain a solution to (2.8) for any ϵ∈(0,1), contradicting the fact that λ1 is an isolated point in the spectrum of -Δp in W01,p(Ω) (see Anane [5]). This proves the claim and completes the proof of the Step 2.

Step 3 (there exists at least one positive-weak solution for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M154"><mml:mi>λ</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></inline-formula> to (<xref ref-type="disp-formula" rid="EEq1.3">1.3</xref>)<sub>+</sub>).

Let {λk}k∈ℕ be such that λk↑Λ as k→∞. Then, from Step 1, there exists uk=uλk≥u_λk to a weak positive solution to (1.3)_{+} for λ=λk. Therefore, for any ϕ∈Cc∞(Ω), we have
(2.22)∫Ω|∇uk|p-2∇uk∇ϕdx=λkk(x)∫Ω(uk)qϕdx+h(x)∫Ωukrϕdx.
Since uk∈W01,p(Ω) and uk≥u_λk it is easy to see that (2.22) holds also for ϕ∈W01,p(Ω). Moreover, from above
(2.23)Eλk(uk)≤Eλk(u_λk)<1p∫Ω|∇u_λk|pdx-λkk(x)q+1∫Ωu_λkq+1dx<0,
it follows that
(2.24)supk‖uk‖p<∞.

Hence, there exists uΛ≥u_λk such that uk⇀uΛ in W01,p(Ω) as k→∞ and then by Sobolev imbedding and using the fact that k,h∈L∞(Ω):
(2.25)uk⇀uinLq(Ω)andpointwisea.e.ask⟶∞.
From (2.22), (2.24), and (2.25), we get for any ϕ∈W01,p(Ω)(2.26)∫Ω|∇uΛ|p-2∇uΛ∇ϕdx=λ∫Ωk(x)uΛqϕdx+∫Ωh(x)uΛrϕdx
which completes the proof of the Step 3 and gives the proof of Theorem 1.1.

3. Proof of Theorem <xref ref-type="statement" rid="thm1.2">1.2</xref>

At first, we introduce some notation which will be used throughout the proof. The norm in W01,p(Ω) will be denoted by
(3.1)‖u‖p=def(∫Ω|∇u|pdx)1/p.
The norm in Lq+1(Ω) will be denoted by
(3.2)‖u‖q+1=def(∫Ω|u|q+1dx)1/q+1.
The norm in Lr+1(Ω) will be denoted by
(3.3)‖u‖r+1=def(∫Ω|u|r+1dx)1/r+1.
Let us define the energy functional Jλ:W01,p(Ω)→ℝ defined by
(3.4)Jλ(u)=1p∫Ω|∇u|pdx-λq+1∫Ωk(x)uq+1dx+1r+1∫Ωh(x)ur+1dx
in the Sobolev space W01,p(Ω).

The proof of the theorem is organized in several steps.

Step 1 (coercivity of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M184"><mml:mrow><mml:msub><mml:mrow><mml:mi>J</mml:mi></mml:mrow><mml:mrow><mml:mi>λ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>:).

For any u∈W01,p(Ω) and all λ>0(3.5)Jλ(u)≥1p‖u‖p-C1‖u‖q+1q+1+C2‖u‖r+1r+1,
where C1=λ∥k∥L∞/(q+1) and C2=(r+1)-1essinfx∈Ωh(x) are positive constants. We call
(3.6)ϕ(T)=ATq+1-p-BTr+1-p.
Then
(3.7)limT→0+ϕ(T)=limT→∞ϕ(T)=∞,
because q+1-p<0<r+1-p; then φ attains a minimum m<0 in [0,∞). By elementary computations shows that this function attains its minimum for T=[A(q+1-p)/(Br+1-p)]1/(r-q).

Returning to (3.5), we deduce that
(3.8)Jλ(u)≥1p‖u‖p+m.
Hence, from (3.8), we get that
(3.9)Jλ(u)⟶+∞as‖u‖⟶∞.

Let n↦un be a minimizing sequence of Jλ in W01,p(Ω), which is bounded in W01,p(Ω) by Step 1. Without loss of generality, we may assume that (un)n is nonnegative, converges weakly to some u in W01,p(Ω), and converges also pointwise. Moreover, by the weak lower semicontinuity of the norm ∥·∥ and the boundedness of (un)n in W01,p(Ω) we get
(3.10)Jλ(u)≤limn→∞infJλ(un).
Hence u is a global minimizer of Jλ in W01,p(Ω), which completes the proof of the Step 1.

Step 2 (the weak limit <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M213"><mml:mrow><mml:mi>u</mml:mi></mml:mrow></mml:math></inline-formula> is a nonnegative weak solution of (<xref ref-type="disp-formula" rid="EEq1.3">1.3</xref>)<sub>-</sub> if <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M214"><mml:mi>λ</mml:mi><mml:mo>></mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math></inline-formula> is sufficiently large).

Firstly, observe that Jλ(0)=0. Thus, to prove that the nonnegative solution is nontrivial, it suffices to prove that there exists λ*>0 such that
(3.11)infu∈W01,p(Ω)Jλ(u)<0∀λ>λ*.
For this, we consider the constrained minimization problem
(3.12)λ*=definf{1p∫Ω|∇w|pdx+1r+1∫Ωh(x)|w|r+1dx:w∈W01,p(Ω)and1q+1×∫Ωk(x)|w|q+1dx=1}.

Let n↦vn be a minimizing sequence of (3.12) in W01,p(Ω), which is bounded in W01,p(Ω), so that we can assume, without loss of generality, that it converges weakly to some v∈W01,p(Ω), with
(3.13)1q+1∫Ωk(x)|v|q+1dx=1,λ*=1p∫Ω|∇v|pdx+1r+1∫Ωh(x)|v|r+1dx.
Thus, Jλ(v)=λ*-λ<0 for any λ>λ*.

Now put
(3.14)Λ=definf{λ>0:(1.3)-admitsanontrivialweaksolution}.
From above λ*≥Λ and that problem (1.3)_{−} has a solution for all λ>λ*. The proof of the Step 2 is now completed.

Step 3 (problem (<xref ref-type="disp-formula" rid="EEq1.3">1.3</xref>)<sub>−</sub> has a weak solution for any <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M229"><mml:mi>λ</mml:mi><mml:mo>></mml:mo><mml:mi mathvariant="normal">Λ</mml:mi></mml:math></inline-formula>).

By the definition of Λ, there exists μ∈(Λ,λ) such that Jμ has a nontrivial critical point uμ∈W01,p(Ω). Since μ<λ,uμ is a subsolution of the problem (1.3)_{−}. In order to find a super-solution of the problem (1.3)_{−} which dominates uμ, we consider the constrained minimization problem
(3.15)inf{Jλ(w);w∈W01,p(Ω)andw≥uμ.}.
Arguments similar to those used in Step 2 show that the above minimization problem has a solution uλ≥uμ which is also a weak solution of problem (1.3)_{−}, provided λ>Λ.

Using similar arguments as in [6]. Thus, from Theorem 2.2 in Pucci and Servadei [7], based on the Moser iteration, it is clear that u∈Lloc∞. Next, again by bootstrap regularity [Corollary on p. 830] due to DiBenedetto, [8] shows that the weak solution u∈C1,α(Ω) where α∈(0,1). Finally, the nonnegative follows immediately by the strong maximum principle since u is a C1 nonnegative weak solution of the differential inequality ∇(|∇u|p-2∇u)-h(x)ur≤0 in Ω, with p-1<r, see, for instance, Section 4.8 of Pucci and Serrin [9]. Thus, u>0 in Ω. The proof of the Step 3 is now completed.

Step 4 (nonexistence for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M250"><mml:mi>λ</mml:mi><mml:mo>></mml:mo><mml:mn mathvariant="normal">0</mml:mn></mml:math></inline-formula> is small).

The same monotonicity arguments as in Step 3 show that (1.3)_{−} does not have any solution if λ<Λ, which completes the proof of the Theorem 1.2.

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