# Existence and Multiplicity of Solutions for Semipositone Problems Involvingp-Laplacian

Authors: Guowei Dai; Chunfeng Yang
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

---

## Abstract

We prove existence and multiplicity of positive solutions for semipositone problems involvingp-Laplacian in a bounded smooth domain of ℝN under the cases of sublinear and superlinear nonlinearities term.


---

## Body

## 1. Introduction

In this paper, we shall study the following semipositone problem involving thep-Laplacian:
(1.1)-div⁡(|∇u|p-2∇u)=λf(u)inΩ,u=0on⁡∂Ω,
where Ω⊂ℝN is a smooth bounded domain, λ is a positive parameter, and f:[0,+∞)→ℝ is a continuous function satisfying the condition(F0)
f(0)=-a<0.Such problems are usually referred in the literature as semipositone problems. We refer the reader to [1], where Castro and Shivaji initially called them nonpositone problems, in contrast with the terminology positone problems, coined by Keller and Cohen in [2], when the nonlinearity f was positive and monotone.Under the case ofp≡2, a novel variational approach is presented by Costa et al. [3] to the question of existence and multiplicity of positive solutions to problem (1.1), where they consider both the sublinear and superlinear cases. The aim of this paper is to extend their results to the case of p-Laplacian. The main difficulty is in verifying (PS)c-condition because of the operator -Δp is not self-adjoint linear.We define the discontinuous nonlinearityg(s) by
(1.2)g(s)={0,ifs≤0,f(s),ifs>0.
We shall consider the modified problem
(1.3)-div⁡(|∇u|p-2∇u)=λg(u)inΩ,u=0on⁡∂Ω.
We note that the set of positive solutions of (1.1) and (1.3) do coincide.Our main results concerning problems (1.1) and (1.3) are the following:Theorem 1.1.
Assume (F0) and the following assumptions:(F1)
lims→+∞(f(s)/sp-1)=0 (the sublinear case);(F2)
F(δ)>0 for some δ>0, where F(u)=∫0u‍f(s)ds.Then, there exist 0<λ0≤λ* such that (1.3) has no nontrivial nonnegative solution for 0<λ<λ0, and has at least two nontrivial nonnegative solutions uλ, vλ for all λ>λ*. Moreover, when Ω is a ball BR=BR(0), these two solutions are non-increasing, radially symmetric and, if N≥2, at least one of them is positive, hence a solution of (1.1).Theorem 1.2.
Assume (F0), (F2) and the following assumptions:(F3)
lims→+∞(f(s)/sp-1)=+∞, lims→+∞(f(s)/sp*-2)=0 (the sublinear, subcritical case);(F4)
θF(s)≤f(s)sfor all s≥K and some θ>p (the AR-condition).Then, (1.3) has at least one nonnegative solution vλ for all λ>0. IfΩ=BR then vλ is nonincreasing, radially symmetric and one of the two alternatives occurs. (i)
There existsλ*>0 such that, for all 0<λ<λ*, vλ is a positive solution of (1.1) having negative normal derivative on ∂BR.(ii)
For some sequenceμn→0, problem (1.1) with λ≡μn has a positive solution wμn with zero normal derivative on ∂BR.


## 2. Preliminaries

We start by recalling some basic results on variational methods for locally Lipschitz functionals. Let(X,∥·∥) be a real Banach space and X* is its topological dual. A function f:X→ℝ is called locally Lipschitz if each point u∈X possesses a neighborhood Ωu such that |f(u1)-f(u2)|≤L∥u1-u2∥ for all u1,u2∈Ωu, for a constant L>0 depending on Ωu. The generalized directional derivative of f at the point u∈X in the direction v∈X is
(2.1)f0(u,v)=limsup⁡w→u,t→0+1t(f(w+tv)-f(w)).
The generalized gradient of f at u∈X is defined by
(2.2)∂f(u)={u*∈X*:〈u*,φ〉≤f0(u;φ)∀φ∈X},
which is a nonempty, convex, and w*-compact subset of X, where 〈·,·〉 is the duality pairing between X* and X. We say that u∈X is a critical point of f if 0∈∂f(u). For further details, we refer the reader to Chang [4].We list some fundamental properties of the generalized directional derivative and gradient that will be possibly used throughout the paper.Proposition 2.1 (see [4, 5]).
(1)  Letj:X→ℝ be a continuously differentiable function. Then ∂j(u)={j′(u)}, j0(u;z) coincides with 〈j′(u),z〉X, and (f+j)0(u,z)=f0(u;z)+〈j′(u),z〉Xfor all u,z∈X.
(2)∂(λf)(u)=λ∂(u) for all λ∈ℝ.
(3)  Iff is a convex functional, then  ∂f(u) coincides with the usual subdifferential of  f in the sense of convex analysis.
(4)  Iff has a local minimum (or a local maximum) at  u0∈X, then 0∈∂f(u0).
(5)∥ξ∥X*≤L  for all  ξ∈∂f(u).
(6)f0(u,h)=max{〈ξ,h〉:ξ∈∂f(u)} for all  h∈X.
(7)  The function(2.3)m(u)=min⁡w∈∂f(u)‖w‖X*,
exists and is lower semicontinuous; that is, limu→u0infm(u)≥m(u0).In the following we need the nonsmooth version of the Palais-Smale condition.Definition 2.2.
One saysφ that nonsmooth satisfies the (PS)c-condition if any sequence {un}⊂X such that φ(un)→c and m(un)→0.
In what follows we write the(PS)c-condition as simply as the PS-condition if it holds for every level c∈ℝ for the Palais-Smale condition at level c.
We note that property (4) above says that a local minimum (or local maximum) ofφ is a critical point of φ.Finally, we point out that many of the results of the classical critical point theory have been extended by Chang [4] to this setting of locally Lipschitz functionals. For example, one has the following celebrated theorem.Theorem 2.3 (nonsmooth mountain pass theorem; see [4, 5]).
IfX is a reflexive Banach space, φ:X→ℝ is a locally Lipschitz function which satisfies the nonsmooth (PS)c-condition, and for some r>0 and e1∈X with ∥e1∥>r, max{φ(0),φ(e1)}≤inf{φ(u):∥u∥=r}. Then φ has a nontrivial critical u∈X such that the critical value c=φ(u) is characterized by the following minimax principle:
(2.4)c=inf⁡γ∈Γmax⁡t∈[0,1]φ(γ(t)),
where Γ={γ∈C([0,1],X):γ(0)=0,γ(1)=e1}.We would like to point out that we can obtain the same results for problem (1.3) through approximation of the discontinuous nonlinearity by a sequence of continuous functions. Variational methods were then applied to the corresponding sequence of problems and limits were taken. For the rest of this paper, we write X=W01,p(Ω) with the norm by ∥u∥=(∫Ω‍|∇u|pdx)1/p and denote by c and ci the generic positive constants for simplicity.


## 3. Proof of the Main Results

Now, having listed some basic results on critical point theory for the Lipschitz functionals, let us consider the functional(3.1)φλ(u)=1p∫Ω‍|∇u|dx-λ∫Ω‍G(u)dx,
where G(u)=∫0u‍g(s)ds and g(s) were defined in (1.2). Clearly G:ℝ→ℝ is a locally Lipschitz continuous function and satisfies G(s)=0 for s≤0. In view of [3, Theorems 2.1 and 2.2], the above formula for φλ(u) defines a locally Lipschitz functional on X whose critical points are solutions of the differential inclusion
(3.2)-div⁡(|∇u|p-2∇u)∈∂G(u)a.e.inΩ.
In our present case, it follows that ∂G(s)=f(s) for s>0, ∂G(s)=0 for s<0, and ∂G(s)=[-a,0] for s=0.We start with some preliminary lemmas.Lemma 3.1.
Assume (F0), (F1), and (F2). Then there existsλ0 such that problem (1.3) has no nontrivial solution 0≤u∈X for 0<λ<λ0.Proof .
Ifu≥0 is a solution of problem (1.3), then, multiplying the equation by u and integrating over Ω yields
(3.3)‖u‖p=∫Ω‍|∇u|pdx=λ∫Ω‍g(u)udx=λ(∫{x∈Ω∣u(x)≥δ0}‍ug(u)dx+∫{x∈Ω∣u(x)≥δ0}‍ug(u)dx),
hence
(3.4)‖u‖p≤λ∫{x∈Ω∣u(x)≥δ0}‍ug(u)dx,
where we have chosen δ0>0 so that g(s)≤0 for 0≤s≤δ0 (such a δ0 exists in view of (F0)). Now, since (F1) implies the existence of c>0 such that
(3.5)sg(s)≤c(1+sp),
for all s>0, we obtain from (3.4) that
(3.6)‖u‖p≤λc∫{x∈Ω∣u(x)≥δ0}‍(1+up)dx≤λc(1+1δ0p)∫{x∈Ω∣u(x)≥δ0}‍updx≤λc1∫Ω‍updx,
so that
(3.7)‖u‖p≤λc2‖u‖p,
where this last constant c2>0 is independent of both u and λ. Therefore we must have
(3.8)λ≥1c2:=λ0>0.Lemma 3.2.
Assume (F0) and either (F1) or (F3). Thenu=0 is a strict local minimum of the functional φλ.Proof .
Since (F1) or (F3) implies the existence ofc3>0 such that
(3.9)G(s)≤c3(1+|s|p*)∀s∈R,
recall also that G(s)=0 for s≤0. Then, with δ0>0 as in the proof of Lemma 3.1 and noticing that G(s)≤0 for all -∞<s≤δ0, we can write for an arbitrary u∈X,
(3.10)φλ(u)=1p‖u‖p-λ∫Ω‍G(u)dx≥1p‖u‖p-λ∫{x∈Ω∣u(x)≥δ0}‍G(u)dx≥1p‖u‖p-λc3∫{x∈Ω∣u(x)≥δ0}‍(1+up*)dx≥1p‖u‖p-λc3(1+1δ0p*)∫{x∈Ω∣u(x)≥δ0}‍up*dx,
so that, using the Sobolev embedding theorem in the last inequality and with a constant c4>0 independent of u and Ω, we obtain
(3.11)φλ(u)≥1p‖u‖p-λc4‖u‖p*=1p‖u‖p(1-pλc4‖u‖p*-p).
Therefore, for each 0<ρ<ρλ:=1/(pλc4)p*-p, it follows that φλ(u)≥αρ>0 if ∥u∥=ρ. This shows that u=0 is a strict local minimum of φλ.Lemma 3.3.
Under the same assumptions as in Lemma3.2, let u^∈X be a critical point of φλ. Then, u^∈C1,α(Ω) and u^ is a nonnegative solution of problem (1.3), where 0<α<1.Proof .
We shall follow some of the arguments in [3]. As mentioned earlier, if u^ is a critical point of φλ, then it is shown in [3] that u^ is a solution of the differential inclusion
(3.12)-div⁡(|∇u^|p-2∇u^)∈∂G(u^)a.e.inΩ.
Since g is only discontinuous at s=0, the above differential inclusion reduces to an equality, except possibly on the subset A⊂Ω where u^=0. Since f is subcritical, using the C1,α regularity results for quasilinear elliptic equations with p-growth condition (see, for example, [6]), we have u^∈C1,α(Ω¯∖A). And, in this latter case, -div(|∇u^|p-2∇u^) takes on values in the bounded interval [-a,0]. Therefore, by the Michael selection theorem (see Theorem  1.2.5 of [5]), we see that -Δp:u↦[-a,0] admits a continuous selection. Using the C1,α regularity results for quasilinear elliptic equations with p-growth condition again, it follows that u^∈C1,α(Ω), 0<α<1.
Next, using a Morrey-Stampacchia theorem [7, Theorem  3.2.2, page 69], we have that -div(|∇u^|p-2∇u^)=0 a.e. in A. Therefore, since we defined g(0)=0, it follows that
(3.13)-div⁡(|∇u^|p-2∇u^)=g(u^)a.e.inΩ.
Replacing the inclusion (3.2) on u^, we conclude that u^∈C1,α(Ω) is a solution of (1.3). Finally, recalling that g(s)=0 for s≤0, it is clear that u^≥0. The proof of Lemma 3.3 is complete.Lemma 3.4.
Assume either (F1) or (F3), (F4). Thenφλ satisfies the nonsmooth (PS)c-condition at every c∈ℝ.Proof.
Let{un}n≥1⊆X be a sequence such that |φλ(un)|≤c for all n≥1 and m(un)→0 as n→∞. In the superlinear and subcritical case, from (F4), we have
(3.14)c≥φλ(un)=1p∫Ω‍|∇un|pdx-λ∫Ω‍G(un)dx≥‖un‖pp-λθ∫Ω‍〈ξ(un),un〉dx-c≥(1p-1θ)‖un‖p+∫Ω‍1θ(‖un‖p-λ〈ξ(un),un〉)dx-c≥(1p-1θ)‖un‖p-λθ‖ξ‖X*‖un‖-c,
where ξ(un)∈∂G(un). Hence {un}n≥1⊆X is bounded.
Thus by passing to a subsequence if necessary, we may assume thatun⇀u in X as n→+∞. We have
(3.15)〈J′(un),un-u〉-∫Ω‍ξ(un)(un-u)dx≤ɛn‖un-u‖
with ɛn↓0, where ξ(un)∈∂Ψ(un) and J=(1/p)∫Ω‍|∇un|pdx. From (F3) and Chang [4] we know that ξ(un(x))∈L(p*-1)′(Ω) ((p*-1)′=(p*-1)/(p*-2)). Since X is embedded compactly in Lp*-1(Ω), we have that un→u as n→∞ in Lp*-1(Ω). So using the Hölder inequality, we have
(3.16)∫Ω‍ξn(x)(un-u)dx⟶0asn⟶+∞.
Therefore, we obtain thatlimn→∞sup〈J′(un),un-u〉≤0. But we know that J′ is a mapping of type (S+). Thus we have
(3.17)un⟶uinX.
Using the similar method, we can more easily get the nonsmooth (PS)c-condition in the case of sublinear.Remark 3.5.
Note that we cannot directly apply the results of [4] because the operator -Δp is not self-adjoint linear.Lemma 3.6.
Under assumptions (F0), (F1), and (F2), letΩ=BR⊂ℝN with N≥2, and let u∈C1(BR¯) be a radially symmetric, nonincreasing function such that u≥0 and u is a minimizer of φλ(u) with φλ(u)=m<0. Then, u does not vanish in BR; that is, u(x)>0 for all x∈BR.Proof.
Sinceg is discontinuous at zero, we note that the conclusion does not follow directly from uniqueness of solution for the Cauchy problem with data at r=R (in fact, writing u=u(r), r=|x|, we may have u(R)=u′(R)=0 and u≢0).
Now, sinceu≢0 by assumption, R0:=inf{r≤R:u(s)=0forr≤s≤R} satisfies 0<R0≤R. If R0=R there is nothing to prove in view of the fact that u is non-increasing. On the other hand, if R0<R then u′(R0)=0 and u(r)>0 for r∈[0,R0). It is not hard to prove that this contradicts that u is a minimizer of φλ. Indeed, if R0<Rthen
(3.18)φλ(u)=1p∫BR0‍|∇u|pdx-λ∫BR0‍G(u)dx=m<0.
A simple calculation shows that the rescaled function v(r)=u(R0r/R)∈W01,p(BR)∩C1(BR¯) satisfies
(3.19)φλ(v)=δp-N[1p∫BR0‍|∇u|pdx-δ-pλ∫BR0‍G(u)dx],
where δ:=R0/R is less than 1. Therefore, since we are assuming 1<p≤N, we would reach the contradiction φλ(v)<m.Remark 3.7.
Note that the condition (F2) is necessary because of which guaranteeG can have positive values.Proof of Theorem1.1.
We observe that the functionalφλ is weakly lower semicontinuous on X. Moreover, the sublinearity assumption (F1) on g(u) implies that φλ is coercive. Therefore, the infimum of φλ is attained at some uλ:
(3.20)inf⁡u∈Xφλ(u)=φλ(uλ).
And, in view of Lemma 3.3, uλ∈C1,α(Ω) is a nonnegative solution of (1.3). We now claim that uλ is nonzero for all λ>0 large.Claim 1.
There existsΛ>0 such that φλ(uλ)<0 for all λ≥Λ.
In order to prove the claim it suffices to exhibit an elementw∈X such that φλ(w)<0 for all λ>0 large. This is quite standard considering that G(δ)>0 by (F2). Indeed, letting Ωɛ:={x∈Ω:dist(x,∂Ω)>ɛ} for ɛ>0 small, define w so that w(x)=δ for x∈Ωɛ and 0≤w(x)≤δ for x∈Ω∖Ωɛ. Then
(3.21)φλ(w)=1p‖w‖p-λ(∫Ωɛ‍G(w)dx+∫Ω∖Ωɛ‍G(w)dx)≤1p‖w‖p-λ(G(δ)meas⁡(Ωɛ)-c(1+δp)meas⁡(Ω∖Ωɛ)),
where we note that the expression in the above parenthesis is positive if we choose ɛ>0 sufficiently small. Therefore, there exists Λ>0 such that φλ(w)<0 for all λ≥Λ, which proves the claim.
On the other hand, whenΩ=BR, let uλ* denote the Schwarz symmetrization of uλ, namely, uλ* is the unique radially symmetric, nonincreasing, nonnegative function in X which is equi-measurable with uλ. As is well known,
(3.22)∫BR‍G(uλ*)dx=∫BR‍G(uλ)dx,
and ∥uλ*∥≤∥uλ∥ (the Faber-Krahn inequality; see [8]), so that φλ(uλ*)≤φλ(uλ). Therefore, we necessarily have φλ(uλ*)=φλ(uλ) and may assume that uλ=uλ*. Moreover, uλ>0 in Ω by Lemma 3.6. Therefore, uλ is a positive solution of both problems (1.1) and (1.3).
Next, we recall thatu=0 is a strict local minimum of φλ by Lemma 3.2. Therefore, since φλ satisfies the nonsmooth Palais-Smale condition by Lemma 3.4, we can use the minima u=0 and u=uλ of φλ to apply the nonsmooth mountain pass theorem and conclude that there exists a second nontrivial critical point vλ with φλ(vλ)>0. Again, vλ is a nonnegative solution of problem (1.3) in view of Lemma 3.3. In addition, when Ω=BR, arguments similar to above passage show that we may assume vλ=vλ*. The proof of Theorem 1.1 is complete.Proof of Theorem1.2.
As is well-known, the AR condition (F4) readily implies the existence of an elementeλ∈X such that φλ(eλ)≤0. On the other hand, Lemma 3.2 says that u=0 is a (strict) local minimum of φλ and Lemma 3.4 says that φλ satisfies nonsmooth (PS)c for every c∈ℝ. Therefore, an application of the nonsmooth mountain pass theorem stated in section 2 yields the existence of a critical point vλ such that
(3.23)φλ(vλ)>0.
In particular, vλ≠0, and it follows that vλ is a nonnegative solution of problem (1.3) by Lemma 3.3. As in the proof of Theorem 1.1, we may assume that vλ=vλ* in the case of a ball Ω=BR.
Finally, still in the case of a ballΩ=BR, we claim that there exists λ*>0 such that, for all 0<λ<λ*, vλ=vλ* is a positive solution of problem (1.3) (hence of problem (1.1)) having negative normal derivative on ∂BR. Indeed, if that is not the case, then, for any given λ>0, we can find 0<μ=μ(λ)<λ such that the nonnegative solution vμ=vμ* of problem (1.1) with λ=μ obtained a bove satisfies
(3.24)vμ(r)>0forr∈[0,R0),vμ′(R0)=0,vμ(r)=0forr∈[R0,R],
for some 0<R0≤R. Therefore, the rescaled function wμ(r):=vμ(R0r/R) is a positive solution of (1.1) with λ=μ0 (again in the ball BR), with μ0:=μR0p/Rp≤μ. This shows that we can always construct a decreasing sequence μn>0 satisfying alternative (ii) of Theorem 1.2, in case alternative (i) does not hold.



---

*Source: 595429-2012-07-19.xml*