A class of semipositone $p$-Laplacian problems with a critical growth reaction term

We prove the existence of ground state positive solutions for a class of semipositone $p$-Laplacian problems with a critical growth reaction term. The proofs are established by obtaining crucial uniform $C^{1,\alpha}$ a priori estimates and by concentration compactness arguments. Our results are new even in the semilinear case $p = 2$.


Introduction
Consider the p-superlinear semipositone p-Laplacian problem in Ω u = on ∂Ω, where Ω is a smooth bounded domain in R N , < p < N, p < q ≤ p * , µ > is a parameter, and p * = Np/(N − p) is the critical Sobolev exponent. The scaling u → µ /(q− ) u transforms the rst equation in (1.1) into −∆p u = µ (q−p)/(q− ) u q− − , so in the subcritical case q < p * , it follows from the results in Castro et al. [1] and Chhetri et al. [2] that this problem has a weak positive solution for su ciently small µ > when p > (see also Unsurangie [3], Allegretto et al. [4], Ambrosetti et al. [5], and Caldwell et al. [6] for the case when p = ). On the other hand, in the critical case q = p * , it follows from a standard argument involving the Pohozaev identity for the p-Laplacian (see Guedda and Véron [7, Theorem 1.1]) that problem (1.1) has no solution for any µ > when Ω is star-shaped. The purpose of the present paper is to show that this situation can be reversed by the addition of lower-order terms, as was observed in the positone case by Brézis and Nirenberg in the celebrated paper [8]. However, this extension to the semipositone case is not straightforward as u = is no longer a subsolution, making it much harder to nd a positive solution as was pointed out in Lions [9]. The positive solutions that we obtain here are ground states, i.e., they minimize the energy among all positive solutions.
We study the Brézis-Nirenberg type critical semipositone p-Laplacian problem where λ, µ > are parameters. Let W ,p (Ω) be the usual Sobolev space with the norm given by For a given λ > , the energy of a weak solution u ∈ W ,p (Ω) of problem (1.2) is given by and clearly all weak solutions lie on the set We will refer to a weak solution that minimizes Iµ on Nµ as a ground state. Let be the rst Dirichlet eigenvalue of the p-Laplacian, which is positive. We will prove the following existence theorem.
The scaling u → µ − /(p * −p) u transforms the rst equation in the critical semipositone p-Laplacian problem so as an immediate corollary we have the following existence theorem for problem (1.4).
We would like to emphasize that Theorems 1.1 and 1.2 are new even in the semilinear case p = .
The outline of the proof of Theorem 1.1 is as follows. We consider the modi ed problem where u+(x) = max u(x), and Weak solutions of this problem coincide with critical points of the C -functional where |·| denotes the Lebesgue measure in R N . Recall that Iµ satis es the Palais-Smale compactness condition at the level c ∈ R, or the (PS)c condition for short, if every sequence u j ⊂ W ,p (Ω) such that Iµ(u j ) → c and I µ (u j ) → , called a (PS)c sequence for Iµ, has a convergent subsequence. As we will see in Lemma 2.1 in the next section, it follows from concentration compactness arguments that Iµ satis es the (PS)c condition for all where S is the best Sobolev constant (see (2.1)). First we will construct a mountain pass level below this threshold for compactness for all su ciently small µ > . This part of the proof is more or less standard. The novelty of the paper lies in the fact that the solution uµ of the modi ed problem (1.5) thus obtained is positive, and hence also a solution of our original problem (1.2), if µ is further restricted. Note that this does not follow from the strong maximum principle as usual since −µ f ( ) < . This is precisely the main di culty in nding positive solutions of semipositone problems (see Lions [9]). We will prove that for every sequence µ j → , a subsequence of uµ j is positive in Ω. The idea is to show that a subsequence of uµ j converges in C (Ω) to a solution of the limit problem This requires a uniform C ,α (Ω) estimate of uµ j for some α ∈ ( , ). We will obtain such an estimate by showing that uµ j is uniformly bounded in W ,p (Ω) and uniformly equi-integrable in L p * (Ω), and applying a result of de Figueiredo et al. [10]. The proof of uniform equi-integrability in L p * (Ω) involves a second (nonstandard) application of the concentration compactness principle. Finally, we use the mountain pass characterization of our solution to show that it is indeed a ground state.

Remark 1.3. Establishing the existence of solutions to the critical semipositone problem
be the best constant in the Sobolev inequality, which is independent of Ω. The proof of Theorem 1.1 will make use of the following compactness result.
Proof. Let u j be a (PS)c sequence. First we show that u j is bounded. We have and Taking v = u j in (2.4), dividing by p, and subtracting from (2.3) gives and it follows from this, (2.3), and the Hölder inequality that u j is bounded in W ,p (Ω). Since u j is bounded, so is u j+ , a renamed subsequence of which then converges to some v ≥ weakly in W ,p (Ω), strongly in L q (Ω) for all q ∈ [ , p * ) and a.e.in Ω, and in the sense of measures, where κ and ν are bounded nonnegative measures on Ω (see, e.g., Folland [11]). By the concentration compactness principle of Lions [12,13], then there exist an at most countable index set I and points x i ∈ Ω, i ∈ I such that where κ i , ν i > and ν p/p * i ≤ κ i /S. We claim that I = ∅. Suppose by contradiction that there exists i ∈ I. Let φ : R N → [ , ] be a smooth function such that φ(x) = for |x| ≤ and φ(x) = for |x| ≥ . Then set for i ∈ I and ρ > , and note that φ i,ρ : R N → [ , ] is a smooth function such that φ i,ρ (x) = for |x − x i | ≤ ρ and φ i,ρ (x) = for |x − x i | ≥ ρ. The sequence φ i,ρ u j+ is bounded in W ,p (Ω) and hence taking v = φ i,ρ u j+ in (2.4) gives By (2.6), Denoting by C a generic positive constant independent of j and ρ, where So passing to the limit in (2.8) gives Letting ρ ↘ and using (2.7) now gives κ i ≤ ν i , which together with ν i > and ν p/p * i ≤ κ i /S then gives ν i ≥ S N/p . On the other hand, passing to the limit in (2.5) and using (2.6) and (2.7) gives Passing to a further subsequence, u j converges to some u weakly in W ,p (Ω), strongly in L q (Ω) for all q ∈ [ , p * ), and a.e.in Ω. Since by (2.9) and the dominated convergence theorem. Then taking v = u j − u in (2.4) gives so u j → u in W ,p (Ω) for a renamed subsequence (see, e.g., Perera et al. [14,Proposition 1.3]).

The in mum in (2.1) is attained by the family of functions
Without loss of generality, we may assume that ∈ Ω. Let r > be so small that B r ( ) ⊂ Ω, take a function ψ ∈ C ∞ (B r ( ), [ , ]) such that ψ = on Br( ), and set Then we have the well-known estimates where C = C(N, p) > is a constant (see, e.g., Drábek and Huang [15]).

Proof of Theorem 1.1
First we show that Iµ has a uniformly positive mountain pass level below the threshold for compactness given in Lemma 2.1 for all su ciently small µ > . Let vε be as in the last section.
Proof. By (1.3) and (2.1), and (i) follows from this for su ciently small ρ, c , µ > since λ < λ . and Taking v = uµ in (3.4), dividing by p, and subtracting from (3.3) gives by (3.1), and (i) follows from this, (3.4) with v = uµ, and the Hölder inequality. If (ii) does not hold, then there exist sequences µ j → and E j with E j → such that Since uµ j is bounded by (i), so is uµ j + , a renamed subsequence of which then converges to some v ≥ weakly in W ,p (Ω), strongly in L q (Ω) for all q ∈ [ , p * ) and a.e.in Ω, and in the sense of measures, where κ and ν are bounded nonnegative measures on Ω. By Lions [12,13], then there exist an at most countable index set I and points x i ∈ Ω, i ∈ I such that where κ i , ν i > and ν p/p * i ≤ κ i /S. Suppose I is nonempty, say, i ∈ I. An argument similar to that in the proof of Lemma 2.1 shows that κ i ≤ ν i , so ν i ≥ S N/p . On the other hand, passing to the limit in (3.5) with µ = µ j and using (3.7) and (3.8) gives ν i ≤ Nβ < S N/p , a contradiction. Hence I = ∅ and Ω u p * µ j + dx → Ω v p * dx.
As in the proof of Lemma 2.1, a further subsequence of uµ j then converges to some u in W ,p (Ω), and hence also in L p * (Ω), and a.e.in Ω. Then We are now ready to prove Theorem 1.1.
Proof of Theorem 1.1. We claim that uµ is positive in Ω, and hence a weak solution of problem (1.2), for all su ciently small µ ∈ ( , µ * ). It su ces to show that for every sequence µ j → , a subsequence of uµ j is since w ∈ Nµ, and the last two integrals are positive since λ < λ and w > , so g (t) > for ≤ t < , g ( ) = , and g (t) < for t > . Hence max t≥ Iµ(tw) = Iµ(w) > since g( ) = . In view of Lemma 3.1 (ii), now it su ces to observe that there exists R > max { , R} such that for all u on the line segment joining w to vε since all norms on a nite dimensional space are equivalent.