ON THE LOCATION OF THE PEAKS OF LEAST-ENERGY SOLUTIONS TO SEMILINEAR DIRICHLET PROBLEMS WITH CRITICAL GROWTH

We study the location of the peaks of solution for the critical growth problem − e 2 Δ u + u = f ( u ) + u 2 * − 1 , 0$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"> u > 0 in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain; 2 * = 2 N / ( N − 2 ) , N ≥ 3 , is the critical Sobolev exponent and f has a behavior like u p , 1 p 2 * − 1 .


Introduction
In this paper, we will study the location of the peaks of least-energy solution for the problem where Ω is a bounded domain in R N , ε > 0, and f is a function satisfying some subcritical conditions.Here 2 * = 2N/(N − 2), N ≥ 3, is the critical Sobolev exponent.
By least-energy solution for problem (1.1) we mean a critical point at the Mountain-Pass level of the associated energy functional (where u + = max{u,0}), defined on the Hilbert space H 1 o (Ω) endowed with the norm (1. 3) The Mountain-Pass level of J ε is defined by J ε g(t) , (1.4) where Γ is the set of all continuous paths joining the origin and a fixed nonzero element e in H 1 o (Ω), such that e = 0 and J ε (e) ≤ 0. Under suitable hypothesis (e.g., (f 1 ), (f 4 ), (f 5 ) below), it is not hard to check that c ε > 0 does not depend on the element 0 = v ∈ H 1 o (Ω) and u is a least-energy solution if and only if J ε (u) = c and J ε (u) = 0, and J ε (u) ≤ J ε (v) for all v = 0 such that J ε (v) = 0.
The existence of least-energy solution of problem (1.1) was given in Brézis and Nirenberg in [3, Theorem 2.1] (see Lemma 2.4

in this paper).
In this paper, we will study some properties of the least-energy solution u ε of problem (1.1) when ε is small.In order to describe these properties, we introduce the hypotheses on the function f .Suppose that f : (f 3 ) there are q 2 ∈ (1,(N + 2)/(N − 2)) and λ > 0 such that f (s) ≥ λs q2 , ∀s > 0 (1.6) (when N = 3, we need q 2 > 2, otherwise we require a sufficiently large λ); Since our interest is on positive solutions we define f (s) = 0, in s ≤ 0. Now we will state our main result.
Theorem 1.1.Suppose that Ω is a bounded domain in R N ; f satisfies (f 1 ), (f 2 ), (f 3 ), (f 4 ), (f 5 ); and let u ε be the least-energy solution of (1.1).Then, there is a ε o > 0 such that (i) u ε attains only one local maximum at some z ε ∈ Ω (hence global maximum), for all ε ∈ (0,ε o ]; (ii) u ε converges uniformly to zero over compact subsets of This statement is analogous to the one given by Ni and Wei in [8], in the subcritical case where h satisfies the following hypothesis: (i) (f 1 ), (f 2 ), (f 4 ), and (f 5 ) hold; (ii) the global problem has a unique positive solution in H 1 (R N ); (iii) this solution is nondegenerate in the sense that has no nontrivial spherically symmetric solution in L 2 (R N ).In [8], Ni and Wei also have described the asymptotic profile (in ε) of u ε , giving a detailed description for ε small.Here in the critical case, the solutions have the same profile.
In this work we will show that a ground state solution of the critical problem (1.1) is also solution of a subcritical problem (1.8) by showing that for small ε we have a uniform bound for the L ∞ norm of u ε .
The difficulty here lies in finding an upper bound for u ε L ∞ (Ω) by obtaining a bound for u ε in L p (Ω) norm, for all p ≥ 2. In the subcritical case this boundedness is obtained since the family u ε is bounded in H 1 (Ω) but this argument does not work in the critical case.Here, we obtain an L ∞ -bound for u ε through the estimate below, which is based on Moser's iteration technique (see [11]) and is essentially due to Brézis and Kato [2].Proposition 1.2.Let Λ be an open subset and q ∈ L N/2 (Λ).Suppose that g : Λ × R → R is a Caratheodory function satisfying g(x,s) ≤ q(x) + C g |s|, ∀s ∈ R, x ∈ Λ and for some C g > 0. (1.11) Remark 1.3.The dependence on q of C p can be given uniformly on a family of functions {q ε } ε>0 such that q ε converges in L N/2 (see the appendix).
We have organized this paper as follows: the next section contains the proof of Theorem 1.1.This proof consists in a series of lemmas which show the L ∞bound for u ε , where these functions are solutions of a class of subcritical problems (1.8).The third section is an appendix proving Proposition 1.2, for the sake of completeness.

Proof of Theorem 1.1
Before proving Theorem 1.1, let us fix some notation and preliminaries.
Remark 2.1.Throughout this section, we use the equivalent characterization of c ε , which is more adequate to our purposes, given by (see Willem [13,Theorem 4.2]).
We denote by J : H 1 (R N ) → R the functional given by where associated with the problem It is known that under assumptions (f 1 ), (f 2 ), (f 3 ), (f 4 ), (f 5 ), and (2.4) possesses a ground state solution ω in the level (see [1]).
Remark 2.2.It is easy to check that for each nonzero v in H 1 (R N ), there is a unique t o = t(v) such that Indeed, since Marco A. S. Souto 551 the maximum point t o of J(tv) is given by We assume, without loss of generality that 0 ∈ Ω.
associated with the problem (2.12) It is easy to check that b ε = ε −N c ε and from the definition of c it follows that b ε ≥ c for all ε > 0.
We will start with the following property of {b ε } ε>0 .
Proof.Fix ω a ground state solution of problem (2.4) and let ) From (2.8) and condition (f 3 ) it follows that Proof.For each h > 0, consider the function (2.17) We recall that φ h satisfies the problem Talenti [12]). (2.18) , where ϕ is the function defined in the proof of Lemma 2.3.From condition (f 3 ) we have Using arguments as in [7], there exists h > 0 such that and the proof of the lemma is completed.
Notice that the same proof of Lemma 2.4 can be used to show that b ε < (1/N)S N/2 , for all ε > 0. Using [3, Theorem 2.1], this inequality implies the existence of v ε and then the existence of u ε .

.27)
Let ≥ 0 be such that v εn 2 → .Passing to the limit in J(v εn ) = b εn and using (2.26) we have = Nc (2.28) and hence > 0. Now, using the definition of the constant S, we have (2.29) Taking the limit in the above inequalities, as n → ∞, we achieve that Finally, to establish (2.23), suppose the contrary.That is, there exist ε n → 0 and R > 0 such that dist(y εn ,∂Ω εn ) ≤ R, hence dist(ε n y εn ,∂Ω) ≤ ε n R. Without loss of generality, we have ε n y εn → y o for some y o ∈ ∂Ω.The arguments that follow can be found in [8].
Let ν be the unit interior normal to ∂Ω at y o , and δ > 0 such that B δ ( and we have that w n converges weakly to some w in H 1 (R N ).Let R N +,ν be the half space {x ∈ R N : and then we can prove that for all compacts , and w ≡ 0 in R N −,ν .Theorem I.1, due to Esteban and Lions in [4], shows that w ≡ 0 which contradicts (2.34) This completes the proof of the lemma.Now we will consider the translation of v ε , defined by From Lemma 2.5, Consider a sequence ε n 0 and set Ω n = Ω εn , ω n = ω εn , v n = v εn , y ε = y εn .We will prove that ω n is bounded in the L ∞ norm.In that case, u ε is also bounded in L ∞ (Ω) norm and the proof of Theorem 1.1 follows from the subcritical case, as Lemma 2.8 will show.
Since the sequence ω n a translation of v n , we have a uniform bound for ω n and there is a ω o ∈ H 1 (R N ) which is weak limit of ω n in H 1 (R N ).From (2.35) we have ω o = 0. We can write limit (2.23) in the following form (2.36) Then for each φ ∈ C ∞ o (R N ) and large n such that supp φ ⊂ Ω n , we have Proof.This fact comes from Lemma 2.5 and Fatou's lemma applied in the positive sequence ω n f (ω n ) − θF(ω n ).Observe that (2.38) From (2.38) (2.39) We have proved that J(ω o ) = c and then (2.39) becomes an equality.
Combining (2.39) with the three following inequalities: we conclude that ω n → ω o and then We are ready to conclude the proof of our main result.From Proposition 1.2 and Remark 1.3 with q where C depends on N, t, and R.
Lemma 2.8.(i) If cε is the minimax level of Jε , then cε = c ε ; (ii) each u ε is a critical point of Jε in the minimax level and satisfies (1.8).

Appendix
Let Λ be some general domain in R N (bounded or unbounded).We will start with the following lemma due to Brézis and Kato [2].