Ground-State Solutions for a Class of N-Laplacian Equation with Critical Growth

and Applied Analysis 3 In this paper, we complement some results 4 from subcritical case to the critical case. Furthermore, the ground-state solution to the problem 1.1 is obtained without assuming that the function s → f s |s|N−1 1.8 is increasing for s > 0 see 5 , and the so-called Ambrosetti-Rabinowitz condition: there exists θ > N, such that for all x ∈ R , 0 < θF x, u ≤ uf x, u . 1.9 The paper is organized as follows. Section 2 contains some technical results which allows us to give a variational approach for our results. In Section 3, we prove our main results. 2. The Variational Framework For 1 ≤ p ≤ ∞, L R denotes the Lebesgue spaces with the norm ‖u‖Lp RN ∫ RN |u|pdx , W1,p R denotes the Sobolev spaces with the norm ‖u‖W1,p RN ∫ RN |∇u|p |u|p dx . As p N, we have the following version of Trudinger-Moser inequality. Lemma 2.1 see 6 . IfN ≥ 2, α > 0 and u ∈W1,N R , then ∫ RN [ exp ( α|u| N−1 ) − SN−2 α, u ] dx <∞. 2.1 Moreover, if ‖∇u‖N LN RN ≤ 1, ‖u‖LN RN ≤ M < ∞ and α < αN , then there exists a constant C, which depends only onN,M, and α, such that ∫ RN [ exp ( α|u| N−1 ) − SN−2 α, u ] dx ≤ C N,M,α , 2.2 where αN Nω 1/ N−1 N−1 and ω 1/ N−1 is the measure of the unit sphere in R . In the sequel, since we seek positive solutions, and assume that f s 0 for s ≤ 0. Consider the following minimization problem: min { 1 N ∫ RN |∇u|dx : ∫ RN G u dx 0 } , 2.3 where g s f s − s|s|N−2, G s ∫s 0 g t dt F s − 1/N s . Since the problem 1.1 is an autonomous problem, under the Schwarz symmetric process, we can minimize the problem 4 Abstract and Applied Analysis 2.3 on the space W rad R N , the subspace of W1,N R formed by radially symmetric functions. Indeed, let u∗ be the Schwarz symmetrization of u, we have ∫ RN G u∗ dx ∫ RN G u dx, ∫ RN |∇u∗|Ndx ≤ ∫ RN |∇u|dx. 2.4 Hence, we canminimize the problem 2.3 on the spaceW rad R N see 7 . Now, we defined the following notations m inf { I u : u is nontrivial solution of the problem 1.1 } A inf { 1 N ∫ RN |∇u|dx : ∫ RN G u dx 0 } b inf γ∈Γ max t∈ 0,1 I ( γ t ) , 2.5 where Γ {γ ∈ C 0, 1 ,W rad R : γ 0 0, I γ 1 < 0}. We recall that Pohozaev-Pucci-Serrin identity shows that any solutions u of the problem 1.1 should satisfies the Pohozaev-Pucci-Serrin identity: ( N − p) ∫ RN |∇u|dx Np ∫ RN G u dx. 2.6 Then, as p N, we have ∫ RN G u dx 0. Hence, we have the Pohozaev identity manifold: P { u ∈W1,N ( R N ) \ {0} : (N − p) ∫ RN |∇u|dx Np ∫ RN G u dx } { u ∈W1,N ( R N ) \ {0} : ∫ RN G u dx 0 } . 2.7


Introduction
Consider the following N-Laplacian equation: where N ≥ 2. Δ N u div |∇u| N−2 ∇u is the N-Laplacian, the nonlinear term f u has critical growth.
The interest in these problems lies in that fact that the order of the Laplacian is the same as the dimension N of the underlying space.The classical case of this problem that N 2, and the the problem 1.1 reduces to 1.2 has been treated by Atkinson and Peletier 1 and Berestycki and Lions 2 .They obtained the existence of ground-state solution which the nonlinear term f u is subcritical growth.
where g : R → R has a subcritical growth and obtain a mountain pass characterization of the ground-state solution for the problem 1.3 .In the present paper, we will improve and complement some of the results cited above.Assume the function f : R → R is continuous and satisfies the following conditions: There exist λ > 0 and q > N such that f s ≥ λs q−1 , for every s ≥ 0.
Remark 1.1.Condition g 2 implies that f has a critical growth with critical exponent α 0 .
Consider the energy functional where F s s 0 f t dt.By a ground-state solution, we mean a solution such ω ∈ W 1,N R N such that I ω ≤ I u for every nontrivial solution u of the problem 1.1 .Let C q > 0 denote the best constant of Sobolev embeddings: Now we state our main theorem in this paper.
Theorem 1.2.If f satisfies g 1 , g 2 , and g 3 , with then the problem 1.1 possesses a nontrivial ground-state solution.
In this paper, we complement some results 4 from subcritical case to the critical case.Furthermore, the ground-state solution to the problem 1.1 is obtained without assuming that the function is increasing for s > 0 see 5 , and the so-called Ambrosetti-Rabinowitz condition: there exists θ > N, such that for all x ∈ R N , 0 < θF x, u ≤ uf x, u .1.9 The paper is organized as follows.Section 2 contains some technical results which allows us to give a variational approach for our results.In Section 3, we prove our main results.

The Variational Framework
then there exists a constant C, which depends only on N, M, and α, such that where α N Nω and ω 1/ N−1 is the measure of the unit sphere in R N .
In the sequel, since we seek positive solutions, and assume that f s 0 for s ≤ 0. Consider the following minimization problem: where Since the problem 1.1 is an autonomous problem, under the Schwarz symmetric process, we can minimize the problem 2.3 on the space W 1,N rad R N , the subspace of W 1,N R N formed by radially symmetric functions.Indeed, let u * be the Schwarz symmetrization of u, we have Hence, we can minimize the problem 2.3 on the space W 1,N rad R N see 7 .Now, we defined the following notations where We recall that Pohozaev-Pucci-Serrin identity shows that any solutions u of the problem 1.1 should satisfies the Pohozaev-Pucci-Serrin identity: Then, as p N, we have R N G u dx 0. Hence, we have the Pohozaev identity manifold:

2.7
So, we have where In what follows, we will show that A is attained, and afterwards we prove that m A b, 2.9 thereby proving that the problem 1.1 has a ground-state solution.

The Proof of Theorem 1.2
In this section, we prove that A is attained, the equality 2.9 is satisfied.Hence the proof of Theorem 1.2 is obtained.
In the following, we consider the following minimax value: Now, we show a sufficient condition, on a sequence {v n } to get a convergence like Lemma 3.1.Assume that f satisfies g 1 and g 2 , and let {v n } be a sequence in Proof.Without loss of generality, we assume that there exist

3.3
Let v * is the Schwarz symmetrization of v, then we have

3.4
From g 1 , we obtain that for > 0, there exists δ > 0, such that f s ≤ |s| N−1 , for |s| < δ, 3.5 so, we have From g 2 , we obtain There two estimates yield On one hand, from Lemma 2.1, we obtain that there exists a constant C, which depends only on N, M, and α such that
Then, by Dominated convergence theorem, we obtain

3.11
On the other hand,

3.12
Abstract and Applied Analysis 7 where v * n is the Schwarz symmetrization of v n .Notice that the estimate

3.13
for all j ≥ N − 1, together with the Radial Lemma 4 leads to

3.15
Which implies that

3.16
Using the estimate Hence, we obtain that

3.19
Lemma 3.2.The numbers A and c satisfy the inequality A ≤ c.
Proof.For each v ∈ W 1,N R N \ {0}, since we only consider the nontrivial solutions of the problem 1.1 , we divide them into two cases to consider.
Case 1.Let v max{v, 0} / 0, we define the function h : R → R by

3.22
we obtain that h t < 0 for t small enough.On the other hand, by g 2 , we obtain that h t > 0 for t large enough.In this way, there exists t 0 > 0 such that h t 0 0, That is, t 0 v ∈ P. Hence

3.24
As a consequence, A ≤ c.

3.25
Combining Cases 1 and 2, we obtain that A ≤ c.

Lemma 3.3.
The number A defined by 2.8 is positive, that is, A > 0.
Proof.Clearly, A ≥ 0. Assume by contradiction that A 0 and let {u n } be a minimizing sequence in W 1,N R N to A, that is,

3.26
For each λ n > 0, set v n x u n x/λ n satisfying

3.27
Similarly, we have

3.29
In what follows, we study in the space W 1,N rad R N .Firstly, we assume that there exists v ∈ W 1,N rad R N such that v n v in W 1,N rad R N .On one hand, since 1/N R N |∇v n | N → A 0, then ∃N 0 > 0, for all 0 < < 1, when n > N 0 , we have R N |∇v n | N dx < < 1 and we also know that Hence, we have

3.33
It implies that v / 0. On the other hand, since v n v in W 1,N rad R N , and the space W 1,N rad R N is a reflexible Banach space, we have lim From which it follows that v 0, we have an absurd.Hence, we have Proof.From g 3 , we have f s > λs q−1 , for all s ≥ 0. Now we choose ψ ∈ W 1,N rad R N such that

3.38
Hence, we have

3.39
Let K t t N /N − λt q /q R N ψ q dx, then K t is continuous function, we have ψ q dx 0.

3.40
By a simple calculation, when t 0 1/λ R N ψ q dx 1/ q−N > 0, we have max t>0 K t K t 0 1 N 1 λ R N ψ q dx N/ q−N − λ q R N ψ q dx 1 λ R N ψ q dx q/ q−N q − N Nq λ −N/ q−N C q/N • N/ q−N q < q − N Nq q − N q q−N /N • −N/ q−N C q/N −N/ q−N q C q/ q−N q 1 N .

3.42
Lemma 3.5.The number A is attained, that is, there exists u ∈ W 1,N rad R N such that A R N |∇u| N dx and R N G u dx 0.
Proof.Let {u n } be a minimizing sequence in W 1,N rad R N for A, that is, where u n u in W 1,N R N , as n → ∞.By 3.43 and 3.45 , we have

2
Abstract and Applied AnalysisAlves et al. 3 extend their results to the critical growth.As N / 2, do Ó and Medeiros 4 consider the following N-Laplacian equation problem: as in Lemma 3.3, we assume that R N |u n | N dx 1.From 3.43 , Lemmas 3.3 and 3.4, we obtain lim n → ∞ R N |∇u n | N dx NA ≤ Nc < 1.
1/p .As p N, we have the following version of Trudinger-Moser inequality.