We consider a p-Laplacian system with critical nonlinearity in ℝ^N. Under the proper assumptions, we obtain the existence of nontrivial solutions to perturbed p-Laplacian system by using the variational approach.

MR Subject Classification: 35B33; 35J60; 35J65.

This article is concerned with the existence of solutions to the following nonlinear perturbed p-Laplacian system

where Δ_pu = div(|∇u|^p-2∇u) is the p-Laplacian operator, 1 < p < N and p* = Np/(N − p) is the critical exponent.

Throughout the article, we will assume that:

(V₀) V ∈ C(ℝ^N), V (0) = inf V (x) = 0 and there exists b > 0 such that the set ν^b := {x ∈ ℝ^N : V (x) < b} has finite Lebesgue measure;

(K₀) K(x) ∈ C(ℝ^N), 0 < inf K ≤ sup K < ∞;

(H₁) H ∈ C¹(ℝ²) and H_s, H_t = o(|s|^p-1 + |t|^p-1) as |s| + |t| → 0;

(H₂) there exist c > 0 and p < q < p* such that

(H₃) There are a₀ > 0, θ ∈ (p, p*) and α, β > p such that H(s, t) ≥ a₀(|s|^α + |t|^β) and 0 < θH(s, t) ≤ sH_s + tH_t.

Under the above mentioned conditions, we will get the following result.

Theorem 1. If (V₀), (K₀) and (H₁)-(H₃) hold, then for any σ > 0, there is ε_σ > 0 such that if ε < ε_σ, the problem (1.1) has at least one positive solution (u_ε, v_ε) which satisfy

The scalar form of the problem (1.1) is as follows

The Equation (1.2) has been studied in many articles. The case p = 2 was investigated extensively under various hypotheses on the potential and the nonlinearity by many authors including Brézis and Nirenberg [1], Ambrosetti [2] and Guedda and Veron [3] (see also their references) in bounded domains. As far as unbounded domains are concerned, we recall the work by Benci and Cerami [4], Floer and Weistein [5], Oh [6], Clapp [7], Del Pino and Felmer [8], Cingolani and Lazzo [9], Ding and Lin [10]. Especially, in [10], the authors studied the Equation (1.2) in the case p = 2. In that article, they made the following assumptions:

(A₁) V ∈ C(ℝ^N), min V = 0 and there is b > 0 such that the set ν^b := {x ∈ ℝ^N : V (x) < b} has finite Lebesgue measure;

(A₂) K(x) ∈ C(ℝ^N), 0 < inf K ≤ sup K < ∞

(B₁) h ∈ C(ℝ^N × ℝ) and h(x, u) = o(|u|) uniformly in x as |u| → 0;

(B₂) there are c₀ > 0, q < 2* such that |h(x, u)| ≤ c₀(1 + |u|^q-1) for all (x, u);

(B₃) there are a₀ > 0, p > 2 and µ > 2 such that H(x, u) = a₀|u|^p and µH(x, u) ≤ h(x, u)u for all (x, u), where .

That article obtained the existence of at least one positive solution u_ε of least energy if the assumptions (A₁)-(A₂) and (B₁)- (B₃) hold.

For the Equation (1.2) in the case p ≠ 2, we recall some works. Garcia Azorero and Peral Alonso [11] considered (1.2) with ε ≤ 1, V (x) = µ, K(x) = 1, h(x, u) = 0 and proved that (1.2) has a solution if p² ≤ N and µ ∈ (0, λ₁), where λ₁ is the first eigenvalue of the p-Laplacian. In [12], Alves and Ding studied the same problem of [11] and obtained the multiplicity of positive solutions in bounded domain Ω ⊂ ℝ^N. Moreover, Liu and Zheng [13] investigated (1.2) in ℝ^N with ε = 1 and K(x) = 0. Under the sign-changing potential and subcritical p-superlinear nonlinearity, the authors got the existence result.

Motivated by some results found in [10,11,13], a natural question arises whether existence of nontrivial solutions continues to hold for the p-Laplacian system with the critical nonlinearity in ℝ^N.

The main difficulty in the case above mentioned is the lack of compactness of the energy functional associated to the system (1.1) because of unbounded domain ℝ^N and critical nonlinearity. To overcome this difficulty, we make careful estimates and prove that there is a Palais-Smale sequence that has a strongly convergent sequence. The method or idea here is similar to the one of [10]. We can prove that the functional associated to (1.1) possesses (PS)_c condition at some energy level c. Furthermore, we prove the existence result by using the mountain pass theorem due to Rabinowitz [14].

The main result in the present article concentrates on the existence of positive solutions to the system (1.1) and can be seen as a complement of the results developed in [10,11,13].

This article is organized as follows. In Section 2, we give the necessary notations and preliminaries. Section 3 is devoted to the behavior of (PS)_c sequence and the mountain geometry structure. Finally, in Section 4, we prove the existence of nontrivial solution.

Let denote the collection of smooth functions with compact support and D^1,p(ℝ^N) be the completion of under

We introduce the space

equipped with the norm

and the space

under

Observe that ‖ · ‖_E is equivalent to the one ‖ · ‖_λ for each λ > 0. It follows from (V₀) that E(ℝ^N, V) continuously embeds in W^1,p(ℝ^N).

Set B = E_λ × E_λ and for any (u, v) ∈ B. Let λ = ε^-p in the system (1.1), then (1.1) is changed into

In order to prove Theorem 1, we only need to prove the following result.

Theorem 2. Let (V₀), (K₀) and (H₁)-(H₃) be satisfied. Then for any σ > 0, there exists Λ_σ > 0 such that if λ ≥ Λ_σ , the system (2.1) has at least one least energy solution (u_λ, v_λ) satisfying

The energy functional associated with (2.1) is defined by

where .

From the assumptions of Theorem 2, standard arguments [14] show that I_λ ∈ C¹(B, ℝ) and its critical points are the weak solutions of (2.1).

In this section, we will recall and prove some lemmas which are crucial in the proof of the main result.

Lemma 3.1. Let the assumptions of Theorem 2 be satisfied. If the sequence {(u_n, v_n)} ⊂ B is a (PS)_c sequence for I_λ, then we get that c ≥ 0 and {(u_n, v_n)} is bounded in the space B.

Proof. One has

By the assumptions (K₀) and (H₃), we have

Together with I_λ(u_n, v_n) → c and as n → ∞, we easily obtain that the (PS)_c sequence is bounded in B and the energy level c ≥ 0. □

From Lemma 3.1, there exists (u, v) ∈ B such that (u_n, v_n) ⇀ (u, v) in B. Furthermore, passing to a subsequence, we have u_n → u and v_n → v in for any d ∈ [p, p*) and u_n → u, v_n → v a.e. in ℝ^N.

Lemma 3.2. Let d ∈ [p, p*). There exists a subsequence such that for any ε > 0, there is r_ε > 0 with

for any r ≥ r_ε , where B_r := {x ∈ ℝ^N : |x| ≤ r}.

Proof. The proof of Lemma 3.2 is similar to the one of Lemma 3.2 of [10], so we omit it. □

Let η ∈ C^∞(ℝ⁺) be a smooth function satisfying 0 ≤ η(t) ≤ 1, η(t) = 1 if t ≤ 1 and η(t) = 0 if t ≥ 2. Define , . It is obvious that

Lemma 3.3. One has

and

uniformly in (φ, ψ) ∈ B with ‖(φ, ψ‖_B ≤ 1.

Proof. From the assumptions (H₁)-(H₂) and Lemma 3.2, we have

By Hölder inequality and Lemma 3.2, it follows that

and

Similarly, we get

and

Thus

From the similar argument, we also get

□

Lemma 3.4. One has along a subsequence

and

Proof. From the Lemma 2.1 of [15] and the argument of [16], we have

By (3.1) and the similar idea of proving the Brézis-Lieb Lemma [17], it is easy to get

and

In connection with the fact I_λ (u_n, v_n) → c and , we obtain

In the following, we will verify the fact .

For any (φ, ψ) ∈ B, it follows that

Standard argument shows that

and

uniformly in ‖φ, ψ)‖_B ≤ 1.

By Lemma 3.3, we have

and

uniformly in ‖(φ, ψ)‖_B ≤ 1. From the facts above mentioned, we obtain

□

Let , , then , . From (3.1), we get (u_n, v_n) → (u, v) in B if and only if in B.

Observe that

where .

Thus by Lemma 3.4, we get

Now, we consider the energy level of the functional I_λ below which the (PS)_c condition hold.

Let V_b(x):= max{V (x), b}, where b is the positive constant in the assumption (V₀). Since the set ν_b has finite measure and , in , we get

From (K₀), (H₁)-(H₃) and Young inequality, there is C_b > 0 such that

Let S be the best Sobolev constant of the immersion

Lemma 3.5. Let the assumptions of Theorem 2 be satisfied. There exists α₀ > 0 independent of λ such that, for any (PS)_c sequence {(u_n, v_n)} ⊂ B for I_λ with (u_n, v_n) ⇀ (u, v), either (u_n, v_n) → (u, v) or .

Proof. Assume that (u_n, v_n) ↛ (u, v), then

and

By the Sobolev inequality, (3.4) and (3.5), we get

This, together with and (3.3), gives

Set , then

This proof is completed. □

Since is not compact, I_λ does not satisfy the (PS)_c condition for all c > 0. But Lemma 3.5 shows that I_λ satisfies the following local (PS)_c condition.

Lemma 3.6. From the assumptions of Theorem 2, there exists a constant α₀ > 0 independent of λ such that, if a (PS)_c sequence {(u_n, v_n)} ⊂ B for I_λ satisfies , the sequence {(u_n, v_n)} has a strongly convergent subsequence in B.

Proof. By the fact , we have

This, together with I_λ(u, v) ≥ 0 and Lemma 3.5, gives the desired conclusion. □

Next, we consider λ = 1. From the following standard argument, we get that I_λ possesses the mountain-pass structure.

Lemma 3.7. Under the assumptions of Theorem 2, there exist α_λ, ρ_λ > 0 such that

Proof. By (3.5), we get that for any δ > 0, there is C_δ > 0 such that

Thus

Note that . If δ ≤ (2pλC₁)^-1, then

The fact p* > p implies the desired conclusion. □

Lemma 3.8. Under the assumptions of Lemma 3.7, for any finite dimensional subspace

F ⊂ B, we have

Proof. By the assumption (H₃), it follows that

Since all norms in a finite-dimensional space are equivalent and α, β > p, we prove the result of this Lemma. □

By Lemma 3.6, for λ larger enough and c_λ small sufficiently, I_λ satisfies (PS)_cλ condition.

Thus, we will find special finite-dimensional subspaces by which we establish sufficiently small minimax levels.

Define the functional

It is apparent that Φ_λ ∈ C¹(B) and I_λ(u, v) ≤ Φ_λ (u, v) for all (u, v) ∈ B.

Observe that

and

For any δ> 0, there are φ_δ, with and suppφ_δ, such that .

Let , then . For t ≥ 0, we get

where

We easily prove that

Together with V (0) = 0 and , this implies that there is Λ_δ > 0 such that for all λ ≥ Λ_δ, we have

It follows from (3.6) that

Lemma 3.9. Under the assumptions of Lemma 3.7, for any ⊂ > 0, there is Λ_σ > 0 such that λ ≥ Λ_σ, there exists with and

where ρ_λ is defined in Lemma 3.7.

Proof. This proof is similar to the one of Lemma 4.3 in [10], it can be easily proved. □

In the following, we will give the proof of Theorem 2.

Proof. From Lemma 3.9, for any σ > 0 with 0 < σ < α₀, there is Λ_σ > 0 such that for λ ≥ Λ_σ, we obtain

where

Furthermore, Lemma 3.6 implies that I_λ satisfies (PS)_cλ condition. Hence, by the mountain-pass theorem, there is (u_λ, v_λ) ∈ B satisfying I_λ (u_λ, v_λ) = c_λ and This shows (u_λ, v_λ) is a weak solution of (2.1). Similar to the argument in [10], we also get that (u_λ, v_λ) is a positive least energy solution.

Finally, we prove (u_λ, v_λ) satisfies the estimate (2.2). Observe that and we have

This shows that (u_λ, v_λ) satisfies the estimate (2.2). The proof is complete. □

The authors declare that they have no competing interests.

The authors contributed equally in this article. They read and approved the final manuscript.

The authors would like to appreciate the referees for their precious comments and suggestions about the original manuscript. This research is supported by the National Natural Science Foundation of China (10771212) and the Fundamental Research Funds for the Central Universities (2010LKSX09).

1. Brézis, H, Nirenberg, L: Some variational problems with lack of compactness. Proc Symp Pure Math. 45, 165–201 (1986)

2. Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J Funct Anal. 14, 349–381 (1973)

3. Guedda, M, Veron, L: Quasilinear elliptic equations involving critical Sobolev exponents. Non Anal. 12, 879–902 (1989)

4. Benci, V, Cerami, G: Existence of positive solutions of the equation in ℝ^N. J Funct Anal. 88, 90–117 (1990)

5. Floer, A, Weinstein, A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J Funct Anal. 69, 397–408 (1986)

6. Oh, YG: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun Math Phys. 131, 223–253 (1990)

7. Clapp, M, Ding, YH: Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential. Diff Integr Equ. 16, 981–992 (2003)

8. Del Pino, M, Felmer, P: Semi-classical states for nonlinear Schrödinger equations. Annales Inst H Poincaré. 15, 127–149 (1998)

9. Cingolani, S, Lazzo, M: Multiple positive solutions to nonlinear Schrödinger equation with competing potential functions. J Diff Equ. 160, 118–138 (2000)

10. Ding, YH, Lin, FH: Solutions of perturbed Schrödinger equations with critical nonlinearity. Calc Var. 30, 231–249 (2007)

11. Garcia Azoero, J, Peral Alonso, I: Existence and non-uniqueness for the p-Laplacian: Nonlinear eigenvalues. Commun Partial Diff Equ. 12, 1389–1430 (1987)

12. Alves, CO, Ding, YH: Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity. J Math Anal Appl. 279, 508–521 (2003)

13. Liu, CG, Zheng, YQ: Existence of nontrivial solutions for p-Laplacian equations in ℝ^N. J Math Anal Appl. 380, 669–679 (2011)

14. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems. Springer-Verlag, New York (1989)

15. Li, YY, Guo, QQ, Niu, PC: Global compactness results for quasilinear elliptic problems with combined critical Sobolev-Hardy terms. Non Anal. 74, 1445–1464 (2011)

16. Ghoussoub, N, Yuan, C: Multiple solutions for quasi-linear PDES involving the critical Sobolev and Hardy exponents. Trans Am Math Soc. 352, 5703–5743 (2000)

17. Brézis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functional. Proc Am Math Soc. 88, 486–490 (1983)