Abstract
We consider a pLaplacian system with critical nonlinearity in ℝ^{N}. Under the proper assumptions, we obtain the existence of nontrivial solutions to perturbed pLaplacian system by using the variational approach.
MR Subject Classification: 35B33; 35J60; 35J65.
Keywords:
pLaplacian system; critical nonlinearity; variational methods.1 Introduction
This article is concerned with the existence of solutions to the following nonlinear perturbed pLaplacian system
where Δ_{p}u = div(∇u^{p2}∇u) is the pLaplacian operator, 1 < p < N and p* = Np/(N − p) is the critical exponent.
Throughout the article, we will assume that:
(V_{0}) 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;
(K_{0}) K(x) ∈ C(ℝ^{N}), 0 < inf K ≤ sup K < ∞;
(H_{1}) H ∈ C^{1}(ℝ^{2}) and H_{s}, H_{t }= o(s^{p1 }+ t^{p1}) as s + t → 0;
(H_{2}) there exist c > 0 and p < q < p* such that
(H_{3}) There are a_{0 }> 0, θ ∈ (p, p*) and α, β > p such that H(s, t) ≥ a_{0}(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 (V_{0}), (K_{0}) and (H_{1})(H_{3}) 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:
(A_{1}) 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;
(A_{2}) K(x) ∈ C(ℝ^{N}), 0 < inf K ≤ sup K < ∞
(B_{1}) h ∈ C(ℝ^{N }× ℝ) and h(x, u) = o(u) uniformly in x as u → 0;
(B_{2}) there are c_{0 }> 0, q < 2* such that h(x, u) ≤ c_{0}(1 + u^{q1}) for all (x, u);
(B_{3}) there are a_{0 }> 0, p > 2 and µ > 2 such that H(x, u) = a_{0}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 (A_{1})(A_{2}) and (B_{1}) (B_{3}) 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 p^{2 }≤ N and µ ∈ (0, λ_{1}), where λ_{1 }is the first eigenvalue of the pLaplacian. 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 signchanging potential and subcritical psuperlinear 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 pLaplacian 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 PalaisSmale 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.
2 Notations and preliminaries
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 (V_{0}) 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 (V_{0}), (K_{0}) and (H_{1})(H_{3}) 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
From the assumptions of Theorem 2, standard arguments [14] show that I_{λ }∈ C^{1}(B, ℝ) and its critical points are the weak solutions of (2.1).
3 Technical lemmas
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 (K_{0}) and (H_{3}), 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 (H_{1})(H_{2}) 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 BrézisLieb 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
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 (V_{0}). Since the set ν_{b }has finite measure and , in , we get
From (K_{0}), (H_{1})(H_{3}) 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 α_{0 }> 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
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 α_{0 }> 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.
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 mountainpass 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 ≤ (2pλC_{1})^{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 (H_{3}), it follows that
Since all norms in a finitedimensional 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 finitedimensional subspaces by which we establish sufficiently small minimax levels.
Define the functional
It is apparent that Φ_{λ }∈ C^{1}(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. □
4 Proof of the main result
In the following, we will give the proof of Theorem 2.
Proof. From Lemma 3.9, for any σ > 0 with 0 < σ < α_{0}, there is Λ_{σ }> 0 such that for λ ≥ Λ_{σ}, we obtain
Furthermore, Lemma 3.6 implies that I_{λ }satisfies (PS)_{cλ }condition. Hence, by the mountainpass 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. □
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors contributed equally in this article. They read and approved the final manuscript.
Acknowledgements
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).
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