Infinitely Many Solutions for p $p$ -Laplacian Equation in R N
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Infinitely Many Solutions for p-Laplacian Equation in RN Without the Ambrosetti-Rabinowitz Condition Caisheng Chen1 · Lijuan Chen1,2
Received: 31 August 2015 / Accepted: 25 February 2016 © Springer Science+Business Media Dordrecht 2016
Abstract Using variational arguments, we establish the existence of infinitely many solutions for p-Laplacian equation −p u + V (x)|u|p−2 u = f (x, u), x ∈ RN , where the potential V (x) ∈ C(RN ) and 0 < infx∈RN V (x) ≤ supx∈RN V (x) < ∞, and f (x, u) fails to satisfy Ambrosetti-Rabinowitz condition. A major point is that we develop a new technique to verify the boundedness and compactness of Cerami sequence in W 1,p (RN ). Keywords Variational methods · p-Laplacian equation · Cerami sequence · Ambrosetti-Rabinowitz condition Mathematics Subject Classification 35J20 · 35J70 · 35J92
1 Introduction and Preliminaries In this paper, we study the existence of infinitely many solutions for p-Laplacian equation −p u + V (x)|u|p−2 u = f (x, u),
x ∈ RN ,
(1.1)
where 1 < p < N . The potential functions V (x) is positive and bounded in RN , and f (x, u) is a suplinear and subcritical function without Ambrosetti-Rabinowitz condition. The function f (x, u) is said to satisfy Ambrosetti-Rabinowitz condition ((AR) condition for short) if there exist the constants M > 0 and θ > p such that u f (x, s)ds. (1.2) 0 < θ F (x, u) ≤ uf (x, u), x ∈ RN , |u| ≥ M, where F (x, u) = 0
B C. Chen
[email protected]
1
College of Science, Hohai University, Nanjing 210098, P.R. China
2
Yancheng Institute of Technology, Yancheng 224051, P.R. China
C. Chen, L. Chen
It is well known that many superlinear functions exist that do not satisfy the (AR) condition and these functions have attracted much interest in recent years. We refer the reader to previous studies by [2, 3, 5, 8–14] as well as the references therein, which consider problems that involve the p-Laplacian, sometimes with p = 2, and (p, q)-Laplacian equations. Denote |∇u|2 + V (x)|u|2 dx < ∞ . (1.3) X = u ∈ H 1 RN : RN
Recently, Tang in [17] proved the existence of infinitely many solutions in X for the problem −u + V (x)u = f (x, u),
x ∈ RN ,
(1.4)
where N ≥ 3 and the potential V (x) and the nonlinearity f (x, u) satisfy (V1 ) V (x) ∈ C(RN ) and infx∈RN V (x) > 0. Moreover, there exists a constant d0 > 0 such that lim meas x ∈ RN : |x − y| ≤ d0 , V (x) ≤ M = 0, ∀M > 0. |y|→∞
(S1 ) f (x, u) ∈ C(RN × R), f (x, −u) = −f (x, u) and ∃c1 , c2 > 0 and p ∈ (2, 2∗ ) such that f (x, u) ≤ c1 |u| + c2 |u|p−1 , ∀(x, u) ∈ RN × R. (x,u)| (S2 ) lim|u|→∞ |F|u| = ∞ a.e. x ∈ RN , and ∃r0 > 0 such that F (x, u) ≥ 0, ∀x ∈ RN , 2 |u| ≥ r0 . (S3 ) F (x, u) := 12 uf (x, u) − F (x, u) ≥ 0 for x ∈ RN , |u| ≥ r0 , and there exist c0 > 0 and k > max{1, N/2} such that
F (x, u) k ≤ c0 |u|2k F (x, u),
∀(x, u) ∈ RN × R, |u| ≥ r0 .
Here we note that the V (x) is unbounded in RN under assumption (V1 ). Furthermore, the coercivity of V (x) is essential in the proof of existence of solutions in [17] since it 2N , and compact ensures that the e
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