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Multiplicity for nonlinear elliptic boundary value problems of p-Laplacian type without Ambrosetti-Rabinowitz condition

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Abstract

In this paper, we study the existence of multiple solutions to the following nonlinear elliptic boundary value problem of p-Laplacian type:

$\left\{ \begin{gathered} - \Delta _p u = f(x,u),x \in \Omega , \hfill \\ u = 0,x \in \partial \Omega , \hfill \\ \end{gathered} \right. $
((*))

where 1 < p < ∞, Ω ⊆ ℝN is a bounded smooth domain, Δ p u = div(|Du|p−2 Du) is the p-Laplacian of u and f: Ω × ℝ → ℝ satisfies \(\mathop {\lim }\limits_{|t| \to \infty } \tfrac{{f(x,t)}} {{|t|^{p - 2} t}} = l \) uniformly with respect to x ∈ Ω, and l is not an eigenvalue of −Δ p in W 1,p0 (Ω) but f(x, t) dose not satisfy the Ambrosetti-Rabinowitz condition. Under suitable assumptions on f(x, t), we have proved that (*) has at least four nontrivitial solutions in W 1,p0 (Ω) by using Nonsmooth Mountain-Pass Theorem under (C) c condition. Our main result generalizes a result by N. S. Papageorgiou, E. M. Rocha and V. Staicu in 2008 (Calculus of Variations and Partial Differential Equations, 33: 199–230(2008)) and a result by G. B. Li and H. S. Zhou in 2002 (Journal of the London Mathematical Society, 65: 123–138(2002)).

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Authors and Affiliations

  1. Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Gong-bao Li & Ying-hong Li

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  1. Gong-bao Li
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  2. Ying-hong Li
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Correspondence to Gong-bao Li.

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Supported in part by the National Natural Science Foundation of China under Grant No: 11071095, Grant No. 11371159 and Program for Changjiang Scholars and Innovative Research Team in University # IRT13066.

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Li, Gb., Li, Yh. Multiplicity for nonlinear elliptic boundary value problems of p-Laplacian type without Ambrosetti-Rabinowitz condition. Acta Math. Appl. Sin. Engl. Ser. 31, 157–180 (2015). https://doi.org/10.1007/s10255-015-0458-4

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  • DOI: https://doi.org/10.1007/s10255-015-0458-4

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