Annales Academić Scientiarum Fennicć
Mathematica
Volumen 36, 2011, 461-480

THE EXISTENCE OF A NONTRIVIAL SOLUTION TO A NONLINEAR ELLIPTIC PROBLEM OF LINKING TYPE WITHOUT THE AMBROSETTI-RABINOWITZ CONDITION

Gongbao Li and Chunhua Wang

Central China Normal University, School of Mathematics and Statistics
Wuhan 430079, P.R. China; ligb 'at' mail.ccnu.edu.cn

Central China Normal University, School of Mathematics and Statistics
Wuhan 430079, P.R. China; wch5923 'at' yahoo.com.cn

Abstract. In this paper, we study the existence of a nontrivial solution to the following nonlinear elliptic problem:

-\Delta u - a(x)u = f(x,u), x \in\Omega,
u|_{\partial\Omega} = 0, (0.1)

where \Omega is a bounded domain of R^N and a \in L^N/2(\Omega), N \geq 3, f \in C⁰(\bar{\Omega} x R¹,R¹) is superlinear at t = 0 and subcritical at t = \infty. Under suitable conditions, (0.1) possesses the so-called linking geometric structure. We prove that the problem (0.1) has at least one nontrivial solution without assuming the Ambrosetti-Rabinowitz condition. Our main result extends a recent result of Miyagaki and Souto given in [14] for (0.1) with a(x) = 0 and possessing the mountain-pass geometric structure.

2010 Mathematics Subject Classification: Primary 35A15, 35D05, 35J20.

Key words: Deformation lemma, minimax theorem under C_c condition, linking geometric structure, without the Ambrosetti-Rabinowitz condition, nontrivial solutions.

Reference to this article: G. Li and C. Wang: The existence of a nontrivial solution to a nonlinear elliptic problem of linking type without the Ambrosetti-Rabinowitz condition. Ann. Acad. Sci. Fenn. Math. 36 (2011), 461-480.

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doi:10.5186/aasfm.2011.3627