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Multiple positive solutions for a class of concave–convex elliptic problems in ℝ^N involving sign-changing weight, II

Yisheng Huang

Department of Mathematics, Soochow University, Suzhou 215006, P. R. China

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Tsung-Fang Wu

Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan

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, and

Yuanze Wu

Department of Mathematics, Soochow University, Suzhou 215006, P. R. China

College of Sciences, China University of Mining and Technology, Xuzhou 221116, P. R. China

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https://doi.org/10.1142/S021919971450045XCited by:19 (Source: Crossref)

Abstract

In this paper, we study the following concave–convex elliptic problems:

where N ≥ 3, 1 < q < 2 < p < 2* = 2N/(N - 2), λ > 0 and μ < 0 are two parameters. By using several variational methods and a perturbation argument, we obtain three positive solutions to this problem under the predefined conditions of f_λ(x) and g_μ(x), which simultaneously extends the result of [T. Hsu, Multiple positive solutions for a class of concave–convex semilinear elliptic equations in unbounded domains with sign-changing weights, Bound. Value Probl. 2010 (2010), Article ID 856932, 18pp.; T. Wu, Multiple positive solutions for a class of concave–convex elliptic problems in ℝ^N involving sign-changing weight, J. Funct. Anal. 258 (2010) 99–131]. We also study the concentration behavior of these three solutions both for λ → 0 and μ → -∞.

Keywords:

AMSC: 35B38, 35B40, 35J10, 35J20

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