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May/June 2018 Existence and multiplicity of solutions for equations of $p(x)$-Laplace type in $\mathbb R^{N}$ without AR-condition
Jae-Myoung Kim, Yun-Ho Kim, Jongrak Lee
Differential Integral Equations 31(5/6): 435-464 (May/June 2018).

Abstract

We are concerned with the following elliptic equations with variable exponents \begin{equation*} -\text{div}(\varphi(x,\nabla u))+V(x)|u|^{p(x)-2}u=\lambda f(x,u) \quad \text{in} \quad \mathbb R^{N}, \end{equation*} where the function $\varphi(x,v)$ is of type $|v|^{p(x)-2}v$ with continuous function $p: \mathbb R^{N} \to (1,\infty)$, $V: \mathbb R^{N}\to(0,\infty)$ is a continuous potential function, and $f: \mathbb R^{N}\times \mathbb R \to \mathbb R$ satisfies a Carathéodory condition. The aims of this paper are stated as follows. First, under suitable assumptions, we show the existence of at least one nontrivial weak solution and infinitely many weak solutions for the problem without the Ambrosetti and Rabinowitz condition, by applying mountain pass theorem and fountain theorem. Second, we determine the precise positive interval of $\lambda$'s for which our problem admits a nontrivial solution with simple assumptions in some sense.

Citation

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Jae-Myoung Kim. Yun-Ho Kim. Jongrak Lee. "Existence and multiplicity of solutions for equations of $p(x)$-Laplace type in $\mathbb R^{N}$ without AR-condition." Differential Integral Equations 31 (5/6) 435 - 464, May/June 2018.

Information

Published: May/June 2018
First available in Project Euclid: 23 January 2018

zbMATH: 06861586
MathSciNet: MR3749216

Subjects:
Primary: 35D30, 35J20, 35J60, 35J92, 47J30

Rights: Copyright © 2018 Khayyam Publishing, Inc.

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Vol.31 • No. 5/6 • May/June 2018
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