Differential and Integral Equations

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, and Jongrak Lee

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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.

Article information

Differential Integral Equations, Volume 31, Number 5/6 (2018), 435-464.

First available in Project Euclid: 23 January 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35D30: Weak solutions 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 35J92: Quasilinear elliptic equations with p-Laplacian 47J30: Variational methods [See also 58Exx]


Kim, Jae-Myoung; Kim, Yun-Ho; Lee, Jongrak. Existence and multiplicity of solutions for equations of $p(x)$-Laplace type in $\mathbb R^{N}$ without AR-condition. Differential Integral Equations 31 (2018), no. 5/6, 435--464. https://projecteuclid.org/euclid.die/1516676437

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