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February, 2019 Critical Points Theorems via the Generalized Ekeland Variational Principle and its Application to Equations of $p(x)$-Laplace Type in $\mathbb{R}^{N}$
Jung-Hyun Bae, Yun-Ho Kim
Taiwanese J. Math. 23(1): 193-229 (February, 2019). DOI: 10.11650/tjm/181004

Abstract

In this paper, we investigate abstract critical point theorems for continuously Gâteaux differentiable functionals satisfying the Cerami condition via the generalized Ekeland variational principle developed by C.-K. Zhong. As applications of our results, under certain assumptions, we show the existence of at least one or two weak solutions for nonlinear elliptic equations with variable exponents \[ -\operatorname{div} (\varphi(x, \nabla u)) + V(x) |u|^{p(x)-2} u = \lambda f(x,u) \quad \textrm{in } \mathbb{R}^{N}, \] where the function $\varphi(x,v)$ is of type $|v|^{p(x)-2}v$ with a continuous function $p \colon \mathbb{R}^{N} \to (1,\infty)$, $V \colon \mathbb{R}^{N} \to (0,\infty)$ is a continuous potential function, $\lambda$ is a real parameter, and $f \colon \mathbb{R}^{N} \times \mathbb{R} \to \mathbb{R}$ is a Carathéodory function. Especially, we localize precisely the intervals of $\lambda$ for which the above equation admits at least one or two nontrivial weak solutions by applying our critical points results.

Citation

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Jung-Hyun Bae. Yun-Ho Kim. "Critical Points Theorems via the Generalized Ekeland Variational Principle and its Application to Equations of $p(x)$-Laplace Type in $\mathbb{R}^{N}$." Taiwanese J. Math. 23 (1) 193 - 229, February, 2019. https://doi.org/10.11650/tjm/181004

Information

Received: 12 March 2018; Accepted: 30 September 2018; Published: February, 2019
First available in Project Euclid: 22 October 2018

zbMATH: 07021724
MathSciNet: MR3909996
Digital Object Identifier: 10.11650/tjm/181004

Subjects:
Primary: 35D30, 35J15, 35J60, 58E05, 58E30

Rights: Copyright © 2019 The Mathematical Society of the Republic of China

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Vol.23 • No. 1 • February, 2019
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