Taiwanese Journal of Mathematics

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 and Yun-Ho Kim

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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.

Article information

Source
Taiwanese J. Math., Volume 23, Number 1 (2019), 193-229.

Dates
Received: 12 March 2018
Accepted: 30 September 2018
First available in Project Euclid: 22 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1540195384

Digital Object Identifier
doi:10.11650/tjm/181004

Mathematical Reviews number (MathSciNet)
MR3909996

Zentralblatt MATH identifier
07021724

Subjects
Primary: 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.) 35D30: Weak solutions 35J15: Second-order elliptic equations 35J60: Nonlinear elliptic equations 58E30: Variational principles

Keywords
critical points theorems Ekeland's variational principle mountain pass theorem $p(x)$-Laplace type operator variable exponent Lebesgue-Sobolev spaces weak solutions

Citation

Bae, Jung-Hyun; Kim, Yun-Ho. 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 (2019), no. 1, 193--229. doi:10.11650/tjm/181004. https://projecteuclid.org/euclid.twjm/1540195384


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