Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2021; 58(6): 1461-1484

Online first article September 30, 2021      Printed November 1, 2021

https://doi.org/10.4134/JKMS.j210099

Copyright © The Korean Mathematical Society.

Multiplicity of solutions for quasilinear Schr\"{o}dinger type equations with the concave-convex nonlinearities

In Hyoun Kim, Yun-Ho Kim, Chenshuo Li, Kisoeb Park

Incheon National University; Sangmyung University; Boston University; Seoul Theological University

Abstract

We deal with the following elliptic equations: \begin{equation*} \left\{ \begin{array}{ll} \displaystyle -\text{div}(\varphi^{\prime}(|\nabla z|^2)\nabla z) +V(x)|z|^{\alpha-2}z=\lambda \rho(x)|z|^{r-2}z + h(x,z), \\ \vspace{-3mm}\\ \displaystyle z(x) \rightarrow 0, \quad \mbox{as} \ |x| \rightarrow \infty, \end{array}\right. \mbox{in} \,\R^N, \end{equation*} where $N \geq 2$, $1 < p < q < N$, $1 < \alpha \leq p^*q^{\prime}/p^{\prime}$, $\alpha < q$, $1 < r < \min\{p,\alpha\}$, $\varphi(t)$ behaves like $t^{q/2}$ for small $t$ and $t^{p/2}$ for large $t$, and $p^{\prime}$ and $q^{\prime}$ the conjugate exponents of $p$ and $q$, respectively. Here, $V:\mathbb R^{N} \to (0,\infty)$ is a potential function and $h:\mathbb R^{N}\times\mathbb R \to \mathbb R$ is a Carath\'eodory function. The present paper is devoted to the existence of at least two distinct non-trivial solutions to quasilinear elliptic problems of Schr\"{o}dinger type, which provides a concave--convex nature to the problem. The primary tools are the well-known mountain pass theorem and a variant of Ekeland's variational principle.

Keywords: Quasilinear elliptic equations, concave-convex nonlinearities, variational methods, Orlicz-Sobolev spaces

MSC numbers: 35J50, 35J62, 46E30, 46E35

Supported by: The first author was supported by the Incheon National University Research Grant in 2017. The authors gratefully thank to the Referee for the constructive comments and recommendations which definitely help to improve the readability and quality of the paper.