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.
In Hyoun Kim, Yun-Ho Kim, Chenshuo Li, Kisoeb Park
Incheon National University; Sangmyung University; Boston University; Seoul Theological University
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.
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