## Taiwanese Journal of Mathematics

### Existence and Multiplicity of Solutions for a Quasilinear Elliptic Inclusion with a Nonsmooth Potential

#### Abstract

This paper is concerned with a nonlinear elliptic inclusion driven by a multivalued subdifferential of nonsmooth potential and a nonlinear inhomogeneous differential operator. We obtain two multiplicity theorems in the Orlicz-Sobolev space. In the first multiplicity theorem, we produce three nontrivial smooth solutions. Two of these solutions have constant sign (one is positive, the other is negative). In the second multiplicity theorem, we derive an unbounded sequence of critical points for the problem. Our approach is variational, based on the nonsmooth critical point theory. We also show that $C^1$-local minimizers are also local minimizers in the Orlicz-Sobolev space for a large class of locally Lipschitz functions.

#### Article information

Source
Taiwanese J. Math., Volume 22, Number 3 (2018), 635-660.

Dates
Received: 23 December 2016
Revised: 16 May 2017
Accepted: 22 August 2017
First available in Project Euclid: 4 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1507082431

Digital Object Identifier
doi:10.11650/tjm/170809

Mathematical Reviews number (MathSciNet)
MR3807330

Zentralblatt MATH identifier
06965390

#### Citation

Yuan, Ziqing; Huang, Lihong; Wang, Dongshu. Existence and Multiplicity of Solutions for a Quasilinear Elliptic Inclusion with a Nonsmooth Potential. Taiwanese J. Math. 22 (2018), no. 3, 635--660. doi:10.11650/tjm/170809. https://projecteuclid.org/euclid.twjm/1507082431

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