Taiwanese Journal of Mathematics

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

Ziqing Yuan, Lihong Huang, and Dongshu Wang

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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

Subjects
Primary: 35R70: Partial differential equations with multivalued right-hand sides 35J70: Degenerate elliptic equations 49J52: Nonsmooth analysis [See also 46G05, 58C50, 90C56]

Keywords
locally Lipschitz nonsmooth critical point Orlicz-Sobolev space nonsmooth fountain theorem second deformation theorem

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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