Advances in Differential Equations

Weak solvability for Dirichlet partial differential inclusions in Orlicz-Sobolev spaces

Nicuşor Costea, Gheorghe Moroşanu, and Csaba Varga

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Abstract

We study PDI's of the type $-\Delta_\Phi u\in \partial_C f(x,u)$ subject to Dirichlet boundary condition in a bounded domain $\Omega\subset\mathbb{R}^N$ with Lipschitz boundary $\partial\Omega$. Here, $\Phi:\mathbb{R}\rightarrow [0,\infty)$ is the $N$-function defined by $\Phi(t):=\int_0^t a(|s|)s\,ds$, with $a:(0,\infty)\rightarrow (0,\infty)$ a prescribed function, not necessarily differentiable, and $\Delta_\Phi u:={\rm div}(a(|\nabla u|)\nabla u)$ is the $\Phi$-Laplacian. In addition, $f:\Omega\times\mathbb{R}\rightarrow \mathbb{R}$ is a locally Lipschitz function with respect to the second variable and $\partial_C$ denotes the Clarke subdifferential. Using a direct variational method and a nonsmooth version of the Mountain Pass Theorem the existence of nontrivial weak solutions is established. A multiplicity alternative is also proved without imposing an Ambrosetti-Rabinowitz type condition. More precisely, we show that our problem possesses either at least two nontrivial weak solutions or a rich family of negative eigenvalues. Several examples which highlight the applicability of our theoretical results are also provided.

Article information

Source
Adv. Differential Equations, Volume 23, Number 7/8 (2018), 523-554.

Dates
First available in Project Euclid: 11 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.ade/1526004065

Mathematical Reviews number (MathSciNet)
MR3801830

Zentralblatt MATH identifier
06889036

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations 49J52: Nonsmooth analysis [See also 46G05, 58C50, 90C56] 35D30: Weak solutions 35B38: Critical points

Citation

Costea, Nicuşor; Moroşanu, Gheorghe; Varga, Csaba. Weak solvability for Dirichlet partial differential inclusions in Orlicz-Sobolev spaces. Adv. Differential Equations 23 (2018), no. 7/8, 523--554. https://projecteuclid.org/euclid.ade/1526004065


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