July/August 2018 Weak solvability for Dirichlet partial differential inclusions in Orlicz-Sobolev spaces
Nicuşor Costea, Gheorghe Moroşanu, Csaba Varga
Adv. Differential Equations 23(7/8): 523-554 (July/August 2018). DOI: 10.57262/ade/1526004065


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.


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Nicuşor Costea. Gheorghe Moroşanu. Csaba Varga. "Weak solvability for Dirichlet partial differential inclusions in Orlicz-Sobolev spaces." Adv. Differential Equations 23 (7/8) 523 - 554, July/August 2018. https://doi.org/10.57262/ade/1526004065


Published: July/August 2018
First available in Project Euclid: 11 May 2018

zbMATH: 06889036
MathSciNet: MR3801830
Digital Object Identifier: 10.57262/ade/1526004065

Primary: 35B38 , 35D30 , 35J20 , 49J52

Rights: Copyright © 2018 Khayyam Publishing, Inc.


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Vol.23 • No. 7/8 • July/August 2018
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