Advances in Differential Equations

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

Nicuşor Costea, Gheorghe Moroşanu, and Csaba Varga

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 11 May 2018

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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.

Export citation