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Multiple solutions for a class of semilinear elliptic variational inclusion problems. (English) Zbl 1231.35327

From the text: In this paper, by using a local linking theorem, we obtain the existence of multiple solutions for a class of semilinear elliptic variational inclusion problems at non-resonance. The system under study, also called a hemivariational inequality problem, is of the form \[ \begin{cases} -\Delta u (x) - c(x)u(x) \in \partial j(x, u(x)), \qquad \text{a.e.}\qquad x\in\Omega,\\u=0, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x\in\Omega,\end{cases} \tag{1} \] where \(\Omega \subseteq {\mathbb R}^N\) \((N \geq 3)\) is a bounded domain with the smooth boundary \(\partial \Omega\), \(j(x, \zeta) : \Omega \times {\mathbb R} \to {\mathbb R}\) and \(j\) is a locally Lipschitz function in the \(\zeta\)-variable, \(\partial j(x, \zeta)\) denotes the generalized gradient of \(\zeta \mapsto j(x, \zeta)\), \(c(x)\in L^\infty (\Omega)\). We obtain that there are at least two nontrivial solutions for the semilinear elliptic variational inclusion problem (1) at non-resonance by using the local linking theorem.

MSC:

35R70 PDEs with multivalued right-hand sides
35J50 Variational methods for elliptic systems
35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
35R45 Partial differential inequalities and systems of partial differential inequalities
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