Robin problems involving the \(p(x)\)-Laplacian. (English) Zbl 1427.35072
Summary: By applying Mountain Pass Lemma and Ekeland’s variational principle, we prove two different situations of the existence of solutions for the following Robin problem \[\begin{alignedat}{2}
&-\Delta_{p(x)} u = \lambda V(x) | u |^{q(x) - 2} u &\qquad&\text{in } \Omega, \\
&|\nabla u|^{p(x) - 2} \frac{\partial u}{\partial \nu} + \beta(x) | u |^{p(x) - 2} u = 0 \& &\text{on } \partial \Omega,
\end{alignedat}\] where \(\Omega \subset \mathbb{R}^N (N \geq 2)\) is a bounded smooth domain, \(V\) is an indefinite weight function which can change sign in \(\Omega\) and \(p, q : \overline{\Omega} \to(1, +\infty)\) are continuous functions.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
| 35J66 | Nonlinear boundary value problems for nonlinear elliptic equations |
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