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Nontrivial solutions for Kirchhoff-type problems involving the \(p(x)\)-Laplace operator. (English) Zbl 1401.35061

Summary: In this article, we study the existence of nontrivial solutions for the following \(p(x)\) Kirchhoff-type problem \[ \begin{cases} -M\bigg(\int_\Omega A(x,\nabla u)dx\bigg)\mathrm{div}(a(x,\nabla u)) =\lambda h(x)(\partial F/\partial u (x,u),\qquad &\text{in}\,\,\Omega \\ u=0, \qquad &\text{on}\,\,\partial \Omega, \end{cases} \] where \(\Omega \subset \mathbb R^n\), \(n\geq 3\), is a smooth bounded domain, \(\lambda >0\), \(h\in C(\Omega)\), \(F:\overline {\Omega}\times \mathbb R\to\mathbb R\) is continuously differentiable and \(a, A:\Omega \times \mathbb R^n\to \mathbb R^n\) are continuous. The proof is based on variational arguments and the theory of variable exponent Sobolev spaces.

MSC:

35J35 Variational methods for higher-order elliptic equations
35J50 Variational methods for elliptic systems
35J60 Nonlinear elliptic equations
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