## 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

### Keywords:

$$p(x)$$-Laplacian; Kirchhoff-type problems
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### References:

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