## Positive solutions for a quasilinear elliptic equation of Kirchhoff type.(English)Zbl 1130.35045

The authors consider the existence of positive solutions to the class of boundary value problems of the type \begin{aligned} -M\Biggl(\int_\Omega|\nabla u|^2\, dx\Biggr)\Delta_x u= f(x,u)\quad &\text{in }\Omega,\\ u= 0\quad &\text{on }\partial\Omega,\end{aligned}\tag{1} where $$\Omega\subset \mathbb R^N$$ is a bounded domain, $$f:\overline\Omega\times\mathbb R\to \mathbb R$$ and $$M: \mathbb R\to\mathbb R$$ are given continuous functions. The goal of the authors is to present conditions on $$M$$ and $$f$$, providing a positive solution for (1) using a variational method.

### MSC:

 35J60 Nonlinear elliptic equations 35J25 Boundary value problems for second-order elliptic equations 47J30 Variational methods involving nonlinear operators

### Keywords:

Variational methods; Nonlocal problems; Kirchhoff equation
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### References:

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