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.


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


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