Existence results for an elliptic equation of Kirchhoff-type with changing sign data. (English) Zbl 1248.35065
Summary: Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) \((N>2)\). We are concerned with the existence and nonexistence of solutions for the following nonlocal problem
\[
-M\Biggl(\int_\Omega |\nabla u(x)|^2dx\Biggr)\,\Delta u= |u|^{p-1} u+\lambda f(x)\quad\text{in }\Omega,\;u|_{\partial\Omega}= 0,
\]
where \(M\) is a continuous function on \(\mathbb{R}^+\) and \(f\in C^1(\overline\Omega)\) changes sign. \(\lambda\) and \(p\) are positive parameters. By direct variational methods, the Galerkin approach and sub- and super-solutions methods, some results are established.
MSC:
35J25 | Boundary value problems for second-order elliptic equations |
35A15 | Variational methods applied to PDEs |
35A16 | Topological and monotonicity methods applied to PDEs |