Another approach on an elliptic equation of Kirchhoff type (Q360092)

Summary: This paper is concerned with the existence of solutions to the class of nonlocal boundary value problems of the type \[ -M\left(\int_\Omega|\nabla u|^2\right)\Delta u=f(x,u),\quad\text{in }\Omega,\, u=0,\text{ on }\partial \Omega, \] where \(\Omega\) is a smooth bounded domain of \(\mathbb R^N\) and \(M\) is a positive continuous function. By assuming that \(f(x,u)\) is a Carathéodory function which growths at most \(| u|^{N/N-2}\), \(N\geq 3\), and under a suitable growth condition on \(M\), one proves an existence result by applying the Leray-Schauder fixed point theorem.