On a nonresonance condition between the first and the second eigenvalues for the \(p\)-Laplacian (Q1599770)

This paper is devoted to the existence of solution to the following quasilinear elliptic problem: \[ \begin{cases} -\Delta_pu= f(x,u)+h(x) & \text{ in }\Omega\\ u=0 & \text{ on }\partial\Omega, \end{cases}\tag{1} \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\), \(N\geq 1\), \(\Delta_p\) is the \(p\)-Laplacian, \(1<p<\infty\). Under some suitable assumptions on \(f\), which related to the first and second eigenvalues of the \(\Delta_p\), the authors prove existence of solutions of (1). Extensions to more general operators, that is \[ \begin{cases} Au=f(x,u)+ h(x)\;&\text{ in }\Omega\\ u=0 & \text{ on }\partial\Omega, \end{cases}\tag{2} \] where \(A=-\sum^n_1{\partial\over \partial x_i}A_i(x,u,\nabla u)\) are also considered.