Nonresonance conditions on the potential with respect to the Fučík spectrum for semilinear Dirichlet problems (Q638701)

This paper deals with the following semilinear Dirichlet problem \[ -\Delta u = f(x,u) \,\, \text{ in } \,\, \Omega,\quad u=0\,\, \text{ on } \,\,\partial\Omega, \tag{1} \] where \(\Omega\subset R^m\) (\(m\geq 1\)) is a bounded domain with smooth boundary \(\partial\Omega\), \(f\in C(\overline{\Omega}\times R,R)\). Let \(F(x,u)=\int^u_0 f(x,s)\text{d}s\) be the primitive of \(f(x,u)\), and the Fučík spectrum be defined on the set \(\Sigma\subset R^2\) of points \((\alpha, \beta)\) for which there exists a nontrivial solution of the problem \[ -\Delta u =\alpha u^+-\beta u^- \,\, \text{in} \,\, \Omega,\quad u=0\,\, \text{on} \,\,\partial\Omega. \] The authors establish the existence of solutions of problem (1) by requiring that \(f\) is of linear growth at infinity and the ratio \(\frac{2F(x,s)}{s^2} \) stays asymptotically at infinity away from the Fučík spectrum.