Nonresonance below the first eigenvalue for a semilinear elliptic problem
Let \(\Omega\) be a bounded open subset in \({\mathbb{R}}^ N \)with smooth boundary \(\partial \Omega\) and let f(u) be a continuous real-valued function on \({\mathbb{R}}\). We consider the semilinear Dirichlet problem (1) \(-\Delta u=f(u)+h(x)\) in \(\Omega\), \(u=0\) on \(\partial \Omega\). The nonlinearity is assumed to lie on the left side of the first eigenvalue \(\lambda _ 1\) of -\(\Delta\) in the following sense: (2) \(\limsup _{u\to \pm \infty} 2F(u)/u\) \(2\leq \lambda _ 1\), where \(F(u)=\int ^{u}_{0}f\) is the associated potential. We are interested in the conditions to be imposed on f or F in order that (1) be solvable for any h(x). This was considered by \textit{A. Hammerstein} [Acta Math. 54, 117-176 (1930)] when (2) holds with a strict inequality. We deal with the situation where equality holds in (2). We also consider the case where the nonlinearity depends on x: f(x,u). Our results in this case are related to those of \textit{J. Mawhin}, \textit{J. R. Ward} and \textit{M. Willem} [Arch. Ration. Mech. Anal. 95, 269-277 (1986)].
scientific article; zbMATH DE number 4038336