Neumann problems with double resonance (Q446220)

Let \(\Omega\) be a bounded open connected subset of \({\mathbb{R}}^{n}\) of class \(C^{2}\). Let \(\nu\) be the outward unit normal to \(\partial\Omega\). Let \(f\) be a Caratheodory function from \(\Omega\times{\mathbb{R}}\) to \({\mathbb{R}}\) such that \(f(z,0)=0\) for almost all \(z\in\Omega\). Let \(\{\hat{\lambda}_{m}\}_{m=0}^{\infty}\) denote the sequence of Neumann eigenvalues of \(-\Delta\) in \(\Omega\).NEWLINENEWLINEThe authors show that if \(m\geq 1\) and \(\hat{\lambda}_{m}<\hat{\lambda}_{m+1}\) and NEWLINE\[NEWLINE \hat{\lambda}_{m}\leq\liminf_{x\to\infty}\frac{f(z,x)}{x} \leq \limsup_{x\to\infty}\frac{f(z,x)}{x}\leq \hat{\lambda}_{m+1} \qquad {\mathrm{for\;a.a.}}\;z\in\Omega\,, NEWLINE\]NEWLINE and if some other technical assumptions which do not involve the differentiability of \(f(z,\cdot)\) hold, then the Neumann boundary value problem NEWLINE\[NEWLINE -\Delta u(z)=f(z,u(z))\qquad\forall z\in\Omega\,, \qquad\frac{\partial u}{\partial\nu}(z)=0\qquad\forall z\in\partial\Omega\,, NEWLINE\]NEWLINE has at least three nontrivial solutions. In case \(m=0\), the authors prove the existence of at least one nontrivial solution under assumptions of the same type.