On a boundary value problem with nonsmooth jumping nonlinearity (Q1174956)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ N\) (\(N\geq 1\)) with smooth boundary \(\partial\Omega\). Let \(0<\lambda_ 1<\dots<\lambda_ n<\dots\) be the sequence of distinct eigenvalues of the eigenvalue problem \(- \Delta u=\lambda u\) in \(\Omega\), \(u=0\) on \(\partial\Omega\). We denote by \(\theta_ 1\) the eigenfunction corresponding to the first eigenvalue \(\lambda_ 1\) with \(\theta_ 1>0\) on \(\Omega\) and \(\int_ \Omega \theta_ 1^ 2 dx=1\). Let \(g(\cdot)\in C(\mathbb{R},\mathbb{R})\) be given for which \(\lim_{\zeta\to-\infty} g(\zeta)/\zeta=a\) and \(\lim_{\zeta\to+\infty} g(\zeta)/\zeta=b\) exist. For a given \(h_ 1\in L^ p(\Omega\)) with \(p>N\) and \(\int_ \Omega \theta_ 1 h_ 1 dx=0\), we are concerned with the number of solutions of the boundary value problem \[ \Delta u+g(u)=t\theta_ 1+h_ 1 \quad\text{in}\quad \Omega, \qquad u=0 \quad\text{on}\quad \partial\Omega,\tag{1} \] where \(t\) is a parameter. We assume that the interval with end points \(a=\lim_{\zeta\to-\infty} g(\zeta)/\zeta\) and \(b=\lim_{\zeta\to+\infty} g(\zeta)/\zeta\) contains exactly one eigenvalue \(\lambda_ n\) of arbitrary multiplicity, \(n\geq 2\), inside and that neither \(a\) nor \(b\) is an eigenvalue. The following result was proved by \textit{A. C. Lazer} and \textit{P. J. McKenna} [Commun. Partial Differ. Equations 10, 107-150 (1985; Zbl 0572.35036)]. Suppose that \(g(\cdot)\in C^ 1(\mathbb{R},\mathbb{R})\). Given \(a\in ]\lambda_{n-1},\lambda_ n[\) with \(n\geq 2\), there exists \(\overline {b}=\overline{b}(a)\in ]\lambda_ n,\lambda_{n+1}[\) such that if \(g'(\zeta)\to a\) as \(\zeta\to-\infty\), \(g'(\zeta)\to b\in ]\lambda_ n,\overline{b}[\) as \(\zeta\to\infty\), and for some constant \(a_ 1>\lambda_{n-1}\), \(g'(\zeta)\geq a_ 1\) \(\forall \zeta\in\mathbb{R}\) then (1) has at least three solutions for \(t\) positive and large. In this paper we prove among other things that the result above remains valid under the weaker assumption that \(g(\cdot)\in C(\mathbb{R},\mathbb{R})\).