On a nonlinear elliptic eigenvalue problem (Q5892648)

Let \(\Omega\) be a smooth bounded domain in \(\mathbb R^N\), \(N\geq2\), and \(f\in C(\mathbb R)\) satisfy \(\lim_{t\to0}f(t)/| t| ^{p-2}t=1\), where \(p\in(2,2N/(N-2))\). Let \(N(\lambda)\) denote the number of solutions of the problem \(-\Delta u=\lambda f(u)\) in \(\Omega\), \(u=0\) on \(\partial\Omega\). The authors prove that \(N(\lambda)\to+\infty\) as \(\lambda\to+\infty\).