Perturbations of symmetric eigenvalue problems (Q1861818)

The paper is concerned with the existence of multiple solutions \((u,\lambda)\in H^1_0(\Omega)\times\mathbb{R}\) of the elliptic eigenvalue problem \[ -\Delta u= \lambda(f(x,u)+\varepsilon g(x,u))\text{ in }\Omega,\;u|_{\partial\Omega}= 0,\;\int_\Omega|\nabla u|^2 dx=r^2,\tag{1} \] where \(f\) and \(g\) are continuous functions from \(\overline\Omega\times\mathbb{R}\) to \(\mathbb{R}\), \(\Omega\subset \mathbb{R}^N\) is a bounded domain with smooth boundary \(\partial\Omega\), \(r> 0\) is a fixed number and \(\varepsilon\geq 0\) is a parameter. The authors prove that under some assumptions concerning \(f\) and \(g\), for every \(n\in\mathbb{N}\), there exists \(\varepsilon_n> 0\) such that the problem (1) possesses at least \(n\) distinct solutions if \(0\leq\varepsilon\leq\varepsilon_n\). They use a cut-off technique.