Nonlinear elliptic eigenvalue problems with discontinuities
Existence of nontrivial solutions to two nonlinear eigenvalue problems with discontinuous nonlinearities is proved in this paper. The first one is given by \(-\Delta_p\in \lambda[f_0(x,u), f_1(x,u)]\) in \(D\), \(u=0\) on \(\partial D\), where \(D\) is a smooth bounded domain in \(\mathbb{R}^N\), \(p\geq 2\), \(\lambda\) is a real parameter and \(f_0(x,u)\) and \(f_1(x,u)\) are the inferior and superior limits in \(u\) for fixed \(x\). Existence of at least two nontrivial solutions is proved if \(f\) has growth less than \(|u|^a\) with \(0<a<p-1\), is ``sublinear'' at the origin and satisfies some further conditions. The result is proved by using non-smooth critical point theory by Chang and the corresponding version of the Mountain Pass Theorem. The second one is an associated semilinear elliptic problem. Here the existence, under different assumptions, of at least one solution follows from a linking theorem. We point out that, contrary to the claim of the authors that ``in all these works existence of only one nontrivial solution (eigenvalue) is proved'', page 407, existence of at least two nontrivial positive solutious is proved for \(p=2\) in Theorems 4.1 and 4.2 of [\textit{K. C. Chang}, Sci. Sin. 21, 139-158 (1978; Zbl 0397.35057)]. However, the method of proof (namely the fixed point index) is different.
scientific article; zbMATH DE number 1289970
scientific article