Remarks on a resonant problem with an unbounded nonlinearity
The semilinear elliptic boundary value problem \[ -\Delta u=K(x)u +g(x,u) \text{ in }\Omega, \quad u=0 \text{ on } \partial\Omega, \] is studied, where \(\Omega\) is a smooth, bounded domain in \(\mathbb{R}^N\), \(K\in L^p (\Omega)\), \(p>N/2\) and \(K>0\) in \(\Omega\), with \(g\) a Carathéodory function satisfying \(|g(x,z)|\leq\gamma|s|^\alpha +b(x)\) with \(b\in L^2 (\Omega)\), \(b\geq 0\) and \(0<\alpha\leq l\), \(\gamma>0\). Existence of a solution is proved if \(\mu_\ell (K)=l\) for some \(l=1,2,3 \dots\) (these are the eigenvalues of the linear part) and some integral condition involving the primitive \(G\) of \(g\) is satisfied. Under some additional assumption existence of a nontrivial solution is also proved. These results extend in particular the previous work by \textit{D. Costa} [A note on unbounded perturbations of linear resonant problems, University of Brasilia (1989)]. The first theorem is proved by using the Saddle Point Theorem by Rabinowitz and the second one follows from the Mountain Pass Theorem by Ambrosetti and Rabinowitz.
scientific article; zbMATH DE number 1031567
scientific article