Nonselfadjoint resonance problems with nonlinearities of superlinear growth
It is considered the Dirichlet problem \(Au+ \lambda_1 u+ g(x, u)= h\) in \(\Omega\), \(u|_{\partial\Omega}= 0\), where \(\Omega\subset \mathbb{R}^N\) \((N\geq 2)\) is a bounded domain with a \(C^2\) boundary \(\partial\Omega\) and \[ Au= \sum^N_{i, j= 1} {\partial\over \partial x_j} \Biggl(a_{ij} {\partial u\over \partial x_i} \Biggr)+ \sum^N_{j= 1} a_j {\partial u\over \partial x_j}+ a_0 u, \] is elliptic in \(\overline\Omega\) with \(a_{ij}= a_{ji}\), \(a_j\in C^1(\overline\Omega)\) for \(1\leq i\), \(j\leq N\), \(a_0\in L^\infty(\Omega)\), \(h\in L^p(\Omega)\) with \(p> N\) and \(g: \Omega\times \mathbb{R}\to \mathbb{R}\) is a Carathéodory function. For the nonlinear term \(g\) it is assumed that there exist \(a> 0\) and \(b\in L^p(\Omega)\) such that \(b\geq 0\) in \(\Omega\) and for almost all \(x\in \Omega\) and \(u\in \mathbb{R}\), \(- b(x)\leq g(x, u)\leq a|u|^{p/(p- 1)}+ b(x)\). The authors obtain solvability under assumptions either with or without a Landesmann-Lazer condition.
scientific article; zbMATH DE number 856959