Existence and multiplicity of solutions for resonant-superlinear problems
The authors study existence and multiplicity of nontrivial solutions for semilinear Dirichlet problems of the form\N\begin{align*}\N-\Delta u(z)=\hat{\lambda}_1u(z)+f(z,u^+(z))+\theta(z) \quad \text{in }\Omega, \quad u|_{\partial\Omega}=0,\N\end{align*}\Nwhere \(\Omega\subseteq \mathbb{R}^N\) with \(N\geq 2\) is a bounded domain with a \(C^2\)-boundary \(\partial\Omega\), \(\hat{\lambda}_1>0\) is the principal eigenvalue of \((-\Delta,H^1_0(\Omega))\), \(u^+=\max\{u,0\}\), \(\theta \in L^\infty(\Omega)\) with \(\theta(z) \leq 0\) for a.a.\,\(z\in\Omega\) and \(f\colon \Omega \times \mathbb{R}\to\mathbb{R}\) is a Carathéodory function which exhibits superlinear growth. By applying variational tools from critical point theory together with truncation and comparison techniques as well as critical groups, the authors prove two multiplicity theorems producing two and three nontrivial solutions. In addition, it is shown that the problem cannot have negative solutions.
scientific article; zbMATH DE number 7846665