Non-homogeneous Dirichlet problems with concave-convex reaction
Let \(\Omega \subseteq \mathbb{R}^N\) (\(N \geq 2\)) be a bounded domain with a smooth boundary \(\partial \Omega\). The authors consider a Dirichlet problem of the form \[\operatorname{div} A(x,\nabla u)=\lambda f(x,u) \mbox{ in } \Omega,\quad u|_{\partial \Omega}=0, \tag{\(P_\lambda\)}\] where \(A:\Omega \times \mathbb{R}^N \to \mathbb{R}^N\) is a function satisfying suitable properties and covering the behavior of a non-homogeneous differential operator of \(p\)-Laplacian type, \(\lambda >0\) is a positive parameter and \(f:\Omega \times \mathbb{R}\to \mathbb{R}\) is a Caratheodory function which is \((p-1)\)-superlinear at infinity. The main result establishes an exhaustive property, which describes the interval values of the parameter \(\lambda>0\) for which the problem \((P_\lambda)\) has at least two nontrivial weak solutions. Additionally, the positivity of the solutions is obtained when \(f\) satisfies a sign condition. The proofs combine variational methods and critical point theorems.
scientific article; zbMATH DE number 7490886
scientific article