Multiple solutions for resonant semilinear elliptic problems in \(\mathbb R^N\)
The authors study the existence of the nontrivial weak solutions of the problem \[ -\Delta u= h(x)(\lambda u+ g(u)),\quad x\in\mathbb{R}^N,\quad u\in D^{1,2}, \] where \(\lambda\) is a real parameter, \(h: \mathbb{R}^N\to \mathbb{R}\) and \(g: \mathbb{R}\to \mathbb{R}\) are functions satisfying natural assumptions. Here \(D^{1,2}\) denotes the Hilbert space obtained through completion of \(C^\infty_c(\mathbb{R}^N)\), where \(N\geq 3\) with respect to the norm \(\| u\|:= (\int|\nabla u|^2\, dx)^{1/2}\). Existence of multiple solutions, for certain values of \(g'(0)\), is obtained by imposing a generalized Landesman-Lazer type condition. To this end, the authors use the saddle point theorem, the mountain pass theorem, as well as a Morse-index result with a combination a Leray-Schauder index theorem for mountain pass type critical points. The results of this paper are based upon multiplicity results for resonant problems on bounded domains.
scientific article; zbMATH DE number 2159795
scientific article