Multiplicity of solutions of semilinear elliptic boundary value problems with jumping nonlinearities at zero
This paper deals with the elliptic boundary value problem \[ -\Delta u= f(u),\quad x\in\Omega;\quad u=0,\quad x\in\partial\Omega,\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\) and \(f: \mathbb{R}\to\mathbb{R}\) is a Lipschitz continuous function. By using the \(L^p\) theory of elliptic operators and Sobolev imbedding theorems, the mountain pass lemma and fixed point index, the author proves that problem (1) has a minimal positive solution \(u_1\), a maximal negative solution \(u_2\) and a sign-changing solution \(u_3\), \(u_2< u_3< u_1\), under some conditions. Furthermore, under some additional assumptions, the author proves that problem (1) has at least nontrivial solutions \(u_1\), \(u_2\), \(u_3\), \(u_4\), and \(u_1\) is the minimal positive solution, \(u_2\) is the maximal negative solution, both \(u_3\) and \(u_4\) are sign-changing solutions, \(u_2< u_3< u_1\), \(u_2< u_4< u_1\).
scientific article; zbMATH DE number 1741845
scientific article