Existence of Positive Solution to Schrödinger-type Semipositone Problems with Mixed Nonlinear Boundary Conditions

Abstract

We studied the existence of a positive solution to Schrödinger-type semipositone problems with mixed nonlinear boundary conditions. By considering the cases when the reaction term with a parameter satisfies a superlinear and a sublinear growth condition at infinity, we established the existence of a positive solution for the large and small values of the parameter, respectively. The proofs are mainly based on the sub- and supersolution method for the sublinear case and the mountain pass lemma with $C^{1,\alpha}(\overline{\Omega})$-regularity for the superlinear case.