On elliptic equations with singular potentials and nonlinear boundary conditions

Abstract

We consider the Laplace equation in the half-space satisfying a nonlinear Neumann condition with boundary potential. This class of problems appears in a number of mathematical and physics contexts and is linked to fractional dissipation problems. Here the boundary potential and nonlinearity are singular and of power-type, respectively. Depending on the degree of singularity of potentials, first we show a nonexistence result of positive solutions in D 1 , 2 ( R + n ) \mathcal {D}^{1,2}(\mathbb {R}^n_+) with a L p L^p -type integrability condition on ∂ R + n \partial \mathbb {R}^n_{+} . After, considering critical nonlinearities and conditions on the size and sign of potentials, we obtain the existence of positive solutions by means of minimization techniques and perturbation methods.