Semilinear elliptic Schrödinger equations with singular potentials and absorption terms

Abstract

AbstractLet () be a bounded domain and be a compact, submanifold without boundary, of dimension with . Put in , where and is a parameter. We investigate the boundary value problem (P) in with condition on , where is a nondecreasing, continuous function, and and are positive measures. The complex interplay between the competing effects of the inverse‐square potential , the absorption term and the measure data discloses different scenarios in which problem (P) is solvable. We provide sharp conditions on the growth of for the existence of solutions. When is a power function, namely with , we show that problem (P) admits several critical exponents in the sense that singular solutions exist in the subcritical cases (i.e. is smaller than a critical exponent) and singularities are removable in the supercritical cases (i.e. is greater than a critical exponent). Finally, we establish various necessary and sufficient conditions expressed in terms of appropriate capacities for the solvability of (P).