Large solutions of semilinear elliptic equations with a Hardy potential

Abstract

Semilinear elliptic problems with solutions blowing up at the boundary are considered. The effect of a Hardy potential with a boundary singularity is discussed. Positive potentials reinforce the solution to blow up whereas negative prevent it. For the standard nonlinearities and sufficiently large potentials there exist solutions which are comparable to the blowing up solutions of the problem without Hardy potential. Near the boundary they depend only on the distance to the boundary, and the first order approximation is independent of the geometry. The precise estimates imply that those solutions are unique. The main tools used in this paper are the method of upper and lower solutions and boundary estimates for the blowup solutions without Hardy potential.

Keywords

• boundary blow-up
• singular coefficients
• singular solutions
• sub and super solutions
• comparison principle