Weak solutions of semilinear elliptic equation involving Dirac mass

Abstract

In this paper, we study the elliptic problem with Dirac mass

{ -Delta u = Vu(p) + k delta(0) in R-N, (1)

lim(vertical bar x vertical bar ->+infinity) u(x) = 0,

where N > 2, p > 0, k > 0, delta(0) is the Dirac mass at the origin and the potential V is locally Lipchitz continuous in R-N \ {0}, with non-empty support and satisfying

0 <= V(x) <= sigma(1)/vertical bar x vertical bar(a0)(1 +vertical bar x vertical bar (a infinity-a0)),

with a(0) < N, a(0) < a(infinity) and sigma(1) > 0. We obtain two positive solutions of (1) with additional conditions for parameters on a(infinity), a(0), p and k. The first solution is a minimal positive solution and the second solution is constructed via Mountain Pass Theorem.