Existence of solutions for a nonhomogeneous semilinear elliptic equation

For a bounded domain Omega, a bounded Caratheodory function g in Omega x R, p > 1 and a nonnegative locally integrable function h in Omega which is strictly positive in a set of positive measure we prove that, contrarily with the case h equivalent to 0, there exists a solution of the semilinear elliptic problem [-Delta u = lambda u + g(x, u) - h vertical bar u vertical bar(p-1)u + f, in Omega u = 0, on partial derivative Omega, for every lambda is an element of R and f is an element of L-2(Omega). (C) 2019 Elsevier Ltd. All rights reserved. (AU)