Global positive solutions of semilinear elliptic problems

Abstract

The existence of bounded positive solutions of semilinear elliptic boundary value problems of the type (1.1) Lu = f(x,u), ce, (1.2) W(JC) = 0, JCG3, will be proved in unbounded domains c, n > 2, with boundary e C 2+ , 0 < a < 1, where is a positive constant and n (1.3) Lw=- D l [a lJ (x)D J u] + m(x)u, x e , Z), = /x / , = 1,..., n. The existence of a bounded positive solution of (1.1) in the entire space R n is proved also by the same procedure. The regularity and additional hypotheses H1-H5 to be imposed on L and / are stated in 2. In particular, the assumption /(JC,O) = 0 for all x e implies that the boundary value problem (1.1), (1.2) always has the trivial solution.