Existence of positive solutions of elliptic mixed boundary value problem

Guofa Li

doi:10.1186/1687-2770-2012-91

Abstract

In this paper, we use variational methods to prove two existence of positive solutions of the following mixed boundary value problem: One deals with the asymptotic behaviors of near zero and infinity and the other deals with superlinear of at infinity. MSC:35M12, 35D30.

Highlights

1 Introduction and preliminaries This paper is concerned with the existence of positive solutions of the following elliptic mixed boundary value problem:
F (x,t) t is nondecreasing with respect to t
We study superlinear of f at infinity with q(x) ≡ +∞ in (S ), which is weaker than the (AR)

Summary

Which implies that

Lemma If conditions (S ) to (S ) hold, < , φ (x) > is defined by Lemma , J(tφ (x)) → –∞ as t → +∞. Lemma Let conditions (S ) and (S ) hold. As n → +∞, there exists a subsequence of {un}, still denoted by {un} such that J(tun) ≤. Lemma (see [ ]) Suppose E is a real Banach space, J ∈ C (E, R) satisfies the following geometrical conditions:. This completes the proof of Theorem (i). (ii) By Lemma , there exists β, ρ > such that J|∂Bρ( ) ≥ β with u = ρ. C ≥ β > and by Lemma , there exists {un} ⊂ V such that.

If w

The conclusion follows from