Eigenvalue problems of nonhomogeneous semilinear elliptic equations in Esteban-Lions domains with holes

摘要

In this article, we consider the following eigenvalue problems -del u + u = lambda(f(u) + h(x)) in Omega, u > 0 in Omega, u is an element of H-0(1)(Omega), where lambda > 0, N >= 2 and Omega subset of R-N is the upper semi-strip domain with a hole in RN. Under some suitable conditions on f and h, we show that there exists a positive constant lambda* such that Eq. (*)(lambda) has at least two solutions if lambda is an element of (0, lambda*), a unique positive solution lambda = lambda*, and no positive solution if lambda > lambda*. We also obtain some further properties of the positive solutions of (*)(lambda). (c) 2006 Elsevier Inc. All rights reserved.