Eigenvalue problems and bifurcation of nonhomogeneous semilinear elliptic equations on unbounded cylinder domains

In this article, we consider the eigenvalue problems of Dirichlet problem -Delta u + u = lambda(f(u) + h(x)) in Omega, u > 0 in Omega, u is an element of H-0(1)(Omega), (*)lambda where lambda > 0, N >= 2, and Omega is an unbounded cylinder domain in R-N. Under some suitable conditions on f and h, we show that there exists a positive constant lambda* such that (*)lambda has at least two solutions if lambda E (0, lambda*), a unique positive solution if lambda = lambda* and no solution if lambda > lambda*. We also obtain some bifurcation results of the solutions at lambda = lambda*.