Abstract
In this thesis we study the existence and the symmetry of the solutions of the semilinear elliptic equations -Δu+u=g(u) in infinite strip domains and flask domains. In Chapter 1, we prove the existence of the solutions of the semilinear elliptic equations -Δu+u=g(u) in infinite strip domains by the Decomposition Lemma, and then apply the improved“moving plane”method to prove the symmetry of the solutions. In Chapter 2, consider the semilinear elliptic equations -Δu+au=b|u|^{p-2}u in unbounded domainΩ, we prove the following assertions: (1) αI=αM=αΓ=αΓˊ(2) Let Ω0=Ω1∪Ω2 where Ω1∩Ω2 is bounded, andαi=α(Ωi) the index of J in Ωi for i=0,1,2. We assert that J satisfies the (PS) αi -condition if and only if the inequality α0<min{α1,α2} holds. (3) There is s0 > 0 such that α∞(D_{s}^{r}) admits a minimizer if s> s0, but α∞(D_{s}^{r}) does not admit any minimizer if s< s0, where D_{s}^{r} is an interior flask domain.