Multiple And Sign-Changing Solutions For Some Semilinear Elliptic Equations

In chapter two, we study the existence of multiple and sign-changing solutions for the following eigenvalue problem: where a We prove problem (1) has at least one positive solution , one negative solution and one sign-changing solution , by using of the concentration-compactness principle to overcome the compactness embeddings and combining variational methods.In chapter three, we consider the following Schrodinger equation with Neumann boundary value problem:where Ω R^N is a bounded domain with smooth boundary, v denotes the unit outward normal vector, a ∈ Z^∞(Ω), a(x) > 0.we study the existence of multiple and sign-changing sloutions for problem (2) when the nonlinearity is under the asymptotically linear conditions, by using of the improved mountain pass lemmas and invariant sets of descending flow, we get two sign-changing solutions except positive and negative solutions.In chapter four, we consider elliptic problem :where N ≥ 3, λ > 0,f ∈ C(R,R),a ∈ C(R^N) chang sign and 0 < h(x) ∈ L^N/2(R^N) ∩ L^∞(R^N)∩C¹(R^N).we prove the existence of sign-changing solution for problem (3) when μ_k(h) < λ <...