Critical Groups And Existence Of Multiple Solutions For Superlinear Elliptic Equations

In this thesis, we study the existence of nontrivial solutions for p-Laplacian equations with Drichlet boundary value by applying the Morse theory. The nonlinearity is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz condition (AR condition) or its dual form near zero. We obtain nontrivial solution in the cases that the nonlinearity is asymptotically linear near zero and superlinear near infinity, or it is superlinear near zero and asymptotically linear near infinity. The key point in the proofs is to compute the critical groups at infinfity and at zero, under the new assumptions.In the case that p = 2, we obtain a positive solution, a negative solution and a sign-changing solution via the truncated technique and Morse inequality.