Existence Of Weak Solutions For The Dirichlet Problems Of Semilinear Elliptic Equations

The Dirichlet problems of semilinear elliptic equations have been deeply and extensively researched. Profounded results have been made. We know that the weak solution for this problem can be attributed to seek the critical point of the corresponding functional, and one method of finding the critical point is Mountain Pass Lemma, but in order to apply the lemma, (AR) conditions are required, but this condition is a little harsh, in order to weaken this condition, many results have been obtained. Status of such researches are briefly outlined in Chapter 1 of this thesis.(AR) conditions in the second chapter which are inspired by literature will be weakened. According to the research of (AR) conditions of semilinear elliptic Dirichlet problems at home and abroad, we find that (AR) conditions and weakened (AR) conditions are restricted to some parameters, such as ?,?>2 are required, for this question, we will weaken (AR) conditions and consider the relationship between F(x,t) and f(x, s) when ? is 2 and ? is less than 2.Different from the references, we use some other existence theorems of critical point in the third chapter to prove the existence of non-trivial weak solutions of Dirichlet problems when nonlinear function f satisfies Holder-type conditions.