The Existence Of A Class Of Quasi Linear Elliptic Equations Dirichlet Boundary Value Problem Changing Solutions

In this paper, we study the existence of sign-changing and multiple nontrivial solutions for a kind of quasilinear elliptic equations with Dirichlet boundary value by applying minimax methods and Morse theory.Consider the boundary value problemwhere is a bounded open domain with smooth boundary We assume the following conditionwhereIt is well known that the weak solutions of the problem are exactly the critical points of the C~1 functionalin , where If f(x, t)=0, then the problem has a trivial solution u = 0. We are interested in finding nontrivial solution of the problem by imposing various conditions on f(x, t) or its primitive F(x,t) near the origin and near infinity. So we assume that f(x,t) satisfies the following conditions:...