On Problems Of Multiple Solutions For Nonlinear Elliptic Equations With Potential

In this thesis, by using variational methods, we discuss the eigenvalue problem of a class of Schr(?)dinger operators, multiple solutions for nonlinear Schr(?)dinger equations with deepening potential well and biharmonic equations. This thesis is divided into four chapters, the main contents are as follows:In Chapter 1, we consider the following eigenvalue problemwhereλ,α∈R, and g satisfies the condition (G):for some bounded open setΩ(?)R^N. For givenλ＞0, we discuss the existence and some properties of the second eigenvalueα₂(λ) for the above eigenvalue problem, and we also give several equivalent formulations of the second eigenvalueα₂(λ).In Chapter 2, we improve an existence theorem of multiple critical points obtained by using the invariant set of descending flow method. This result will be an important tool to deal with the existence of multiple solutions of asymptotically linear elliptic equations, in particular, the existence of sign-changing solutions.In Chapter 3, we are concerned with the following nonlinear Schr(?)dinger equationwhere V_λ(x)=1+λg(x), A is a positive parameter, the function g satisfies the above-mentioned condition (G), f is asymptotically linear at infinity, and asymptotic constant is 1 +α. By using the theorem we obtained in Chapter 2 and the concentration-compactness principle, we obtain a sign-changing solution, a positive solution and a negative solution for this equation ifλandαsatisfy appropriate relations.In Chapter 4, we consider the following fourth-order nonlinear elliptic equationwhereΩ(?)R^N is a bounded domain with smooth boundary, and c∈R is a parameter, the nonlinear term f is asymptotically linear at both zero and infinity. By using the variational methods and the theorem in Chapter 2, we obtain a sign-changing solution, a positive solution and a negative solution for this fourth-order equation.