Sign-changing Solutions For Nonlinear Elliptic Equation And Systems

In this thesis, by using the methods of perturbation and invariant sets of de-scending flow, we obtain the existence of infinitely many sign-changing solutions of nonlinear elliptic equation and systems. This thesis is divided into four chapters, the main contents are as follows:In Chapter 1, we give the research problem and its backgrounds, and give our main results.In Chapter 2, we consider the following quasilinear elliptic equation-△u-u△u2+u= a(x)|u|r-2u, x ∈ RN, N≥ 3, where a(x) satisfies the condition: (A) a(x) ≥∈ LS(RN), s ∈ [2·2*/2·2*-r,+∞), r ∈ (4,2 · 2*). The equation only has a variational structure formally, there is no suitable space in which the variational functional enjoys both smoothness and compact properties. By adding a 4-Laplacian operator and a coercive potential term, and by using the methods of perturbation and invariant sets of descending flow, we obtain the existence of a )ositive solution, a negative solution and infinitely many sign-changing solutions.In Chapter 3, we are concerned with the following semilinear systems where b(x)、c(x) are potential functional and Fu, Fv are subcritical and superlinear. By using the method of invariant sets of descending flow, we obtain the existence of infinitely many sign-changing solutions.In Chapter 4, we consider the following quasilinear system where Fu, Fv are subcritical and superlinear. In the same way, by adding a 4-Laplacian operator and a coercive potential term, and by combining the methods of invariant sets of descending flow, we also obtain the existence of infinitely many sign-changing solutions. Different from a single equation, in order to prove that each component of solutions are sign-changing, we need some technical approach when we define and estimate the critical values.