Existence And Behavior Of Solutions For A Class Of Nonlocal Elliptic Equations

Kirchhoff equation was originally the D’Alembert wave equation discovered by the German physicist Kirchhoff in his study of free string vibra-tions.In the last decade,scholars have done lots of research on the existence and behavior of the solutions of Kirchhoff type equations.However,there are few studies on its nodal solutions in higher dimensions.Hence,we study the nodal solutions of Kirchhoff equation in high dimensions.Firstly,we review the latest research results for Kirchhoff equations and introduce preliminary results.Then,we consider the following Kirchhoff type problems:where a,b>0,N≥4,2<P<2~*=2N/N-2 and function f∈L~∞(R~N)is positive and continuous.Combining constraint Nehari manifold method with the detailed energy estimates,we prove that the above problem admits a nodal solution when N=4 while two nodal solutions with opposite energy level for corresponding action functional are achieved when N≥5 by controlling the parameter(6 sufficiently small.Furthermore,we consider the following Kirchhoff type equations affect-ed by weight function:where N≥4,(?)>0 is a small parameter,M(t)=kt+b(k,b>0) and2<p<2~*=2N/(N-2).We assume that the weight function g∈(?R~N,R~+)has m maximum points in R~N.By using constraint Nehari manifold method as well as the barycenter map,it is proved under appropriate conditions that the equation has m~2 nodal solutions when N≥4 for (?),k sufficiently small.Finally,we pose some interesting questions for further exploration.