Existence Of Solutions To Two Classes Of Partial Differential Equations With Nonlocal Term

With the development of the applied subjects,such as physics,biology,chemistry,nonlinear differential equations?especially?nonlinear partial differential equations have been studied by many scholars.Nonlinear partial differential equations can be applied to solve many problems which are very important in the field of natural science and engineering.They can also be used to describe the mathematical models in daily life.In the present paper,the existence of solutions to two classes of differential equations with nonlocal term is studied.The two classes of equations are derived from the physical phenomenons,and have significant physical meaning and research value.In the first chapter,we studied the Kirchhoff equation:where is a bounded domain with a smooth boundary in R~2,a,b>0 are positive constants and the nonlinearity f satisfies an exponential growth.We proved that the problem(1.1.3)possesses one least energy sign-changing solution.It is worth noticing that the usual variational approach seems not to be available because of the appearance of nonlocal term(b??|?u|~2dx)?u.We use the Brouwer's fixed point theorem and quantitative deformation lemma to solve the problem.In the second chapter,we studied the second harmonic equation:where 2 ? N ? 5 and V satisfies the following hypotheses:(V1)V(·)?C(R~N,R)is nonnegative,V(x)?(?)V(x)= 1 for every x?R~N,and the inequality is strict in a subset of positive Lebesgue measure;(V2)there exists a constant C>0 such thatIn problem(2.1.1),if we solve the second equation for given v?H~1(R~N)as Schrodinger-Poisson system,then we get w =?_v.Substituting into the first equation,we change the problem(2.1.1)into a single equation with nonlocal term?_vv.In this chapter,we do not use the usual variational approach,and consider directly the critical points of functional of two variables corresponding to equation(2.1.1).By using the existence of solution to the corresponding limit equationcombining with global compactness lemma,we proved the existence of solution to variable coefficient second harmonic equation(2.1.1).