Existence Of Infinitely Many Spherically Symmetric Solutions Of A Nonlinear Elliptic Problem On The Entire Plane

In this thesis we study the existence of infinitely many symmetric solutions for a class of nonlinear elliptic problems on the entire plane.By mountain pass theory and symmetric mountain pass theory,we can find a Palais-Smale sequence with a extra property related to Pohozaev identity,thus showing that the elliptic problem has a nonnegative least energy solution and infinitely many symmetric solutions.The content of this thesis is as follows.In Chapter 1,we introduce the specific form of the elliptic equation we are about to study,review the research status of related problems,and state the main results about this problem.In Chapter 2,we introduce some of the mathematical notations used in the thesis,define some Banach Spaces and Sobolev Spaces and related embedding theorems.We state Egoroff theorem,Lebesgue dominated convergence theorem,Fatou lemma,and Moser-Trudinger type inequality on the plane.We also introduce Krasnoselski’s genus and related properties.In Chapter 3,we obtain the continuity of I(v)by using the Trudinger-Moser type inequality we developed here,Egoroff theorem and Lebesgue dominated convergence theorem.By proving that the G(?)teaux derivative of energy functional I(v)exists and is continuous,we obtain that I(v)is continuously differentiable.We also give some compactness results,which will be used later.In Chapter 4,we show that the energy functional I(v)has mountain pass structure around v=0.From standard deformation result,for any minimax value cn,there is a corresponding Palais-Smale sequence.Due to the lack of enough information,we can not get the boundedness and compactness of the Palais-Smale sequence.So we construct an auxiliary functional Ie(t,v)and use deformation lemma to obtain a special PalaisSmale sequence.By using Fatou lemma and Lebesgue dominated convergence theorem,we manage to show the boundedness and compactness of this sequence,which implies the existence of a sequence of solutions for our problem.In Chapter 5,we construct a comparison functional Ic(v),which is smaller than I(v)and satisfies the Ambrosetti-Rabinowitz condition.By proving that Ic(v)has mountain pass structure around v=0,and the boundedness and compactness of the Palais-Smale sequence,we obtain the unboundedness of the minimax values of Ic(v)through Krasnoselskii principle.So we obtain the unboundedness of the minimax values cn of I(v).In Chapter 6,by constructing a continuous curve,we obtain that the minimax value c1 is the least energy level.We also prove the existence of one nonnegative solution corresponding to c1,which is one nonnegative least energy solution.