Costa Has A Small Perturbation Of Non-quadratic Bit Homoclinic Solution Research Potential Equation

In recent30years, when people are studying the existence of nonzeor solution of oridinary differential equations, they often trsform it into the study of the existence of nontrival critical point of related functional and obtained good result (Referces[1]-[10] and Referces[20]-[30])In the yesr of2009Li and Costa[10] transformed the study of the existence of nonzeor solution of a class of equation into the study of the existence of notrivial critical point of the related functional: And they proofed the existence of nontrivial critical point of y/on some conditions. This paper study the existence of nontrivial critical point of functional^with small disturbance which is based on [10]. Function F(x,u)∈C1(Rk×Rn,R) satisfies conditions (V0)-(V4),and conditon (V3) is called nonquadratic condition of Costa type. That is: u·Vu(x,u)>2V(x,u)(?)x∈Rk,u∈Rn\{0} and,for some M2,u,δ>0, u·Vu(x,u)-2V(x,u)≥M2|u|v>0(?)x∈Rk,|\u|>δ;This paper is separated into3ChaptersThe first chapter is introduction, it introduced the background knowledge(that is the theory base of variation method which was used in this paper), the object of the paper, the main research work by former people, the source of the subject and the three innovation points.The second chapter introduced the theroy of critical theory related to this paper, and the proof of key theorms are given.The third chapter is about the main result of this paper, and we proofed them using Mountain pass theorm of Brezis-Niernberg type, Lions Lemer and other Variatiional Methods. This part is divided into four parts1. We prove functional φ possess nontrivial critical point when disturbance g(x)∈Lq0(Rk), there exists a bounded (PS)d sequence, and the condition (N1)-(N2) are satisfied(Theory3.1).This theory is based on the periodical invariant of functional φ and Lions Lemma.2. We deduced that the functional φ ppossess (PS)d sequence after proofing that the functional φ satisfying the geometric condition of Mountain Pass Theorm, which is based on conditions.(V0),(V1),(V4)3. We prove that this (PS)d sequence is bounded based on conditions (V0),(V1),(V3)、interpolation inequality、embeding inequality and Yong inequality etc.4. We prove our Theory3.2after proving that the functional φ satisfying the conditions in theory3.1.