Infinitely Many Solutions For Non-compactness Elliptic Equations And Systems

In this paper,we mainly study three kinds of non-compactness elliptic problems:that is,quasilinear Schr(?)dinger system in R~N?Brezis-Nirenberg type quasilinear system and the prescribed curvature problem for fractional Laplacian operator,we will prove the existence of infinitely many solutions for the above three kinds of equations(systems).Since the problems we are concerned are either defined on the whole space R~Nor with critical nonlinear term,when we try to apply the Critical Point Theory to prove the exis-tence of the solutions to these problems,the first difficulty we have to overcome is that the corresponding energy functional does not satisfy Palais-Smale condition(P.S.condition in short).On the other hand,when we consider the generalized quasilinear equations,duo to the appearance of the quasilinear term,the energy functional of these equations are not smooth.Hence the second main difficulty we have to overcome is to find a suitable work-ing space such that the functional is smooth and satisfies some compactness condition.More precisely:When we study the quasilinear Schr(?)dinger system in R~N,in order to prove the existence of infinitely many solutions,we proceed two approximation schemes:one is adding a coercive potential to overcome the lack of compactness;another is q-Laplacian regularization approximation to overcome the non-smoothness of the functional.When we study the Brezis-Nirenberg type quasilinear system,we apply the appro-ximation of subcritical solutions.By analyzing the asymptotic behavior and the concen-tration compactness of the approximation solutions for the subcritical problems,there is a subsequence of this approximation solutions converges strongly to a solution of the critical problem.In order to prove the strongly converges,the key steps are proving the uniform boundness of approximation solutions.By doing so,we have to estimate the behavior of the approximation solutions near the blow-up points carefully,meanwhile,we have to establish the local Pohozave identity which will help us to prove the final results.Finally,we need to prove the critical value tending to infinite to show that the Brezis-Nirenberg type quasilinear system has infinitely many solutions.When we study the prescribed curvature problem for fractional Laplacian operator,we will apply the method of finite dimensional reduction to construct infinitely many non-radially symmetric solutions.By this method,we do not need to check(P.S.)condition any more.Instead,we have to find good approximation solution,and reduce the infinite dimensional original problem to a finite dimensional problem.