The Research On The Existence Of Concentrated Solutions To Two Nonlinear Elliptic Equations

In this thesis,we mainly use the Lyapunov-Schmidt reduction method to study the exist.ence of concentrated solutions to two nonlinear elliptic equations,including the singularly perturbed problem and prescribed scalar curvature problem.The present thesis consists of three chapters:In chapter one,we summarize the background of the related problems and state the main results of the present thesis.In chapter two,we study the existence of the concentrated solution of the following singularly perturbed problem:(?)Here ? is a bounded domain in Rd with smooth boundary,p>1,?>0 is a small parameter,V is a uniformly positive,smooth potential on S2,and v denotes the outward normal of ?.On the existence of interior concentration phenomena intersecting the boundary of ?,a conjecture is proposed by A.Ambrosetti,A.Malchiodi and W.-M.Ni(p.327,Indiana Univ.Math.J.,2004),which can be stated as:Let 1C be a k-dimensional manifold inte'rsecting ? perpendicularly,which is also stationary and non-degenerate with respect to the following functional(?)Then there exists a solution to(0.0.3)concentrating near K,at least along a sub-sequence ?j?0.0.In this chapter,we will prove the above conjecture for the two dimensional case.Precisely,let ? be a curve intersecting orthogonally with ?? at exactly two points and dividing ? into two parts.Moreover,? satisfies the station-ary and non-aeoeneracy cocnditions with respect to the functional ??Vp+1/p-1-1/2.By using the infinite Lyapunov-Schmidt reduction method,we prove the existence of a solution u,concentrating along the,whole of T.In chapter three,we deal with the following prescribed scalar curvature problem:-?u = Q(|y'|,y")uN+2/N-2,u>0,y=(y',y")?R2×RN-2.where Q(y)is nonnegative and bounded.By combining a finite Lyapunov-Schmidt reduction argument and local Pohozaev type of identities,we prove that if N?5 and Q(r,y")has a stable critical point(r0,y0")with r0>0 and Q(r0,y0")>0,then the above problem has infinitely ma.ny non-radial positive solutions,whose energy can be made arbitrarily large.Here,we apply some local Pohozaev type of identities to locate the concentration points of the bump solutions.Moreover,the concentration points of the bump solutions include a saddle point of Q(y).