On Ambrosetti-Malchiodi-Ni Conjecture On Two-dimensional Smooth Bounded Domains:Clustering Concentration Layers

In this paper,we consider the problem??u-V(y)u +up=0,u>0 in ?,(?)u/(?)v=0 on(?)?,where ? is a bounded domain in R2 with smooth boundary,the exponent p>1,?>0 isa small parameter,V is a uniformly positive,smooth potential on ?,and v denotes the outward normal of(?)?.Let ? be a curve intersecting orthogonally with(?)? at exactly two points and dividin,? into two parts.Moreover,? satisfies stationary and non-degeneracy conditions with respect to the functional ??V? where ?=p+1/p-1-1/2? and V satisfy the compatibility condition.We prove the existence of a solution u? with clustering concentration layers directed along ?,exponentially small in ? at any positive distance from it,provided that ? is small and away from certain critical numbers.In particular,this establishes the validity of the two dimensional case of a conjecture by A.Ambrosetti,A.Malchiodi and W.-M.Ni(p.327,Indiana Univ.Math.J.,vol 53,2004.).