The Concentration Solutions Of The Nonlinear Magnetic Schrodinger Equation On Two-dimensional Smooth Bounded Domains

In this paper,we study the two-dimensional stationary magnetic nonlinear Schrodinger equation(?)where ? is a bounded domain in R2 with smooth boundary,the exponent p>1,?>0 is a small parameter,V is a uniformly positive,smooth potential on ?,and v denotes the outward normal of(?)?,the magnetic potential A is related to the magnetic field B by the relation B =(?)y1Ay2-(?)y2Ay1.Let ?be a smooth curve intersecting orthogonally with(?)? at exactly two points.Moreover,? satisfies stationary and non-degeneracy conditions with respect to the functional ??V?,where ?=p+1/p-1-1/2.We prove the existence of a solution u? with having a concentration layer orthogonal to(?)? directed along r,exponentially small in ? at any positive distance from it,provided that ? is small and away from certain critical numbers.In a small neighborhood of ?,|u?| has exponential decay property.