Existence of a nontrivial solution is established, via variational methods, for a system of weakly coupled nonlinear Schrödinger equations. The main goal is to obtain a positive solution, of minimal action if possible, with all vector components not identically zero. Generalizations for nonautonomous systems are considered.

SUMMARYMotivated by the study of a two-dimensional point vortex model, we analyse the following Emden-Fowler type problem with singular potential:We first extend various results, already known in case 0, to cover the case ∈ (0, 1). In particular, we study the concentration-compactness problem and the mass quantization properties, obtaining some existence results. Then, by a special choice of K , we include the effect of the angular momentum in the system and obtain the existence of axially symmetric one peak non-radial blow-up solutions.

Existence results for radially symmetric oscillating solutions for a class of nonlinear autonomous Helmholtz equations are given and their exact asymptotic behaviour at infinity is established. Some generalizations to nonautonomous radial equations as well as existence results for nonradial solutions are found. Our theorems prove the existence of standing waves solutions of nonlinear Kleinâ\u80\u93Gordon or Schrödinger equations with large frequencies

In the framework of the nonsmooth critical point theory for lower semi-continuous functionals, we propose a direct variational approach to investigate the existence of infinitely many weak solutions for a class of semi-linear elliptic equations with logarithmic nonlinearity arising in physically relevant situations. Furthermore, we prove that there exists a unique positive solution which is radially symmetric and nondegenerate.

Abstract. We consider systems of weakly coupled Schrödinger equations with nonconstant potentials and investigate the existence of nontrivial nonnegative solutions which concentrate around local minima of the potentials. We obtain sufficient and necessary conditions for a sequence of least energy solutions to concentrate.