In the present paper we survey some recents results concerning existence of semiclassical standing waves solutions for nonlinear Schrödinger equations. Furthermore, from Maxwell's equations we derive a nonlinear Schrödinger equation which represents a model of propagation of an electromagnetic field in optical waveguides.
We present critical groups estimates for a functional defined on the Banach space , bounded domain in , , associated to a quasilinear elliptic equation involving -laplacian. In spite of the lack of an Hilbert structure and of Fredholm property of the second order differential of in each critical point, we compute the critical groups of in each isolated critical point via Morse index.
We prove existence of a positive, radial solution for a semilinear elliptic problem with a discontinuous nonlinearity. We use an approximating argument which requires no monotonicity assumptions on the nonlinearity.
In this work we consider the magnetic NLS equation where , is a magnetic potential, possibly unbounded, is a multi-well electric potential, which can vanish somewhere, is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution to (0.1), under conditions on the nonlinearity which are nearly optimal.
Using some perturbation results in critical point theory, we prove that a class of nonlinear Schrödinger equations possesses semiclassical states that concentrate near the critical points of the potential .
In this work we consider the magnetic NLS equation where , is a magnetic potential, possibly unbounded, is a multi-well electric potential, which can vanish somewhere, is a subcritical nonlinear term. We prove the existence of a semiclassical multi-peak solution to (0.1), under conditions on the nonlinearity which are nearly optimal.
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