In this paper we study the existence of radially symmetric solitary waves for nonlinear Klein–Gordon equations and nonlinear Schrödinger equations coupled with Maxwell equations. The method relies on a variational approach and the solutions are obtained as mountain-pass critical points for the associated energy functional.

In this paper we obtain some non-existence results for the Klein-Gordon equation coupled with the electrostatic field. The method relies on the deduction of some suitable Pohožaev identity which provides necessary conditions to get existence of nontrivial solutions. The case of Maxwell-Schrödinger type coupled equations is also considered.

We consider a nonlinear elliptic equation driven by the sum of a -Laplacian and a -Laplacian, where <!-- MATH $1<q\leq 2\leq p<\infty$ --> <img width="172" height="37" align="MIDDLE" border="0" src="images/img4.gif" alt="$ 1<q\leq 2\leq p<\infty $"> with a -superlinear Carathéodory reaction term which doesn't satisfy the usual Ambrosetti-Rabinowitz condition. Using variational methods based on critical point theory together with techniques from Morse theory, we show that the problem has at least three nontrivial solutions; among them one is positive and one is negative.

We establish Carleman estimates for singular/degenerate parabolic Dirichlet problems with degeneracy and singularity occurring in the interior of the spatial domain. Our results are completely new, since this situation is not covered by previous contributions for degeneracy and singularity on the boundary. In addition, we consider non-smooth coefficients, thus preventing the use of standard calculations in this framework.

We study the existence of radially symmetric solitary waves for a nonlinear Schrödinger-Poisson system. In contrast to all previous results, we consider the presence of a positive potential, of interest in physical applications.

The American Mathematical Society