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.

IntroductionIn the last recent years an increasing interest has been devoted to degenerate parabolic equations. Indeed, many problems coming from physics (boundary layer models in [13], models of Kolmogorov type in [7], . . . ), biology (Wright-Fisher models in [50] and Fleming-Viot models in [29]), and economics (Black-MertonScholes equations in [23]) are described by degenerate parabolic equations, whose linear prototype iswith the associated desired boundary conditions, whereIn this paper we concentrate on a special topic related to this field of research, i.e. Carleman estimates for the adjoint problem to (1.1). Indeed, they have so many applications that a large number of papers has been devoted to prove some forms of them and possibly some applications. For example, it is well known that they are a fundamental tool to prove observability inequalities, which lead to global null controllability results for (1.1) also in the non degenerate case: for all T > 0 and for all initial data u 0 ∈ L 2 ((0, T ) × (0, 1)) there is a suitable control h ∈ L 2 ((0, T ) × (0, 1)), supported in a subset ω of [0, 1], such that the solution u of (1.1) satisfies u(T, x) = 0 for all x ∈ [0, 1] (see, for instance, The common point of all the previous papers dealing with degenerate equations, is that the function a degenerates at the boundary of the domain. For example, as a, one can consider the double power functionwhere k and α are positive constants. For related systems of degenerate equations we refer to [1], [2] and [14]. However, the papers cited above deal with a function a that degenerates at the boundary of the spatial domain. To our best knowledge, [51] is the first paper treating the existence of a solution for the Cauchy problem associated to a parabolic equation which degenerates in the interior of the spatial domain, while degenerate parabolic problems modelling biological phenomena and related optimal control problems are later studied in [43] and [9]. Recently, in [32] the authors analyze in detail the degenerate operator A in the space L 2 (0, 1), with or without weight,

We consider a nonlinear elliptic equation driven by the sum of a p-Laplacian and a q-Laplacian, where 1 < q ≤ 2 ≤ p < ∞ with a (p − 1)-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 consider a parabolic problem with degeneracy in the interior of\ud the spatial domain, and we focus on Carleman estimates for the\ud associated adjoint problem. The novelty of interior degeneracy does\ud not let us adapt previous Carleman estimate to our situation. As an\ud application, observability inequalities are established

Abstract: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 consider a general nonlinear elliptic problem of the second order whose associated functional presents two linking structures and we prove the existence of three nontrivial solutions to the problem.2000 Mathematics Subject Classification: 35J65, 35J20, 49J40.

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