Marino Badiale

Homoclinics : Poincaré-Melnikov type results via a variational approach

We introduce a variational approach to obtain some Poincare-Melnikov type results on the existence and multiplicity of homoclinics.

Variational perturbative methods and bifurcation of bound states from the essential spectrum

This paper consists of two main parts. The first deals with a perturbative method in critical point theory and can be seen as the generalisation and completion of some earlier results. The second part is concerned with applications of the abstract setup to the existence of bound states of a class of elliptic differential equations that branch off from the infimum of the essential spectrum.

Semilinear Elliptic Equations for Beginners

Introduction and basic results.- Minimization techniques: compact problems.- Minimization techniques: lack of compactness.- Introduction to minimax methods.- Index of the main assumptions

Concentration around a sphere for a singularly perturbed Schrödinger equation

In this paper we study the existence of concentrated solutions for the following nonlinear elliptic equation −h∆v + V (x)v = |v|p−2v where v : RN → R, N ≥ 3 and 2 < p < 2N N−2 . We assume that the potential V is radially symmetric and bounded below away from zero. Existence results are established provided that h is sufficiently small and we are able to find positive solutions with spherical symmetry which exhibit a concentration behaviour near a sphere centred in zero. The proofs of our results are variational and rely on a constrained minimization method; furthermore a penalization-type technique is developed and permits to single out the desired solutions by means of a suitable modification of the variational problem.

Multiplicity results for the supercritical Henon equation

Abstract We study the Dirichlet problem for the Hénon equation ¡Δu = |x|αup-1 in the unit ball B ⊂ RN. For N ≥ 4 and α large we prove the existence of positive nonradial solutions for a range of ps including supercritical values.

A nonlinear elliptic equation with singular potential and applications to nonlinear field equations

We study existence and asymptotic properties of solutions to a semilinear elliptic equation in the whole space. The equation has a cylindrical symmetry and we find cylindrical solutions. The main features of the problem are that the potential has a large set of singularities (i.e. a subspace), and that the nonlinearity has a double power-like behaviour, subcritical at infinity and supercritical near the origin. We also show that our results imply the existence of solitary waves with nonvanishing angular momentum for nonlinear evolution equations of Schrodinger and Klein-Gordon type.

A note on nonlinear elliptic problems with singular potentials

We deal with the semi-linear elliptic problem −4u + V (|x|)u = f (u) , u ∈ D(R ;R) where the potential V > 0 is measurable, singular at the origin and may also have a continuous set of singularities. The nonlinearity is continuous and has a super-linear power-like behaviour; both sub-critical and super-critical cases are considered. We prove the existence of positive radial solutions. If f is odd, we show that the problem has infinitely many radial solutions. Nonexistence results for particular potentials and nonlinearities are also given.

Compactness and existence results in weighted Sobolev spaces of radial functions. Part I: compactness

Given two measurable functions

A nonexistence result for a nonlinear elliptic equation with singular and decaying potential

V(r )\ge 0