Denis Bonheure

Bound state solutions for a class of nonlinear Schrödinger equations

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrodinger equations of the form -epsilon(2)Delta u + V(x)u = K(x)u(p), x is an element of R-N, where V, K are positive continuous functions and p > 1 is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential V is allowed to vanish at infinity and the competing function K does not have to be bounded. In the semi-classical limit, i.e. for epsilon similar to 0, we prove the existence of bound state solutions localized around local minimum points of the auxiliary function A = (VK-2/p-1)-K-theta, where theta = (p + 1)/(p - 1) - N/2. A special attention is devoted to the qualitative properties of these solutions as e goes to zero.

Chapter 2 Heteroclinic Orbits for Some Classes of Second and Fourth Order Differential Equations

Publisher Summary This chapter describes Heteroclinic orbits for some classes of second and fourth order differential equations. In qualitative theory of differential equations, a prominent role is played by special classes of solutions, like periodic solutions or solutions to some kind of boundary value problem. When a system of ordinary differential equations has equilibria (i.e. constant solutions) whose stability properties are known, it becomes significant to study the connections between them by trajectories of solutions of the given system. These are called homoclinic or heteroclinic solutions (sometimes pulses or kinks) according to whether they just describe a loop based at one single equilibrium or they “start” and “end” at two distinct equilibria.

Ground state and non-ground state solutions of some strongly coupled elliptic systems

We study an elliptic system of the form Lu = ⌊v⌋p&1 v and Lv = ⌊u⌋ q&1 u in Ω with homogeneous Dirichlet boundary condition, where Lu:= &Δu in the case of a bounded domain and Lu:= &Δu + u in the cases of an exterior domain or the whole space RN. We analyze the existence, uniqueness, sign and radial symmetry of ground state solutions and also look for sign changing solutions of the system. More general non-linearities are also considered.

Increasing radial solutions for Neumann problems without growth restrictions

Abstract We study the existence of positive increasing radial solutions for superlinear Neumann problems in the ball. We do not impose any growth condition on the nonlinearity at infinity and our assumptions allow for interactions with the spectrum. In our approach we use both topological and variational arguments, and we overcome the lack of compactness by considering the cone of nonnegative, nondecreasing radial functions of H 1 ( B ) .

Hamiltonian elliptic systems: a guide to variational frameworks

Consider a Hamiltonian elliptic system of type 8 < : u = Hv(u;v) in v = Hu(u;v) in u;v = 0 on @ where H is a power-type nonlinearity, for instance H(u;v) =juj p+1 =(p + 1) +jvj q+1 =(q + 1); having subcritical growth, and is a bounded domain of R N , N 1. The aim of this paper is to give an overview of the several variational frameworks that can be used to treat such a system. Within each approach, we address existence of solutions, and in particular of ground state solutions. Some of the available frameworks are more adequate to derive certain qualitative proper- ties; we illustrate this in the second half of this survey, where we also review some of the most recent literature dealing mainly with symmetry, concentra- tion, and multiplicity results. This paper contains some original results as well as new proofs and approaches to known facts.

On the Electrostatic Born–Infeld Equation with Extended Charges

AbstractIn this paper, we deal with the electrostatic Born–Infeld equation

Concentration on circles for nonlinear Schrödinger-Poisson systems with unbounded potentials vanishing at infinity

\left\{\begin{array}{ll}-\operatorname{div} \left(\displaystyle\frac{\nabla\phi}{\sqrt{1-|\nabla \phi|^2}} \right)= \rho \quad{in} \mathbb{R}^N, \\ \displaystyle\lim_{|x|\to \infty} \phi(x)= 0,\end{array}\right. \quad \quad \quad \quad ({\mathcal{BI}})

Erratum to: “Multiple critical points of perturbed symmetric strongly indefinite functionals” [http://dx.doi.org/10.1016/j.anihpc.2008.06.002]

-div∇ϕ1-|∇ϕ|2=ρinRN,lim|x|→∞ϕ(x)=0,(BI)where