Abstract. In this paper, we deal with the electrostatic Born-Infeld equationwhere ρ is an assigned extended charge density. We are interested in the existence and uniqueness of the potential φ and finiteness of the energy of the electrostatic field −∇φ. We first relax the problem and treat it with the direct method of the Calculus of Variations for a broad class of charge densities. Assuming ρ is radially distributed, we recover the weak formulation of (BI) and the regularity of the solution of the Poisson equation (under the same smootheness assumptions).In the case of a locally bounded charge, we also recover the weak formulation without assuming any symmetry. The solution is even classical if ρ is smooth. Then we analyze the case where the density ρ is a superposition of point charges and discuss the results in [17]. Other models are discussed, as for instance a system arising from the coupling of the nonlinear Klein-Gordon equation with the Born-Infeld theory.

We study the mixed dispersion fourth order nonlinear Schrödinger equationwhere γ, σ > 0 and β ∈ R. We focus on standing wave solutions, namely solutions of the form ψ(x, t) = e iαt u(x), for some α ∈ R. This ansatz yields the fourth-order elliptic equationWe consider two associated constrained minimization problems: one with a constraint on the L 2 -norm and the other on the L 2σ+2 -norm. Under suitable conditions, we establish existence of minimizers and we investigate their qualitative properties, namely their sign, symmetry and decay at infinity as well as their uniqueness, nondegeneracy and orbital stability.

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form −ε 2 ∆u + V (x)u = K(x)u p , x ∈ 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 ε ∼ 0, we prove the existence of bound state solutions localized around local minimum points of the auxiliary function A = V θ K − 2 p−1 , where θ = (p + 1)/(p − 1) − N/2. A special attention is devoted to the qualitative properties of these solutions as ε goes to zero.

No abstract

No abstract

Abstract. Consider a Hamiltonian elliptic system of type    −∆u = Hv (u, v) in Ω −∆v = Hu (u, v) in Ω u, v = 0 on ∂Ω where H is a power-type nonlinearity, for instancehaving subcritical growth, and Ω 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 properties; we illustrate this in the second half of this survey, where we also review some of the most recent literature dealing mainly with symmetry, concentration, and multiplicity results. This paper contains some original results as well as new proofs and approaches to known facts.

In this paper, we consider the Lane–Emden problem [Formula: see text] where Ω is a bounded domain in ℝN and p > 2. First, we prove that, for p close to 2, the solution is unique once we fix the projection on the second eigenspace. From this uniqueness property, we deduce partial symmetries of least energy nodal solutions. We also analyze the asymptotic behavior of least energy nodal solutions as p goes to 2. Namely, any accumulation point of sequences of (renormalized) least energy nodal solutions is a second eigenfunction that minimizes a reduced functional on a reduced Nehari manifold. From this asymptotic behavior, we also deduce an example of symmetry breaking. We use numerics to illustrate our results.

We discuss existence and multiplicity of positive solutions of the \ud one-dimensional prescribed curvature problem \ud $$\ud -\left(\ud {u'}/{\sqrt{1+{u'}^2}}\right)' = \lambda f(t,u),\ud \quad\ud u(0)=0,\,\,u(1)=0, \ud $$\ud depending on the behaviour at the origin and at infinity of the potential $\int_0^u f(t,s)\,ds$. Besides solutions in $W^{2,1}(0,1)$, we also consider solutions in $W_{loc}^{2,1}(0,1)$ which are possibly discontinuos at the endpoints of $[0,1]$. Our approach is essentially variational and is based on a regularization of the action functional associated with the curvature problem