We consider the problemHere g is a superlinear, subcritical, possibly nonhomogeneous, odd nonlinearity. We deal with the case where the associated functional is not bounded below on the L 2 -unit sphere, and we show the existence of infinitely many solutions.MSC 2010: Primary: 35J60; Secondary: 35P30, 58E05

Abstract. Steady vortices for the three-dimensional Euler equation for inviscid incompressible flows and for the shallow water equation are constructed and showed to tend asymptotically to singular vortex filaments. The construction is based on a study of solutions to the semilinear elliptic problemfor small values of ε > 0.

We prove existence and multiplicity results for sign‐changing solutions of a prescribed mean curvature equation with Dirichlet boundary conditions. Our arguments involve a perturbation of the degenerate part \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$1/\sqrt{1+s}$\end{document}, which allows us to use classical variational techniques and to localize small regular solutions.

In classical pension design, there are essentially two kinds of pension schemes: defined benefit (DB) or defined contribution (DC) plans. Each corresponds to a different philosophy of spreading risk between the stakeholders: in a DB the main risks are taken by the sponsor of the plan while in a DC the active members must bear all the risks. Especially when applied to social security pension systems, this traditional view can in both cases lead to unfair intergenerational equilibrium. The purpose of this paper, which focuses on social security, is twofold. First, we present alternative architectures based on a mix between DB and DC in order to achieve both financial sustainability and social adequacy. An example of this approach is the socalled Musgrave rule, but other risk-sharing approaches will be developed in a payas-you-go philosophy. More precisely, we build convex and log-convex families of hybrid pension schemes whose extremal points correspond to DB and DC. Second, we study these new architectures in a stochastic environment, and present different rules to select the most efficient ones. To do so, we search for optimality from a risksharing point of view among the new pension plans.

In this paper we prove the almost everywhere convergence of the gradient of Palais-Smale sequences, allowing us to apply the Brezis-Lieb lemma. This leads us to show that infima are attained, and thus to prove the existence of optimal solutions for some critical problems. Our method does not use the concentration- compactness principle.