On the finite space blow up of the solutions of the Swift-Hohenberg equation

The aim of this paper is to study the finite space blow up of the solutions for a class of fourth order differential equations. Our results answer a conjecture in [F. Gazzola and R. Pavani. Wide oscillation finite time blow up for solutions to nonlinear fourth order differential equations. Arch. Ration. Mech. Anal., 207(2):717--752, 2013] and they have implications on the nonexistence of beam oscillation given by traveling wave profile at low speed propagation.Comment: 24 pages, 2 figure

H\'enon type equations and concentration on spheres

In this paper we study the concentration profile of various kind of symmetric solutions of some semilinear elliptic problems arising in astrophysics and in diffusion phenomena. Using a reduction method we prove that doubly symmetric positive solutions in a $2m$-dimensional ball must concentrate and blow up on $(m-1)$-spheres as the concentration parameter tends to infinity. We also consider axially symmetric positive solutions in a ball in $\mathbb{R}^N$, $N \geq 3$, and show that concentration and blow up occur on two antipodal points, as the concentration parameter tends to infinity

Hamiltonian elliptic systems: a guide to variational frameworks

Consider a Hamiltonian system of type $$-\Delta u=H_{v}(u,v),\ -\Delta v=H_{u}(u,v) \ \ \text{ in } \Omega, \qquad u,v=0 \text{ on } \partial \Omega$$ where $H$ is a power-type nonlinearity, for instance $H(u,v)= |u|^p/p+|v|^q/q$, having subcritical growth, and $\Omega$ is a bounded domain of $\mathbb{R}^N$, $N\geq 1$. 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.Comment: 78 pages, 7 figures. This corresponds to the second version of this paper. With respect to the original version, this one contains additional references, and some misprints were correcte

Sapo Campus: what users really think about an institutionally supported PLE

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Local minimizers in spaces of symmetric functions and applications

We study $H^1$ versus $C^1$ local minimizers for functionals defined on spaces of symmetric functions, namely functions that are invariant by the action of some subgroups of $\mathcal{O}(N)$. These functionals, in many cases, are associated with some elliptic partial differential equations that may have supercritical growth. So we also prove some results on classical regularity for symmetric weak solutions for a general class of semilinear elliptic equations with possibly supercritical growth. We then apply these results to prove the existence of a large number of classical positive symmetric solutions to some concave-convex elliptic equations of H\'enon type

Periodic solutions and torsional instability in a nonlinear nonlocal plate equation

A thin and narrow rectangular plate having the two short edges hinged and the two long edges free is considered. A nonlinear nonlocal evolution equation describing the deformation of the plate is introduced: well-posedness and existence of periodic solutions are proved. The natural phase space is a particular second order Sobolev space that can be orthogonally split into two subspaces containing, respectively, the longitudinal and the torsional movements of the plate. Sufficient conditions for the stability of periodic solutions and of solutions having only a longitudinal component are given. A stability analysis of the so-called prevailing mode is also performed. Some numerical experiments show that instabilities may occur. This plate can be seen as a simplified and qualitative model for the deck of a suspension bridge, which does not take into account the complex interactions between all the components of a real bridge.Comment: 34 pages, 4 figures. The result of Theorem 6 is correct, but the proof was not correct. We slightly changed the proof in this updated versio