On a fourth order nonlinear Helmholtz equation

In this paper, we study the mixed dispersion fourth order nonlinear Helmholtz equation $\Delta^2 u -\beta \Delta u + \alpha u= \Gamma|u|^{p-2} u$ in $\mathbb R^N$ for positive, bounded and $\mathbb Z^N$-periodic functions $\Gamma$. Using the dual method of Evequoz and Weth, we find solutions to this equation and establish some of their qualitative properties

The logarithmic Choquard equation: sharp asymptotics and nondegeneracy of the groundstate

We derive the asymptotic decay of the unique positive, radially symmetric solution to the logarithmic Choquard equation and we establish its nondegeneracy. For the corresponding three-dimensional problem, the nondegeneracy property of the positive ground state to the Choquard equation was proved by E. Lenzmann (2009)

One-dimensional symmetry and Liouville type results for the fourth order Allen-Cahn equation in RN^N

In this paper, we prove an analogue of Gibbons' conjecture for the extended fourth order Allen-Cahn equation in R N , as well as Liouville type results for some solutions converging to the same value at infinity in a given direction. We also prove a priori bounds and further one-dimensional symmetry and rigidity results for semilinear fourth order elliptic equations with more general nonlinearities

Equilibrium measures and equilibrium potentials in the Born-Infeld model

In this paper, we consider the electrostatic Born-Infeld model \begin{equation*} \tag{$\mathcal{BI}$} \left\{ \begin{array}{rcll} -\operatorname{div}\left(\displaystyle\frac{\nabla \phi}{\sqrt{1-|\nabla \phi|^2}}\right)&=& \rho & \hbox{in }\mathbb{R}^N, \\[6mm] \displaystyle\lim_{|x|\to \infty}\phi(x)&=& 0 \end{array} \right. \end{equation*} where $\rho$ is a charge distribution on the boundary of a bounded domain $\Omega\subset \mathbb{R}^N$. We are interested in its equilibrium measures, i.e. charge distributions which minimize the electrostatic energy of the corresponding potential among all possible distributions with fixed total charge. We prove existence of equilibrium measures and we show that the corresponding equilibrium potential is unique and constant in $\overline \Omega$. Furthermore, for smooth domains, we obtain the uniqueness of the equilibrium measure, we give its precise expression, and we verify that the equilibrium potential solves $\mathcal{BI}$. Finally we characterize balls in $\mathbb{R}^N$ as the unique sets among all bounded $C^{2,\alpha}$-domains $\Omega$ for which the equilibrium distribution is a constant multiple of the surface measure on $\partial \Omega$. The same results are obtained also for Taylor approximations of the electrostatic energy

Existence of positive solutions of a superlinear boundary value problem with indefinite weight

We deal with the existence of positive solutions for a two-point boundary value problem associated with the nonlinear second order equation $u''+a(x)g(u)=0$. The weight $a(x)$ is allowed to change its sign. We assume that the function $g\colon\mathopen{[}0,+\infty\mathclose{[}\to\mathbb{R}$ is continuous, $g(0)=0$ and satisfies suitable growth conditions, so as the case $g(s)=s^{p}$, with $p>1$, is covered. In particular we suppose that $g(s)/s$ is large near infinity, but we do not require that $g(s)$ is non-negative in a neighborhood of zero. Using a topological approach based on the Leray-Schauder degree we obtain a result of existence of at least a positive solution that improves previous existence theorems.Comment: 12 pages, 4 PNG figure

Resonance of isochronous oscillators

An oscillator such that all motions have the same minimal period is called isochronous. When the isochronous is forced by a time-dependent perturbation with the same natural frequency as the oscillator the phenomenon of resonance can appear. This fact is well understood for the harmonic oscillator and we extend it to the nonlinear scenario

Existence, regularity and boundary behaviour of bounded variation solutions of a one-dimensional capillarity equation

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A concentration phenomenon for semilinear elliptic equations

For a domain \Omega\subset\dR^N we consider the equation -\Delta u + V(x)u = Q_n(x)\abs{u}^{p-2}u with zero Dirichlet boundary conditions and $p\in(2,2^*)$. Here $V\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $\Omega$ and negative outside, and such that the sets $\{Q_n>0\}$ shrink to a point $x_0\in\Omega$ as $n\to\infty$. We show that if $u_n$ is a nontrivial solution corresponding to $Q_n$, then the sequence $(u_n)$ concentrates at $x_0$ with respect to the $H^1$ and certain $L^q$-norms. We also show that if the sets $\{Q_n>0\}$ shrink to two points and $u_n$ are ground state solutions, then they concentrate at one of these points

Stationary solutions of the nonlinear Schr\"odinger equation with fast-decay potentials concentrating around local maxima

We study positive bound states for the equation $$- \epsilon^2 \Delta u + Vu = u^p, \qquad \text{in $\mathbf{R}^N$},$$ where $\epsilon > 0$ is a real parameter, $\frac{N}{N-2} < p < \frac{N+2}{N-2}$ and $V$ is a nonnegative potential. Using purely variational techniques, we find solutions which concentrate at local maxima of the potential $V$ without any restriction on the potential.Comment: 25 pages, reformatted the abstract for MathJa

Quantitative symmetry breaking of groundstates for a class of weighted Emden–Fowler equations

We prove that symmetrybreaking occurs in dimensions N ≥ 3 for the groundstate solutions to a class of Emden-Fowler equa-tions on the unit ball, with Dirichlet boundary conditions. We show that this phenomenon occurs forlarge values of a certain exponent for a radial weight function appearing in the equation. The problemreads as a possibly large perturbation of the classical H ́enon equation. In particular we consider aweight function having a spherical shell of zeroes centred at the origin and of radius R. A quantitativecondition on R for this phenomenon to occur is given by means of universal constants, such as thebest constant for the subcritical Sobolev embedding