We prove $L^p$ lower bounds for Coulomb energy for radially symmetric functions in $dot H^s(R^3)$ with $frac 12 <s<frac{3}{2}$. In case $frac 12 <s leq 1$ we show that the lower bounds are sharp.
In this paper we study the existence of maximizers for two families of interpolation inequalities, namely a generalized Gagliardo-Nirenberg inequality and a new inequality involving the Riesz energy. Two basic tools in our argument are a generalization of Liebs Translation Lemma and a Riesz energy version of the Brezis--Lieb lemma.
We consider the nonlinear Klein-Gordon equation in $R^d$. We call multi-solitary waves a solution behaving at large time as a sum of boosted standing waves. Our main result is the existence of such multi-solitary waves, provided the composing boosted standing waves are stable. It is obtained by solving the equation backward in time around a sequence of approximate multi-solitary waves and showing convergence to a solution with the desired property. The main ingredients of the proof are finite speed of propagation, variational characterizations of the profiles, modulation theory and energy estimates.
In this paper we prove the existence of vortices, namely standing waves with non null angular momentum, for the nonlinear Klein-Gordon equation in dimension $Ngeq 3$. We show with variational methods that the existence of these kind of solutions, that we have called emph{hylomorphic vortices}, depends on a suitable energy-charge ratio. Our variational approach turns out to be useful for numerical investigations as well. In particular, some results in dimension N=2 are reported, namely exemplificative vortex profiles by varying charge and angular momentum, together with relevant trends for vortex frequency and energy-charge ratio. The stability problem for hylomorphic vortices is also addressed. In the absence of conclusive analytical results, vortex evolution is numerically investigated: the obtained results suggest that, contrarily to solitons with null angular momentum, vortex are unstable.
In this paper we study the existence and the instability of standing waves with prescribed $L^2$-norm for a class of Schrodinger-Poisson-Slater equations in $R^{3}$ %orbitally stable standing waves with arbitray charge for the following Schrodinger-Poisson type equation label{evolution1} ipsi_{t}+ Delta psi - (|x|^{-1}*|psi|^{2}) psi+|psi|^{p-2}psi=0 % text{in} R^{3}, when $p in (10/3,6)$. To obtain such solutions we look to critical points of the energy functional $$F(u)=1/2| triangledown u|_{L^{2}(mathbb{R}^3)}^2+1/4int_{mathbb{R}^3}int_{mathbb{R}^3}frac{|u(x)|^2| u(y)|^2}{|x-y|}dxdy-frac{1}{p}int_{mathbb{R}^3}|u|^pdx $$ on the constraints given by $$S(c)= {u in H^1(mathbb{R}^3) :|u|_{L^2(R^3)}^2=c, c>0}.$$ For the values $p in (10/3, 6)$ considered, the functional $F$ is unbounded from below on $S(c)$ and the existence of critical points is obtained by a mountain pass argument developed on $S(c)$. We show that critical points exist provided that $c>0$ is sufficiently small and that when $c>0$ is not small a non-existence result is expected. Concerning the dynamics we show for initial condition $u_0in H^1(R^3)$ of the associated Cauchy problem with $|u_0|_{2}^2=c$ that the mountain pass energy level $gamma(c)$ gives a threshold for global existence. Also the strong instability of standing waves at the mountain pass energy level is proved. Finally we draw a comparison between the Schrodinger-Poisson-Slater equation and the classical nonlinear Schrodinger equation.
We prove the existence of ground states for the semi-relativistic Schrodinger-Poisson-Slater energy $$I^{alpha,beta}(rho)=inf_{substack{uin H^frac 12(R^3) int_{R^3}|u|^2 dx=rho}} frac{1}{2}|u|^2_{H^frac 12(R^3)} +alphaintint_{R^{3}timesR^{3}} frac{| u(x)|^{2}|u(y)|^2}{|x-y|}dxdy-betaint_{R^{3}}|u|^{frac{8}{3}}dx$$ $alpha,beta>0$ and $rho>0$ is small enough. The minimization problem is $L^2$ critical and in order to characterize of the values $alpha, beta>0$ such that $I^{alpha, beta}(rho)>-infty$ for every $rho>0$, we prove a new lower bound on the Coulomb energy involving the kinetic energy and the exchange energy. We prove the existence of a constant $S>0$ such that $$frac{1}{S}frac{|varphi|_{L^frac 83(R^3)}}{|varphi|_{dot H^frac 12(R^3)}^frac 12}leq left (intint_{R^3times R^3} frac{|varphi(x)|^2|varphi(y)|^2}{|x-y|}dxdyright)^frac 18 $$ for all $varphiin C^infty_0(R^3)$. Eventually we show that similar compactness property fails provided that in the energy above we replace the inhomogeneous Sobolev norm $|u|^2_{H^frac 12(R^3)}$ by the homogeneous one $|u|_{dot H^frac 12(R^3)}$.
