Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence

We prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation $$-\triangle u+V\left( \left| x\right| \right) u=g\left( \left| x\right| ,u\right) \quad \textrm{in }\Omega \subseteq \mathbb{R}^{N},\ N\geq 3,$$ where $\Omega$ is a radial domain (bounded or unbounded) and $u$ satisfies $u=0$ on $\partial \Omega$ if $\Omega \neq \mathbb{R}^{N}$ and $u\rightarrow 0$ as $\left| x\right| \rightarrow \infty$ if $\Omega$ is unbounded. The potential $V$ may be vanishing or unbounded at zero or at infinity and the nonlinearity $g$ may be superlinear or sublinear. If $g$ is sublinear, the case with $g\left( \left| \cdot \right| ,0\right) \neq 0$ is also considered.Comment: 29 pages, 8 figure

A Computer-Assisted Uniqueness Proof for a Semilinear Elliptic Boundary Value Problem

A wide variety of articles, starting with the famous paper (Gidas, Ni and Nirenberg in Commun. Math. Phys. 68, 209-243 (1979)) is devoted to the uniqueness question for the semilinear elliptic boundary value problem -{\Delta}u={\lambda}u+u^p in {\Omega}, u>0 in {\Omega}, u=0 on the boundary of {\Omega}, where {\lambda} ranges between 0 and the first Dirichlet Laplacian eigenvalue. So far, this question was settled in the case of {\Omega} being a ball and, for more general domains, in the case {\lambda}=0. In (McKenna et al. in J. Differ. Equ. 247, 2140-2162 (2009)), we proposed a computer-assisted approach to this uniqueness question, which indeed provided a proof in the case {\Omega}=(0,1)x(0,1), and p=2. Due to the high numerical complexity, we were not able in (McKenna et al. in J. Differ. Equ. 247, 2140-2162 (2009)) to treat higher values of p. Here, by a significant reduction of the complexity, we will prove uniqueness for the case p=3

Desingularization of vortices for the Euler equation

We study the existence of stationary classical solutions of the incompressible Euler equation in the plane that approximate singular stationnary solutions of this equation. The construction is performed by studying the asymptotics of equation -\eps^2 \Delta u^\eps=(u^\eps-q-\frac{\kappa}{2\pi} \log \frac{1}{\eps})_+^p with Dirichlet boundary conditions and $q$ a given function. We also study the desingularization of pairs of vortices by minimal energy nodal solutions and the desingularization of rotating vortices.Comment: 40 page

Singular solutions of fully nonlinear elliptic equations and applications

We study the properties of solutions of fully nonlinear, positively homogeneous elliptic equations near boundary points of Lipschitz domains at which the solution may be singular. We show that these equations have two positive solutions in each cone of $\mathbb{R}^n$, and the solutions are unique in an appropriate sense. We introduce a new method for analyzing the behavior of solutions near certain Lipschitz boundary points, which permits us to classify isolated boundary singularities of solutions which are bounded from either above or below. We also obtain a sharp Phragm\'en-Lindel\"of result as well as a principle of positive singularities in certain Lipschitz domains.Comment: 41 pages, 2 figure

A Precise Measurement of the Weak Mixing Angle in Neutrino-Nucleon Scattering

We report a precise measurement of the weak mixing angle from the ratio of neutral current to charged current inclusive cross-sections in deep-inelastic neutrino-nucleon scattering. The data were gathered at the CCFR neutrino detector in the Fermilab quadrupole-triplet neutrino beam, with neutrino energies up to 600 GeV. Using the on-shell definition, ${\rm sin ^2\theta_W} \equiv 1 - \frac{{\rm M_W} ^2}{{\rm M_Z} ^2}$, we obtain ${\rm sin ^2\theta_W} = 0.2218 \pm 0.0025 ({\rm stat.}) \pm 0.0036 ({\rm exp.\: syst.}) \pm 0.0040 ({\rm model})$.Comment: 10 pages, Nevis Preprint #1498 (Submitted to Phys. Rev. Lett.

Global bifurcation for asymptotically linear Schr\"odinger equations

We prove global asymptotic bifurcation for a very general class of asymptotically linear Schr\"odinger equations \begin{equation}\label{1} \{{array}{lr} \D u + f(x,u)u = \lam u \quad \text{in} \ {\mathbb R}^N, u \in H^1({\mathbb R}^N)\setmimus\{0\}, \quad N \ge 1. {array}. \end{equation} The method is topological, based on recent developments of degree theory. We use the inversion $u\to v:= u/\Vert u\Vert_X^2$ in an appropriate Sobolev space $X=W^{2,p}({\mathbb R}^N)$, and we first obtain bifurcation from the line of trivial solutions for an auxiliary problem in the variables (\lambda,v) \in {\mathbb R} \x X. This problem has a lack of compactness and of regularity, requiring a truncation procedure. Going back to the original problem, we obtain global branches of positive/negative solutions 'bifurcating from infinity'. We believe that, for the values of $\lambda$ covered by our bifurcation approach, the existence result we obtain for positive solutions of \eqref{1} is the most general so fa

Determination of the Strange Quark Content of the Nucleon from a Next-to-Leading-Order QCD Analysis of Neutrino Charm Production

We present the first next-to-leading-order QCD analysis of neutrino charm production, using a sample of 6090 $\nu_\mu$- and $\bar\nu_\mu$-induced opposite-sign dimuon events observed in the CCFR detector at the Fermilab Tevatron. We find that the nucleon strange quark content is suppressed with respect to the non-strange sea quarks by a factor \kappa = 0.477 \: ^{+\:0.063}_{-\:0.053}, where the error includes statistical, systematic and QCD scale uncertainties. In contrast to previous leading order analyses, we find that the strange sea $x$-dependence is similar to that of the non-strange sea, and that the measured charm quark mass, $m_c = 1.70 \pm 0.19 \:{\rm GeV/c}^2$, is larger and consistent with that determined in other processes. Further analysis finds that the difference in $x$-distributions between $xs(x)$ and $x\bar s(x)$ is small. A measurement of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cd}|=0.232 ^{+\:0.018}_{-\:0.020}$ is also presented. uufile containing compressed postscript files of five Figures is appended at the end of the LaTeX source.Comment: Nevis R#150

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

Non-existence and uniqueness results for supercritical semilinear elliptic equations

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich-Pohozaev type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local and non-local settings.Comment: Annales Henri Poincar\'e (2009) to appea