Infinitely many positive solutions for nonlinear equations with non-symmetric potential

We consider the following nonlinear Schrodinger equation [{l} \Delta u-(1+\delta V)u+f(u)=0 in \R^N, u>0 in \R^N, u\in H^1(\R^N).] where $V$ is a potential satisfying some decay condition and $f(u)$ is a superlinear nonlinearity satisfying some nondegeneracy condition. Using localized energy method, we prove that there exists some $\delta_0$ such that for $0<\delta<\delta_0$, the above problem has infinitely many positive solutions. This generalizes and gives a new proof of the results by Cerami-Passaseo-Solimini (CPAM to appear). The new techniques allow us to establish the existence of infinitely many positive bound states for elliptic systems.Comment: 43 page

Boundary concentrations on segments

We consider the following singularly perturbed Neumann problem \begin{eqnarray*} \ve^2 \Delta u -u +u^p = 0 \, \quad u>0 \quad {\mbox {in}} \quad \Omega, \quad {\partial u \over \partial \nu}=0 \quad {\mbox {on}} \quad \partial \Omega, \end{eqnarray*} where $p>2$ and $\Omega$ is a smooth and bounded domain in $\R^2$. We construct a new class of solutions which consist of large number of spikes concentrating on a {\bf segment} of the boundary which contains a local minimum point of the mean curvature function and has the same mean curvature at the end points. We find a continuum limit of ODE systems governing the interactions of spikes and show that the mean curvature function acts as {\em friction force}.Comment: 36 page

Wave equations associated to Liouville-type problems: global existence in time and blow up criteria

We are concerned with wave equations associated to some Liouville-type problems on compact surfaces, focusing on sinh-Gordon equation and general Toda systems. Our aim is on one side to develop the analysis for wave equations associated to the latter problems and second, to substantially refine the analysis initiated in [11] concerning the mean field equation. In particular, by exploiting the variational analysis recently derived for Liouville-type problems we prove global existence in time for the sub-critical case and we give general blow up criteria for the super-critical and critical case. The strategy is mainly based on fixed point arguments and improved versions of the Moser-Trudinger inequality

Uniqueness and nondegeneracy of sign-changing radial solutions to an almost critical elliptic problem

We study sign-changing radial solutions for the following semi-linear elliptic equation \begin{align*} \Delta u-u+|u|^{p-1}u=0\quad{\rm{in}}\ \mathbb{R}^N,\quad u\in H^1(\mathbb{R}^N), \end{align*} where $1<p<\frac{N+2}{N-2}$, $N\geq3$. It is well-known that this equation has a unique positive radial solution and sign-changing radial solutions with exactly $k$ nodes. In this paper, we show that such sign-changing radial solution is also unique when $p$ is close to $\frac{N+2}{N-2}$. Moreover, those solutions are non-degenerate, i.e., the kernel of the linearized operator is exactly $N$-dimensional.Comment: 37 page

An optimal bound on the number of interior spike solutions for Lin-Ni-Takagi problem

We consider the following singularly perturbed Neumann problem {eqnarray*} \ve^2 \Delta u -u +u^p = 0 \quad {{in}} \quad \Omega, \quad u>0 \quad {{in}} \quad \Omega, \quad {\partial u \over \partial \nu}=0 \quad {{on}} \quad \partial \Omega, {eqnarray*} where $p$ is subcritical and $\Omega$ is a smooth and bounded domain in $\R^n$ with its unit outward normal $\nu$. Lin-Ni-Wei \cite{LNW} proved that there exists \ve_0 such that for 0<\ve<\ve_0 and for each integer $k$ bounded by {equation} 1\leq k\leq \frac{\delta(\Omega,n,p)}{(\ve |\log \ve |)^n} {equation} where $\delta(\Omega,n,p)$ is a constant depending only on $\Omega$, $p$ and $n$, there exists a solution with $k$ interior spikes. We show that the bound on $k$ can be improved to {equation} 1\leq k\leq \frac{\delta(\Omega,n,p)}{\ve^n}, {equation} which is optimal

Nondegeneracy of nonradial sign-changing solutions to the nonlinear Schr\"odinger equations

We prove that the non-radial sign-changing solutions to the nonlinear Schr\"odinger equation \begin{equation*} \Delta u-u+|u|^{p-1}u=0 \mbox{ in }\R^N, \quad u \in H^1 (\R^N ) \end{equation*} constructed by Musso, Pacard and Wei is non-degenerate. This provides the first example of non-degenerate sign-changing solution with finite energy to the above nonlinear Schr\"odinger equation

On Non-topological Solutions of the G2{\bf G}_2 Chern-Simons System

For any rank 2 of simple Lie algebra, the relativistic Chern-Simons system has the following form: \begin{equation}\label{e001} \left\{\begin{array}{c} \Delta u_1+(\sum_{i=1}^2K_{1i}e^{u_i} -\sum_{i=1}^2\sum_{j=1}^2e^{u_i}K_{1i}e^{u_j}K_{ij})=4\pi\displaystyle \sum_{j=1}^{N_1}\delta_{p_j}\\ \Delta u_2+ (\sum_{i=1}^2K_{2i}e^{u_i}-\sum_{i=1}^2\sum_{j=1}^2e^{u_i}K_{2i}e^{u_j}K_{ij})=4\pi\displaystyle \sum_{j=1}^{N_2}\delta_{q_j} \end{array} \right.\mbox{in}\; \mathbb{R}^2, \end{equation} where $K$ is the Cartan matrix of rank $2$. There are three Cartan matrix of rank 2: ${\bf A}_2$, ${\bf B}_2$ and ${\bf G}_2$. A long-standing open problem for \eqref{e001} is the question of the existence of non-topological solutions. In a previous paper \cite{ALW}, we have proven the existence of non-topological solutions for the ${\bf A}_2$ and ${\bf B}_2$ Chern-Simons system. In this paper, we continue to consider the ${\bf G}_2$ case. We prove the existence of non-topological solutions under the condition that either $N_2\displaystyle\sum_{j=1}^{N_1} p_j=N_1\displaystyle \sum_{j=1}^{N_2} q_j$ or $N_2\displaystyle\sum_{j=1}^{N_1}p_j \not =N_1\displaystyle \sum_{j=1}^{N_2} q_j$ and $N_1,N_2>1$, $|N_1-N_2|\neq 1$. We solve this problem by a perturbation from the corresponding ${\bf G}_2$ Toda system with one singular source. Combining with \cite{ALW}, we have proved the existence of non-topological solutions to the Chern-Simons system with Cartan matrix of rank $2$.Comment: 40 page

Bound state solutions for the supercritical fractional Schr\"odinger equation

We prove the existence of positive solutions for the supercritical nonlinear fractional Schr\"odinger equation $(-\Delta)^s u+V(x)u-u^p=0$ in $\mathbb R^n$, with $u(x)\to 0$ as $|x|\to +\infty$, where $p>\frac{n+2s}{n-2s}$ for $s\in (0,1), \ n>2s$. We show that if $V(x)=o(|x|^{-2s})$ as $|x|\to +\infty$, then for $p>\frac{n+2s-1}{n-2s-1}$, this problem admits a continuum of solutions. More generally, for $p>\frac{n+2s}{n-2s}$, conditions for solvability are also provided. This result is the extension of the work by Davila, Del Pino, Musso and Wei to the fractional case. Our main contributions are: the existence of a smooth, radially symmetric, entire solution of $(-\Delta)^s w=w^p$ in $\mathbb R^n$, and the analysis of its properties. The difficulty here is the lack of phase-plane analysis for a nonlocal ODE; instead we use conformal geometry methods together with Schaaf's argument as in the paper by Ao, Chan, DelaTorre, Fontelos, Gonz\'alez and Wei on the singular fractional Yamabe problem.Comment: Minor changes from previous versio

Boundary connected sum of Escobar manifolds

Let $(X_1, \bar g_1)$ and $(X_2, \bar g_2)$ be two compact Riemannian manifolds with boundary $(M_1,g_1)$ and $(M_2,g_2)$ respectively. The Escobar problem consists in prescribing a conformal metric on a compact manifold with boundary with zero scalar curvature in the interior and constant mean curvature of the boundary. The present work is the construction of a connected sum $X=X_1 \sharp X_2$ by excising half ball near points on the boundary. The resulting metric on $X$ has zero scalar curvature and a CMC boundary. We fully exploit the nonlocal aspect of the problem and use new tools developed in recent years to handle such kinds of issues. Our problem is of course a very well-known problem in geometric analysis and that is why we consider it but the results in the present paper can be extended to other more analytical problems involving connected sums of constant fractional curvatures

On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program

We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, and we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of an ODE, and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. No traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we examine a fractional Schr\"{o}dinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case