74 research outputs found
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 is a
potential satisfying some decay condition and is a superlinear
nonlinearity satisfying some nondegeneracy condition. Using localized energy
method, we prove that there exists some such that for
, 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 and is a smooth and
bounded domain in . 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
, . It is well-known that this equation has a
unique positive radial solution and sign-changing radial solutions with exactly
nodes. In this paper, we show that such sign-changing radial solution is
also unique when is close to . Moreover, those solutions
are non-degenerate, i.e., the kernel of the linearized operator is exactly
-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 is subcritical and is a smooth
and bounded domain in with its unit outward normal . Lin-Ni-Wei
\cite{LNW} proved that there exists \ve_0 such that for 0<\ve<\ve_0 and for
each integer bounded by {equation} 1\leq k\leq
\frac{\delta(\Omega,n,p)}{(\ve |\log \ve |)^n} {equation} where
is a constant depending only on , and ,
there exists a solution with interior spikes. We show that the bound on
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 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 is the Cartan matrix of rank . There are three
Cartan matrix of rank 2: , and . 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 and
Chern-Simons system. In this paper, we continue to consider the
case. We prove the existence of non-topological solutions under the condition
that either or and , . We solve this problem by a perturbation from the
corresponding 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 .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 in ,
with as , where for . We show that if as , then
for , this problem admits a continuum of solutions.
More generally, for , 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 in , 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 and be two compact Riemannian
manifolds with boundary and 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 by excising half ball near points on the boundary. The resulting
metric on 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
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