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Register a new userA construction of two different solutions to an elliptic system
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The paper aims at constructing two different solutions to an elliptic system $$ u cdot abla u + (-Delta)^m u = lambda F $$ defined on the two dimensional torus. It can be viewed as an elliptic regularization of the stationary Burgers 2D system. A motivation to consider the above system comes from an examination of unusual propetries of the linear operator $lambda sin y partial_x w + (-Delta)^{m} w$ arising from a linearization of the equation about the dominant part of $F$. We argue that the skew-symmetric part of the operator provides in some sense a smallness of norms of the linear operator inverse. Our analytical proof is valid for a particular force $F$ and for $lambda > lambda_0$, $m> m_0$ sufficiently large. The main steps of the proof concern finite dimension approximation of the system and concentrate on analysis of features of large matrices, which resembles standard numerical analysis. Our analytical results are illustrated by numerical simulations.
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We study the existence of multi-bubble solutions for the following skew-symmetric Chern--Simons system begin{equation}label{e051} left{ begin{split} &Delta u_1+frac{1}{varepsilon^2}e^{u_2}(1-e^{u_1})=4pisum_{i=1}^{2k}delta_{p_{1,i}} &Delta u_2+frac{1}{varepsilon^2}e^{u_1}(1-e^{u_2})=4pisum_{i=1}^{2k}delta_{p_{2,i}} end{split} text{ in }quad Omegaright., end{equation} where $kgeq 1$ and $Omega$ is a flat tours in $mathbb{R}^2$. It continues the joint work with Zhangcite{HZ-2015}, where we obtained the necessary conditions for the existence of bubbling solutions of Liouville type. Under nearly necessary conditions(see Theorem ref{main-thm}), we show that there exist a sequence of solutions $(u_{1,varepsilon}, u_{2,varepsilon})$ to eqref{e051} such that $u_{1,varepsilon}$ and $u_{2,varepsilon}$ blow up simultaneously at $k$ points in $Omega$ as $varepsilonto 0$.
We prove the existence of positive solutions to a sys- tem of k non-linear elliptic equations corresponding to standing- wave k-uples solutions to a system of non-linear Klein-Gordon equations. Our solutions are characterised by a small energy/charge ratio, appropriately defined.
In this paper we find viscosity solutions to a coupled system composed by two equations, the first one is parabolic and driven by the infinity Laplacian while the second one is elliptic and involves the usual Laplacian. We prove that there is a two-player zero-sum game played in two different boards with different rules in each board (in the first one we play a Tug-of-War game taking the number of plays into consideration and in the second board we move at random) whose value functions converge uniformly to a viscosity solution to the PDE system.
We prove the existence of infinitely many nonnegative solutions to the following nonlocal elliptic partial differential equation involving singularities begin{align} (-Delta)_{p(cdot)}^{s} u&=frac{lambda}{|u|^{gamma(x)-1}u}+f(x,u)~text{in}~Omega, onumber u&=0~text{in}~mathbb{R}^NsetminusOmega, onumber end{align} where $Omegasubsetmathbb{R}^N,, Ngeq2$ is a smooth, bounded domain, $lambda>0$, $sin (0,1)$, $gamma(x)in(0,1)$ for all $xinbar{Omega}$, $N>sp(x,y)$ for all $(x,y)inbar{Omega}timesbar{Omega}$ and $(-Delta)_{p(cdot)}^{s}$ is the fractional $p(cdot)$-Laplacian operator with variable exponent. The nonlinear function $f$ satisfies certain growth conditions. Moreover, we establish a uniform $L^{infty}(bar{Omega})$ estimate of the solution(s) by the Moser iteration technique.
We use blow up analysis for local integral equations to prove compactness of solutions to higher order critical elliptic equations provided the potentials only have non-degenerate zeros. Secondly, corresponding to Schoens Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow up points of solutions.
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