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Register a new userExistence results for a generalized mean field equation on a closed Riemann surface
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Let $Sigma$ be a closed Riemann surface, $h$ a positive smooth function on $Sigma$, $rho$ and $alpha$ real numbers. In this paper, we study a generalized mean field equation begin{align*} -Delta u=rholeft(dfrac{he^u}{int_Sigma he^u}-dfrac{1}{mathrm{Area}left(Sigmaright)}right)+alphaleft(u-fint_{Sigma}uright), end{align*} where $Delta$ denotes the Laplace-Beltrami operator. We first derive a uniform bound for solutions when $rhoin (8kpi, 8(k+1)pi)$ for some non-negative integer number $kin mathbb{N}$ and $alpha otinmathrm{Spec}left(-Deltaright)setminusset{0}$. Then we obtain existence results for $alpha<lambda_1left(Sigmaright)$ by using the Leray-Schauder degree theory and the minimax method, where $lambda_1left(Sigmaright)$ is the first positive eigenvalue for $-Delta$.
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Based on a recent work of Mancini-Thizy [28], we obtain the nonexistence of extremals for an inequality of Adimurthi-Druet [1] on a closed Riemann surface $(Sigma,g)$. Precisely, if $lambda_1(Sigma)$ is the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition, then there exists a positive real number $alpha^ast<lambda_1(Sigma)$ such that for all $alphain (alpha^ast,lambda_1(Sigma))$, the supremum $$sup_{uin W^{1,2}(Sigma,g),,int_Sigma udv_g=0,,| abla_gu|_2leq 1}int_Sigma exp(4pi u^2(1+alpha|u|_2^2))dv_g$$ can not be attained by any $uin W^{1,2}(Sigma,g)$ with $int_Sigma udv_g=0$ and $| abla_gu|_2leq 1$, where $W^{1,2}(Sigma,g)$ denotes the usual Sobolev space and $|cdot|_2=(int_Sigma|cdot|^2dv_g)^{1/2}$ denotes the $L^2(Sigma,g)$-norm. This complements our earlier result in [39].
Let $(Sigma,g)$ be a closed Riemannian surface, $textbf{G}={sigma_1,cdots,sigma_N}$ be an isometric group acting on it. Denote a positive integer $ell=inf_{xinSigma}I(x)$, where $I(x)$ is the number of all distinct points of the set ${sigma_1(x),cdots,sigma_N(x)}$. A sufficient condition for existence of solutions to the mean field equation $$Delta_g u=8piellleft(frac{he^u}{int_Sigma he^udv_g}-frac{1}{{rm Vol}_g(Sigma)}right)$$ is given. This recovers results of Ding-Jost-Li-Wang (Asian J Math 1997) when $ell=1$ or equivalently $textbf{G}={Id}$, where $Id$ is the identity map.
In this note, we study symmetry of solutions of the elliptic equation begin{equation*} -Delta _{mathbb{S}^{2}}u+3=e^{2u} hbox{on} mathbb{S}^{2}, end{equation*} that arises in the study of rigidity problem of Hawking mass in general relativity. We provide various conditions under which this equation has only constant solutions, and consequently imply the rigidity of Hawking mass for stable constant mean curvature (CMC) sphere.
We are concerned with existence results for a critical problem of Brezis-Nirenberg Type involving an integro-differential operator. Our study includes the fractional Laplacian. Our approach still applies when adding small singular terms. It hinges on appropriate choices of parameters in the mountain-pass theorem
Let $(Sigma,g)$ be a closed Riemannian surface, $W^{1,2}(Sigma,g)$ be the usual Sobolev space, $textbf{G}$ be a finite isometric group acting on $(Sigma,g)$, and $mathscr{H}_textbf{G}$ be a function space including all functions $uin W^{1,2}(Sigma,g)$ with $int_Sigma udv_g=0$ and $u(sigma(x))=u(x)$ for all $sigmain textbf{G}$ and all $xinSigma$. Denote the number of distinct points of the set ${sigma(x): sigmain textbf{G}}$ by $I(x)$ and $ell=inf_{xin Sigma}I(x)$. Let $lambda_1^textbf{G}$ be the first eigenvalue of the Laplace-Beltrami operator on the space $mathscr{H}_textbf{G}$. Using blow-up analysis, we prove that if $alpha<lambda_1^textbf{G}$ and $betaleq 4piell$, then there holds $$sup_{uinmathscr{H}_textbf{G},,int_Sigma| abla_gu|^2dv_g-alpha int_Sigma u^2dv_gleq 1}int_Sigma e^{beta u^2}dv_g<infty;$$ if $alpha<lambda_1^textbf{G}$ and $beta>4piell$, or $alphageq lambda_1^textbf{G}$ and $beta>0$, then the above supremum is infinity; if $alpha<lambda_1^textbf{G}$ and $betaleq 4piell$, then the above supremum can be attained. Moreover, similar inequalities involving higher order eigenvalues are obtained. Our results partially improve original inequalities of J. Moser cite{Moser}, L. Fontana cite{Fontana} and W. Chen cite{Chen-90}.
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