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Register a new userMean field equations on a closed Riemannian surface with the action of an isometric group
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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.
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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}.
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 $-Delta$.
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].
We investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in $Mtimes mathbb R_+$, where $M$ is a complete noncompact Riemannian manifold. Under specific assumptions, we establish existence of solutions satisfying prescribed conditions at infinity, depending on the direction along which infinity is approached. Moreover, the large-time behavior of such solutions is studied. We consider also elliptic equations on $M$ with similar conditions at infinity.
In this paper we provide a characterization of second order fully nonlinear CR invariant equations on the Heisenberg group, which is the analogue in the CR setting of the result proved in the Euclidean setting by A. Li and the first author (2003). We also prove a comparison principle for solutions of second order fully nonlinear CR invariant equations defined on bounded domains of the Heisenberg group and a comparison principle for solutions of a family of second order fully nonlinear equations on a punctured ball.
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