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Register a new userA remark on energy estimates concerning extremals for Trudinger-Moser inequalities on a disc
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Yunyan Yang
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In this short note, we generalized an energy estimate due to Malchiodi-Martinazzi (J. Eur. Math. Soc. 16 (2014) 893-908) and Mancini-Martinazzi (Calc. Var. (2017) 56:94). As an application, we used it to reprove existence of extremals for Trudinger-Moser inequalities of Adimurthi-Druet type on the unit disc. Such existence problems in general cases had been considered by Yang (Trans. Amer. Math. Soc. 359 (2007) 5761-5776; J. Differential Equations 258 (2015) 3161-3193) and Lu-Yang (Discrete Contin. Dyn. Syst. 25 (2009) 963-979) by using another method.
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Though much work has been done with respect to the existence of extremals of the critical first order Trudinger-Moser inequalities in $W^{1,n}(mathbb{R}^n)$ and higher order Adams inequalities on finite domain $Omegasubset mathbb{R}^n$, whether there exists an extremal function for the critical higher order Adams inequalities on the entire space $mathbb{R}^n$ still remains open. The current paper represents the first attempt in this direction. The classical blow-up procedure cannot apply to solving the existence of critical Adams type inequality because of the absence of the P{o}lya-Szeg{o} type inequality. In this paper, we develop some new ideas and approaches based on a sharp Fourier rearrangement principle (see cite{Lenzmann}), sharp constants of the higher-order Gagliardo-Nirenberg inequalities and optimal poly-harmonic truncations to study the existence and nonexistence of the maximizers for the Adams inequalities in $mathbb{R}^4$ of the form $$ S(alpha)=sup_{|u|_{H^2}=1}int_{mathbb{R}^4}big(exp(32pi^2|u|^2)-1-alpha|u|^2big)dx,$$ where $alpha in (-infty, 32pi^2)$. We establish the existence of the threshold $alpha^{ast}$, where $alpha^{ast}geq frac{(32pi^{2})^2B_{2}}{2}$ and $B_2geq frac{1}{24pi^2}$, such that $Sleft( alpharight) $ is attained if $32pi^{2}-alpha<alpha^{ast}$, and is not attained if $32pi^{2}-alpha>alpha^{ast}$. This phenomena has not been observed before even in the case of first order Trudinger-Moser inequality. Therefore, we also establish the existence and non-existence of an extremal function for the Trudinger-Moser inequality on $mathbb{R}^2$. Furthermore, the symmetry of the extremal functions can also be deduced through the Fourier rearrangement principle.
Let $M$ be a complete, simply connected Riemannian manifold with negative curvature. We obtain some Moser-Trudinger inequalities with sharp constants on $M$.
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}.
We study existence of maximizer for the Trudinger-Moser inequality with general nonlinearity of the critical growth on $R^2$, as well as on the disk. We derive a very sharp threshold nonlinearity between the existence and the non-existence in each case, in asymptotic expansions with respect to growth and decay of the function. The expansions are explicit, using Aperys constant. We also obtain an asymptotic expansion for the exponential radial Sobolev inequality on $R^2$.
We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $displaystyle Lu:=-r^{-theta}(r^{alpha}vert u(r)vert^{beta}u(r))$, where $theta, betageq 0$ and $alpha>0$, are constants satisfying some existence conditions. It worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Polya-Szeg{o} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.
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