Maximizers of a Trudinger-Moser-type inequality with the critical growth on the whole plane (Analysis on Shapes of Solutions to Partial Differential Equations)

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(1)89. 数理解析研究所講究録第2046巻 2017年 89-91. Maximizers of a nudinger‐Moser‐type critical growth on the whole plane. inequality. with the. Kenji Nakanishi Department of Pure and Applied Mathematics Technology Osaka University. Graduate School of Information Science and. In this. I. talk,. reported. joint work with Slim Ibrahim, Trudinger‐Moser inequality on \mathbb{R} 2:. recent progress in the. Masmoudi and Feberica Sani. The. Nader. \displaystyle\sup_{u\inH_{0}^{1}($\Omega$),|\nabla\mathrm{u}\Vert_{2}^{2}\leq2$\pi$}\int_{$\Omega$}e^{2u^{2}dx\leqC|$\Omega$|. (1). embedding H1( \mathbb{R} 2) \not\subset L ( \mathbb{R} 2) for bounded domains. Henceforth \Vert $\varphi$\Vert_{p} denotes the U_{x} norm. Moser [6] proved the critical case \Vert\nabla u\Vert_{2}^{2}=2 $\pi$ as well as the optimality of the growth e^{2u^{2}} as u\rightarrow\infty, and loss of compactness by concentration for some bounded sequences in H1( \mathbb{R} 2) Carleson and Chang [2] proved however that the concentration cannot occur for maximizing sequences of the inequality, hence existence of maximizers. We consider a similar question on the whole plane \mathbb{R} 2. Cao [1] proved a variant is. a. celebrated substitute for the forbidden Sobolev. \infty. ,. .. (1). of. on. \mathbb{R}. 2. in the subcritical.case:. $\alpha$<2$\pi$\displaystyle\Rightar ow\sup_{\Vert\nablau\Vert_{2}^{2}\leq$\alpha$}\int_{\mathb {R}^{2} (e^{2\mathrm{u}^{2} -1)d_{X}\leqC_{$\alpha$}\Vertu\Vert_{2}^{2} It is indeed. a. natural formulation for the nonlinear. (2). .. Schrödinger equation. i\dot{u}- $\Delta$ u=f'(u) similar Klein‐Gordon. or. equation. on. \mathbb{R}. 1+. 2,. (3). where L2‐invariance. plays. crucial roles. It is easy to observe by Mosers sequence of functions [6] that the critical exponent $\alpha$= 2 $\pi$ is prohibited on \mathbb{R} 2. Motivated by the dynamical in the. dynamics,. problems,. the. finite if. [3].. optimal. Theorem 1. Let. is. cf.. and. version of. f:\mathbb{R}\rightarrow[0, \infty). (2). was. obtained in. be any continuous. [5]:. in. of. f. ,. is. functions u_{n} satisfying \Vert\nabla u_{n}\Vert_{2}^{2} \leq 2 $\pi$ and weakly strongly convergent in L^{1}(\mathbb{R}^{2}) if and only if ,. S(f). attained, Schrödinger equation:. is smooth and. static nonlinear. (5). .. radial. H^{1}(\mathbb{R}^{2}) f(u_{n}). \displaystyle\lim_{|\mathrm{u}|\rightar ow\infty}|f(u)|\frac{|u^{2} {e^{2|u^{2} +\lim_{|u\rightar ow0}\frac{f(u)}{|u^{2} =0 If. (4). only if. For every sequence. convergent. (i). S(f):=\displaystyle\sup_{\Vert\nabla\mathrm{u}\Vert_{2}^{2}\leq2$\pi$}\Vertu\Vert_{2}^{-2}\int_{\mathb {R}^{2} f(u)dx \displaystyle\lim_{|u\rightar ow}\sup_{\infty}f(u)\frac{|u^{2}{e^{2|u^{2} +\mathrm{h}\mathrm{m}\sup_{|\mathrm{u}|\rightar ow0}\frac{f(u)}{|u^{2}<\infty. (ii). Then. function.. is. then each maximizer. - $\Delta$ u+ $\omega$ u=f^{f}(u). (6). .. u\in H^{1}(\mathbb{R}^{2}). solves the. (7).

(2) 90. with. $\omega$. 2S(f). =. The above compactness (ii), implies that S(f) is attained if (6) decay for |u| \rightarrow 0 can generally be assumed after subtracting. .. holds. Note that the the limit. mass:. f(u)-u^{2}\displaystyle \lu\rightarrow im\under0linu^{2}e{f(u)} at least for smooth. f. compactness issue in critical one,. .. Hence the. our. namely. (8). . vanishing (or spreading) phenomenon. setting. On the other hand, if f(u). \displaystyle \foral $\epsilon$>0, f(u)=o(e^{(2+ $\epsilon$)|u|^{2} ) , \lim_{|u|\rightar ow}\sup_{\infty}f(u)\frac{|u|^{2} {e^{2|u|^{2} =\infty then for every energy. 0< $\omega$<\infty,. \Vert\nabla u\Vert_{2}^{2}<2 $\pi$. essentially. the. hence. ,. $\omega$=2\displaystyle \Vert u\Vert_{2}^{-2}\int f(u)dx is reached with. (7). has. only remaining. a. case. right. subcritical kinetic result. Then. (10). .. hand side is. of maximizers, which differs from the state it more precisely, let f_{h}^{*} : \mathbb{R} \rightarrow. parameter h>0 , defined. by. a sharp threshold also for the existence original Trudinger‐Moser inequality (1). To. [0, \infty ). be the model function with. f_{h}^{*}(u)=\left\{ begin{ar ay}{l} |u^{-2}e^{2u^{2} &(|u>h),\ 0&(|u\leqh). \end{ar ay}\right. Theorem 2. Let $\delta$>0 and let. f:\mathbb{R}\rightarrow[0, \infty). be. a. continuous. is attained. if f satisfies, for. is not attained. if f satisfies, for. function satisfying. \dot{u} attained. if f satisfies, for. some. is not attained. (13). some. words,. inequality (1). .. .. nonlinearity f_{h}^{*}(u). .. Note that for the. the existence of maximizer is. (15). (16). by the sign of small order perturbations original Trudinger‐Moser stable for small perturbations.. the existence is determined. around the critical. (14). sufficiently large h\gg 1,. f(u)\leq f_{h}^{*}(u)+mu^{2}- $\delta$ u^{4} In other. .. h>0,. some. if f satisfies, for. (12). sufficiently large h\gg 1,. f(u)\geq f_{h}^{*}(u)+mu^{2}+ $\delta$ u^{4} (iv) S(f). .. .. f(u)\leq f_{h}^{*}(u)(1- $\delta$ u^{-2})+mu^{2} (iii) S(f). cut‐off. h>0,. some. f(u)\geq f_{h}^{*}(u)(1+ $\delta$ u^{-2}) (ii) S(f). a. (11). \displaystyle \lim_{|u|\rightarrow\infty}f(u)/f_{1}^{*}(u)=1, \lim_{|u|\rightarrow 0}f(u)/u^{2}=m\in[0, \infty) (i) S(f). a. (9). ,. by the above compactness exactly critical growth:. f(u)\displaystyle \sim\frac{e^{2|u|^{2} {|u|^{2} (|u|\rightar ow\infty) It turned out that the. a. solution. is the. is not. grows faster than the.

(3) 91. Our. proof of Theorem. follows.. as. inspired by [2], based on asymptotic expansion of S(f) precisely, we introduce another height parameter H>0 radially symmetric decreasing u\in H^{1}(\mathbb{R}^{2}) let. 2 is. concentration. More. along. X_{H}. Restricting. to. ,. :=\{u|\exists R>0, \Vert\nabla u\Vert_{L^{2}(|x|<R)}^{2}\leq $\pi$, \Vert\nabla u\Vert_{L^{2}(|x|>R)}^{2}\leq $\pi$, u(R)=H\},. S_{H}(f). :=\displaystyle \sup_{u\in X_{H} \Vert u\Vert_{2}^{-2}\int_{\mathb {R}^{2} f(u)dx. Theorem 2 follows from the. (17). .. asymptotic expansion. h\rightarrow\infty of. as. S_{h}(f_{h}^{*})=2e^{- $\gamma$}+O(h^{-4}). (18). ,. where $\gamma$ denotes Eulers constant. The vanishing of the order O(h^{-2}) is the special property of f_{h}^{*} (besides the critical growth for Theorem 1), which allows the small order. perturbations to determine the existence in Theorem 2. asymptotic expansion of an exponential radial Sobolev inequality:. We also need. \displaystyle \inf\{\Vert u\Vert_{L^{2}(|x|>1)}^{2};u(1)=h, \Vert\nabla u\Vert_{L^{2}(|x|>1)}^{2}\leq 2 $\pi$\} whose. S(f). growth. order. < \infty can. =\displaystyle \frac{ $\pi$ e^{2h^{2}+ $\gamma$-1} {4h^{2} \{1+\frac{1}{2h^{2} +O(h^{-4})\}. was. obtained in. noteworthy that original Trudinger‐Moser inequality. (1) using be derived from Theorem 1 using the Hardy‐type. be derived from the. the above. inequality, inequality:. while. (1). can. [5]. (19) ,. to prove Theorem 1. It is. 0<r<R \Rightar ow |u(r)|-|u(R)|\leq \sqrt{\frac{1}{2 $\pi$}\Vert\nabla u\Vert_{L^{2}(|x|<R)}^{2}\log(R/r)}. At the time of the of. S_{H}(f_{H}^{*}). conference,. with that of. S_{H}(f_{h}^{*}). we were. missing. a. logic. to connect the. for fixed h \gg 1 and H. \rightarrow \infty ,. which. .. (20). asymptotics requires us. to control the tail left behind the concentration.. Using carefully the above two inequalities, however, we are now able to prove that maximizers of S_{H}(f_{h}^{*}) should almost optimize the height H in the exterior region |x| >R and the nonlinear energy f_{H}^{*}(u) in the interior region |x| <R. A full proof will be published elsewhere for Theorem 2, as well as for (19). REFERENCES. [1]. [2] [3] [4] [5] [6]. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R^{2}, Equations 17 (1992), no. 3‐4, 407‐435. L. Carleson and A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sc. Math. 110 (1986), 113‐127. S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein‐Gordon equation, Anal. PDE 4 (2011), no. 3, 405‐460. S. Ibrahim, N. Masmoudi and K. Nakanishi, Correction to Scattering threshold for the focus‐ ing nonlinear Klein‐Gordon equahon Anal. PDE 9 (2016), no. 2, 503‐514. S. Ibrahim, N. Masmoudi and K. Nakanishi, Trudinger‐Moser inequality on the whole plane with the exact growth condition. J. Eur. Math. Soc. 17 (2015), no. 4, 819‐835. J. Moser, A sharp form of an inequality of N. Trudinger, Ind. Univ. Math. J. 20 (1971), D. M.. Comm. Partial Differential. 1077‐1092..

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