{"id":1149,"date":"2011-12-09T20:57:00","date_gmt":"2011-12-09T12:57:00","guid":{"rendered":"http:\/\/vanabel.sinaapp.com\/?p=1149"},"modified":"2011-12-09T20:57:00","modified_gmt":"2011-12-09T12:57:00","slug":"pohozaves-equality-and-non-existence-results","status":"publish","type":"post","link":"https:\/\/lttt.vanabel.cn\/?p=1149","title":{"rendered":"Pohozave’s equality and non-existence results"},"content":{"rendered":"

This lecture note is totally based on professor Guo Yuxia’s lecture of\u00a0 non-linear functional analysis<\/em>.<\/p>\n

Let $\\Omega$ be a bounded domain in $R^{N}$, $N\\geq 3$,we consider the following N-B problem<\/strong>:
\n\\[\\left\\{\\begin{array}{rll}
\n-\\Delta u+\\lambda u&=u|u|^{p-2},&\\quad\\text{in }\\Omega;\\\\
\nu&>0,&\\quad\\text{in }\\Omega;\u00a0\\\\
\nu&=0,&\\quad\\text{on }\\partial \\Omega.\\tag{A}
\n\\end{array}\\right.
\n\\]
\nwhere $p=2^\\star=\\frac{2N}{N-2}$.
\n
\n\\begin{thm}
\nSuppose $\\Omega \\neq R^{N}$ is smooth (possibly unbounded) domain in $R^{N}$,$N\\geq 3$, which is strictly star shaped with respect to the origin in $R^{N}$,$\\lambda \\leq 0$,then any solution $u\\in H_{0}^{1}(\\Omega)$ of the problem (A) vanishes.
\n\\end{thm}
\n\\begin{proof}
\nFirstly, we need the following
\n\\begin{lem}
\nLet g:$R\\rightarrow R$ be continuous with primitive $G(u)=\\int_{0}^{u}g(v)dv$, $u \\in C^{2}(\\Omega)\\cap C^{1}(\\Omega)$ be a solution of the problem:
\n\\[\\left\\{\\begin{array}{rll}
\n -\\Delta u&=g(u), &\\quad\\text{in }\\Omega;\\\\
\n u&=0,&\\quad \\text{on }\\partial \\Omega;
\n\\end{array}\\right.\\]
\nin a domain in $R^{N}(N\\geq 3)$, then there holds
\n\\[
\n\\frac{N-2}{2}\\int_{\\Omega}|\\nabla u|^{2}dx-N\\int_{\\Omega}G(u)dx+\\int_{\\partial \\Omega}\\frac{1}{2}|\\frac{\\partial u}{\\partial \\nu}|^{2}x\\cdot n d\\sigma=0.
\n\\]
\n\\end{lem}
\nLet $G(u)=\\frac{\\lambda u^{2}}{2}+\\frac{|u|^{2^{\\star}}}{2^{\\star}}$, by standard elliptic discussion, any weak solution of (A) is smooth on $\\bar{\\Omega}$, hence by Lemma 2, we infer that
\n\\[
\n\\frac{N-2}{2}\\int_{\\Omega}|\\nabla u|^{2}dx-N\\int_{\\Omega}G(u)dx+\\int_{\\partial \\Omega }\\frac{1}{2}|\\frac{\\partial u}{\\partial \\nu}|^{2}x\\cdot n d\\sigma=0,
\n\\]
\ni.e.,
\n\\[
\n\\int_{\\Omega}|\\nabla u|^{2}dx-2^{\\star}\\int_{\\Omega}G(u)dx+\\int_{\\partial \\Omega}\\frac{1}{N-2}|\\frac{\\partial u}{\\partial \\nu}|^{2}x\\cdot n d\\sigma=0.
\n\\]
\nThus,
\n\\[
\n\\int_{\\Omega}(|\\nabla u|^{2}-|u|^{2^{\\star}})dx+\\frac{2^{\\star}}{2}\\lambda\\int_{\\Omega}|u|^{2}dx+\\int_{\\partial \\Omega}\\frac{1}{N-2}|\\frac{\\partial u}{\\partial \\nu}|^{2}x\\cdot n d\\sigma=0.
\n\\]
\nOn the other hand, testing the equation with $u$, we have
\n\\[
\n\\int_{\\Omega}|\\nabla u|^{2}-\\lambda u^{2}-|u|^{2^{\\star}}dx=0,
\n\\]
\nhence
\n\\[
\n2\\lambda\\int_{\\Omega}|u|^{2}dx+\\int_{\\partial \\Omega}|\\frac{\\partial u}{\\partial \\nu}|^{2}x\\cdot n d\\sigma=0.
\n\\]
\nSince $\\Omega$ is strictly star shaped with respect to $x\\cdot n>0,\\forall x \\in \\Omega
\n\\Rightarrow u=0$.\\end{proof}<\/p>\n","protected":false},"excerpt":{"rendered":"

This lecture note is totally based on pr …<\/p>\n

Pohozave’s equality and non-existence results<\/span> \u9605\u8bfb\u66f4\u591a »<\/a><\/p>\n","protected":false},"author":3,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12],"tags":[],"class_list":["post-1149","post","type-post","status-publish","format-standard","hentry","category-mathnotes"],"_links":{"self":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1149","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1149"}],"version-history":[{"count":0,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=\/wp\/v2\/posts\/1149\/revisions"}],"wp:attachment":[{"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1149"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1149"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lttt.vanabel.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1149"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}