The effect of the graph topology on a semilinear elliptic equation with critical exponent
The authors consider the elliptic problem \[ -\Delta u=Q(x)u^{2^*-1}+\varepsilon u\text{ in }\Omega,\;u>0 \text{ in }\Omega,\;u=0\text{ on } \partial \Omega, \] where \(\varepsilon\geq 0\), \(2^*\) is the critical Sobolev exponent, \(\Omega\) is bounded in \(\mathbb{R}^N\), \(N\geq 4\), with a smooth boundary, and \(Q\) is a smooth positive function in \(\overline\Omega\). From Pohozaev identity it follows that this problem has no solution for \(\varepsilon=0\) if \(\Omega\) is starshaped and \((DQ(x),x)\leq 0\). For sufficiently small \(\varepsilon >0\) the authors construct a solution which concentrates at a local maximum of \(Q\) as \(\varepsilon\to 0\) and give a lower bound for the number of such solutions, using the category of the set \(M\) where \(Q\) attains its local maxima. Moreover, let \(N\geq 5\), \(Q_{\max}=\max_{x\in \Omega}Q(x)\), and \(t\in(2^{-{2\over N-2}} Q_{\max}, Q_{\max})\) such that \(\max_{x\in\partial \Omega}Q(x)<t\) and \(M\) is not contractible in a small neighborhood of itself but in \(\{x:Q(x)\geq t\}\). Suppose each critical point \(x\) of \(Q\) with \(Q_{\max} >Q(x)\geq t\) satisfies \(\Delta Q(x)>0\). The authors then prove the existence of a solution converging (up to a subsequence) strongly in \(H^1(\Omega)\) as \(\varepsilon\to 0\).
scientific article; zbMATH DE number 1279182
scientific article