Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent
The authors interest is directed to the existence and behavior of multi-peak solutions of the semilinear Neumann problem with the critical Sobolev exponent \[ d\Delta u- u+ u^{2^*}= 0,\quad u>0\quad\text{in }\Omega,\quad \partial u/\partial\nu= 0\quad\text{on }\partial\Omega, \] which, through a proper scaling, is changed to the equivalent problem \[ \Delta u-\lambda u+ u^{2^*}= 0,\quad u>0\quad\text{in }\Omega,\quad \partial u/\partial\nu= 0\quad\text{on }\partial\Omega,\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^n\), \(\nu\) is the unit outer normal to \(\partial\Omega\), \(2^*= (n+ 2)/(n- 2)\), \(d>0\) is a small parameter and \(\lambda= 1/d\). After introducing the mean curvature \(H(y)\) of \(\partial\Omega\) at \(y\in \partial\Omega\), and functions \(U(x)\), \(U_{\varepsilon, y}(x)\) with \(\varepsilon> 0\) such as \[ U(x)= \Biggl({n(n- 2)\over n(n- 2)+|x|^2}\Biggr)^{{n- 2\over 2}},\;U_{\varepsilon, y}(x)= \varepsilon^{-{n- 2\over 2}} U\Biggl({x- y\over \varepsilon}\Biggr),\;x,y\in \mathbb{R}^n \] and the norm \[ \|u\|_\lambda= \int_\Omega (|\nabla u|^2+ \lambda|u|^2) dx,\quad u\in H^1(\Omega), \] the main result is stated as follows: In the case \(n\geq 5\), if there exist \(k\) disjoint smooth patches, i.e. the closure of a connected open set on \(\partial\Omega\) with a smooth boundary \(\Lambda_i\), \(i= 1,2,\dots, k\) such that \(H(y)\geq 0\) for \(y\in\Lambda_i\) and \[ M_i:= \max_{y\in\Lambda_i} H(y)> m_i:= \min_{y\in \partial\Lambda_i} H(y),\quad i= 1,2,\dots, k, \] then for \(\lambda\) sufficiently large, there exist a classical positive solution \(u_\lambda\) of (1), \(\varepsilon_{\lambda, i}\) and \(y_{\lambda,i}\in \Lambda_i\), \(i= 1,2,\dots, k\), such that \[ \Biggl\|u_\lambda- \sum^k_{i=1} U_{\varepsilon_{\lambda,i},y_{\lambda, i}}\Biggr\|_\lambda\to 0,\quad \varepsilon_{\lambda, i}= q_n M_i{1\over\lambda}+ o\Biggl({1\over \lambda}\Biggr)\quad\text{as }\lambda\to \infty, \] where \(q_n\) is a positive constant depending only on \(n\).
scientific article; zbMATH DE number 1336834