Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents
We study the behavior of least-energy solutions, as \(d\to 0\), of the singularly perturbed semilinear Neumann problem \[ d\Delta u-u+u^ \tau=0\text{ in }\Omega,\;u>0\text{ in }\Omega,\;{\partial u\over\partial\nu}=0\text{ on }\partial\Omega, \tag{1.1} \] where \(\Delta=\sum^ n_{j=1}\partial^ 2/\partial x^ 2_ j\) is the Laplace operator, \(\Omega\) is a bounded smooth domain in \(\mathbb{R}^ n\), \(n\geq 3\), \(v\) is the unit outer normal to \(\partial\Omega\), \(\tau=(n+2)/(n-2)\), and \(d>0\) is a constant. By a least-energy solution of (1.1) we mean a (classical) solution of (1.1) which minimizes the ``energy'' functional \[ J_ d(u)=\int\left\{{1\over 2}(d|\nabla u|^ 2+u^ 2)-{1\over\tau+1}u_ +^{\tau+1}\right\}, \] where \(u_ +=\max(u,0)\), among all the solutions of (1.1). The purpose of this paper is to investigate the behavior of the least-energy solution \(u_ d\) of (1.1) as \(d\to 0\). We establish that \(u_ d\to 0\) in \(\Omega\) as \(d\to 0\) and the (global) maximum of \(u_ d\) in \(\overline\Omega\) is assumed at exactly one point \(P_ d\) which must lie on the boundary \(\partial\Omega\). However, in contrast to the case of \(\tau<(n+2)/(n-2)\) we now have \(\| u_ d\|_{L^ \infty(\Omega)}\to\infty\) as \(d\to 0\). Moreover, after a change of variables and a suitable rescaling, the function \(u_ d/\| u_ d\|_{L^ \infty(\Omega)}\) converges in a small neighborhood of \(P_ d\) to a positive solution of \(\Delta U+U^ \tau=0\) in \(\mathbb{R}^ n\).
singularly perturbed semilinear Neumann problem
scientific article; zbMATH DE number 78550
scientific article