Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents.(English)Zbl 0785.35041

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 $$\mathbb{R}^ n$$.

MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs 35B40 Asymptotic behavior of solutions to PDEs
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