Concentration of low energy extremals
The authors investigate the asymptotic behaviour of solutions and approximate solutions \(u_\varepsilon\) of the variational problem \[ \sup\Big\{\int_\Omega F(u) : \int_\Omega |\nabla u|^2 \leq \varepsilon^2, \;\;u=0 \;\;on \;\;\partial\Omega\Big\} \] in the limit \(\varepsilon \rightarrow 0\). The integrand \(F\) (that can be nonconvex and discontinuous) is supposed to satisfy \(0\leq F(t)\leq c|t|^{2n/(n-2)}\) on a domain of dimension \(n\geq 3\). For smooth \(F\) the extremals \(u_\varepsilon\) satisfy the corresponding Euler-Lagrange equation \(-\Delta u_\varepsilon=\lambda_\varepsilon f(u_\varepsilon)\) in \(\Omega, u_\varepsilon=0\) on \(\partial\Omega\) with \(f=F'\). The main result says 1) The extremals \((u_\varepsilon)\) concentrate at a single point \(x_0 \in \overline{\Omega}\). 2) On a microscopic scale near the concentration point they tend to an entire extremal, i.e. to a solution of the variational problem on \({\mathbb R}^n\) with \(\varepsilon = 1\). These results extend some work by P. L. Lions, using a new proof based on a local generalized Sobolev inequality. The identification of the concentration point \(x_0\) will be discussed in a forthcoming paper.
scientific article; zbMATH DE number 1287843