A concentration phenomenon for semilinear elliptic equations. (English) Zbl 1266.35074

In the very interesting paper under review the authors consider the equation \[ -\Delta u+V(x)u=Q_n(x)|{u}|^{p-2}u \] in a domain \(\Omega \subset \mathbb R^N\) with zero Dirichlet boundary conditions and where \(p\in(2,2^\ast).\) It is assumed that \(V\geqq 0\) and \(Q_n\) are bounded functions which are positive in a sub-region of \(\Omega\) and negative outside, and such that the sets \(\{Q_n>0\}\) shrink to a point \(x_0\in\Omega\) as \(n\to \infty.\) The main result of the paper states that if \(u_n\) is a nontrivial solution corresponding to \(Q_n,\) then the sequence \((u_n)\) concentrates at \(x_0\) with respect to the \(H^1\) and certain \(L^q\)-norms. It is also shown that if the sets \(\{Q_n>0\}\) shrink to two points and \(u_n\) are ground state solutions, then they concentrate at one of these points.


35J61 Semilinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
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