Concentration of solutions to elliptic equations with critical nonlinearity (Q1199280)

Let \(\Omega\) be a smooth, bounded domain in \(\mathbb{R}^ N(N>2)\), \(p:=(N+2/N-2)\) and \(\varphi\) be the regular part of Green's function for \(-\Delta\) on \(\Omega\) under homogeneous Dirichlet boundary conditions. Assume that \(f\in L^ 2(\Omega)\) allows a solution \(w\) of \(-\Delta w=f\) on \(\Omega\), \(w|\partial\Omega=0\), in \(C^ 2(\Omega)\). The author shows: Let \(x_ 0\in\Omega\) with \(w(x_ 0)>0\) be a nondegenerate critical point of \(w/\varphi^{1/2}\), then there exists a family of solutions \(u_ \eta\) of \[ -\Delta u=| u|^{p-1}u+\eta f\quad\text{on }\Omega,\quad u=0\quad\text{on }\partial\Omega \] with \(|\text{grad} u_ \eta|^ 2\to S^{N/2}\delta_{x_ 0}\), \(| u_ \eta|^{p-1}\to S^{N/2}\delta_{x_ 0}\) as \(\eta\to 0\), where convergence is meant in the sense of measures and \(S\) is the Sobolev constant. Moreover, if \(f\neq 0\) is nonnegative and \(\eta\) is sufficiently small, then one finds at least as many positive solutions as \(\text{cat}(\Omega)\), and each of these solution families concentrates at a critical point of \(w/\varphi^{1/2}\) in the way described above. A corresponding result holds for \(-\Delta u=| u|^{p-1}u\) on \(\Omega\), \(u=\eta g\) on \(\partial\Omega\).