文摘
Given a smooth bounded domain \(\Omega \) of \(\mathbb {R}^n\) and consider the problem $$\begin{aligned} \left\{ \begin{array} {ll} - \Delta u = |u|^p - t\,\psi &{}\quad \hbox { in } \ \Omega \\ u = 0 &{}\quad \hbox { on }\ \partial \Omega \end{array}\right. \end{aligned}$$ (0.1)where t is a large positive parameter, \(p>1\) and \(\psi \) is an eigenfunction of \(- \Delta \) with Dirichlet boundary condition corresponding to the first eigenvalue \(\lambda _1\). Assuming that \(\Omega \) contains a k-dimensional compact submanifold K which is stationary and non-degenerate for the weighted functional $$\begin{aligned} \int _K \psi ^{(1 - \frac{1}{p})(\frac{p+1}{p-1}-\frac{n-k}{2})}dvol \end{aligned}$$such that \(\mathrm{dist}(K,\partial \Omega )>\delta _0>0\), then for \(1< p < \frac{n+2-k}{n-2-k}\) we prove the existence of a sequence \(t=t_j\rightarrow \infty \) and solutions \(u_t\) that concentrate along K. This result proves in particular the validity of a conjecture by Hollman–Mckenna in full generality, see Hollman and McKenna (Commun Pure Appl Anal 10(2):785–802, 2011), extending the result by Manna and Santra (Discrete Contin Dyn Syst 36(10):5595–5626, 2016) where the case \(n=2\) and \(k=1\) has been considered.Mathematics Subject Classification35J2535J2035B3335B40Communicated by A. Malchiodi.