Bubbling on boundary submanifolds for the Lin-Ni-Takagi problem at higher critical exponents
In this paper, the problem NEWLINE\[NEWLINE \begin{cases} d^2 \Delta u -u + u^{\frac{n-k+2}{n-k-2}} = 0 & \text{in} \;\Omega, \\ \frac{\partial u}{\partial \nu} = 0 & \text{on} \;\partial\Omega \end{cases}NEWLINE\]NEWLINE is considered, where \(\Omega \subset {\mathbb R}^n\) is a bounded domain with a smooth boundary \(\partial \Omega\) and \(d\) is a small positive parameter. It is assumed that there exists a \(k\)-dimensional closed, embedded minimal submanifold \(K\) of \(\partial \Omega\) which is nondegenerate, and a certain weighted average of sectional curvatures of \(\partial \Omega\) is positive along \(K\). Then it is shown the existence of a sequence \(d=d_j \to 0\) and a positive solution \(u_d\) such that NEWLINE\[NEWLINE d^2 |\Delta u_d|^2 \rightharpoonup S\delta_k \quad \text{as} \;d \to 0 NEWLINE\]NEWLINE in the sense of measures, where \(\delta_K\) stands for the Dirac measure supported \(K\) and \(S\) is a positive constant.
scientific article; zbMATH DE number 6363151
scientific article