Bubbling on Boundary Submanifolds for the Lin-Ni-Takagi Problem at Higher Critical Exponents

TL;DR

This paper constructs solutions to a boundary value problem involving higher critical exponents that concentrate along a minimal submanifold of the boundary as a small parameter tends to zero, revealing bubbling phenomena.

Contribution

It establishes the existence of boundary-concentrating solutions for a higher critical exponent problem on domains with minimal submanifolds, extending previous bubbling results.

Findings

Solutions concentrate along the submanifold as the parameter tends to zero.

The gradient squared of solutions converges to a measure supported on the submanifold.

The solutions exhibit bubbling behavior near the boundary submanifold.

Abstract

We consider the equation $d^2\Delta u - u+ u^{\frac{n-k+2}{n-k-2}} =0\,\hbox{in}\Omega $, under zero Neumann boundary conditions, where $\Omega$ is open, smooth and bounded and $d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\partial\Omega$, which is non-degenerate, and certain weighted average of sectional curvatures of $\partial\Omega$ is positive along $K$. Then we prove the existence of a sequence $d=d_j\to 0$ and a positive solution $u_d$ such that $$ d^2 |\nabla u_{d} |^2 \rightharpoonup S, \delta_K \ass d \to 0 $$ in the sense of measures, where $\delta_K$ stands for the Dirac measure supported on $K$ and $S$ is a positive constant.