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

We consider the equation $d^2\Delta u - u+ u^{\frac{n-k+2}{n-k-2}} =0,\hbox{in}\Omega $,underzeroNeumannboundaryconditions,where$\Omega$isopen,smoothandboundedand$d$ is a small positive parameter. We assume that there is a $k$-dimensional closed, embedded minimal submanifold $K$ of $\mathrm{\partial }\mathrm{\Omega }$, which is non-degenerate, and certain weighted average of sectional curvatures of $\mathrm{\partial }\mathrm{\Omega }$ 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 $$d^2\mathrm{\mid }\mathrm{\nabla }{u}_d{\mathrm{\mid }}^2\rightharpoonup S,{\delta }_K\text{ass}d\to 0$$ in the sense of measures, where ${\delta }_K$ stands for the Dirac measure supported on $K$ and $S$ is a positive constant.