Interior bubbling solutions for the critical Lin-Ni-Takagi problem in dimension 3

Abstract

We consider the problem of finding positive solutions of the problem $\Delta u-\lambda u+u^5=0$ in a bounded, smooth domain $\Omega$ in ${\mathbb{R}}^3$, under zero Neumann boundary conditions. Here $\lambda$ is a positive number. We analyze the role of Green's function of $-\Delta +\lambda$ in the presence of solutions exhibiting single bubbling behavior at one point of the domain when $\lambda$ is regarded as a parameter. As a special case of our results, we find and characterize a positive value ${\lambda }_{\ast }$ such that if $\lambda -{\lambda }^{\ast }>0$ is sufficiently small, then this problem is solvable by a solution $u_{\lambda }$ which blows-up by bubbling at a certain interior point of $\Omega$ as $\lambda \downarrow {\lambda }_{\ast }$.

26 pages, 1 figure

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