The Lin-ni's Problem For Mean Convex Domains (memoirs Of The American Mathematical Society)

The authors prove some refined asymptotic estimates for positive blow-up solutions to $\Delta u+\epsilon u=n(n-2)u^{\frac{n+2}{n-2}}$ on $\Omega$, $\partial\_\nu u=0$ on $\partial\Omega$, $\Omega$ being a smooth bounded domain of $\mathbb{R}^n$, $n\geq 3$. In particular, they show that concentration can occur only on boundary points with nonpositive mean curvature when $n=3$ or $n\geq 7$. As a direct consequence, they prove the validity of the Lin-Ni's conjecture in dimension $n=3$ and $n\geq 7$ for mean convex domains and with bounded energy. Recent examples by Wang-Wei-Yan show that the bound on the energy is a necessary condition Introduction 8 Chapter 1. L-bounded solutions 12 Chapter 2. Smooth domains and extensions of solutions to elliptic equations 14 Chapter 3. Exhaustion of the concentration points 18 Chapter 4. A first upper-estimate 26 Chapter 5. A sharp upper-estimate 34 Chapter 6. Asymptotic estimates in C1() 50 Chapter 7. Convergence to singular harmonic functions 52 1. Convergence at general scale 52 2. Convergence at appropriate scale 58 Chapter 8. Estimates of the interior blow-up rates 64 Chapter 9. Estimates of the boundary blow-up rates 76 Chapter 10. Proof of Theorems 1 and 2 88 Appendix A. Construction and estimates on the Green's function 90 Appendix B. Projection of the test functions 104 Bibliography 110