Uniqueness, multiplicity and nondegeneracy of positive solutions to the Lane-Emden problem

In this paper, we study the nearly critical Lane-Emden equations(⁎)$\{\begin{array}{ccc}& -\Delta u=u^{p-\epsilon }& \text{in}\Omega ,\\ & u>0& \text{in}\Omega ,\\ & u=0& \text{on}\partial \Omega ,\end{array}$ where $\Omega \subset R^N$ with $N\ge 3$, $p=\frac{N+2}{N-2}$ and $\epsilon >0$ is small. Our main result is that when Ω is a smooth bounded convex domain and the Robin function on Ω is a Morse function, then for small ε the equation (⁎) has a unique solution, which is also nondegenerate. As for non-convex domain, we also obtain exact number of solutions to (⁎) under some conditions.

In general, the solutions of (⁎) may blow-up at multiple points $a_1,\cdots ,a_k$ of Ω as $\epsilon \to 0$. In particular, when Ω is convex, there must be a unique blow-up point (i.e., $k=1$). In this paper, by using the local Pohozaev identities and blow-up techniques, even having multiple blow-up points (non-convex domain), we can prove that such blow-up solution is unique and nondegenerate. Combining these conclusions, we finally obtain the uniqueness, multiplicity and nondegeneracy of solutions to (⁎).