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

In this paper, we study then early critical Lane-Emden equations { -Delta u=up-epsilon in Omega, u>0 in Omega, u=0 on partial derivative Omega,where Omega subset of R(N)with N >= 3,p=N+2N-2and epsilon>0 is small.Our main result is that when Omega is a smooth bounded convex domain and the Robin function on Omega is a Morse function, then for small epsilon the equation(& lowast;) has a unique solution, which is also non degenerate.As for non-convex domain, we also obtain exact number of solutions to(& lowast;) under some conditions .In general, the solutions of (& lowast;) may blow-up at multiple points a1,,a k of Omega as epsilon -> 0.In particular, when Omega 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 non degenerate. Combining these conclusions, we finally obtain the uniqueness, multiplicity and nondegeneracy of solutions to(& lowast;) (c) 2023 Elsevier Masson SAS. All rights reserved.

Automated summary

In this paper, we study the properties of solutions to the early critical Lane-Emden equations in both convex and non-convex domains. We show that in a smooth bounded convex domain with a Morse Robin function, the equation has a unique non-degenerate solution for small epsilon. In non-convex domains, we obtain the exact number of solutions under certain conditions. The solutions of the equation may blow-up at multiple points, but in convex domains, there is only one blow-up point.