Nondegeneracy properties and uniqueness of positive solutions to a class of fractional semilinear equations

Abstract

With the help of a new Picone-type identity, we prove that positive solutions $u\in H^s({\mathbb{R}}^N)$ to the equation $(-\Delta {)}^{s}u+u=u^p$ in ${\mathbb{R}}^N$ are nonradially nondegenerate, for all $s\in (0,1)$, $N\ge 1$ and $p>1$ strictly smaller than the critical Sobolev exponent. By this we mean that the linearized equation $(-\Delta {)}^{s}w+w-p{u}^{p-1}w=0$ does not admit nonradial solutions besides the directional derivatives of $u$. Letting $B$ be the unit centered ball and ${\lambda }_1(B)$ the first Dirichlet eigenvalue of the fractional Laplacian $(-\Delta {)}^s$, we also prove that positive solutions to $(-\Delta {)}^{s}u+\lambda u=u^p$ in $B$, with $u=0$ on ${\mathbb{R}}^N\setminus B$, are nonradially nondegenerate for any $\lambda >-{\lambda }_1(B)$ in the sense that the linearized equation does not admit nonradial solutions. From these results, we then deduce uniqueness and full nondegeneracy of positive solutions in some special cases. In particular, in the case $N=1$, we prove that the equation $(-\Delta {)}^{s}u+u=u^2$ in $\mathbb{R}$ or in $B$, with zero exterior data, admits a unique even solution which is fully nondegenerate in the optimal range $s\in (\frac{1}{6},1)$, thus extending the classical uniqueness result of Amick and Toland on the Benjamin–Ono equation. Moreover, in the case $N=1$, $\lambda =0$, we also prove the uniqueness and full nondegeneracy of positive solutions for the Dirichlet problem in $B$ with arbitrary subcritical exponent $p$. Finally, we determine the unique positive ground state solution of $(-\Delta {)}^{\frac{1}{2}}u+u=u^p$ in ${\mathbb{R}}^N$, $N\ge 1$ with $p=1+\frac{2}{N+1}$and compute the sharp constant in the associated Gagliardo–Nirenberg inequality $\Vert u{\Vert }_{L^{p+1}({\mathbb{R}}^N)}\le C\Vert (-\Delta {)}^{\frac{1}{4}}u{\Vert }_{L^2({\mathbb{R}}^N)}^{\frac{N}{N+2}\Vert u{\Vert }_{L^2({\mathbb{R}}^N)}^{\frac{2}{N+2}}}$.