Non-degeneracy and uniqueness of ground states to nonlinear elliptic equations with mixed local and nonlocal operators

This paper concerns the non-degeneracy and uniqueness of ground states to the following nonlinear elliptic equation with mixed local and nonlocal operators, $$-\mathrm{\Delta }u+(-\mathrm{\Delta }{)}^{s}u+\lambda u=\mathrm{\mid }u{\mathrm{\mid }}^{p-2}uinB,u=0in{\mathbb{R}}^N\mathrm{\backslash }B,$$ where $N\ge 2$, $0<s<1$, $2<p<2^{\ast }:=\frac{2N}{(N-2{)}^{+}}$, $\lambda >-{\lambda }_1$, $(-\mathrm{\Delta }{)}^s$ denotes the fractional Laplacian, ${\lambda }_1>0$ denotes the first Dirichlet eigenvalue of the operator $-\mathrm{\Delta }+(-\mathrm{\Delta }{)}^s$ in $B$ and $B$ denotes the unit ball in ${\mathbb{R}}^N$. We prove that the second eigenvalue to the linearized operator $-\mathrm{\Delta }+(-\mathrm{\Delta }{)}^s-(p-1)u^{p-2}$ in the space of radially symmetric functions is simple, the corresponding eigenfunction changes sign precisely once in the radial direction, where $u$ is a ground state. By deriving a new Hopf type lemma, we then get that $-\lambda$ cannot be an eigenvalue of the linearized operator, which in turns leads to the non-degeneracy of ground states. Moreover, by establishing a Picone type identity with respect to antisymmetric functions, we then derive the non-degeneracy of ground states in the space of non-radially symmetric functions. Relying on the non-degeneracy of ground states and adapting a blow-up argument together with a continuation argument, we then obtain the uniqueness of ground states.