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, $$ -Δu +(-Δ)^s u + λu=|u|^{p-2}u \quad \mbox{in} \,\,\, B, \quad u=0 \quad \mbox{in} \,\,\, \R^N \backslash {B}, $$ where $N \geq 2$, $0<s<1$, $2<p<2^*:=\frac{2N}{(N-2)^+}$, $λ> -λ_1$, $(-Δ)^s$ denotes the fractional Laplacian, $λ_1>0$ denotes the first Dirichlet eigenvalue of the operator $-Δ+(-Δ)^s$ in $B$ and $B$ denotes the unit ball in $\R^N$. We prove that the second eigenvalue to the linearized operator $-Δ+(-Δ)^s -(p-1)u^{p-2}$ 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 $-λ$ 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.