Pictures below is from Weinstein, Michael I., Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16, 472-491 (1985). ZBL0583.35028.
For $0<\sigma<\frac{2}{N-2}$ ($N$ is the dimension of space), $$ L_- = -\Delta +1 -R^{2\sigma} $$ where $R$ is the solution of $$ \Delta u -u + |u|^{2\sigma}u=0\tag{1.4} $$ First, I don't know what is ground state. Seemly, the ground state is the solution of (1.4) with minimal $L^2$ norm. Whether it is right ?
About $R$, we have the properties in first picture below. But why in second picture below, $L_-$ is nonnegative ? In my view, nonnegative means that $$ (L_-v,v)\ge 0 $$ I can't get it from the fact that $R$ is ground.
Besides, what is the mean of ground state is nondegenerate ? I don't know what is nondegenerate. Seemly, in other paper, the nondegerate is the kernel of operator is the partial derivative of solution. But here, I don't whether it is.
