Finite Morse index solutions of a nonlinear Schrödinger equation in half-space with nonlinear boundary value conditions

We are concerned with Liouville-type theorems for the nonlinear Schrödinger equation \(\begin{aligned} -\Delta u+\lambda |x|^\alpha u=|x|^\beta |u|^{p-1}u\,\,\,\text{ in }\,\, {\mathbb {R}}^N_+, \end{aligned}\) $$\begin{array}{c}\hfill -\mathrm{\Delta }u+{\lambda |x|}^{\alpha }u={|x|}^{\beta }{|u|}^{p-1}u\text{in}{\mathbb{R}}_{+}^N,\end{array}$$ with \(\frac{\partial u}{\partial \nu }=|x|^\gamma |u|^{q-1}u\) $\frac{\partial u}{\partial \nu }={|x|}^{\gamma }{|u|}^{q-1}u$ on \(\Sigma _1\) ${\mathrm{\Sigma }}_1$ , where \(\Sigma _1:=\{x=(x_1,\ldots x_N)\in {\mathbb {R}}^N; x_N=0, x_1>0 \}\) ${\mathrm{\Sigma }}_1:=\{x=(x_1,\dots x_N)\in {\mathbb{R}}^N;x_N=0,x_1>0\}$ . Here, \(N\ge 2\) $N\ge 2$ , \(p,q>1\) $p,q>1$ , \(\alpha , \beta , \gamma >-2\) $\alpha ,\beta ,\gamma >-2$ and \(\lambda \) $\lambda$ is a positive real parameter. We prove the nonexistence of weak sign-changing solutions which are stable or with finite Morse index, possibly unbounded. The main ideas we use here are integral estimates, a Pohozaev type identity, and a monotonicity formula.