We study the nonlinear Schrödinger equation with an arbitrary real potential \(V(x)\in (L^1+L^\infty )(\Gamma )\) on a star graph \(\Gamma \). At the vertex an interaction occurs described by the generalized Kirchhoff condition with strength \(-\gamma <0\). We show the existence of ground states \(\varphi _{\omega }(x)\) as minimizers of the action functional on the Nehari manifold under additional negativity and decay conditions on V(x). Moreover, for \(V(x)=-\dfrac{\beta }{x^{\alpha }}\), in the supercritical case, we prove that the standing waves \(e^{i\omega t}\varphi _{\omega }(x)\) are orbitally unstable in \(H^{1}(\Gamma )\) when \(\omega \) is large enough. Analogous result holds for an arbitrary \(\gamma \in {\mathbb {R}}\) when the standing waves have symmetric profile.

我们在星图\（\ Gamma \）上研究具有任意实际势\（V（x）\ in（L ^ 1 + L ^ \ infty）（\ Gamma）\）的非线性Schrödinger方程。在顶点处发生了一种相互作用，该相互作用由强度为\（-\ gamma <0 \）的广义基尔霍夫条件描述。我们显示了基态\（\ varphi _ {\ omega}（x）\）的存在，作为在附加负性和V（x）上的衰减条件下，对Nehari流形起作用的作用的极小值。此外，对于\（V（x）=-\ dfrac {\ beta} {x ^ {\ alpha}} \），在超临界情况下，我们证明了驻波\（e ^ {i \ omega t} \ varphi _ {\ omega}（x）\）在当\（\ omega \）足够大时，\（H ^ {1}（\ Gamma）\）。当驻波具有对称轮廓时，类似结果适用于任意\（\ gamma \ in {\ mathbb {R}} \）。