LAUSR

We consider the problem $$\begin{aligned} -\Delta u+\left( V_{\infty }+V(x)\right) u=|u|^{p-2}u,\quad u\in H_{0} ^{1}(\Omega ), \end{aligned}$$-Δu+V∞+V(x)u=|u|p-2u,u∈H01(Ω),where $$\Omega $$Ω is either $$\mathbb {R}^{N}$$RN or a smooth domain in $$\mathbb {R} ^{N}$$RN with unbounded boundary, $$N\ge 3,$$N≥3,$$V_{\infty }>0,$$V∞>0,$$V\in \mathcal {C} ^{0}(\mathbb {R}^{N}),$$V∈C0(RN),$$\inf _{\mathbb {R}^{N}}V>-V_{\infty }$$infRNV>-V∞ and $$2