Existence of a Positive Solution to a Nonlinear Scalar Field Equation with Zero Mass at Infinity

Abstract

Abstract We establish the existence of a positive solution to the problem - Δ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( u ) , u ∈ D 1 , 2 ⁢ ( ℝ N ) , -\Delta u+V(x)u=f(u),\quad u\in D^{1,2}(\mathbb{R}^{N}), for N ≥ 3 {N\geq 3} , when the nonlinearity f is subcritical at infinity and supercritical near the origin, and the potential V vanishes at infinity. Our result includes situations in which the problem does not have a ground state. Then, under a suitable decay assumption on the potential, we show that the problem has a positive bound state.