Existence and concentration of bound states of a class of nonlinear Schrödinger equations in ℝ² with potential tending to zero at infinity

Abstract

In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schrödinger equation

$$- \varepsilon ^2 \Delta u_\varepsilon + V\left( x \right)u_\varepsilon = K\left( x \right)\left| {u_\varepsilon } \right|^{p - 2} u_\varepsilon e^{\alpha _0 \left| {u_\varepsilon } \right|^\gamma } , u_\varepsilon > 0, u_\varepsilon \in H^1 \left( {\mathbb{R}^2 } \right),$$

where 2 < p < ∞, α ₀ > 0, 0 < γ < 2. When the potential function V (x) decays at infinity like (1 + |x|)^−α with 0 < α ≤ 2 and K(x) > 0 are permitted to be unbounded under some necessary restrictions, we will show that a positive H ¹(ℝ²)-solution u _ɛ exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schrödinger equation $- \Delta u + V\left( \xi \right)u = K\left( \xi \right)\left| u \right|^{p - 2} ue^{\alpha _0 \left| u \right|^\gamma }$ has local minimum points. Furthermore, the concentration property of u _ɛ is also established as ɛ tends to zero.