Existence and Concentration of Bound States of a Class of Nonlinear SchrSdinger Equations in R2 with Potential Tending to Zero at Infinity |
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Authors: | Da Cheng Cui Ji Hui Zhang Ming Wen Fei |
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Institution: | 1. School of Mathematics and Information Technology, Nanjing Xiaozhuang University, Nanjing, 211171, P. R. China 2. School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210046, P. R. China 3. Academy of Mathematics and Systems Science, China Academy of Science, Beijing, 100190, P. R. China 4. School of Mathematical and Computer Sciences, Anhui Normal University, Wuhu, 241003, P. R. China
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Abstract: | In this paper, we establish the existence and concentration of solutions of a class of nonlinear Schr?dinger 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(?2)-solution u ? exists if it is assumed that the corresponding ground energy function G(ξ) of nonlinear Schr?dinger 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 ? is also established as ? tends to zero. |
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Keywords: | Bound state nonlinear SchrSdinger equation Harnack inequality concentration-compact-ness |
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