Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at Infinity
We study the following nonlinear Schrödinger equation −Δu+V(x)u=K(x)f(u), x∈ℝN, u∈H1(ℝN), where the potential V(x) vanishes at infinity. Working in weighted Sobolev space, we obtain the ground states of problem (𝒫) under a Nahari type condition. Furthermore, if V(x),K(x) are radically symmetric wi...
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Wiley
2013-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2013/786736 |
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| author | Hongbo Zhu |
| author_facet | Hongbo Zhu |
| author_sort | Hongbo Zhu |
| collection | DOAJ |
| description | We study the following nonlinear Schrödinger equation −Δu+V(x)u=K(x)f(u), x∈ℝN, u∈H1(ℝN), where the potential V(x) vanishes at infinity. Working in weighted Sobolev space, we obtain the ground states of problem (𝒫) under a Nahari type condition. Furthermore, if V(x),K(x) are radically symmetric with respect to x∈ℝN, it is shown that problem (𝒫) has a positive solution with some more general growth conditions of the nonlinearity. Particularly, if f(u)=up, then the growth restriction σ≤p≤N+2/N-2 in Ambrosetti et al. (2005) can be relaxed to σ~≤p≤N+2/N-2, where σ~<σ if 0<β<α<2. |
| format | Article |
| id | doaj-art-02d6cbdc3b594bccae5291f522f53136 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-02d6cbdc3b594bccae5291f522f531362025-02-03T05:50:55ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2013-01-01201310.1155/2013/786736786736Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at InfinityHongbo Zhu0Faculty of Applied Mathematics, Guangdong University of Technology, Guangzhou 510006, ChinaWe study the following nonlinear Schrödinger equation −Δu+V(x)u=K(x)f(u), x∈ℝN, u∈H1(ℝN), where the potential V(x) vanishes at infinity. Working in weighted Sobolev space, we obtain the ground states of problem (𝒫) under a Nahari type condition. Furthermore, if V(x),K(x) are radically symmetric with respect to x∈ℝN, it is shown that problem (𝒫) has a positive solution with some more general growth conditions of the nonlinearity. Particularly, if f(u)=up, then the growth restriction σ≤p≤N+2/N-2 in Ambrosetti et al. (2005) can be relaxed to σ~≤p≤N+2/N-2, where σ~<σ if 0<β<α<2.http://dx.doi.org/10.1155/2013/786736 |
| spellingShingle | Hongbo Zhu Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at Infinity Discrete Dynamics in Nature and Society |
| title | Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at Infinity |
| title_full | Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at Infinity |
| title_fullStr | Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at Infinity |
| title_full_unstemmed | Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at Infinity |
| title_short | Remarks on a Class of Nonlinear Schrödinger Equations with Potential Vanishing at Infinity |
| title_sort | remarks on a class of nonlinear schrodinger equations with potential vanishing at infinity |
| url | http://dx.doi.org/10.1155/2013/786736 |
| work_keys_str_mv | AT hongbozhu remarksonaclassofnonlinearschrodingerequationswithpotentialvanishingatinfinity |