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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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1026-0226 1607-887X |