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Remarks on a class of nonlinear Schrödinger equations with potential vanishing at infinity. (English) Zbl 1417.35046

Summary: We study the following nonlinear Schrödinger equation \(- \Delta u + V(x) u = K(x) f(u)\), \(x \in \mathbb{R}^N\), \(u \in H^1(\mathbb{R}^N)\), where the potential \(V(x)\) vanishes at infinity. Working in weighted Sobolev space, we obtain the ground states of problem \((\mathcal{P})\) under a Nahari type condition. Furthermore, if \(V(x), K(x)\) are radically symmetric with respect to \(x \in \mathbb{R}^N\), it is shown that problem \((\mathcal{P})\) has a positive solution with some more general growth conditions of the nonlinearity. Particularly, if \(f(u) = u^p\), then the growth restriction \(\sigma \leq p \leq N + 2 / N - 2\) in [A. Ambrosetti et al., J. Eur. Math. Soc. (JEMS) 7, No. 1, 117–144 (2005; Zbl 1064.35175)] can be relaxed to \(\widetilde{\sigma} \leq p \leq N + 2 / N - 2\), where \(\widetilde{\sigma} < \sigma\) if \(0 < \beta < \alpha < 2\).

MSC:

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian

Citations:

Zbl 1064.35175
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References:

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