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Infinitely many positive solutions for the nonlinear Schrödinger equations in \({\mathbb{R}^N}\)

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Abstract

We consider the following nonlinear problem in \({\mathbb {R}^N}\)

$$- \Delta u +V(|y|)u = u^{p},\quad u > 0 \quad {\rm in}\, \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \quad \quad \quad (0.1)$$

where V(r) is a positive function, \({1< p < {\frac{N+2}{N-2}}}\). We show that if V(r) has the following expansion:

$$V(r) = V_0+\frac a {r^m} +O \left(\frac1{r^{m+\theta}}\right),\quad {\rm as} \, r\to +\infty,$$

where a > 0, m > 1, θ > 0, and V 0 > 0 are some constants, then (0.1) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.

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Authors and Affiliations

  1. Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong

    Juncheng Wei

  2. School of Mathematics, Statistics and Computer Science, The University of New England, Armidale, NSW, 2351, Australia

    Shusen Yan

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  1. Juncheng Wei
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  2. Shusen Yan
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Correspondence to Juncheng Wei.

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Wei, J., Yan, S. Infinitely many positive solutions for the nonlinear Schrödinger equations in \({\mathbb{R}^N}\) . Calc. Var. 37, 423–439 (2010). https://doi.org/10.1007/s00526-009-0270-1

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