Abstract In this paper, we investigate the following Schrodinger equation { − Δ u − μ | x | 2 u = g ( u ) in R N ∖ { 0 } , u ∈ H 1 ( R N ) , where N ≥ 3 , μ ( N − 2 ) 2 4 , 1 | x | 2 is called the Hardy potential (the inverse-square potential) and g satisfies the Berestycki-Lions type condition. If 0 μ ( N − 2 ) 2 4 , combining variational methods with analytical skills, we show that the above problem has a positive and radial ground state solution. At the same time, our results suggest that this solution together with its derivatives up to order 2 have exponential decay at infinity while this solution has the possibility of blow-up at the origin. Furthermore, we construct a family of ground state solutions which converges to a ground state solution of the limiting problem as μ → 0 + . If μ 0 , we prove that the mountain pass level in H 1 ( R N ) can not be achieved. Provided further assumption that the above problem in the radial space H r 1 ( R N ) , we obtain the ground state solutions whose energy is strictly greater than the mountain pass level in H 1 ( R N ) . We also construct a family of solutions which converges to the ground state solution of the limiting problem as μ → 0 − .

中文翻译：

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具有哈代势和 Berestycki-Lions 类型条件的薛定谔方程基态解的存在性和渐近行为

摘要 在本文中，我们研究了以下薛定谔方程 { − Δ u − μ | × | 2 u = g ( u ) in RN ∖ { 0 } , u ∈ H 1 ( RN ) , 其中 N ≥ 3 , μ ( N − 2 ) 2 4, 1 | × | 2称为哈代势（平方反比势），g满足Berestycki-Lions型条件。如果 0 μ ( N − 2 ) 2 4 ，结合变分方法和分析技巧，我们证明上述问题有一个正的径向基态解。同时，我们的结果表明，该解及其最高 2 阶的导数在无穷远处具有指数衰减，而该解在原点有可能爆炸。此外，我们构建了一系列基态解，其收敛到极限问题的基态解为 μ → 0 + 。如果 μ 0 ，我们证明了 H 1 ( RN ) 中的山口水平无法达到。进一步假设上述问题在径向空间H r 1 (RN) 中，我们得到能量严格大于H 1 (RN) 中山口水平的基态解。我们还构建了一系列解决方案，其收敛到极限问题的基态解决方案为 μ → 0 − 。