Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency
Groundstates and radial solutions to nonlinear Schrödinger–Poisson–Slater equations at the critical frequency
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DOI：
10.1007/s00526-016-1079-3
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发表时间：
2015-07
影响因子：
2.1
通讯作者：
C. Mercuri;Vitaly Moroz;Jean Van Schaftingen
中科院分区：
文献类型：
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作者：
C. Mercuri;Vitaly Moroz;Jean Van Schaftingen
本文研究了非局部Schrödinger-Poisson-斯莱特型方程$$\开始{aligned} - \Delta u +（I_\alpha *\vert u\vert ^p）\vert u\vert ^{p - 2} u= \vert u\vert ^{q-2}u\quad \text {in}\quad \mathbb {R}^N，\end{aligned}$$其中，是Riesz势的阶数，我们引入并研究了库仑势，Sobolev函数空间中的能量泛函，并建立了相应的最优插值不等式族。我们证明了不等式的最优解的存在性，这意味着在一定范围内的参数方程的解的存在性。我们还研究了解的正则性和一些定性性质。最后，我们推导出径向施特劳斯型估计，并使用它们来证明方程在一定参数范围内径向解的存在性，该参数范围通常比通过插值不等式获得的存在参数范围更宽。
We study the nonlocal Schrödinger–Poisson–Slater type equation $$\begin{aligned} - \Delta u + (I_\alpha *\vert u\vert ^p)\vert u\vert ^{p - 2} u= \vert u\vert ^{q-2}u\quad \text {in}\quad \mathbb {R}^N, \end{aligned}$$where,,andis the Riesz potential of orderWe introduce and study the Coulomb–Sobolev function space which is natural for the energy functional of the problem and we establish a family of associated optimal interpolation inequalities. We prove existence of optimizers for the inequalities, which implies the existence of solutions to the equation for a certain range of the parameters. We also study regularity and some qualitative properties of solutions. Finally, we derive radial Strauss type estimates and use them to prove the existence of radial solutions to the equation in a range of parameters which is in general wider than the range of existence parameters obtained via interpolation inequalities.