Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity

Sitong Chen; Junping Shi; Xianhua Tang

doi:10.3934/dcds.2019257

Article Contents

2019, Volume 39, Issue 10: 5867-5889. Doi: 10.3934/dcds.2019257

This issue Previous Article Uniqueness and nondegeneracy of solutions for a critical nonlocal equation Next Article Multiplicative combinatorial properties of return time sets in minimal dynamical systems

Ground state solutions of Nehari-Pohozaev type for the planar Schrödinger-Poisson system with general nonlinearity

1.
School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China
2.
Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795, USA

^* Corresponding author: Junping Shi
^* Corresponding author: Junping Shi

Received: September 30, 2018

Revised: February 28, 2019

Published: July 2019

This work is partially supported by the National Natural Science Foundation of China (No: 11571370)

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Abstract

It is shown that the planar Schrödinger-Poisson system with a general nonlinear interaction function has a nontrivial solution of mountain-pass type and a ground state solution of Nehari-Pohozaev type. The conditions on the nonlinear functions are much weaker and flexible than previous ones, and new variational and analytic techniques are used in the proof.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35Q55.

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Table 1. Examples of nonlinear functions $ f(u) $ satisfying conditions in Theorems 1.1 and 1.2. Here $ b_0 = \frac{q(p-2)}{(q-1)(3-q)}\left[\frac{(p-1)(p-3)}{p(q-2)}\right]^{\frac{q-2}{p-2}}[2(p-q)]^{\frac{q-p}{p-2}}. $

$ f(u) $	(F4)	(F5)	(F6)
$ f_1(u)=\|u\|^{p-2}u $	$ 3\le p $	$ 2 <p\le 3 $	$ 3\le p $
$ f_2(u)=\left(\|u\|^{p-2}+b\|u\|^{q-2}\right)u $	$ 2<q<3< p $	$ 2<q<p\le 3 $	$ 2<q<3\le p $
			$ 0\le b\le b_0 $
$ f_3(u)=u\left[1-\frac{1}{\ln(e+u^2)}\right] $	YES	YES	NO
$ f_4(u)=u\ln (1+u^2) $	NO	YES	NO
$ f_5(u)=\|u\|u\ln (1+u^2) $	YES	NO	YES
$ f_6(u)=3\|u\|u\ln\left(1+u^2\right)+\frac{2\|u\|^3u}{1+u^2} $	YES	NO	YES
$ f_7(u)=4u^3\int_{0}^{u}\|s\|^{1+\sin s}s\mathrm{d}s+\|u\|^{5+\sin u}u $	YES	NO	NO

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