Nehari type ground state solutions for periodic Schrödinger–Poisson systems with variable growth

Abstract

In this paper, we deal with the variable growth Schrödinger–Poisson Systems in ${\mathbb{R}}^3$: $\begin{array}{l}-\mathrm{d}\mathrm{i}\mathrm{v}\left(|\mathrm{\nabla }u{|}^{p\left(x\right)-2}\mathrm{\nabla }u\right)+\left(V\right(x)\\ +K\left(x\right)\varphi \left(x\right))|u{|}^{p\left(x\right)-2}u=f(x,u),& x\in {\mathbb{R}}^3,\\ -\mathrm{\Delta }\varphi \left(x\right)=K\left(x\right)|u{|}^{p\left(x\right)},& x\in {\mathbb{R}}^3,\\ u\in W^{1,p\left(x\right)}\left({\mathbb{R}}^3\right),\end{array}$ where $p\left(x\right)\le p^{\ast }\left(x\right):=\frac{3p\left(x\right)}{3-p\left(x\right)}$ and $V\left(x\right)$, $K\left(x\right)$ and $f(x,u)$ are periodic in x. We use the non-Nehari manifold approach to establish the existence of the Nehari type ground state solutions, under the condition: $\begin{array}{rl}& \left[{\left(\frac{p^{-}}{p^{+}}\right)}^{{\chi }_{(0,1)}\left(t\right)}\frac{f(x,\tau )}{|\tau {|}^{2\hat{p}-1}}-\frac{f(x,t\tau )}{|t\tau {|}^{2\hat{p}-1}}\right]\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}(1-t)\\ & +{\left(\frac{p^{-}}{p^{+}}\right)}^{{\chi }_{(0,1)}\left(t\right)}{\theta }_{0}V\left(x\right)\frac{|t^{\hat{p}}-1|}{t^{\hat{p}}|\tau {|}^{2\hat{p}-p\left(x\right)}}\ge 0\end{array}$ for all $x\in {\mathbb{R}}^3$, t>0, $\tau \ne 0$ and the constant ${\theta }_0\in (0,1)$, where $\hat{p}=p^{+}$ as $t\ge 1$ and $\hat{p}=p^{-}$ as t<1. In particular, some new inequalities and tricks are used to overcome the difficulties caused by the variable exponent.

COMMUNICATED BY:

• V. Rădulescu

Keywords:

• Variable growth
• non-Nehari manifold
• Schrödinger–Poisson systems

AMS Subject Classifications:

• 35J60
• 35J20

Acknowledgements

The authors thank the anonymous referees for their valuable suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

X.H. Tang was partly supported by the National Natural Science Foundation of China (grant number 11971485).