Ground state solution of p-Laplacian equation with finite many critical nonlinearities

Abstract

In this paper, we consider the following problem: $-{\mathrm{\Delta }}_{p}u-\zeta \frac{|u{|}^{p-2}u}{|x{|}^p}=\sum _{i=1}^k\left(I_{{\alpha }_i}\ast |u{|}^{p_{{\alpha }_i}^{\ast }}\right)|u{|}^{p_{{\alpha }_i}^{\ast }-2}u+|u{|}^{p^{\ast }-2}u,\mathrm{i}\mathrm{n}{\mathbb{R}}^N,$ where $N=3,4,5,p\in (1,2],\zeta \in \left[0,\mathrm{\Lambda }\right),\mathrm{\Lambda }={\left(\frac{N-p}{p}\right)}^p,{\mathrm{\Delta }}_p:=\mathrm{d}\mathrm{i}\mathrm{v}\left(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u\right)$ is the p-Laplacian operator, $p^{\ast }=\frac{Np}{N-p}$ is the critical Sobolev exponent, $p_{{\alpha }_i}^{\ast }=\frac{p}{2}\left(\frac{N+{\alpha }_i}{N-p}\right)$ are the Hardy–Littlewood–Sobolev critical upper exponents, the parameters ${\alpha }_i$ satisfy some assumptions. First, we establish the refined Sobolev inequality with Coulomb norm, and show the corresponding best constant is achieved in ${\mathbb{R}}^N$ by a nonnegative function. Second, by using the refined Sobolev inequality with Coulomb norm, the refined Sobolev inequality with Morrey norm and variational methods, we establish the existence of nonnegative ground state solution for the above problem.

COMMUNICATED BY:

• V. Moroz

Keywords:

• p-Laplacian equation
• refined Sobolev inequality with Coulomb norm
• refined Sobolev inequality with Morrey norm
• critical Sobolev exponent
• critical Hardy–Littlewood–Sobolev exponent

2010 AMS SUBJECT CLASSIFICATIONS:

• 35J92
• 35B33

Acknowledgments

Yu Su would like to thank Professor J. Bellazzini, M. Ghimenti, C. Mercuri and J. Van Schaftingen for their very valuable comments on endpoint refined Sobolev inequality. Yu Su also would like to thank Professor Xianwen Fang for his help.

Disclosure statement

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

Additional information

Funding

This research was supported by the National Natural Science Foundation of China, Grant No. 11671403, and by the University-level key projects of Anhui University of science and technology, Grant No. QN2019101.