Ground state solutions of polyharmonic equations with potentials of positive low bound

Caifeng Zhang, Jungang Li and Lu Chen

Vol. 305 (2020), No. 1, 353–384

DOI: 10.2140/pjm.2020.305.353

Abstract

The purpose of this paper is threefold. First, we establish the critical Adams inequality on the whole space with restrictions on the norm

$${\left(\parallel {\nabla }^{m}u{\parallel }_{\frac{n}{m}}^{\frac{n}{m}}+\tau \parallel u{\parallel }_{\frac{n}{m}}^{\frac{n}{m}}\right)}^{\frac{m}{n}}$$

for any $\tau >0$. Second, we prove a sharp concentration-compactness principle for singular Adams inequalities and a new Sobolev compact embedding in $W^{m,2}\left({ℝ}^{2m}\right)$. Third, based on the above results, we give sufficient conditions for the existence of ground state solutions to the following polyharmonic equation with singular exponential nonlinearity

$${\left(-\Delta \right)}^{m}u+V\left(x\right)u=\frac{f\left(x,u\right)}{|x{|}^{\beta }}\text{ in }{ℝ}^{2m},$$

(1)

where $0<\beta <2m$, $V\left(x\right)$ has a positive lower bound and $f\left(x,t\right)$ behaves like $exp\left(\alpha |t{|}^2\right)$ as $t\to +\infty$. Furthermore, when $\beta =0$, in light of the principle of the symmetric criticality and the radial lemma, we also derive the existence of nontrivial weak solutions by assuming $f\left(x,t\right)$ and $V\left(x\right)$ are radially symmetric with respect to $x$ and $f\left(x,t\right)=o\left(t\right)$ at origin. Thus our main theorems extend the recent results on bi-Laplacian in ${ℝ}^4$ by Chen, Li, Lu and Zhang (2018) to ${\left(-\Delta \right)}^m$ in ${ℝ}^m$.

Keywords

ground state solutions, Adams inequality, concentration-compactness principle, exponential growth

Mathematical Subject Classification 2010

Primary: 26D10, 35A23, 46E35

Milestones

Received: 31 May 2018

Revised: 13 September 2018

Accepted: 9 October 2019

Published: 17 March 2020