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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 concentrationcompactness principle for
singular Adams inequalities and a new Sobolev compact embedding in
${W}^{m,2}\left({\mathbb{R}}^{2m}\right)$.
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
$${\left(\Delta \right)}^{m}u+V\left(x\right)u=\frac{f\left(x,u\right)}{x{}^{\beta}}\phantom{\rule{1em}{0ex}}\text{in}{\mathbb{R}}^{2m},$$  (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 biLaplacian in
${\mathbb{R}}^{4}$ by Chen, Li, Lu
and Zhang (2018) to
${\left(\Delta \right)}^{m}$
in
${\mathbb{R}}^{m}$.

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Keywords
ground state solutions, Adams inequality,
concentrationcompactness 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

