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Abstract
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The purpose of this paper is threefold. First, we establish the critical Adams
inequality on the whole space with restrictions on the norm
for any
.
Second, we prove a sharp concentration-compactness principle for
singular Adams inequalities and a new Sobolev compact embedding in
.
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
| (1) |
where
,
has a positive
lower bound and
behaves like
as
. Furthermore,
when
,
in light of the principle of the symmetric criticality and the radial lemma,
we also derive the existence of nontrivial weak solutions by assuming
and
are radially symmetric
with respect to
and
at origin. Thus our main theorems extend the recent results on bi-Laplacian in
by Chen, Li, Lu
and Zhang (2018) to
in
.
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Keywords
ground state solutions, Adams inequality,
concentration-compactness principle, exponential growth
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Mathematical Subject Classification 2010
Primary: 26D10, 35A23, 46E35
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Milestones
Received: 31 May 2018
Revised: 13 September 2018
Accepted: 9 October 2019
Published: 17 March 2020
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