#### Vol. 305, No. 1, 2020

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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
##### 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

 (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