# On Nonlinear Biharmonic Problems on the Heisenberg Group

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## Abstract

**:**

## 1. Introduction

- $\left({f}_{1}\right)$
- For some ${c}_{1},{c}_{2}>0$,$$\left|f(x,\xi )\right|\le {c}_{1}+{c}_{2}{\left|\xi \right|}^{s}\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{a}.\mathrm{e}.\phantom{\rule{4pt}{0ex}}x\in \Omega \phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{4pt}{0ex}}\xi \in \mathbb{R},$$$$2<s<\frac{2Q}{Q-2}-1;$$
- $\left({f}_{2}\right)$
- $\underset{\left|\xi \right|\to 0}{lim}\frac{f(x,\xi )}{\left|\xi \right|}=0$ uniformly in $x\in \Omega $;
- $\left({f}_{3}\right)$
- For some $\mu >2$ and $r>0$,$$0<\mu F(x,\xi )<\xi f(x,\xi )\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{a}.\mathrm{e}.\phantom{\rule{4pt}{0ex}}x\in \Omega \phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{4pt}{0ex}}\xi \in \mathbb{R},$$
- $\left({f}_{4}\right)$
- $f(x,\xi )\in C(\overline{\Omega},\mathbb{R})$.

**Example**

**1.**

**Theorem**

**1.**

## 2. Preliminaries

**Theorem**

**2**

- (i)
- If $k<\frac{Q}{p},$ then ${D}_{0}^{k,p}(\Omega )$ is continuously embedded in ${L}^{{p}^{*}}(\Omega )$ for $\frac{1}{{p}^{*}}=\frac{1}{p}-\frac{k}{Q}$;
- (ii)
- If $k=\frac{Q}{p},$ then ${D}_{0}^{k,p}(\Omega )$ is continuously embedded in ${L}^{r}(\Omega )$ for all $r\in [1,\infty )$;
- (iii)
- If $k>\frac{Q}{p},$ then ${D}_{0}^{k,p}(\Omega )$ is continuously embedded in ${C}^{0,\gamma}\left(\overline{\Omega}\right)$ for all $0\le $$\gamma <k-\frac{Q}{p}$.

- (i)
- There exist ${c}_{3},{c}_{4}>0$ such that$${c}_{3}{\left|\xi \right|}^{\mu}-{c}_{4}\le F(x,\xi )\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{4pt}{0ex}}x\in \Omega ,$$
- (ii)
- For any $\xi \in \mathbb{R},$ we have$$\begin{array}{c}\hfill \left|f(x,\xi )\right|\le \epsilon \left|\xi \right|+{(s+1)\kappa \left(\epsilon \right)\left|\xi \right|}^{s}\end{array}$$$$\begin{array}{c}\hfill \left|F(x,\xi )\right|\le {\epsilon \left|\xi \right|}^{2}+\kappa \left(\epsilon \right){\left|\xi \right|}^{s+1},\end{array}$$

## 3. Proof of Theorem 1

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**3**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Proof.**

**Remark**

**1.**

## 4. Epilogue

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Zuo, J.; Taarabti, S.; An, T.; Repovš, D.D.
On Nonlinear Biharmonic Problems on the Heisenberg Group. *Symmetry* **2022**, *14*, 705.
https://doi.org/10.3390/sym14040705

**AMA Style**

Zuo J, Taarabti S, An T, Repovš DD.
On Nonlinear Biharmonic Problems on the Heisenberg Group. *Symmetry*. 2022; 14(4):705.
https://doi.org/10.3390/sym14040705

**Chicago/Turabian Style**

Zuo, Jiabin, Said Taarabti, Tianqing An, and Dušan D. Repovš.
2022. "On Nonlinear Biharmonic Problems on the Heisenberg Group" *Symmetry* 14, no. 4: 705.
https://doi.org/10.3390/sym14040705