On the existence of positive solutions to semilinear elliptic systems involving gradient term

ABSTRACT

In this work, we analyse the existence of solutions to the nonlinear elliptic system: $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill -\mathrm{\Delta }u& =v^q+\alpha g& \text{in}\mathrm{\Omega },\hfill \\ \hfill -\mathrm{\Delta }v& ={|\mathrm{\nabla }u|}^p+\lambda f& \text{in}\mathrm{\Omega },\hfill \\ \hfill u=v& =0& \text{on}\partial \mathrm{\Omega },\hfill \\ \hfill u,v& \ge & 0\hfill & \text{in}\mathrm{\Omega },\hfill \end{array}\right.\end{array}$$

where $\mathrm{\Omega }$ is a bounded domain of ${\mathbb{R}}^N$ and $p\ge 1$, $q>0$ with $pq>1$. f, g are nonnegative measurable functions with additional hypotheses and $\alpha ,\lambda \ge 0$. As a consequence we show that the fourth-order problem $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill {\mathrm{\Delta }}^{2}u& ={|\mathrm{\nabla }u|}^p+\tilde{\lambda }\tilde{f}& \text{in}\mathrm{\Omega },\hfill \\ \hfill u=\mathrm{\Delta }u& =0& \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right.\end{array}$$

has a solution for all $p>1$, under suitable conditions on $\tilde{f}$ and $\tilde{\lambda }$.

KEYWORDS:

• Elliptic system
• nonlinear gradient terms
• Bi-Laplacien operator

AMS SUBJECT CLASSIFICATIONS:

• 35J55
• 35D10
• 35J60

Acknowledgements

The authors would like to thank Prof. Michel PIERRE for his helpful suggestions and fruitful discussions during the preparation of this work. Part of this work was realized while the first author was visiting the Institute Elie Cartan, Université Lorraine. He would like to thanks the institute for the warm hospitality.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author is partially supported by [project MTM2016-80474-P], MINECO, Spain.