Existence results for zero mass polyharmonic system

ABSTRACT

In this paper, we deal with the following polyharmonic system in the zero-mass case $\begin{array}{rl}(-\mathrm{\Delta }{)}^{m}u=K(x\left)f\right(v)& \mathrm{i}\mathrm{n}{\mathbb{R}}^{\mathrm{N}},\\ (-\mathrm{\Delta }{)}^{m}v=K(x\left)g\right(u)& \mathrm{i}\mathrm{n}{\mathbb{R}}^{\mathrm{N}},\end{array}$ where K is a positive weight function and f, g are two continuous functions with critical or supercritical growth at zero and satisfying a quasicritical growth at infinity. We give a global condition which is weaker than the Ambrosetti–Rabinowitz condition and we point out its importance for checking the Palais Smale compactness condition. By applying a corollary of Theorem 2.1 in [Li GB, Szulkin A. An asymptically periodic Schrödinger equation with indefinite linear part. Commun Contemp Math. 2002;4:763–776] for strongly indefinite functionals, we prove the existence of at least one nontrivial pair solution $(u,v)$ under weaker hypotheses on the nonlinear terms. The present paper extend previous results of He [Nonlinear Schrödinger equations with sign-changing potential. Adv Nonlinear Stud. 2012;12:237–253] and Li-Ye [Li GB, Ye H. Existence of positive solutions to semilinear elliptic systems in RN with zero mass. Acta Math Sci. 2013;33(4):913–928].

KEYWORDS:

• Polyharmonic system
• zero mass case
• Palais Smale condition
• linking theorem

AMS SUBJECT CLASSIFICATIONS:

• 35J55
• 35J65
• 35B65

Acknowledgments

The first author would like to express his deepest gratitude to the Military School of Aeronautical Specialities, Sfax (ESA) for providing an excellent atmosphere for work. Also, second and third authors greatly indebted to the deanship of Scientific Research at Northern Border University for its funding of the present work through the research project No. SCI-2018-3-9-F-7867. Finally, the authors wish to acknowledge the referees for useful comments and valuable suggestions which have helped improve the presentation and also they wish to thank Professor Dong Ye for stimulating discussions on the subject.

Disclosure statement

No potential conflict of interest was reported by the authors.