Multiple positive solutions for a critical fractional p-Laplacian system with concave nonlinearities

Abstract

The aim of this paper is to study the following nonlinear fractional p-Laplacian system with critical exponents: $\{\begin{array}{ll}(-\mathrm{\Delta }{)}_p^{s}u+|u{|}^{p-2}u=\mathrm{\lambda g}\left(x\right)|u{|}^{q-2}u+\frac{\alpha }{\alpha +\beta }f(x\left)\right|u{|}^{\alpha -2}u|v{|}^{\beta }& \text{ in }\mathrm{\Omega },\\ (-\mathrm{\Delta }{)}_p^{s}v+|v{|}^{p-2}v=\mathrm{\mu h}\left(x\right)|v{|}^{q-2}v+\frac{\beta }{\alpha +\beta }f(x\left)\right|u{|}^{\alpha }|v{|}^{\beta -2}v& \text{ in }\mathrm{\Omega },\\ u=v=0& \text{ in }{\mathbb{R}}^N\mathrm{\setminus \Omega },\end{array}$ where Ω is a smooth bounded set in ${\mathbb{R}}^N$, 0<s<1, $\lambda ,\mu >0$ are two parameters, $1<q<p<p_s^{\ast }$, N>ps, $\alpha ,\text{β}>1$ satisfy $\alpha +\text{β}=p_s^{\ast }$ with $p_s^{\ast }=\frac{\mathrm{np}}{n-\mathrm{ps}}$ is the fractional Sobolev critical exponent and $(-\mathrm{\Delta }{)}_p^s$ is the fractional p-Laplacian operator. Using the Nehari manifold and Ljusternik–Schnirelmann category, we study the topology of the global maximum set Θ of $f\left(x\right)$, and show that the system has at least at least $\mathrm{ca}{t}_{{\mathrm{\Theta }}_{\delta }}\left(\mathrm{\Theta }\right)+1$ distinct positive solutions.

Keywords:

• Fractional p-Laplacian
• critical elliptic sysytem
• multiplicity
• concave-convex nonlinearities
• Lusternik–Schnirelmann category

AMS SUBJECT CLASSIFICATIONS:

• 35J50
• 35B33
• 47G20

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data Availability Statement

Data sharing not applicable to this article as no data sets were generated or analysed during the current study.