Existence of solutions for fractional p&q-Laplacian system involving critical sandwich-type nonlinearities

Abstract

In this paper, we consider the following fractional $p\mathrm{\&}q$-Laplacian system involving critical sandwich-type nonlinearities: $\{\begin{array}{ll}(-\mathrm{\Delta }{)}_p^{s_1}u+(-\mathrm{\Delta }{)}_q^{s_2}u=\lambda |u{|}^{r-2}u+\theta \frac{\alpha }{\alpha +\beta }|u{|}^{\alpha -2}u|v{|}^{\beta },& \mathrm{in}\mathrm{\Omega };\\ (-\mathrm{\Delta }{)}_p^{s_1}v+(-\mathrm{\Delta }{)}_q^{s_2}v=\mu |v{|}^{r-2}v+\theta \frac{\beta }{\alpha +\beta }|u{|}^{\alpha }|v{|}^{\beta -2}v,& \mathrm{in}\mathrm{\Omega };\\ u=v=0,& \mathrm{in}{\mathbb{R}}^n\setminus \mathrm{\Omega },\end{array}$where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded domain with smooth boundary, $0<s_2<s_1<1<q<r<p<p_{s_1}^{\ast }$, $\lambda ,\mu ,\theta >0$ are three parameters, $\alpha >1$, $\beta >1$ satisfy $\alpha +\beta =p_{s_1}^{\ast }$, $p_{s_1}^{\ast }=\frac{np}{n-p{s}_1}(p<\frac{n}{s_1})$ is the fractional critical Sobolev exponent. A nontrivial solution is obtained by variational methods and some analytical techniques.

Keywords:

• Fractional $p\mathrm{\&}q$-Laplacian system
• critical exponent
• existence of solutions

2010 Mathematics Subject Classifications:

• 35A15
• 35R11
• 35B33
• 46E35

Disclosure statement

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

Additional information

Funding

The project was supported by NSFC [grant numbers 11501252, 11571176].