Anisotropic system with singular and regular nonlinearities

ABSTRACT

We consider in this paper the following system: $\begin{array}{rl}-L_{\left(p_i\right)}u& =-\sum _{i=1}^N{\mathrm{\partial }}_i\left[{\left|{\mathrm{\partial }}_{i}u\right|}^{p_i-2}{\mathrm{\partial }}_{i}u\right]=\frac{1}{v^{{\alpha }_1}}+v^{{\beta }_1}\\ -L_{\left(q_i\right)}v& =-\sum _{i=1}^N{\mathrm{\partial }}_i\left[{\left|{\mathrm{\partial }}_{i}v\right|}^{q_i-2}{\mathrm{\partial }}_{i}v\right]=\frac{1}{u^{{\alpha }_2}}+u^{{\beta }_2}\\ & u=v=0\mathrm{on}\mathrm{\partial \Omega },\\ & u>0,v>0\mathrm{in}\mathrm{\Omega }.\end{array}$ where Ω is a bounded regular domain in ${\mathbb{R}}^N$, we will assume that ${\beta }_1<{\overline{p}}^{\ast }-1$, ${\beta }_2<{\overline{q}}^{\ast }-1,$ ${\alpha }_1>0$, ${\alpha }_2>0,$ $2<p_1\le p_2\le \dots \le p_N$ and $2<q_1\le q_2\le \dots \le q_N$. Using approximation techniques, a result of Rabinowitz, and Minty–Browder Theorem, we obtain the existence of positive solutions to the considered problem.

COMMUNICATED BY:

• V. Radulescu

KEYWORDS:

• Anisotropic problem
• anisotropic system
• singular-regular nonlinearity
• approximation

AMS SUBJECT CLASSIFICATIONS:

• 35J65
• 35B65
• 35J70

Acknowledgments

The authors are deeply grateful to the anonymous referees and the associate editor for carefully reading of the paper and for their valuable comments.

Disclosure statement

No potential conflict of interest was reported by the authors.