On Lyapunov-type inequalities for a certain class of partial differential equations

ABSTRACT

A Lyapunov-type inequality is established for the partial differential equation: $\begin{array}{rl}-G_{\gamma }u(x,y)& =w\left(x\right)u(x,y),(x,y)\in \mathrm{\Omega },\\ u(x,y)& =0,(x,y)\in \mathrm{\partial \Omega },\end{array}$ where $\mathrm{\Omega }=]a,b[\times \mathcal{O}$, $(a,b)\in {\mathbb{R}}^2$, a<b, $\mathcal{O}$ is an open bounded subset in ${\mathbb{R}}^N$, $N\ge 1$, $G_{\gamma }$ is the differential operator given by $G_{\gamma }u(x,y):=\left({\mathrm{\partial }}^{2}u/\mathrm{\partial }{x}^2\right)+x^{2\gamma }{\mathrm{\Delta }}_{y}u$, $\gamma \ge 0$, $(x,y)\in \mathrm{\Omega }$, and $w\in C\left(\right[a,b\left]\right)$. Next, some sign-changing criteria are obtained for the nontrivial solutions to the considered problem.

KEYWORDS:

• Lyapunov-type inequalities
• partial differential equations
• sign-changing criteria
• first eigenvalue

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35A23
• 47F05
• 34A12

Acknowledgements

The first and third authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the Research Group Project No RGP-1436-034.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by King Saud University.