On a variational inequality for a plate equation with p-Laplacian end memory terms

ABSTRACT

In this paper we investigate the unilateral problem for a plate equation with memory terms and lower order perturbation of p-Laplacian type $u{}^{″}+{\mathrm{\Delta }}^{2}u-{\mathrm{\Delta }}_{p}u+{\int }_0^{t}g(t-s)\mathrm{\Delta }u\left(s\right)\mathrm{d}s+\mathrm{\Delta }{u}^{\prime }+f\left(u\right)=0\mathrm{i}\mathrm{n}\mathrm{\Omega }\times {\mathbb{R}}^{+},$ where Ω is a bounded domain of ${\mathbb{R}}^n$, g>0 is a memory kernel and $f\left(u\right)$is a nonlinear perturbation. Using the penalty and Faedo–Galerkin's methods, we establish results on existence and uniqueness of weak solutions.

Keywords:

• Plate equations
• p-Laplacian
• memory terms
• variational inequality

2010 Mathematics Subject Classifications:

• 35K60
• 35L86
• 35F30

Acknowledgments

The authors are very grateful to all referees for their valuable suggestions and comments.

Disclosure statement

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