Multiple solutions for polyharmonic equations with exponential nonlinearity

Abstract

In this paper, we study the following polyharmonic equation: $\{\begin{array}{ll}-{\mathrm{\Delta }}_{{\displaystyle \frac{n}{m}}}^{m}u=Q_{\lambda }(x)|u{|}^{q-1}u+\left({\displaystyle {\int }_{\mathrm{\Omega }}{\displaystyle \frac{F(u)}{|x-y{|}^{\mu }|y{|}^{\alpha }}}\mathrm{d}y}\right){\displaystyle \frac{f(u)}{|x{|}^{\alpha }}}& \text{in}\mathrm{\Omega },\\ u=\mathrm{\nabla }u=\cdots ={\mathrm{\nabla }}^{m-1}u=0& \text{on}\mathrm{\partial }\mathrm{\Omega }.\end{array}$Using the Nehari manifold and some analytical techniques, we show the existence of two weak solutions of the problem with respect to parameter λ.

Keywords:

• Polyharmonic equation
• multiple solutions
• Nehari manifold
• fibering maps

2020 Mathematics Subject Classifications:

• 35J35
• 35J60

Disclosure statement

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

Data availability statement

Date sharing is not applicable to this article as no new data were created or analyzed in this study.