Existence and Multiplicity of a Nonhomogeneous Polyharmonic Equation With Critical Exponential Growth in Even Dimension

TL;DR

This paper investigates conditions for the existence of multiple positive solutions to a nonhomogeneous polyharmonic equation with critical exponential growth in even dimensions, identifying parameter bounds for solution multiplicity and non-existence.

Contribution

It provides new bounds on the bifurcation parameter that guarantee multiple solutions or non-existence for a critical growth polyharmonic problem.

Findings

Established bounds for the bifurcation parameter $\lambda$

Proved existence of at least two positive solutions within certain parameter ranges

Identified parameter ranges leading to non-existence of solutions

Abstract

In this paper we study the existence of at least two positive weak solutions for an inhomogeneous fourth order equation with Navier boundary data involving nonlinearities of critical growth with a bifurcation parameter $\lambda$ in $\mathbb{R}^{2m}$. We establish here the lower and upper bound for $\lambda$ which determine multiplicity and non-existence respectively.