# Existence and multiplicity of solutions for a class of quasilinear elliptic field equation on $\mathbb{R}^{N}$

## TL;DR

This paper proves the existence and multiplicity of solutions for a class of quasilinear elliptic field equations on ^N, involving singular and positive potential functions, expanding understanding of such nonlinear PDEs.

## Contribution

It establishes new results on the existence and multiplicity of solutions for a broad class of quasilinear elliptic equations with singular nonlinearities.

## Key findings

- Proved existence of solutions under certain conditions.
- Established multiplicity of solutions.
- Analyzed equations with singular nonlinearities and positive potentials.

## Abstract

In this paper, we establish existence and multiplicity of solutions for the following class of quasilinear field equation $$ -\Delta u+V(x)u-\Delta_{p}u+W'(u)=0, \,\,\, \mbox{in} \,\,\, \mathbb{R}^{N}, \eqno{(P)} $$ where $u=(u_1,u_2,...,u_{N+1})$, $p>N \geq 2$, $W$ is a singular function and $V$ is a positive continuous function.