Multiple solutions for a class of nonhomogeneous elliptic systems with Dirichlet boundary or Neumann boundary

TL;DR

This paper proves the existence of at least three solutions for a class of nonhomogeneous quasilinear elliptic systems with boundary conditions, using a method that involves parameter intervals and monotone operator theory.

Contribution

It introduces a novel approach to establish multiple solutions for nonhomogeneous elliptic systems with boundary conditions, extending previous methods by exploiting parameter intervals and monotone operator techniques.

Findings

Established at least three non-trivial solutions for the systems.

Derived concrete open intervals for the parameter λ.

Applied monotone operator theory and dimensional augmentation in proofs.

Abstract

In this paper, we mainly establish the existence of at least three non-trivial solutions for a class of nonhomogeneous quasilinear elliptic systems with Dirichlet boundary value or Neumann boundary value in a bounded domain $\Omega\subset\mathbb{R}^N $ and $N\geq 1$. We exploit the method which is based on [6]. This method let us obtain the concrete open interval about the parameter $\lambda$. Since the quasilinear term depends on $u$ and $\nabla u$, it is necessary for our proofs to use the theory of monotone operators and the skill of adding one dimension to space.