Multiple solutions of the superlinear elliptic equations with Dirichlet boundary condition
Abstract
In this paper, we investigate the existence and the multiplicity of solutions to the elliptic equation with Dirichlet boundary value condition$ \begin{cases}-\Delta u = f(u), & \text{in} \ \Omega, \\ u = 0, & \text{on} \ \partial \Omega, \end{cases} $where $ \Omega\subset\mathbb{R}^{N} $ is an open, bounded, smooth domain. We show that under suitable conditions, the elliptic equation with superlinear terms possesses at least nine nontrivial solutions. Our approach combines the sub-supersolution method, the mountain pass theorem in order intervals, bifurcation theory, degree theory, Morse theory, and the invariance of descending flow method.
Fields of Study & Subjects
Nonlinear Partial Differential Equations
Nonlinear Differential Equations Analysis
Advanced Differential Equations and Dynamical Systems
Publication details
- Journal / source
- Communications on Pure & Applied Analysis
- Volume
- 29
- Issue
- 0
- Pages
- 81–96
- Publication date
- 1 January 2026