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On an elliptic Kirchhoff-type problem depending on two parameters
Biagio Ricceri
TL;DR
The paper studies a Dirichlet problem on a bounded smooth domain involving a continuous function K and seeks a multi-solution result. Using a variational functional framework and a parameterized three-solution theorem, it proves at least three weak solutions for specified parameter ranges, while leaving broader interval and three-dimensional cases open.
Problem
The paper considers a Dirichlet problem on a bounded smooth domain involving a given continuous function K.
Method
The approach applies a three-solution variational theorem to a C1 functional with compact derivative and an associated parameterized equation.
Results
For specified compact parameter intervals and sufficiently small µ, the problem has at least three weak solutions with H1_0(Ω) norms below r.
Takeaways & Limitations
The result establishes a parameter-dependent multiplicity conclusion for the Dirichlet problem under the paper's stated assumptions.
Takeaways & Limitations
The paper leaves open whether the conclusion extends to intervals ]θ∗, b] and whether Theorem 2 holds when n = 3.
Abstract
from arXiv · showhide
In this paper, we consider the Dirichlet problem associated to an elliptic Kirchhoff-type equation depending on two parameters. Under rather general and natural assumptions, we prove that, for certain values of the parameters, the problem has at least three solutions.