Kirchhoff-type problems involving subcritical and superlinear nonlinearities satisfying no further condition

TL;DR

This paper investigates Kirchhoff-type boundary value problems with subcritical and superlinear nonlinearities, establishing the existence of multiple solutions using a new multiplicity theorem without requiring conditions on the nonlinearity's behavior at zero.

Contribution

It introduces a novel multiplicity result applied to Kirchhoff problems with minimal assumptions on the nonlinearity, proving the existence of at least three solutions.

Findings

Existence of at least three solutions to the Kirchhoff problem.

Two solutions are identified as global minima of the energy functional.

No conditions on the nonlinearity at zero are needed.

Abstract

In this note, we deal with a problem of the type $$\cases {-h\left ( \int_{\Omega}|\nabla u(x)|^2dx\right ) \Delta u=f(u) & in $\Omega$\cr & \cr u_{|\partial\Omega}=0\ .\cr}$$ As an application of a new general multiplicity result, we establish the existence of at least three solutions, two of which are global minima of the related energy functional. The only condition assumed on $f$ is that it be subcritical and superlinear: no condition on the behaviour of $f$ at $0$ is requested.