Positivity for fourth-order semilinear problems related to the Kirchhoff-Love functional

TL;DR

This paper investigates the existence, positivity, and convergence of ground state solutions for a generalized Kirchhoff-Love functional involving fourth-order semilinear problems, with special focus on parameter-dependent behavior and radial solutions.

Contribution

It introduces new techniques to prove positivity of ground states for a range of parameters and analyzes their convergence as the parameter varies, extending understanding of fourth-order Kirchhoff-Love problems.

Findings

Positivity of ground states established for different parameter ranges.

Convergence of ground states with respect to the parameter  analyzed.

Existence of positive radial solutions in spherical domains confirmed.

Abstract

We study the ground states of the following generalization of the Kirchhoff-Love functional, $$J_\sigma(u)=\int_\Omega\dfrac{(\Delta u)^2}{2} - (1-\sigma)\int_\Omega det(\nabla^2u)-\int_\Omega F(x,u),$$ where $\Omega$ is a bounded convex domain in $\mathbb{R}^2$ with $C^{1,1}$ boundary and the nonlinearities involved are of sublinear type or superlinear with power growth. These critical points correspond to least-energy weak solutions to a fourth-order semilinear boundary value problem with Steklov boundary conditions depending on $\sigma$. Positivity of ground states is proved with different techniques according to the range of the parameter $\sigma\in\mathbb{R}$ and we also provide a convergence analysis for the ground states with respect to $\sigma$. Further results concerning positive radial solutions are established when the domain is a ball.