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

Abstract

We study the ground states of the following generalization of the Kirchhoff–Love functional,

$$J_{\sigma }\left(u\right)={\int }_{\Omega }\frac{{\left(\Delta u\right)}^2}{2}-\left(1-\sigma \right){\int }_{\Omega }det\left({\nabla }^{2}u\right)-{\int }_{\Omega }F\left(x,u\right),$$

where $\Omega$ is a bounded convex domain in ${ℝ}^2$ 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 $\sigma \in ℝ$ 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.