  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.
