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

36 pages, accepted for publication in Analysis and PDE

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