#### Vol. 10, No. 4, 2017

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

### Giulio Romani

Vol. 10 (2017), No. 4, 943–982
##### Abstract

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

${J}_{\sigma }\left(u\right)={\int }_{\Omega }\frac{{\left(\Delta u\right)}^{2}}{2}-\left(1-\sigma \right)\phantom{\rule{0.3em}{0ex}}{\int }_{\Omega }det\left({\nabla }^{2}u\right)-{\int }_{\Omega }F\left(x,u\right),$

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.

##### Keywords
biharmonic operator, positivity-preserving property, semilinear problem, positive least-energy solutions, Nehari manifold
##### Mathematical Subject Classification 2010
Primary: 35G30, 49J40