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arXiv · 1605.02504

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

Giulio Romani
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Abstract

We study the ground states of the following generalization of the Kirchhoff-Love functional, $$J_σ(u)=\int_Ω\dfrac{(Δu)^2}{2} - (1-σ)\int_Ωdet(\nabla^2u)-\int_ΩF(x,u),$$ where $Ω$ 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 $σ$. Positivity of ground states is proved with different techniques according to the range of the parameter $σ\in\mathbb{R}$ and we also provide a convergence analysis for the ground states with respect to $σ$. Further results concerning positive radial solutions are established when the domain is a ball.

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Giulio Romani. 2017-03-09. Positivity for fourth-order semilinear problems related to the Kirchhoff-Love functional. https://doi.org/10.2140/apde.2017.10.943

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