## Analysis & PDE

• Anal. PDE
• Volume 10, Number 4 (2017), 943-982.

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

Giulio Romani

#### Abstract

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

$Jσ(u) = ∫ Ω(Δu)2 2 − (1 − σ)∫ Ω det(∇2u) −∫ ΩF(x,u),$

where $Ω$ is a bounded convex domain in $ℝ2$ with $C1,1$ 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 $σ ∈ ℝ$ 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.

#### Article information

Source
Anal. PDE, Volume 10, Number 4 (2017), 943-982.

Dates
Revised: 6 February 2017
Accepted: 7 March 2017
First available in Project Euclid: 19 October 2017

https://projecteuclid.org/euclid.apde/1508432243

Digital Object Identifier
doi:10.2140/apde.2017.10.943

Mathematical Reviews number (MathSciNet)
MR3649372

Zentralblatt MATH identifier
06715608

#### Citation

Romani, Giulio. Positivity for fourth-order semilinear problems related to the Kirchhoff–Love functional. Anal. PDE 10 (2017), no. 4, 943--982. doi:10.2140/apde.2017.10.943. https://projecteuclid.org/euclid.apde/1508432243

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