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

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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 2 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 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

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

Received: 29 June 2016
Revised: 6 February 2017
Accepted: 7 March 2017
First available in Project Euclid: 19 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35G30: Boundary value problems for nonlinear higher-order equations 49J40: Variational methods including variational inequalities [See also 47J20]

biharmonic operator positivity-preserving property semilinear problem positive least-energy solutions Nehari manifold


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.

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