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

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.

biharmonic operator, positivity-preserving property, semilinear problem, positive least-energy solutions, Nehari manifold
Mathematical Subject Classification 2010
Primary: 35G30, 49J40
Received: 29 June 2016
Revised: 6 February 2017
Accepted: 7 March 2017
Published: 9 May 2017
Giulio Romani
Aix Marseille Univ, CNRS, Centrale Marseille
Institute de Mathématique de Marseille (I2M)
13453 Marseille