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,

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.

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