A nonlinear boundary value problem involving the p-biharmonic operator is investigated, where p > 1. It describes various problems in the theory of elasticity, e.g. the shape of an elastic beam where the bending moment depends on the curvature as a power function with exponent p− 1. We prove existence of solutions satisfying a quite general boundary condition that incorporates many particular b...

This work is devoted to study the existence and the regularity of solutions of two nonlinear problems of fourth order governed by p-biharmonic operators in nonresonance cases. In the first problem we establish the nonresonance part of the Fredholm alternative. In the second problem, nonresonance relative to the first eigenvalue is considered for p = 2 at the case where the nonresonance is betwe...

This article shows the existence and multiplicity of weak solutions for the (p, q)-biharmonic type system ∆(|∆u|p−2∆u) = λFu(x, u, v) in Ω, ∆(|∆v|q−2∆v) = λFv(x, u, v) in Ω, u = v = ∆u = ∆v = 0 on ∂Ω. Under certain conditions on F , we show the existence of infinitely many weak solutions. Our technical approach is based on Bonanno and Molica Bisci’s general critical point theorem.

Let denote the unit circle in the complex plane. Given a function , one uses t usual (harmonic) Poisson kernel for the unit disk to define the Poisson integral of , namely . Here we consider the biharmonic Poisson kernel for the unit disk to define the notion of -integral of a given function ; this associated biharmonic function will be denoted by . We then consider the dilations ...

B.Y. Chen introduced biharmonic submanifolds in Euclidean spaces and raised the conjecture ”Any biharmonic submanifold is minimal”. In this article, we show some affirmative partial answers of generalized Chen’s conjecture. Especially, we show that the triharmonic hypersurfaces with constant mean curvature are minimal. M.S.C. 2010: 58E20, 53C43.