Recently, an identity of the Picone type for the weighted p-biharmonic operator was established by the present author [5] and employed in proving comparison theorems and other qualitative results for a pair of fourth-order elliptic partial differential equations of the form ∆ ( a(x)|∆u|p−2∆u ) − c(x)|u|p−2u = 0, (1.1) and ∆ ( A(x)|∆v|p−2∆v ) − C(x)|v|p−2v = 0, (1.2) where p > 1, a,A ∈ C(Ḡ,R+), ...

The biharmonic supercritical equation ∆u = |u|p−1u, where n > 4 and p > (n + 4)/(n − 4), is studied in the whole space R as well as in a modified form with λ(1 + u) as right-hand-side with an additional eigenvalue parameter λ > 0 in the unit ball, in the latter case together with Dirichlet boundary conditions. As for entire regular radial solutions we prove oscillatory behaviour around the expl...

The aim of this article is to establish the existence of at least three‎ ‎solutions for a perturbed $p$-biharmonic equation depending on two‎ ‎real parameters‎. ‎The approach is based on variational methods‎.

In this paper, we investigate the existence of positive solutions for the ellipticequation $Delta^{2},u+c(x)u = lambda f(u)$ on a bounded smooth domain $Omega$ of $R^{n}$, $ngeq2$, with Navier boundary conditions. We show that there exists an extremal parameter$lambda^{ast}>0$ such that for $lambda< lambda^{ast}$, the above problem has a regular solution butfor $lambda> lambda^{ast}$, the probl...

∆p := ∆(|∆u|p−2∆u) is the operator of fourth order, so-called the p-biharmonic (or p-bilaplacian) operator. For p = 2, the linear operator ∆2 = ∆2 = ∆ · ∆ is the iterated Laplacian that to a multiplicative positive constant appears often in the equations of Navier-Stokes as being a viscosity coefficient, and its reciprocal operator noted (∆2)−1 is the celebrated Green’s operator (see [8]). Exis...

In this paper, we consider p-biharmonic submanifolds of aspace form. We give the necessary and sufficient conditions for a submanifold to be in space present some new properties stress p-bienergy tensor.