P-biharmonic Operator Research Articles

Multiple solutions for a class of [formula omitted]-biharmonic problems with potential boundary conditions

We are concerned with multiplicity of solutions for differential inclusions involving the p-biharmonic operator, submitted to a general potential multivalued boundary condition which, among others, covers many particular boundary conditions that are frequently invoked in the literature. Our approach is a variational one and relies on non-smooth critical point theory.

Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities

This article concerns the existence and the nonexistence of solution for the following boundary problem involving the p-biharmonic operator and singular nonlinearities, Δ p 2 u = u γ − 1 u + μ u − 1 − α / x β u in Ω and u = ∂ u / ∂ n = 0 on ∂ Ω , where 4 < 2 p < N , 0 ∈ Ω , − ∞ < μ < μ ∗ , = N − 2 p 1 − α / p N , p < γ < p ∗ = p N / N − 2 p is the critical Sobolev exponent, 0 ≤ β < N γ + α / γ + 1 , 0 < α < 1 . Under some sufficient conditions on coefficients, we prove the existence of at least one nontrivial solutions in E by using variational methods. By using the Pohozaev identity type, we show the nonexistence of positive solution when Ω ⊂ ℝ N be a bounded, smoothandstrictlystar-shapeddomain, β = 0 and γ ≥ γ ∗ , = pN 1 − α / N − 2 p 1 − α − μ Np > p ∗ = pN / N − 2 p .

Multiple Solutions to p-Biharmonic Equations of Kirchhoff Type with Vanishing Potential

In this paper, we study the p-biharmonic equation of Kirchhoff type where is a positive parameter, is the p-Laplacian operator and is the p-biharmonic operator, V, K, g are nonnegative functions, V is vanishing at infinity in the sense that When the nonlinear term satisfies some suitable conditions, we prove that the above problem has at least two nontrivial solutions using the mountain pass theorem combined with the Ekeland variational principle.

Infinitely many solutions for a nonlinear Navier problem involving the p-biharmonic operator

In this paper we establish some results of existence of infinitely many solutions for an elliptic equation involving the p-biharmonic and the p-Laplacian operators coupled with Navier boundary conditions where the nonlinearities depend on two real parameters and do not satisfy any symmetric condition. The nature of the approach is variational and the main tool is an abstract result of Ricceri. The novelty in the application of this abstract tool is the use of a class of test functions which makes the assumptions on the data easier to verify.

On eigenvalues of p-biharmonic operator and associated concave-convex type equation

In this article, we are interested in the simplicity and the existence of the first eigensurface for the third order spectrum of p-biharmonic operator plus potential with weights and the existence of multiple solutions for associated concave-convex type equation under Navier boundary conditions.

Existence and multiplicity of solutions for a singular problem involving the p-biharmonic operator in [formula omitted

In this paper, we are interested in the existence and the multiplicity of solutions for the following singular p-biharmonic problem with Rellich potentialΔp2u−μ|u|p−2u|x|2p−Δpu=λf(x)uq−1+g(x)um−1, in RNu(x)>0, in RN where 1<p<N2, 0<m<1, p<q<pNN−2p, λ is a positive real parameter and 0<μ<γN,p with γN,p=(N(p−1)(N−2p)p2)p. Under some appropriate assumptions on the functions f and g, by using the Nehari manifold combined with the fibering maps, we obtain the existence of two positive entire solutions.

Eigenvalues for the p-Biharmonic Dirichlet-Steklov problem with weights

This note is concerned with a weighted Dirichlet-Steklov problem driven by the p-biharmonic operator. Our approach is based on variational method and Ljusternick-Schnirelmann principle, we establish that the above problem admits a non-decreasing sequence of non-negative eigenvalues.

On the Spectrum of the p-Biharmonic Operator Under the Ricci Flow

This paper is devoted to the study of the behaviour of the spectrum of the p-biharmonic operator on a complete closed Riemannian manifold evolving by the Ricci flow. In particular, evolution formulas, monotonicity properties and differentiability for the least nonzero eigenvalue are derived along the flow. As a by-product, several monotone quantities involving the first eigenvalue are obtained under the flow. These monotone quantities depend on the Euler characteristics of compact surfaces in the case $$n=2$$. Furthermore, the spectrum diverges in a finite time under some geometric condition on the curvature.

A generalized nonlinear Picone identity for the p-biharmonic operator and its applications

A generalized nonlinear Picone identity for the p-biharmonic operator is established in this paper. As applications, a Sturmian comparison principle to the p-biharmonic equation with singular term, a Liouville’s theorem to the p-biharmonic system, and a generalized Hardy–Rellich type inequality are obtained.

ON APPROXIMATION OF STATE-CONSTRAINED OPTIMAL CONTROL PROBLEM IN COEFFICIENTS FOR p-BIHARMONIC EQUATION

We study a Dirichlet-Navier optimal design problem for a quasi-linear mono- tone p-biharmonic equation with control and state constraints. The coecient of the p-biharmonic operator we take as a design variable in BV ( )\L1( ). In order to handle the inherent degeneracy of the p-Laplacian and the pointwise state constraints, we use regularization and relaxation approaches. We derive existence and uniqueness of solutions to the underlying boundary value problem and the optimal control problem. In fact, we introduce a two-parameter model for the weighted p-biharmonic operator and Henig approximation of the ordering cone. Further we discuss the asymptotic behaviour of the solutions to regularized problem on each (; k)-level as the parameters tend to zero and innity, respectively.

Infinitely many solutions for a fourth order singular elliptic problem

Here, a fourth order singular elliptic problem involving p-biharmonic operator with Dirichlet boundary condition is established where the exponent in the singular term is different from that in the p-biharmonic operator. The existence of infinitely many solutions is proved by the variational methods in Sobolev spaces and the critical points principle of Ricceri. Finally, an example is presented.

On Henig Regularization of Material Design Problems for Quasi-Linear &amp;lt;i&amp;gt;p&amp;lt;/i&amp;gt;-Biharmonic Equation

We study a Dirichlet optimal design problem for a quasi-linear monotone p-biharmonic equation with control and state constraints. We take the coefficient of the p-biharmonic operator as a design variable in . In this article, we discuss the relaxation of such problem.

Resonance problems for the weighted p-biharmonic operator

Resonance problems for the weighted p-biharmonic operator

Entire solutions for a class of elliptic equations involving p-biharmonic operator and Rellich potentials

We study existence, multiplicity and qualitative properties of entire solutions for a noncompact problem related to p-biharmonic type equations with weights. More precisely, we deal with the following family of equationsΔp2u=λ|x|−2p|u|p−2u+|x|−β|u|q−2uin RN, where N>2p, p>1, q>p, β=N−qp(N−2p) and λ∈R is smaller than the Rellich constant.

Existence of solutions for a fourth order problem at resonance

In this work, we are interested at the existence of nontrivial solutions of two fourth order problems governed by the weighted p-biharmonic operator. The first is the following$$\Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda_1 m(x)|u|^{p-2}u+f(x,u)-h \mbox{ in }\Omega,\,\, u=\Delta u=0 \mbox{ on }\partial\Omega,$$where $\lambda_1$ is the first eigenvalue for the eigenvalue problem$ \Delta(\rho|\Delta u|^{p-2}\Delta u)=\lambda m(x) |u|^{p-2}u\mbox{in }\Omega, \,\, u=\Delta u=0 \mbox{ on } \partial\Omega.$ \\In the seconde problem, we replace $\lambda_1$ by $\lambda$ suchthat $\lambda_1&lt;\lambda&lt;\bar{\lambda}$, where $\bar{\lambda}$ is given bellow.

On the p-Biharmonic Operator with Critical Sobolev Exponent and Nonlinear Steklov Boundary Condition

We show that this operator possesses at least one nondecreasing sequence of positive eigenvalues. A direct characterization of the principal eigenvalue (the first one) is given that we apply to study the spectrum of the p-biharmonic operator with a critical Sobolev exponent and the nonlinear Steklov boundary conditions using variational arguments and trace critical Sobolev embedding.

On the p-biharmonic operator with critical Sobolev exponent

We study the existence of solutions for a $p$-biharmonic problem with a critical Sobolev exponent and Navier boundary conditions, using variational arguments. We establish the existence of a precise interval of parameters for which our problem admits a no

On the first eigensurface for the third order spectrum of p-biharmonic operator with weight

In this work, we will study the simplicity and the isolation of the first eigensurface for the spectrum of the operator Δ 2u+2β.∇(|Δu| p−2 Δu)+ |β| 2 |Δu| p−2 Δu, where β ∈ IR N under Navier boundary conditions.

On the spectrum of the p-biharmonic operator involving p-Hardy's inequality

On the spectrum of the p-biharmonic operator involving p-Hardy's inequality

The third order spectrum of the p-biharmonic operator with weight

The third order spectrum of the p-biharmonic operator with weight