Fourth-order Elliptic Equations Research Articles

Multiplicity results for fourth order elliptic equations of Kirchhoff-type

In this paper, we concern with the following fourth order elliptic equations of Kirchhoff type {Δ2u-(a+b∫R3|∇u|2dx)Δu=f(x,u),x∈ℝ3,u∈H2(ℝ3), where a,b > 0 are constants and the primitive of the nonlinearity f is of superlinear growth near infinity in u and is also allowed to be sign-changing. By using variational methods, we establish the existence and multiplicity of solutions. Our conditions weaken the Ambrosetti-Rabinowitz type condition.

Numerical methods for fourth-order elliptic equations with nonlocal boundary conditions

This paper is concerned with some numerical methods for a fourth-order semilinear elliptic boundary value problem with nonlocal boundary condition. The fourth-order equation is formulated as a coupled system of two second-order equations which are discretized by the finite difference method. Three monotone iterative schemes are presented for the coupled finite difference system using either an upper solution or a lower solution as the initial iteration. These sequences of monotone iterations, called maximal sequence and minimal sequence respectively, yield not only useful computational algorithms but also the existence of a maximal solution and a minimal solution of the finite difference system. Also given is a sufficient condition for the uniqueness of the solution. This uniqueness property and the monotone convergence of the maximal and minimal sequences lead to a reliable and easy to use error estimate for the computed solution. Moreover, the monotone convergence property of the maximal and minimal sequences is used to show the convergence of the maximal and minimal finite difference solutions to the corresponding maximal and minimal solutions of the original continuous system as the mesh size tends to zero. Three numerical examples with different types of nonlinear reaction functions are given. In each example, the true continuous solution is constructed and is used to compare with the computed solution to demonstrate the accuracy and reliability of the monotone iterative schemes.

Multiple solutions for biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth

The main purpose of this paper is to establish the existence of three nontrivial solutions for a class of fourth-order elliptic equations with subcritical polynomial growth and subcritical exponential growth by using the minimax method and Morse theory.

Compactness properties for geometric fourth order elliptic equations with application to the Q-curvature flow

Abstract We prove the compactness of solutions of general fourth order elliptic equations which are L 1 L^{1} -perturbations of the Q-curvature equation on compact Riemannian 4-manifolds. Consequently, we prove the global existence and convergence of the Q-curvature flow on a generic class of Riemannian 4-manifolds. As a by-product, we give a positive answer to an open question by A. Malchiodi [J. reine angew. Math. 594 (2006), 137–174] on the existence of bounded Palais–Smale sequences for the Q-curvature problem when the Paneitz operator is positive with trivial kernel.

On the dimension of the kernel of the Dirichlet problem for fourth-order equations

We study the dimension of the kernel of the Dirichlet problem in the disk for fourth-order elliptic equations with constant complex coefficients in general position. Special attention is paid to improperly elliptic equations, because the kernel of the Dirichlet problem for properly elliptic equations is finite-dimensional. We single out classes of improperly elliptic equations that have properties similar to those of properly elliptic equations: the kernel of the Dirichlet problem is also finite-dimensional and in some cases even trivial. We consider fourth-order equations in general position, including equations of principal type as well as equations that have multiple characteristics and whose characteristic equation has roots ±i.

Residual‐based a posteriori error control for Space‐time finite element approximations of parabolic optimal control problems

AbstractIn this paper we investigate a space‐time finite element approximation of parabolic optimal control problems. The first order optimality conditions are transformed into an elliptic equation of fourth order in space and second order in time involving only the state or the adjoint state in the space‐time domain. We derive a priori and a posteriori error estimates for the time discretization of the state and the adjoint state. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On a fourth order elliptic equation with supercritical exponent

This paper is concerned with the semi-linear elliptic problem involving nearly critical exponent : in Ω, on ∂ Ω, where Ω is a smooth bounded domain in , , and ε is a positive real parameter. We show that, for ε small, has no sign-changing solutions with low energy which blow up at exactly three points. Moreover, we prove that has no bubble-tower sign-changing solutions. MSC:35J20, 35J60.

Radial solutions of inhomogeneous fourth order elliptic equations and weighted Sobolev embeddings

Abstract We deal with a class of inhomogeneous elliptic problems involving the biharmonic operator Δ2 u + V(|x|)|u| q-2 u = Q(|x|)f(u), u ∈ D 0 2,2(ℝ N ), where Δ2 is the biharmonic operator and V, Q are singular continuous functions. Compact embedding results are established and by using these facts some existence results are obtained.

Ground states for a class of asymptotically linear fourth-order elliptic equations

In this paper, we study the following fourth-order elliptic equation,where is the biharmonic operator, and . We apply the variational methods to obtain the existence of ground-state solutions for asymptotically linear case.

Existence and multiplicity of solutions for fourth-order elliptic Kirchhoff equations with potential term

In this paper, we consider the existence of nodal and multiple solutions for non-linear fourth-order elliptic Kirchhoff equationwhere . Making use of Mountain Pass Theorem, we establish existence of a non-trivial solution when is superlinear and subcritical. We also prove the existence of a positive solution, a negative solution and a nodal solution by using invariant sets of descending flow. Moreover, we show this nodal solution has a least energy and precisely two nodal domains. Under additional assumptions on , we obtain three existence results of infinitely many non-trivial and radial solutions via Fountain Theorem and Dual Fountain Theorem. We also show the existence of infinitely many nodal solutions by means of a sign-changing critical theorem.

Existence and multiplicity of nontrivial solutions for a class of modified nonlinear fourth-order elliptic equations on [formula omitted

In this paper, we study the following fourth-order elliptic equation of the form▵2u-▵u+V(x)u-κ2▵(u2)u=h(x,u),x∈RN,where ▵2≔▵(▵) is the biharmonic operator, κ⩾0, N⩽6, V∈C(RN,R) and h∈C(RN×R,R). Under appropriate assumptions on V(x) and h(x,u), some existence and multiplicity results for nontrivial solutions are obtained via variational methods.

High energy solutions for the fourth-order elliptic equations in R N

In the present paper, we study the following fourth-order elliptic equations: , for , , where , . Under more relaxed assumptions on , by using some special techniques, a new existence result of high energy solutions is obtained via the symmetric mountain pass theorem.

Existence and multiplicity of solutions for fourth-order elliptic equations of Kirchhoff type via genus theory

In this paper, we study the following fourth-order elliptic equations of Kirchhoff type: , in , , where are constants, we have the potential and the nonlinearity . Under certain assumptions on and , we show the existence and multiplicity of negative energy solutions for the above system based on the genus properties in critical point theory. MSC: 35J20, 35J65, 35J60.

Multiple solutions to singular fourth order elliptic equations on compact manifolds

Using the method of Nehari manifolds, we prove the existence of at least two distinct weak solutions to elliptic equation of four order with singularities and with critical Sobolev growth.

Fourth-order Elliptic Equations

Abstract We prove an existence result for a fourth order elliptic equation where the associated functional does not satisfy the Palais-Smale condition. We use some truncations argument and L∞-norm estimates.

On the asymptotics of the solution of the Dirichlet problem for a fourth-order equation in a layer

The Dirichlet problem for a fourth-order elliptic equation with constant coefficients without first derivatives is considered in the region (layer) $$\prod { = \left\{ {(x',x_n ) \in R^n |x' \in R^{n - 1} ,x_n \in (a,b)} \right\}} , - \infty < a < b < + \infty ,n \geqslant 3$$ The first term of the asymptotics of the solution at infinity is obtained.

Biharmonic equations with improved subcritical polynomial growth and subcritical exponential growth

The main purpose of this paper is to establish the existence of two nontrivial solutions and the existence of infinitely many solutions for a class of fourth-order elliptic equations with subcritical polynomial growth and subcritical exponential growth by using a suitable version of the mountain pass theorem and the symmetric mountain pass theorem.

A fourth order elliptic equation with a singular nonlinearity

In this paper, we study the structure of solutions of a fourth orderelliptic equation with a singular nonlinearity. For differentboundary values $\kappa$, we establish the global bifurcationbranches of solutions to the equation. More precisely, we show that$\kappa=1$ is a critical boundary value to change the structure ofsolutions to this problem.

Qualitative properties and standard estimates of solutions for some fourth order elliptic equations

In this paper, first, we make the uniform estimates for a class of fourth-order elliptic system in bounded and smooth domains. Second, we study the qualitative properties of solutions with prescribed integration in \(R^4\). Finally, we also will obtain some radially symmetric results by using moving planes methods.

INFINITELY MANY SOLUTIONS FOR FOURTH-ORDER ELLIPTIC EQUATIONS WITH SIGN-CHANGING POTENTIAL

In this paper, we study the following fourth-order elliptic equation $$ \left\{ \begin{array}{ll} \Delta^{2}u-\Delta u+V(x)u=f(x, u), x\in\mathbb{R}^{N}, u\in H^{2}(\mathbb{R}^{N}), \end{array} \right. $$ where the potential $V\in C(\mathbb{R}^N, \mathbb{R})$ is allowed to be sign-changing. Under the weakest superquadratic conditions, we establish the existence of infinitely many solutions via variational methods for the above equation. Recent results from the literature are extended.