Fourth-order Elliptic Equations Research Articles

On second-order tensor representation of derivatives in shape optimization.

In this article, we study general properties of distributed shape derivatives admitting a volumetric tensor representation of order two. We obtain a general result providing a range of expressions for the shape derivative, with the distributed shape derivative at one end of the range and the standard Hadamard formula at the other end. We further apply this result to a cost functional depending on the solution of a fourth-order elliptic equation, and obtain the distributed shape derivative in the case of open sets, and the Hadamard formula for sets of class [Formula: see text]. We also consider the case of polygons, for which a description of the weak singularities of the solution appearing in the neighbourhood of the vertices is required to obtain the Hadamard formula. This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Sign-changing Solutions for Fourth Order Elliptic Equation with Concave-convex Nonlinearities

In this paper, we study the following fourth order elliptic equation: Δ²u - Δu + V(x)u = κ(x)|u|q-2u + |u|p-2u in RN, where Δ² := Δ(Δ) is the biharmonic operator, 4 > N, 1 < q < 2 < p < 2* := 2N / (N - 4). Assuming that V(x) satisfies a class of coercive conditions and the nonnegative weighted function κ(x) belongs to Lp / (p-q)(RN), we obtain the existence of one sign-changing solution with the help of constraint variational method and quantitative deformation lemma. The novelty of this paper is that when the nonlinearity is the combination of concave and convex functions, we are able to obtain the existence of sign-changing solutions. Some recent results are improved and generalized significantly.

Multiplicity results for nonhomogenous elliptic equation involving the generalized Paneitz-Branson operator

"Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 without boundary ∂M, we consider the multiplicity result of solutions of the following nonhomogenous fourth order elliptic equation involving the generalized Paneitz-Branson operator, Pg (u) = f (x) |u|2 −2 u + h(x). Under some conditions and using critical points theory, we prove the existence of two distinct solutions of the above equation. At the end, we give a geometric example when the equation has negative and positive solutions. Keywords: Riemannian manifold, multiplicity result, nonhomogeneous, Paneitz-Branson operator, critical points theory."

Multiple Solutions to Boundary-Value Problems for Fourth-Order Elliptic Equations

Multiple Solutions to Boundary-Value Problems for Fourth-Order Elliptic Equations

On the completeness of a system of decreasing elementary solutions for one class of fourth-order operator-differential equations on the semi-axis

In this work, we study a homogeneous fourth-order elliptic operator-differential equation with an unbounded operator under non-homogeneous boundary conditions on the semi-axis. We obtain sufficient conditions for the completeness of decreasing elementary solutions of the equation under study in the space of all its solutions from a fourth-order Sobolev type space.

Infinitely many high energy solutions for fourth-order elliptic equations with p-Laplacian in bounded domain

Infinitely many high energy solutions for fourth-order elliptic equations with p-Laplacian in bounded domain

Multiple solutions to boundary-value problems for fourth-order elliptic equations

UDC 517.9 We study the existence of multiple solutions for the biharmonic problem Δ 2 u = f ( x , u ) + g ( x , u ) in Ω , u = ∂ ν u = 0 on ∂ Ω , where Ω is a bounded domain with smooth boundary in ℝ N , N > 4 , f ( x , ξ ) is odd in ξ , and g ( x , ξ ) is a perturbation term. Under certain growth conditions on f and g , we show that there are infinitely many weak solutions to the problem.

Infinitely many small energy solutions for Fourth-Order Elliptic Equations with p-Laplacian in [formula omitted

In this paper, we study the following fourth-order elliptic equation with p-Laplacian, steep potential well and sublinear perturbation: Δ2u−Δpu+μV(x)u=f(x,u)+ξ(x)|u|q−2u,x∈RN,where N≥5, Δ2≔Δ(Δ) is the biharmonic operator, Δpu=div|∇u|p−2∇u with p>2, μ>0 is a parameter, f∈CRN×R,R, ξ∈L22−qRN with 1≤q<2, we have the potential V∈C(RN,R), and V−1(0) has nonempty interior. Under certain assumptions on V and f, we show the existence nontrivial solutions by virtue of variational methods, the existence criteria of infinitely many nontrivial small energy solutions are established.

Infinitely many high energy solutions for nonlocal fourth-order equation with sign-changing potential

This paper is devoted to a class of general nonlocal fourth-order elliptic equation where is the biharmonic operator, are constants and the potential V is allowed to be sign-changing. Under certain assumptions on f, infinitely many high energy solutions are established. Our results enrich that obtained in the literature.

Solvability of boundary-value problem with bounded operator in boundary conditions for fourth-order elliptic operator-differential equation

In the paper, we derive sufficient conditions for well-posed and unique solvability of a boundary-value problem with a bounded operator in the boundary conditions for a class of fourth-order elliptic operator-differential equations. At the same time, we make estimates for the norms of operators of intermediate derivatives closely related to the solvability conditions, which are expressed using the properties of the operator coefficients of the boundary-value problem.

Nontrivial solutions of modified nonlinear fourth-order elliptic equation in [formula omitted

This paper is concerned with the following fourth-order elliptic equation of the form: Δ2u−Δu−uΔ(u2)+u=|u|p−2u,x∈RN,where Δ2u≔Δ(Δu), 4≤N≤6 and p∈2,2∗. By making use of the constrained minimization on a special manifold, we prove that the existence of nontrivial solutions of the above problem for any p∈2,2∗. Our result can be regarded as a partial extension of some related literature with 4<p<2∗.

A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

&lt;abstract&gt;&lt;p&gt;We discuss the design and implementation details of two conforming virtual element methods for the numerical approximation of two partial differential equations that emerge in phase-field modeling of fracture propagation in elastic material. The two partial differential equations are: (i) a linear hyperbolic equation describing the momentum balance and (ii) a fourth-order elliptic equation modeling the damage of the material. Inspired by &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b1"&gt;1&lt;/xref&gt;,&lt;xref ref-type="bibr" rid="b2"&gt;2&lt;/xref&gt;,&lt;xref ref-type="bibr" rid="b3"&gt;3&lt;/xref&gt;]&lt;/sup&gt;, we develop a new conforming VEM for the discretization of the two equations, which is implementation-friendly, i.e., different terms can be implemented by exploiting a single projection operator. We use $ C^0 $ and $ C^1 $ virtual elements for the second-and fourth-order partial differential equation, respectively. For both equations, we review the formulation of the virtual element approximation and discuss the details pertaining the implementation.&lt;/p&gt;&lt;/abstract&gt;

Multiplicity results for some fourth-order elliptic equations with combined nonlinearities

&lt;abstract&gt;&lt;p&gt;In this paper, we regard with some fourth-order elliptic boundary value problems involving subcritical polynomial growth and subcritical (critical) exponential growth. Some new existence and multiplicity results are established by using variational methods combined Adams inequality.&lt;/p&gt;&lt;/abstract&gt;

An efficient Fourier spectral method and error analysis for the fourth order problem with periodic boundary conditions and variable coefficients

&lt;abstract&gt;&lt;p&gt;We propose in this paper an efficient algorithm based on the Fourier spectral-Galerkin approximation for the fourth-order elliptic equation with periodic boundary conditions and variable coefficients. First, by using the Lax-Milgram theorem, we prove the existence and uniqueness of weak solution and its approximate solution. Then we define a high-dimensional $ L^2 $ projection operator and prove its approximation properties. Combined with Céa lemma, we further prove the error estimate of the approximate solution. In addition, from the Fourier basis function expansion and the properties of the tensor, we establish the equivalent matrix form based on tensor product for the discrete scheme. Finally, some numerical experiments are carried out to demonstrate the efficiency of the algorithm and correctness of the theoretical analysis.&lt;/p&gt;&lt;/abstract&gt;

Nontrivial Solutions of a Class of Fourth-Order Elliptic Problems with Potentials

This paper deals with a fourth-order elliptic equation with Dirichlet boundary conditions. Using a variant form of the mountain pass theorem, we prove the existence of nontrivial solutions to this problem. Furthermore, we discuss the fundamental properties of the representation of the solution by considering two cases. Our results not only make previous results more general but also show new insights into fourth-order elliptic problems.

Ground State Solution for a Fourth Order Elliptic Equation of Kirchhoff Type with Critical Growth in ℝ N

In this paper, we consider the following fourth order elliptic Kirchhoff-type equation involving the critical growth of the form Δ 2 u − a + b ∫ ℝ N ∇ u 2 d x Δ u + V x u = I α ∗ F u f u + λ u 2 ∗ ∗ − 2 u , in ℝ N , u ∈ H 2 ℝ N , where a &gt; 0 , b ≥ 0 , λ is a positive parameter, α ∈ N − 2 , N , 5 ≤ N ≤ 8 , V : ℝ N ⟶ ℝ is a potential function, and I α is a Riesz potential of order α . Here, 2 ∗ ∗ = 2 N / n − 4 with N ≥ 5 is the Sobolev critical exponent, and Δ 2 u = Δ Δ u is the biharmonic operator. Under certain assumptions on V x and f u , we prove that the equation has ground state solutions by variational methods.

Existence of Hölder Continuous Solutions for a Class of Degenerate Fourth-Order Elliptic Equations

This paper deals with the existence of bounded and locally Hölder continuous weak solutions of a homogeneous Dirichlet problem related to a class of nonlinear fourth-order elliptic equations with strengthened degenerate ellipticity condition.

On a viscous fourth-order parabolic equation with boundary degeneracy

A viscous fourth-order parabolic equation with boundary degeneracy is studied. By using the variational method, the existence of a time-discrete fourth-order elliptic equation with homogeneous boundary conditions is solved. Moreover, the existence and uniqueness for the corresponding parabolic problem with nondegenerate coefficient is shown by several asymptotic limit processes. Finally, by applying the regularization method, the existence and uniqueness for the problem with degenerate boundary coefficient is obtained by applying the energy method and a small parameter limit process.

On a Boundary Problem for a Fourth-Order Elliptic Equation on a Plane

On a Boundary Problem for a Fourth-Order Elliptic Equation on a Plane

New types of Caffarelli-Kohn-Nirenberg inequalities and applications

&lt;p style='text-indent:20px;'&gt;New types of Caffarelli-Kohn-Nirenberg inequalities are established in some weighted Sobolev spaces. The corresponding embeddings of the weighted Sobolev spaces are presented via such inequalities. Using such embeddings, we obtain existence and regularity of positive solutions of boundary value problems of a weighted fourth order elliptic equation.&lt;/p&gt;