Nonlinear Elliptic Fourth-order Equations Research Articles

On the existence of continuous solutions for nonlinear fourth-order elliptic equations with strongly growing lower-order terms

In this article, we consider nonlinear elliptic fourth-order equations with the monotone principal part satisfying the common growth and coerciveness conditions for Sobolev space $W^{2,p}(\Omega )$, $\Omega \subset \mathbb {R}^{n}$. It is supposed that the lower-order term of the equations admits arbitrary growth with respect to an unknown function and is arbitrarily close to the growth limit with respect to the derivatives of this function. We assume that the lower-order term satisfies the sign condition with respect to the unknown function. We prove the existence of continuous generalized solutions for the Dirichlet problem in the case $n=2p$.

Nonlinear elliptic fourth order equations existence and multiplicity results

This paper deals with the existence of solutions to a class of fourth order nonlinear elliptic equations. The technique used relies on critical points theory. The solutions appeared as critical points of a functional restricted to a suitable manifold. In the case of constant coefficients we obtain the existence of three distinct solutions.

Existence of bounded solutions for a class of nonlinear fourth-order equations

In this article, we consider a class of nonlinear elliptic fourth-order equations with the principal part satisfying a strengthened coercivity condition. It is supposed that the lower- order term of the equations admits an arbitrary growth with respect to unknown function and the growth rates of derivatives of this function coinciding with the exponents of the corresponding energy space. We prove a theorem on existence of bounded generalized solutions of the Dirichlet problem for equations of the given class.

On the sets of boundedness of solutions for a class of degenerate nonlinear elliptic fourth-order equations with L 1-data

In this article, we deal with a class of degenerate, nonlinear, elliptic fourth-order equations in divergence form with coefficients satisfying a strengthened ellipticity condition and right-hand sides of the class L1 depending on the unknown function. We consider the Dirichlet problem for equations of the given class and prove the existence of solutions of this problem bounded on the sets where the behavior of the data of the problem and the weighted functions involved is sufficiently regular.

On regularity up to the boundary of solutions to a system of degenerate nonlinear elliptic fourth-order equations

Under some hypotheses on weighted functions, using the interior regularity results established in (Kovalevsky, A. and Nicolosi, F., 2005, Existence and regularity of solutions to a system of degenerate nonlinear fourth-order equations. Nonlinear Analysis, 61, 281–307) and estimating the oscillation of solutions near the boundary of Ω, we establish results on regularity up to the boundary of a solutions of the system (1.1).

On the sets of [formula omitted]-regularity of solutions for a class of degenerate nonlinear problems with slightly regular data

We consider the Dirichlet problem for a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened ellipticity condition involving two weighted functions. The right-hand sides F ( x , u ) of the equations depend on the unknown function u . Under some conditions, including L m -regularity of F ( ⋅ , 0 ) with m close to 1, we establish that the problem under consideration has a W -solution bounded on the sets where the behaviour of the data of the problem is regular enough.

On the Sets of Regularity of Solutions for a Class of Degenerate Nonlinear Elliptic Fourth-Order Equations with L1 Data

We establish Holder continuity of generalized solutions of the Dirichlet problem, associated to a degenerate nonlinear fourth-order equation in an open bounded set , with data, on the subsets of where the behavior of weights and of the data is regular enough.

On improvement of summability properties of solutions of some degenerate nonlinear fourth-order equations

We consider a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened ellipticity condition involving two weighted functions. The right-hand side F (x, u) of the equations depends on the unknown function u. Under some conditions, including Lm -summability of F(·, 0) with m close to 1 and restrictions on the exponent σ of the growth of F (x, u) with respect to u, we establish existence of W-solutions of the Dirichlet problem for the given equations and describe the dependence of summability properties of these solutions on m and σ.

Summability of solutions of some degenerate nonlinear elliptic fourth-order equations

We consider a class of degenerate nonlinear elliptic fourth-order equations in divergence form. Coefficients of the equations satisfy a strengthened ellipticity condition involving two weighted functions. Existence of solutions of Dirichlet problem for equations of the given class in the case where these equations have L 1-right-hand sides depending on unknown function has been established in our recent article. In the present article we prove some results which show the improvement of summability of solutions of the same problem under additional conditions on right-hand sides of the equations under consideration.

Solvability of Dirichlet problem for a class of degenerate anisotropic equations with L1L1-right-hand sides

We study Dirichlet problem for a class of degenerate anisotropic nonlinear elliptic fourth order equations with L1-right-hand sides. Anisotropic and degenerate character of the equations under consideration is expressed by the presence in their coefficients of different exponents and weighted functions associated to the first and second order derivatives of unknown function. Under some hypotheses on the involved exponents and weighted functions we prove existence of so-called Hν,μ-solution and H-solution of the given problem.

Solvability of Dirichlet problem for a class of degenerate anisotropic equations with [formula omitted]-right-hand sides

We study Dirichlet problem for a class of degenerate anisotropic nonlinear elliptic fourth order equations with L 1 -right-hand sides. Anisotropic and degenerate character of the equations under consideration is expressed by the presence in their coefficients of different exponents and weighted functions associated to the first and second order derivatives of unknown function. Under some hypotheses on the involved exponents and weighted functions we prove existence of so-called H ν , μ -solution and H-solution of the given problem.

On the summability of entropy solutions for the Dirichlet problem in a class of non-linear elliptic fourth-order equations

We present some results on the summability of entropy solutions for the Dirichlet problem in a class of fourth-order non-linear elliptic equations in relation to integrability properties of their right-hand sides. Consideration is given to the cases in which the right-hand sides belong to Lebesgue spaces with exponents close to unity and also to some wider sets contained in L1.

Existence of Solutions of Some Degenerate Nonlinear Elliptic Fourth-order Equations with L 1 -data

This article deals with a class of degenerate nonlinear elliptic fourth-order equations with L 1 -right-hand sides. Equations of the given class have divergence form and their coefficients satisfy a strengthened ellipticity condition with two different weights associated, respectively, to the first- and the second-order derivatives of unknown function. Under suitable hypotheses on the weighted functions involved we establish solvability in a Sobolev space of Dirichlet problem for equations under consideration.

Entropy solutions of the Dirichlet problem for a class of non-linear elliptic fourth-order equations with right-hand sides in

In this paper we introduce and study the notion of an entropy solution of the Dirichlet problem for a class of non-linear elliptic fourth-order equations whose right-hand sides admit arbitrary growth with respect to the variable corresponding to the unknown function and belong to the space for each fixed value of this variable. We prove the existence and uniqueness of an entropy solution. We establish the existence of so-called -solutions and -solutions of the problem and prove that the entropy solutions belong to certain Sobolev spaces.