Anisotropic Elliptic Equations Research Articles

Liouville's type results for singular anisotropic operators

Abstract We present two Liouville-type results for solutions to anisotropic elliptic equations that have a growth of power 2 along the first s s coordinate directions and of power p p , with 1 < p < 2 1\lt p\lt 2 along the other ( N − s ) \left(N-s) ones. First, we begin our investigation by assuming that the solution is bounded only from below, deriving a rigidity result for the range p + ( N − s ) ( p − 2 ) > 0 p+\left(N-s)\left(p-2)\gt 0 of non-degeneration, which is a purely parabolic shade. Then we break free from this constraint at the price of assuming the solution to be bounded also from above.

Multiplicity of solutions for anisotropic Dirichlet problem with variable exponent

We establish some results on the existence of multiple nontrivial solutions for a general anisotropic elliptic equations. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces,combined with adequate variational methods and a variant of the Mountain Pass lemma.

THE DIRICHLET PROBLEM FOR A CLASS OF ANISOTROPIC SCHRÖDINGER-KIRCHHOFF-TYPE EQUATIONS WITH CRITICAL EXPONENT

In this paper, our focus lies in addressing the Dirichlet problem associated with a specific class of critical anisotropic elliptic equations of Schrödinger-Kirchhoff type. These equations incorporate variable exponents and a real positive parameter. Our objective is to establish the existence of at least one solution to this problem.

On the concentration-compactness principle for anisotropic variable exponent Sobolev spaces and its applications

We obtain critical embeddings and the concentration-compactness principle for the anisotropic variable exponent Sobolev spaces. As an application of these results,we confirm the existence of and find infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters. With the groundwork laid in this work, there is potential for future extensions, particularly in extending the concentration-compactness principle to anisotropic fractional order Sobolev spaces with variable exponents in bounded domains. This extension could find applications in solving the generalized fractional Brezis–Nirenberg problem.

Boundedness of solutions to singular anisotropic elliptic equations

Boundedness of solutions to singular anisotropic elliptic equations

Existence and regularity of solutions of nonlinear anisotropic elliptic problem with Hardy potential

In this paper, we are interested in the existence and regularity of solutions for some anisotropic elliptic equations with Hardy potential and L m ( Ω ) data in appropriate anisotropic Sobolev spaces. The aim of this work is to get natural conditions to show the existence and regularity results for the solutions, that is related to an anisotropic Hardy inequality.

An unconditionally stable and $$L^2$$ optimal quadratic finite volume scheme over triangular meshes for anisotropic elliptic equations

An unconditionally stable and $$L^2$$ optimal quadratic finite volume scheme over triangular meshes for anisotropic elliptic equations

Regularity results for local solutions to some anisotropic elliptic equations

In this paper we study the higher integrability of local solutions for a class of anisotropic equations with lower order terms whose growth coefficients lay in Marcinkiewicz spaces. A condition for the boundedness of such solutions is also given.

Liouville-type results for some quasilinear anisotropic elliptic equations

We prove some Liouville-type theorems for stable solutions (and solutions stable outside a compact set) of quasilinear anisotropic elliptic equations. Our results cover the particular case of the pure Finsler p-Laplacian.

Existence and multiplicity of solutions for a class of critical anisotropic elliptic equations of Schrödinger–Kirchhoff‐type

In this article, we obtain the existence and infinitely many nontrivial solutions for a class of nonlinear critical anisotropic elliptic equations involving variable exponents and two real parameters, via combining the variational method, and the concentration‐compactness principle for anisotropic variable exponent under suitable assumptions on the nonlinearities.

On the local behavior of local weak solutions to some singular anisotropic elliptic equations

Abstract We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations of the kind ∑ i = 1 s ∂ i i u + ∑ i = s + 1 N ∂ i ( A i ( x , u , ∇ u ) ) = 0 , x ∈ Ω ⊂ ⊂ R N for 1 ≤ s ≤ ( N − 1 ) , \mathop{\sum }\limits_{i=1}^{s}{\partial }_{ii}u+\mathop{\sum }\limits_{i=s+1}^{N}{\partial }_{i}({A}_{i}\left(x,u,\nabla u))=0,\hspace{1.0em}x\in \Omega \subset \hspace{-0.3em}\subset \hspace{0.33em}{{\mathbb{R}}}^{N}\hspace{1.0em}\hspace{0.1em}\text{for}\hspace{0.1em}\hspace{0.33em}1\le s\le \left(N-1), where each operator A i {A}_{i} behaves directionally as the singular p p -Laplacian, 1 < p < 2 1\lt p\lt 2 . Throughout a parabolic approach to expansion of positivity we obtain the interior Hölder continuity and some integral and pointwise Harnack inequalities.

Hermite Fitted Block Integrator for Solving Second-Order Anisotropic Elliptic Type PDEs

A Hermite fitted block integrator (HFBI) for numerically solving second-order anisotropic elliptic partial differential equations (PDEs) was developed, analyzed, and implemented in this study. The method was derived through collocation and interpolation techniques using the Hermite polynomial as the basis function. The Hermite polynomial was interpolated at the first two successive points, while the collocation occurred at all the suitably chosen points. The major scheme and its complementary scheme were united together to form the HFBI. The analysis of the HFBI showed that it had a convergence order of eight with small error constants, was zero-stable, absolutely-stable, and satisfied the condition for convergence. In order to confirm the usefulness, accuracy, and efficiency of the HFBI, the method of lines approach was applied to discretize the second-order anisotropic elliptic partial differential equation PDE into a system of second-order ODEs and consequently used the derived HFBI to obtain the approximate solutions for the PDEs. The computed solution generated by using the HFBI was compared to the exact solutions of the problems and other existing methods in the literature. The proposed method compared favorably with other existing methods, which were validated through test problems whose solutions are presented in tabular form, and the comparisons are illustrated in the curves.

APFOS-Net: Asymptotic preserving scheme for anisotropic elliptic equations with deep neural network

In this paper, a new asymptotic preserving (AP) scheme is proposed for the anisotropic elliptic equations. Different from previous AP schemes, the actual one is based on first-order system least-squares for second-order partial differential equations, and it is uniformly well-posed with respect to anisotropic strength. The numerical computation is realized by a deep neural network (DNN), where least-squares functionals are employed as loss functions to determine parameters of DNN. Numerical results show that the current AP scheme is easy for implementation and is robust to approximate solutions or to identify the anisotropic parameter in various 2D and 3D tests.

Existence of solutions for unilateral problems associated to some quasilinear anisotropic elliptic equations with measure data

Abstract In this paper, we will study the existence of solutions for some nonlinear anisotropic elliptic equation of the type { A u + g ( x , u , ∇ u ) = μ − d i v φ ( u ) i n Ω , u = 0 o n ∂ Ω , \left\{ {\matrix{{Au + g\left( {x,u,\nabla u} \right) = \mu - div\,\phi \left( u \right)} \hfill & {in\,\Omega ,} \hfill \cr {u = 0} \hfill & {on\,\,\partial \Omega ,} \hfill \cr } } \right. where A u = − ∑ i = 1 N ∂ ∂ x i a i ( x , u , ∇ u ) Au = - \sum\limits_{i = 1}^N {{\partial \over {\partial {x_i}}}{a_i}\left( {x,u,\nabla u} \right)} is a Leray-Lions operator, the Carathéodory function g(x, s, ξ) is a nonlinear lower order term that verify some natural growth and sign conditions, where the data µ = f − div F belongs to L 1−dual and ϕ (·) ∈ C 0(R, R N ).

Infinitely many solutions for anisotropic elliptic equations with variable exponent

In this article, we study the existence and multiplicity of solutions for a class of anisotropic elliptic equations First we establisch that anisotropic space is separable and by using the Fountain theorem, and dual Fountain theorem we prove, under suitable conditions, that the problem (P) admits two sequences of weak solutions.

Existence of Solutions of Anisotropic Elliptic Equations with Variable Indices of Nonlinearity in ℝn

In this paper, we consider a certain class of anisotropic second-order elliptic equations of divergent type with variable indices of nonlinearities. We examine conditions of the solvability in the whole space ℝn, n ≥ 2. We prove the existence of solutions without restrictions to the growth rate as |x| → ∞.

On Entropy Solutions of Anisotropic Elliptic Equations with Variable Nonlinearity Indices in Unbounded Domains

For a class of second-order anisotropic elliptic equations with variable nonlinearity indices and summable right-hand sides, we consider the Dirichlet problem in arbitrary unbounded domains. We prove the existence and uniqueness of entropy solutions in anisotropic Sobolev spaces with variable exponents.

Entropy and renormalized solutions for some nonlinear anisotropic elliptic equations with variable exponents and L 1-data

Abstract We prove in this paper some existence and unicity results of entropy and renormalized solutions for some nonlinear elliptic equations with general anisotropic diffusivities and variable exponents. The data are assumed to be merely integrable.

Regularity of weak supersolutions to elliptic and parabolic equations: Lower semicontinuity and pointwise behavior

We demonstrate a measure theoretical approach to the local regularity of weak supersolutions to elliptic and parabolic equations in divergence form. In the first part, we show that weak supersolutions become lower semicontinuous after redefinition on a set of measure zero. The proof relies on a general principle, i.e. the De Giorgi type lemma, which offers a unified approach for a wide class of elliptic and parabolic equations, including an anisotropic elliptic equation, the parabolic p-Laplace equation, and the porous medium equation. In the second part, we shall show that for parabolic problems the lower semicontinuous representative at an instant can be recovered pointwise from the “ess lim inf” of past times. We also show that it can be recovered by the limit of certain integral averages of past times. The proof hinges on the expansion of positivity for weak supersolutions. Our results are structural properties of partial differential equations, independent of any kind of comparison principle.

The obstacle problem for degenerate anisotropic elliptic equations with variable exponents and $L^1$-data

The obstacle problem for degenerate anisotropic elliptic equations with variable exponents and $L^1$-data