Lazer-McKenna conjecture for higher order elliptic problem with critical growth

<p style='text-indent:20px;'>It is established existence of solutions for subcritical and critical nonlinearities considering a fourth-order elliptic problem defined in the whole space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula>. The work is devoted to study a class of potentials and nonlinearities which can be periodic or asymptotically periodic. Here we consider a general fourth-order elliptic problem where the principal part is given by <inline-formula><tex-math id="M2">\begin{document}$ \alpha \Delta^2 u + \beta \Delta u + V(x)u $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ \alpha > 0, \beta \in \mathbb{R} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ V: \mathbb{R}^N \rightarrow \mathbb{R} $\end{document}</tex-math></inline-formula> is a continuous potential. Hence our main contribution is to consider general fourth-order elliptic problems taking into account the cases where <inline-formula><tex-math id="M5">\begin{document}$ \beta $\end{document}</tex-math></inline-formula> is negative, zero or positive. In order to do that we employ some fine estimates proving the compactness for the associated energy functional.</p>

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A New Estimate for the Homogenization Method for Second-Order Elliptic Problem with Rapidly Oscillating Periodic Coefficients

Journal of Function Spaces ◽

10.1155/2021/8036814 ◽

2021 ◽

Vol 2021 ◽

pp. 1-6

Author(s):

Xiong Liu ◽

Wenming He

Keyword(s):

Elliptic Problem ◽

Homogenization Method ◽

Second Order ◽

Homogenization Theory ◽

High Demand ◽

Periodic Coefficients ◽

Order Elliptic Problem ◽

Multiscale Homogenization

In this paper, we will investigate a multiscale homogenization theory for a second-order elliptic problem with rapidly oscillating periodic coefficients of the form ∂ / ∂ x i a i j x / ε , x ∂ u ε x / ∂ x j = f x . Noticing the fact that the classic homogenization theory presented by Oleinik has a high demand for the smoothness of the homogenization solution u 0 , we present a new estimate for the homogenization method under the weaker smoothness that homogenization solution u 0 satisfies than the classical homogenization theory needs.

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An elimination approach for fast parallel solution of the model fourth-order elliptic problem

Computer Physics Communications ◽

10.1016/0010-4655(82)90129-1 ◽

1982 ◽

Vol 26 (3-4) ◽

pp. 357-361 ◽

Cited By ~ 2

Author(s):

Marian Vajteršic

Keyword(s):

Fourth Order ◽

Elliptic Problem ◽

Parallel Solution ◽

Order Elliptic Problem

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An Iterative Substructuring Method for Mortar Nonconforming Discretization of a Fourth-Order Elliptic Problem in Two Dimensions

Lecture Notes in Computational Science and Engineering - Domain Decomposition Methods in Science and Engineering XVI ◽

10.1007/978-3-540-34469-8_85 ◽

2007 ◽

pp. 683-690

Author(s):

Leszek Marcinkowski

Keyword(s):

Fourth Order ◽

Elliptic Problem ◽

Two Dimensions ◽

Iterative Substructuring ◽

Order Elliptic Problem

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On the error estimates of a hybridizable discontinuous Galerkin method for second-order elliptic problem with discontinuous coefficients

IMA Journal of Numerical Analysis ◽

10.1093/imanum/drz003 ◽

2019 ◽

Vol 40 (2) ◽

pp. 1577-1600

Author(s):

Gang Chen ◽

Jintao Cui

Keyword(s):

Error Estimates ◽

Discontinuous Galerkin ◽

Elliptic Problem ◽

Posteriori Error ◽

Discontinuous Coefficients ◽

Second Order ◽

A Posteriori ◽

A Posteriori Error ◽

Hybridizable Discontinuous Galerkin ◽

Order Elliptic Problem

Abstract Hybridizable discontinuous Galerkin (HDG) methods retain the main advantages of standard discontinuous Galerkin (DG) methods, including their flexibility in meshing, ease of design and implementation, ease of use within an $hp$-adaptive strategy and preservation of local conservation of physical quantities. Moreover, HDG methods can significantly reduce the number of degrees of freedom, resulting in a substantial reduction of computational cost. In this paper, we study an HDG method for the second-order elliptic problem with discontinuous coefficients. The numerical scheme is proposed on general polygonal and polyhedral meshes with specially designed stabilization parameters. Robust a priori and a posteriori error estimates are derived without a full elliptic regularity assumption. The proposed a posteriori error estimators are proved to be efficient and reliable without a quasi-monotonicity assumption on the diffusion coefficient.

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