scholarly journals Noncoercive quasilinear elliptic operators with singular lower order terms

2021 ◽  
Vol 60 (3) ◽  
Author(s):  
Fernando Farroni ◽  
Luigi Greco ◽  
Gioconda Moscariello ◽  
Gabriella Zecca
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Obstacle Problem ◽  
Differential Operators ◽  
Existence Of A Solution ◽  
Marcinkiewicz Spaces ◽  
Higher Integrability ◽  
Elliptic Differential Operators ◽  
Quasilinear Elliptic ◽  
Lower Order Terms

AbstractWe consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular.

Quasilinear elliptic problems with singular and homogeneous lower order terms

2019 ◽  
Vol 179 ◽  
pp. 105-130 ◽  
Author(s):  
José Carmona ◽  
Tommaso Leonori ◽  
Salvador López-Martínez ◽  
Pedro J. Martínez-Aparicio
Keyword(s):  
Lower Order ◽  
Elliptic Problems ◽  
Quasilinear Elliptic Problems ◽  
Quasilinear Elliptic ◽  
Lower Order Terms

scholarly journals Existence of Solutions to Elliptic Problem with Convection Term and Lower-Order Term

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Xiaohua He ◽  
Shuibo Huang ◽  
Qiaoyu Tian ◽  
Yonglin Xu
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Order Term ◽  
Combined Effects ◽  
Convection Term ◽  
Bounded Smooth Domain ◽  
Bounded Smooth ◽  
Lower Order Terms

In this paper, we establish the existence of solutions to the following noncoercivity Dirichlet problem − div M x ∇ u + u p − 1 u = − div u E x + f x , x ∈ Ω , u x = 0 , x ∈ ∂ Ω , where Ω ⊂ ℝ N N > 2 is a bounded smooth domain with 0 ∈ Ω , f belongs to the Lebesgue space L m Ω with m ≥ 1 , p > 0 . The main innovation point of this paper is the combined effects of the convection terms and lower-order terms in elliptic equations.

A LAPLACE INTEGRAL, THE T–Y–Z EXPANSION, AND BEREZIN'S TRANSFORM ON A KÄHLER MANIFOLD

2005 ◽  
Vol 02 (03) ◽  
pp. 359-371 ◽  
Author(s):  
ANDREA LOI
Keyword(s):  
Asymptotic Expansion ◽  
Complex Manifold ◽  
Lower Order ◽  
Differential Operators ◽  
Expansion Formula ◽  
Covariant Derivatives ◽  
Open Set ◽  
Type Integral ◽  
Lower Order Terms ◽  
Laplace Type

Let M be an n-dimensional complex manifold endowed with a C∞ Kähler metric g. We show that a certain Laplace-type integral ℒm(x), when x varies in a sufficiently small open set U ⊂ M, has an asymptotic expansion [Formula: see text], where Cr:C∞(U) → C∞(U) are smooth differential operators depending on the curvature of g and its covariant derivatives. As a consequence we furnish a different proof of Lu's theorem by computing the lower order terms of Tian–Yau–Zelditch expansion in terms of the operator Cj. Finally, we compute the differential operators Qj of the expansion Ber m(f) = Σr≥0m-rQr(f) of Berezin's transform in terms of the operators Cj.

scholarly journals Adams inequalities with exact growth condition : on Rn and the Heisenberg group

2020 ◽  
Author(s):  
◽  
Liuyu Qin
Keyword(s):  
Heisenberg Group ◽  
Growth Condition ◽  
Differential Operators ◽  
Riesz Potential ◽  
Type Inequality ◽  
Integral Operators ◽  
Ground State Solution ◽  
Elliptic Differential Operators ◽  
Exact Growth Condition ◽  
Quasilinear Elliptic

In this thesis we prove sharp Adams inequality with exact growth condition for the Riesz potential as well as the more general strictly Riesz-like potentials on R[superscript n]. Then we derive the Moser-Trudinger type inequality with exact growth condition for fractional Laplacians with arbitrary 0 [less than] [alpha] [less than] n, higher order gradients and homogeneous elliptic differential operators. Next we give an application to a quasilinear elliptic equation, and prove the existence of ground state solution of this equation. Lastly, we extend our result to the Heisenberg group. By applying the same technique used in R[superscript n], we derive a sharp Adams inequality with critical growth condition on H[superscript n] for integral operators whose kernels are strictly Riesz-like on H[superscript n]. As a consequence we then derive the corresponding sharp Moser-Trudinger inequalities with exact growth condition for the powers of sublaplacian -L[subscript 0] [superscript alpha/2] when [alpha] is an even integer, and for the subgradient [del] H[subscript n].

scholarly journals A regularity result for a class of elliptic equations with lower order terms

2020 ◽  
Vol 6 (2) ◽  
pp. 751-771 ◽  
Author(s):  
Claudia Capone ◽  
Teresa Radice
Keyword(s):  
Dirichlet Problem ◽  
Lower Order ◽  
Elliptic Equations ◽  
Order Term ◽  
Regularity Result ◽  
Lower Order Term ◽  
Bounded Solutions ◽  
Differentiability Of Solutions ◽  
Higher Differentiability ◽  
Lower Order Terms

Abstract In this paper we establish the higher differentiability of solutions to the Dirichlet problem $$\begin{aligned} {\left\{ \begin{array}{ll} \text {div} (A(x, Du)) + b(x)u(x)=f &{} \text {in}\, \Omega \\ u=0 &{} \text {on} \, \partial \Omega \end{array}\right. } \end{aligned}$$ div ( A ( x , D u ) ) + b ( x ) u ( x ) = f in Ω u = 0 on ∂ Ω under a Sobolev assumption on the partial map $$x \rightarrow A(x, \xi )$$ x → A ( x , ξ ) . The novelty here is that we take advantage from the regularizing effect of the lower order term to deal with bounded solutions.

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