Global a priori bounds for weak solutions of quasilinear elliptic systems with nonlinear boundary condition

Marino, Greta; Winkert, Patrick

doi:10.1016/j.jmaa.2019.123555

Cited by 10 publications

(3 citation statements)

References 33 publications

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“…Finally, we mention papers which are very close to our topic dealing with certain types of a priori bounds for equations with p-or p(•)-structure. We refer to Ding-Zhang-Zhou [10,11], García Azorero-Peral Alonso [17], Guedda-Véron [21], Marino-Winkert [32,33], Pucci-Servadei [40], Wang [43], Winkert [44,46], Winkert-Zacher [49], Zhang-Zhou [50], Zhang-Zhou-Xue [51] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

Ho¹,

Kim²,

Winkert³

et al. 2020

Preprint

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In this paper we prove the boundedness and Hölder continuity of quasilinear elliptic problems involving variable exponents for a homogeneous Dirichlet and a nonhomogeneous Neumann boundary condition, respectively. The novelty of our work is the fact that we allow critical growth even on the boundary and so we close the gap in the papers of Fan-Zhao [Nonlinear Anal. 36 (1999), no. 3, 295-318.] and Winkert-Zacher [Discrete Contin. Dyn. Syst. Ser. S 5 (2012), no. 4, 865-878.] in which the critical cases are excluded. Our approach is based on a modified version of De Giorgi's iteration technique along with the localization method. As a consequence of our results, the C 1,αregularity follows immediately.

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Section: Introductionmentioning

confidence: 99%

The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

Ho¹,

Kim²,

Winkert³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The proof of existence of solutions to () relies on the theory of pseudomonotone operators and properties of convolution and extension operator. In order to prove a priori estimates for problem () and show the boundedness of its solutions, we develop a modified version of Moser iteration originating in References 5 and 6.…”

Section: Introductionmentioning

confidence: 99%

Existence and L‐estimates for elliptic equations involving convolution

Marino

Motreanu

2020

Comp and Math Methods

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In this article, with a fixed p ∈ (1,+∞) and a bounded domain Ω⊂ℝN, N≥2, whose boundary ∂Ω fulfills the Lipschitz regularity, we study the following boundary value problem prefix−div𝒜false(x,u,normal∇ufalse)+afalse|ufalse|pprefix−2u=ℬfalse(x,ρ.3em∗.3emEfalse(ufalse),normal∇false(ρ.3em∗.3emEfalse(ufalse)false)false)1.3emin.5emnormalΩ,9.3em𝒜false(x,u,normal∇ufalse)·ν=𝒞false(x,ufalse)6.3emon.3em∂normalΩ,2emfalse(Pfalse)where 𝒜:Ω×ℝ×ℝN→ℝN, ℬ:Ω×ℝ×ℝN→ℝ, 𝒞:∂Ω×ℝ→ℝ are Carathéodory functions, a > 0 is a constant, E:W1,p(Ω)→W1,p(ℝN) is an extension operator related to Ω, and ρ is an integrable function on ℝN. This is a novel problem that involves the nonlocal operator assigning to u the convolution ρ∗E(u) of ρ with E(u). Under verifiable conditions, we prove the existence of a (weak) solution to problem (P) by using the surjectivity theorem for pseudomonotone operators. Moreover, through a modified version of Moser iteration up to the boundary, we show that (any) weak solution to (P) is bounded.

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“…Works which are closely related to our paper dealing with certain types of double phase problems, convection terms or elliptic systems can be found in Bahrouni-Rȃdulescu-Repovš [2], Bahrouni-Rȃdulescu-Winkert [3], Cencelj-Rȃdulescu-Repovš [12], Marano-Marino-Moussaoui [23], Marano-Winkert [24], Marino-Winkert [27], Motreanu-Winkert [30], Papageorgiou-Rȃdulescu-Repovš [31], [32], [33], Rȃdulescu [36], Zhang-Rȃdulescu [37], Zheng-Gasiński-Winkert-Bai [38] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and uniqueness of elliptic systems with double phase operators and convection terms

Marino¹,

Winkert²

2020

Preprint

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In this paper we study quasilinear elliptic systems driven by socalled double phase operators and nonlinear right-hand sides depending on the gradients of the solutions. Based on the surjectivity result for pseudomonotone operators we prove the existence of at least one weak solution of such systems. Furthermore, under some additional conditions on the data, the uniqueness of weak solutions is shown.

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Global a priori bounds for weak solutions of quasilinear elliptic systems with nonlinear boundary condition

Cited by 10 publications

References 33 publications

The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

Existence and L‐estimates for elliptic equations involving convolution

Existence and uniqueness of elliptic systems with double phase operators and convection terms

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