Moser iteration applied to elliptic equations with critical growth on the boundary

Marino, Greta; Winkert, Patrick

doi:10.1016/j.na.2018.10.002

Cited by 30 publications

(17 citation statements)

References 19 publications

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“…From this point, we proceed as in Theorem 3.1, Case I.1 of Reference 5, with

‖ u u_{h}^{κ} ‖_{L^{p} (Ω)}

replaced by

‖ u u_{h}^{κ} ‖_{L^{p \tilde{r}} (Ω)}

, which gives us

\begin{align} ‖ u ‖_{L^{false(κ + 1 false) p^{*}} false(normalΩ false)} \leq M_{14} false(κ, u false), \end{align}

for any κ > 0, where M 14 (κ, u ) is a positive constant which depends on κ and on the solution u . Consequently, the claim that u ∈ L r (Ω) for every r ∈ [1, +∞) follows.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Once the L r (Ω)‐bound is reached, the proof of the L r (∂Ω)‐boundedness is straightforward (see Case I.2 of Reference 5).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Taking advantage of (), we fix

q_{1} \in (p \tilde{r}, p^{*})

and q 2 ∈ ( p , p ∗ ). By Hölder's inequality and the obtained L r ‐bounds in Ω and on ∂Ω, we can express () in the form

\begin{align} \frac{κ p + 1}{{false(κ + 1 false)}^{p}} ‖ u u_{h}^{κ} ‖_{W^{1, p} false(normalΩ false)}^{p} & \leq M_{15} ((), \frac{κ p + 1}{{false(κ + 1 false)}^{p}} + κ p + 2) ‖ u u_{h}^{κ} ‖_{L^{q_{1}} false(normalΩ false)}^{p} \\ + M_{16} ‖ u u_{h}^{κ} ‖_{L^{q_{2}} false(\partial normalΩ false)}^{p} + M_{17} false(κ + 1 false) . \end{align}

Then, proceeding as in Case II.1 of Reference 5, arranging the constants and applying Hölder's inequality, the Sobolev embedding, and Fatou's lemma, we achieve

\begin{align} ‖ u ‖_{L^{false(κ_{n} + 1 false) p^{*}} false(normalΩ false)} \leq M_{18}, \end{align}

where M 18 is independent on κ and (κ n +1) p ∗ →∞ as n →∞.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…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%

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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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“…From this point, we proceed as in Theorem 3.1, Case I.1 of Reference 5, with

‖ u u_{h}^{κ} ‖_{L^{p} (Ω)}

replaced by

‖ u u_{h}^{κ} ‖_{L^{p \tilde{r}} (Ω)}

, which gives us

\begin{align} ‖ u ‖_{L^{false(κ + 1 false) p^{*}} false(normalΩ false)} \leq M_{14} false(κ, u false), \end{align}

for any κ > 0, where M 14 (κ, u ) is a positive constant which depends on κ and on the solution u . Consequently, the claim that u ∈ L r (Ω) for every r ∈ [1, +∞) follows.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Once the L r (Ω)‐bound is reached, the proof of the L r (∂Ω)‐boundedness is straightforward (see Case I.2 of Reference 5).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Taking advantage of (), we fix

q_{1} \in (p \tilde{r}, p^{*})

and q 2 ∈ ( p , p ∗ ). By Hölder's inequality and the obtained L r ‐bounds in Ω and on ∂Ω, we can express () in the form

\begin{align} \frac{κ p + 1}{{false(κ + 1 false)}^{p}} ‖ u u_{h}^{κ} ‖_{W^{1, p} false(normalΩ false)}^{p} & \leq M_{15} ((), \frac{κ p + 1}{{false(κ + 1 false)}^{p}} + κ p + 2) ‖ u u_{h}^{κ} ‖_{L^{q_{1}} false(normalΩ false)}^{p} \\ + M_{16} ‖ u u_{h}^{κ} ‖_{L^{q_{2}} false(\partial normalΩ false)}^{p} + M_{17} false(κ + 1 false) . \end{align}

Then, proceeding as in Case II.1 of Reference 5, arranging the constants and applying Hölder's inequality, the Sobolev embedding, and Fatou's lemma, we achieve

\begin{align} ‖ u ‖_{L^{false(κ_{n} + 1 false) p^{*}} false(normalΩ false)} \leq M_{18}, \end{align}

where M 18 is independent on κ and (κ n +1) p ∗ →∞ as n →∞.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Existence and L‐estimates for elliptic equations involving convolution

Marino

Motreanu

2020

Comp and Math Methods

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show abstract

“…No growth condition or any monotonicity condition is needed in our proof which is in contrast to other works in this direction, see for example the recent paper of Gasiński-Papageorgiou [9,Proposition 3.4]. In addition, we present a priori bounds for weak solutions of problem (1.1) which are based on the recent work of Marino-Winkert [16] by applying Moser's iteration.…”

Section: Introductionmentioning

confidence: 99%

Constant sign solutions for double phase problems with superlinear nonlinearity

Gasiński

Winkert

2020

Nonlinear Analysis

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We study parametric double phase problems involving superlinear nonlinearities without supposing any growth condition. Based on truncation and comparison methods the existence of two constant sign solutions is shown provided the parameter is larger than the first eigenvalue of the p-Laplacian. As a result of independent interest we prove a priori estimates for solutions for a general class of double phase problems with convection term.2010 Mathematics Subject Classification. 35J15, 35J62, 35J92, 35P30.

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On the L^∞‐regularity for fractional Orlicz problems via Moser's iteration

Carvalho

Silva

Albuquerque

et al. 2022

Math Methods in App Sciences

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L p estimates for the fractional Φ-Laplacian operator defined in bounded domains are established, where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with Moser's iteration, we prove that any weak solution for fractional Φ-Laplacian operator defined in bounded domains belongs to L ∞ (Ω), under appropriate hypotheses on the N-function Φ. Using the Orlicz space and taking into account the fractional setting for our problem, the main results are stated for a huge class of nonlinear operators and nonlinearities.

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Moser iteration applied to elliptic equations with critical growth on the boundary

Abstract: This paper deals with boundedness results for weak solutions of the equation

Cited by 30 publications

References 19 publications

Existence and L‐estimates for elliptic equations involving convolution

Existence and L‐estimates for elliptic equations involving convolution

Constant sign solutions for double phase problems with superlinear nonlinearity

On the L^∞‐regularity for fractional Orlicz problems via Moser's iteration

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Moser iteration applied to elliptic equations with critical growth on the boundary

Abstract: This paper deals with boundedness results for weak solutions of the equation

Cited by 30 publications

References 19 publications

Existence and L‐estimates for elliptic equations involving convolution

Existence and L‐estimates for elliptic equations involving convolution

Constant sign solutions for double phase problems with superlinear nonlinearity

On the L∞‐regularity for fractional Orlicz problems via Moser's iteration

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On the L^∞‐regularity for fractional Orlicz problems via Moser's iteration