Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces

Bahrouni, Sabri; Salort, Ariel

doi:10.1051/cocv/2020064

Cited by 25 publications

(20 citation statements)

References 35 publications

(28 reference statements)

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“…We point out that this kind of hypothesis has been used before in the literature, see for example [3]. Now, we are in position to prove our first lemma.…”

Section: Solutions In a Sub-supersolution Intervalmentioning

confidence: 76%

Existence and multiplicity of solutions for a Dirichlet problem in Fractional Orlicz- Sobolev spaces

Ochoa¹,

Silva²,

Marziani³

2021

Preprint

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In this paper, we first prove the existence of solutions to Dirichlet problems involving the fractional g-Laplacian operator and lower order terms by appealing to sub-and supersolution methods. Moreover, we also state the existence of extremal solutions. Afterwards, and under additional assumptions on the lower order structure, we establish by variational techniques the existence of multiple solutions: one positive, one negative and one with non-constant sign.

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“…We point out that this kind of hypothesis has been used before in the literature, see for example [3]. Now, we are in position to prove our first lemma.…”

Section: Solutions In a Sub-supersolution Intervalmentioning

confidence: 76%

Existence and multiplicity of solutions for a Dirichlet problem in Fractional Orlicz- Sobolev spaces

Ochoa¹,

Silva²,

Marziani³

2021

Preprint

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“…where σ is given in (6). Moreover, the embedding (3). Then, the definition of the Luxemburg norm together with (3) yields…”

Section: Young Functions An Applicationmentioning

confidence: 99%

“…where k is the constant for which H ≺ G, and C is given in (3). By using (3) and Hölder's inequality for Young functions we get…”

Section: Young Functions An Applicationmentioning

confidence: 99%

“…Moreover, it would be interesting to obtain similar result without assuming the ∆ condition on G and H. Finally, in [11] a fractional version of the g−Laplacian was introduced. Eigenvalues of that operator were recently studied in [3,24,25]. Then, a natural question is to analyze whether an extension of the results of this manuscript can be formulated for the nonlocal version of problem (1).…”

mentioning

confidence: 99%

“…Let G and H be Young functions satisfying(3). Let λ 1 be the first eigenvalue of(8) with Ω = (a, b) ⊂ R a bounded interval.…”

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confidence: 99%

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Lower bounds for Orlicz eigenvalues

Salort¹

2022

DCDS

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In this article we consider the following weighted nonlinear eigenvalue problem for the <inline-formula><tex-math id="M1">\begin{document}$ g- $\end{document}</tex-math></inline-formula>Laplacian<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -{\text{ div}}\left( g(|\nabla u|)\frac{\nabla u}{|\nabla u|}\right) = \lambda w(x) h(|u|)\frac{u}{|u|} \quad \text{ in }\Omega\subset \mathbb R^n, n\geq 1 $\end{document} </tex-math></disp-formula>with Dirichlet boundary conditions. Here <inline-formula><tex-math id="M2">\begin{document}$ w $\end{document}</tex-math></inline-formula> is a suitable weight and <inline-formula><tex-math id="M3">\begin{document}$ g = G' $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ h = H' $\end{document}</tex-math></inline-formula> are appropriated Young functions satisfying the so called <inline-formula><tex-math id="M5">\begin{document}$ \Delta' $\end{document}</tex-math></inline-formula> condition, which includes for instance logarithmic perturbation of powers and different power behaviors near zero and infinity. We prove several properties on its spectrum, being our main goal to obtain lower bounds of eigenvalues in terms of <inline-formula><tex-math id="M6">\begin{document}$ G $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ H $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ w $\end{document}</tex-math></inline-formula> and the normalization <inline-formula><tex-math id="M9">\begin{document}$ \mu $\end{document}</tex-math></inline-formula> of the corresponding eigenfunctions.We introduce some new strategies to obtain results that generalize several inequalities from the literature of <inline-formula><tex-math id="M10">\begin{document}$ p- $\end{document}</tex-math></inline-formula>Laplacian type eigenvalues.

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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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Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces

Cited by 25 publications

References 35 publications

Existence and multiplicity of solutions for a Dirichlet problem in Fractional Orlicz- Sobolev spaces

Existence and multiplicity of solutions for a Dirichlet problem in Fractional Orlicz- Sobolev spaces

Lower bounds for Orlicz eigenvalues

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

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Neumann and Robin type boundary conditions in Fractional Orlicz-Sobolev spaces

Cited by 25 publications

References 35 publications

Existence and multiplicity of solutions for a Dirichlet problem in Fractional Orlicz- Sobolev spaces

Existence and multiplicity of solutions for a Dirichlet problem in Fractional Orlicz- Sobolev spaces

Lower bounds for Orlicz eigenvalues

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