Positive solutions of 1-D half Laplacian equation with singular and exponential nonlinearity

Arora, Rakesh; Giacomoni, Jacques; Goel, Divya; Sreenadh, K.

doi:10.3233/asy-191557

Cited by 8 publications

(18 citation statements)

References 14 publications

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“…The proof uses mainly Theorem 1 to get local uniform bound in Ω\{0}, the asymptotic behavior of f and Theorem 5 (proved in [2] and extending some results in [5]) to get the behavior of solutions near isolated singularities. In [2], we give further applications.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Une étape importante dans la preuve du théorème qui constitue par ailleurs un résultat d'intérêt plus large est l'étude des singularités isolés dans l'esprit du résultat bien connu de Brezis-Lions ( [4]) dans le cas local. Dans ce contexte, le Théorème 5 (voir la preuve dans [2]) étend un résultat récent de Chen and Quaas ( [5]) dans le cas de nonlinéarités exponentielles.…”

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“…From Theorem 1 we know that the solutions are radial and radially decreasing, from this we only need to study the behavior near an isolated singularity. For that we exploit the following result proved in [2]:…”

mentioning

confidence: 99%

See 2 more Smart Citations

Symmetry of solutions to singular fractional elliptic equations and applications

Arora¹,

Giacomoni²,

Goel³

et al. 2020

Comptes Rendus. Mathématique

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Section: Introduction and Main Resultsmentioning

confidence: 99%

unclassified

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Symmetry of solutions to singular fractional elliptic equations and applications

Arora¹,

Giacomoni²,

Goel³

et al. 2020

Comptes Rendus. Mathématique

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“…In the critical case for 0 < q < 1, the question of existence and multiplicity of (weak) solutions of nonlocal singular problems has been answered in [14,26,27], whereas the q ≥ 1 case has been dealt in [13]. We also refer to the recent paper [3] on nonlocal singular problems with exponential nonlinearity. The singular Kirchhoff problems involving the Laplace operator has been investigated in [20,21].…”

mentioning

confidence: 99%

Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems

Mukherjee¹,

Pucci²,

Xiang³

2022

DCDS

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In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} M\left(\|u\|^{{n}/{s}}\right)(-\Delta)^s_{n/s}u = \mu u^{-q}+ u^{r-1}\exp( u^{\beta})\quad\text{in } \Omega,\\ u>0\qquad\text{in } \Omega,\\ u = 0\qquad\text{in } \mathbb R^n \setminus{ \Omega}, \end{cases} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a smooth bounded domain of <inline-formula><tex-math id="M2">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ s\in (0,1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ \mu>0 $\end{document}</tex-math></inline-formula> is a real parameter, <inline-formula><tex-math id="M6">\begin{document}$ \beta <{n/(n-s)} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ q\in (0,1) $\end{document}</tex-math></inline-formula>.The paper covers the so called degenerate Kirchhoff case andthe existence proofs rely on the Nehari manifold techniques.

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“…In the critical case for 0 < q < 1, the question of existence and multiplicity of weak solutions to nonlocal singular problems has been answered in [19,27,28] whereas q ≥ 1 case has been dealt in [18]. We also refer a recent article [5] related to nonlocal singular problem with exponential nonlinearity. Moreover, we refer readers to [1,2] concerning the existence and multiplicity results for the fractional p-Laplacian problems.…”

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confidence: 99%

Quasilinear nonlocal elliptic problems with variable singular exponent

Garain

Mukherjee

2020

Communications on Pure &Amp; Applied Analysis

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In this article, we provide existence results to the following nonlocal equation (−∆) s p u = g(x, u), u > 0 in Ω, u = 0 in R N \ Ω, (P λ) where (−∆) s p is the fractional p-Laplacian operator. Here Ω ⊂ R N is a smooth bounded domain, s ∈ (0, 1), p > 1 and N > sp. We establish existence of at least one weak solution for (P λ) when g(x, u) = f (x)u −q(x) and existence of at least two weak solutions when g(x, u) = λu −q(x) + u r for a suitable range of λ > 0. Here r ∈ (p − 1, p * s − 1) where p * s is the critical Sobolev exponent and 0 < q ∈ C 1 (Ω).

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Positive solutions of 1-D half Laplacian equation with singular and exponential nonlinearity

Cited by 8 publications

References 14 publications

Symmetry of solutions to singular fractional elliptic equations and applications

Symmetry of solutions to singular fractional elliptic equations and applications

Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems

Quasilinear nonlocal elliptic problems with variable singular exponent

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