A Global multiplicity result for a very singular critical nonlocal equation
TLDR
In this paper, the global multiplicity result for the nonlocal singular problem (P$_\lambda) was shown and it was shown that any weak solution to this problem is in (C^\alpha(\mathbb R^n) with α = α(s,q) in (0, 1) where α is a constant.Abstract:
In this article we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (-\Delta)^s u = u^{-q} + \lambda u^{{2^*_s}-1}, \quad u> 0 \quad \text{in } \Omega,\quad u = 0 \quad \mbox{in } \mathbb R^n \setminus\Omega, \tag{ P$_\lambda$} \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $n > 2s$, $ s \in (0,1)$, $ \lambda > 0$, $q> 0$ satisfies $q(2s-1) \Lambda$, where $\Lambda> 0$ is appropriately chosen. We also prove a result of independent interest that any weak solution to (P$_\lambda)$ is in $C^\alpha(\mathbb R^n)$ with $\alpha=\alpha(s,q)\in (0,1)$. The asymptotic behaviour of weak solutions reveals that this result is sharp.read more
Citations
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Regularity results on a class of doubly nonlocal problems
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Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems
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Combined effects of singular and exponential nonlinearities in fractional Kirchhoff problems
TL;DR: In this article , the existence of weak solutions for the degenerate Kirchhoff problem with singular and exponential nonlinearities was established, and the existence proofs rely on the Nehari manifold techniques.
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Singular Doubly Nonlocal Elliptic Problems with Choquard Type Critical Growth Nonlinearities
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Singular doubly nonlocal elliptic problems with Choquard type critical growth nonlinearities
TL;DR: In this paper, a very singular and doubly nonlocal singular problem (P_\lambda) was studied and a weak comparison principle and optimal Sobolev regularity was established using critical point theory of non-smooth analysis and geometry of the energy functional.
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Semilinear problems for the fractional laplacian with a singular nonlinearity
TL;DR: In this article, the authors partially supported by project MTM2013-40846-P MINECO and the third author was also supported for the grant BES-2011-044216 associated to MTM2010-18128.
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Some remarks on the solvability of non-local elliptic problems with the Hardy potential
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