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Journal ArticleDOI

A Global multiplicity result for a very singular critical nonlocal equation

21 Jul 2019-Topological Methods in Nonlinear Analysis (Nicolaus Copernicus University, Juliusz P. Schauder Centre for Nonlinear Studies)-Vol. 54, Iss: 1, pp 345-370

AbstractIn 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.

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Citations
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Journal ArticleDOI
Abstract: The purpose of this article is twofold. First, an issue of regularity of weak solution to the problem (P) (see below) is addressed. Secondly, we investigate the question of H s versus C 0 -weighted minimizers of the functional associated to problem (P) and then give applications to existence and multiplicity results.

11 citations

Journal ArticleDOI
Abstract: The theory of elliptic equations involving singular nonlinearities is well-studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $$(P_\lambda )$$ (See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of nonsmooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.

8 citations

Posted Content
Abstract: The theory of elliptic equations involving singular nonlinearities is well studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $(P_\lambda)$(See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of non-smooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.

2 citations

Journal ArticleDOI
TL;DR: This paper establishes the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities.
Abstract: In this paper we establish the existence of at least two (weak) solutions for the following fractional Kirchhoff problem involving singular and exponential nonlinearities \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} $\end{document} where \begin{document}$ \Omega $\end{document} is a smooth bounded domain of \begin{document}$ \mathbb R^n $\end{document} , \begin{document}$ n\geq 1 $\end{document} , \begin{document}$ s\in (0,1) $\end{document} , \begin{document}$ \mu>0 $\end{document} is a real parameter, \begin{document}$ \beta and \begin{document}$ q\in (0,1) $\end{document} .The paper covers the so called degenerate Kirchhoff case andthe existence proofs rely on the Nehari manifold techniques.

1 citations

Journal ArticleDOI
Abstract: In this paper, we consider the following fractional Schrodinger–Poisson system with singularity \begin{equation*} \left \{\begin{array}{lcl} ({-}\Delta)^s u+V(x)u+\lambda \phi u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ ({-}\Delta)^t \phi = u^2, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. \end{equation*} where 0 0 and 0 3. Under certain assumptions on V and f, we show the existence, uniqueness, and monotonicity of positive solution uλ using the variational method. We also give a convergence property of uλ as λ → 0, when λ is regarded as a positive parameter.

1 citations


References
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Book
01 Jan 2002
Abstract: PrefaceGuide to the ReaderPrologue3IReal-Variable Theory7IIMore About Maximal Functions49IIIHardy Spaces87IVH[superscript 1] and BMO139VWeighted Inequalities193VIPseudo-Differential and Singular Integral Operators: Fourier Transform228VIIPseudo-Differential and Singular Integral Operators: Almost Orthogonality269VIIIOscillatory Integrals of the First Kind329IXOscillatory Integrals of the Second Kind375XMaximal Operators: Some Examples433XIMaximal Averages and Oscillatory Integrals467XIIIntroduction to the Heisenberg Group527XIIIMore About the Heisenberg Group587Bibliography645Author Index679Subject Index685

6,362 citations

Journal ArticleDOI
Abstract: Given a function ϕ and s ∈ (0, 1), we will study the solutions of the following obstacle problem: • u ≥ ϕ in R n , • (−� ) s u ≥ 0i nR n , • (−� ) s u(x) = 0 for those x such that u(x )>ϕ (x), • lim|x|→+∞ u(x) = 0. We show that when ϕ is C 1,s or smoother, the solution u is in the space C 1,α for every α< s. In the case where the contact set {u = ϕ} is convex, we prove the optimal regularity result u ∈ C 1,s . When ϕ is only C 1,β for a β< s, we prove that our solution u is C 1,α for every α< β. c � 2006 Wiley Periodicals, Inc.

1,111 citations

Journal ArticleDOI
Abstract: We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of ( − Δ ) s u = g in Ω, u ≡ 0 in R n \ Ω , for some s ∈ ( 0 , 1 ) and g ∈ L ∞ ( Ω ) , then u is C s ( R n ) and u / δ s | Ω is C α up to the boundary ∂Ω for some α ∈ ( 0 , 1 ) , where δ ( x ) = dist ( x , ∂ Ω ) . For this, we develop a fractional analog of the Krylov boundary Harnack method. Moreover, under further regularity assumptions on g we obtain higher order Holder estimates for u and u / δ s . Namely, the C β norms of u and u / δ s in the sets { x ∈ Ω : δ ( x ) ⩾ ρ } are controlled by C ρ s − β and C ρ α − β , respectively. These regularity results are crucial tools in our proof of the Pohozaev identity for the fractional Laplacian (Ros-Oton and Serra, 2012 [19] , [20] ).

662 citations