Fractional schrödinger–poisson system with singularity: existence, uniqueness, and asymptotic behavior
TL;DR: In this paper, the existence, uniqueness, and monotonicity of positive solution uλ using the variational method were shown for fractional Schrodinger-Poisson systems with singularity under certain assumptions on V and f.
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.
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01 Jan 2021
TL;DR: In this paper, the authors considered the following fractional Kirchhoff problem with singularity, where the singularity is the fractional Laplacian of the problem, and they considered the problem in terms of singularity.
Abstract: In this paper, we consider the following fractional Kirchhoff problem with singularity $ \left \{\begin{array}{lcl} \Big(1+ b\int_{\mathbb{R}^3}\int_{\mathbb{R}^3} \frac{|u(x)-u(y)|^2}{|x-y|^{3+2s}}\mathrm{d}x \mathrm{d}y \Big)(-\Delta)^s u+V(x)u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. $ where $ (-\Delta)^s $ is the fractional Laplacian with $ 0
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TL;DR: In this article, the authors deal with the fractional Sobolev spaces W s;p and analyze the relations among some of their possible denitions and their role in the trace theory.
Abstract: This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.
3,555 citations
01 Jan 2004
TL;DR: Applebaum et al. as discussed by the authors give an introduction to a class of stochastic processes called Levy processes, in honor of the great French probabilist Paul Levy, who first studied them in the 1930s.
Abstract: 1320 NOTICES OF THE AMS VOLUME 51, NUMBER 11 T he theory of stochastic processes was one of the most important mathematical developments of the twentieth century. Intuitively, it aims to model the interaction of “chance” with “time”. The tools with which this is made precise were provided by the great Russian mathematician A. N. Kolmogorov in the 1930s. He realized that probability can be rigorously founded on measure theory, and then a stochastic process is a family of random variables (X(t), t ≥ 0) defined on a probability space (Ω,F , P ) and taking values in a measurable space (E,E) . Here Ω is a set (the sample space of possible outcomes), F is a σ-algebra of subsets of Ω (the events), and P is a positive measure of total mass 1 on (Ω,F ) (the probability). E is sometimes called the state space. Each X(t) is a (F ,E) measurable mapping from Ω to E and should be thought of as a random observation made on E made at time t . For many developments, both theoretical and applied, E is Euclidean space Rd (often with d = 1); however, there is also considerable interest in the case where E is an infinite dimensional Hilbert or Banach space, or a finite-dimensional Lie group or manifold. In all of these cases E can be taken to be the Borel σalgebra generated by the open sets. To model probabilities arising within quantum theory, the scheme described above is insufficiently general and must be embedded into a suitable noncommutative structure. Stochastic processes are not only mathematically rich objects. They also have an extensive range of applications in, e.g., physics, engineering, ecology, and economics—indeed, it is difficult to conceive of a quantitative discipline in which they do not feature. There is a limited amount that can be said about the general concept, and much of both theory and applications focusses on the properties of specific classes of process that possess additional structure. Many of these, such as random walks and Markov chains, will be well known to readers. Others, such as semimartingales and measure-valued diffusions, are more esoteric. In this article, I will give an introduction to a class of stochastic processes called Levy processes, in honor of the great French probabilist Paul Levy, who first studied them in the 1930s. Their basic structure was understood during the “heroic age” of probability in the 1930s and 1940s and much of this was due to Paul Levy himself, the Russian mathematician A. N. Khintchine, and to K. Ito in Japan. During the past ten years, there has been a great revival of interest in these processes, due to new theoretical developments and also a wealth of novel applications—particularly to option pricing in mathematical finance. As well as a vast number of research papers, a number of books on the subject have been published ([3], [11], [1], [2], [12]) and there have been annual international conferences devoted to these processes since 1998. Before we begin the main part of the article, it is worth David Applebaum is professor of probability and statistics at the University of Sheffield. His email address is D.Applebaum@sheffield.ac.uk. He is the author of Levy Processes and Stochastic Calculus, Cambridge University Press, 2004, on which part of this article is based.
348 citations
"Fractional schrödinger–poisson syst..." refers background in this paper
...The fractional Laplacian is the infinitesimal generator of Lévy stable diffusion process and arises in anomalous diffusion in plasma, population dynamics, geophysical fluid dynamics, flames propagation, chemical reactions in liquids, and American options in finance, see [2] for instance....
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TL;DR: In this paper, the existence of ground state solutions for the nonlinear fractional Schrodinger-Poisson system with critical Sobolev exponent was studied and a nontrivial ground state solution was proved under certain assumptions on V ( x ), using the method of Pohozaev-Nehari manifold and the arguments of Brezis-Nirenberg, the monotonic trick and global compactness Lemma.
129 citations
"Fractional schrödinger–poisson syst..." refers background in this paper
...[36], and Teng [29, 34] obtained the existence of a positive (nontrivial) ground state solution and a sign-changing (least energy) solution for system (1....
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...Guo [15], Yu et al. [36], and Teng [29, 34] obtained the existence of a positive (nontrivial) ground state solution and a sign-changing (least energy) solution for system (1.1) involving critical nonlinearities....
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...2) For convenience, we recall the following lemma on the properties of the function φt u from [7, 21, 29, 30], etc....
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...C. Ye and K. Teng, Ground state and sign-changing solutions for fractional Schrödinger-Poisson system with critical growth, Complex Var....
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...K. Teng and R. Agarwal, Existence and concentration of positive ground state solutions for nonlinear fractional Schrödinger-Poisson system with critical growth, Math....
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TL;DR: In this article, a fractional Schrodinger-Poisson system with a general nonlinearity in the subcritical and critical case is considered, and the existence of positive solutions is proved by using a perturbation approach.
Abstract: We consider a fractional Schrodinger-Poisson system with a general nonlinearity in the subcritical and critical case. The Ambrosetti-Rabinowitz condition is not required. By using a perturbation approach, we prove the existence of positive solutions. Moreover, we study the asymptotics of solutions for a vanishing parameter.
113 citations
TL;DR: In this article, the authors considered the degenerate case when the Kirchhoff function is zero at zero and provided the existence of two solutions to the nonlocal nonlocal problem.
Abstract: In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s u = \displaystyle\frac{\lambda}{u^\gamma}+u^{2^*_s-1}&\quad\mbox{in } \Omega,\\ u>0&\quad\mbox{in } \Omega,\\ u=0&\quad\mbox{in } \mathbb{R}^N\setminus\Omega, \end{array}\right. $$ where $\Omega$ is an open bounded subset of $\mathbb R^N$ with continuous boundary, dimension $N>2s$ with parameter $s\in (0,1)$, $2^*_s=2N/(N-2s)$ is the fractional critical Sobolev exponent, $\lambda>0$ is a real parameter, exponent $\gamma\in(0,1)$, $M$ models a Kirchhoff type coefficient, while $(-\Delta)^s$ is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is when the Kirchhoff function $M$ is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.
58 citations