Singular Doubly Nonlocal Elliptic Problems with Choquard Type Critical Growth Nonlinearities
01 May 2021-Journal of Geometric Analysis (Springer Science and Business Media LLC)-Vol. 31, Iss: 5, pp 4492-4530
TL;DR: In this article, a very singular and doubly nonlocal singular problem with singular nonlinearity was studied and a very weak comparison principle and the optimal Sobolev regularity was established.
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.
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TL;DR: In this article, a coupled Schrodinger system with Stein-Weiss type convolution part is considered and the existence and nonexistence of the solutions by variational methods are investigated.
Abstract: In this paper, we are interested in a coupled Schrodinger system with Stein–Weiss type convolution part. Firstly we study the existence and nonexistence of the solutions by variational methods. Second, by changing the system into an equivalent integral form, we study the symmetry, regularity and asymptotic behaviors of the solutions by moving plane arguments.
10 citations
TL;DR: In this article, the existence of a positive sola for the singular critical Choquard problem with fractional power of Laplacian and a critical Hardy potential was discussed, and the authors showed that such a sola can be obtained in the sense of the Hardy-Littlewood-Sobolev inequality.
Abstract: This article concerns about the existence of a positive SOLA (Solutions Obtained as Limits of Approximations) for the following singular critical Choquard problem involving fractional power of Laplacian and a critical Hardy potential. 0.1
$$\begin{aligned} \begin{aligned} (-\Delta )^su-\alpha \frac{u}{|x|^{2s}}&=\lambda u+ u^{-\gamma }+\beta \left( \int _{\Omega }\frac{u^{2_b^*}(y)}{|x-y|^b}dy\right) u^{2_b^*-1}+\mu ~\text {in}~\Omega ,\\ u&>0~\text {in}~\Omega ,\\ u&= 0~\text {in}~\mathbb {R}^N{\setminus }\Omega . \end{aligned} \end{aligned}$$
Here, $$\Omega $$
is a bounded domain of $$\mathbb {R}^N$$
, $$s\in (0,1)$$
, $$\alpha ,\lambda $$
and $$\beta $$
are positive real parameters, $$N>2s$$
, $$\gamma \in (0,1)$$
, $$0
4 citations
TL;DR: In this article, the existence, multiplicity and regularity of positive weak solutions for the following Kirchhoff-Choquard problem were studied, and it was shown that each positive weak solution is bounded and satisfy Holder regularity.
Abstract: In this paper we study the existence, multiplicity and regularity of positive weak solutions for the following Kirchhoff-Choquard problem:
\begin{equation*}
\begin{array}{cc}
\displaystyle M\left( \iint\limits_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}\,dxdy\right) (-\Delta)^s u = \frac{\lambda}{u^\gamma} + \left( \int\limits_{\Omega} \frac{|u(y)|^{2^{*}_{\mu ,s}}}{|x-y|^ \mu}\, dy\right) |u|^{2^{*}_{\mu ,s}-2}u \;\text{in} \; \Omega,
%\quad \quad u > 0\quad \text{in} \; \Omega,
\quad \quad u = 0\quad \text{in} \; \mathbb{R}^{N}\backslash\Omega,
\end{array}
\end{equation*} where $\Omega$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$. $M$ models Kirchhoff-type coefficient in particular, the degenerate case where Kirchhoff coefficient M is zero at zero. $(-\Delta)^s$ is fractional Laplace operator, $\lambda > 0$ is a real parameter, $\gamma \in (0,1)$ and $2^{*}_{\mu ,s} = \frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We prove that each positive weak solution is bounded and satisfy Holder regularity of order $s$. Furthermore, using the variational methods and truncation arguments we prove the existence of two positive solutions.
3 citations
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TL;DR: In this article, the effect of Hardy potential on the existence or non-existence of solutions to a fractional Laplacian problem involving a singular nonlinearity was studied.
Abstract: In this paper, we study the effect of Hardy potential on the existence or non-existence of solutions to a fractional Laplacian problem involving a singular nonlinearity. Also, we mention a stability result.
3 citations
2 citations
References
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1,039 citations
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Book•
01 Mar 2016TL;DR: A thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators can be found in this paper, where the authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of equations, plus their application to various processes arising in the applied sciences.
Abstract: This book provides researchers and graduate students with a thorough introduction to the variational analysis of nonlinear problems described by nonlocal operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations, plus their application to various processes arising in the applied sciences. The equations are examined from several viewpoints, with the calculus of variations as the unifying theme. Part I begins the book with some basic facts about fractional Sobolev spaces. Part II is dedicated to the analysis of fractional elliptic problems involving subcritical nonlinearities, via classical variational methods and other novel approaches. Finally, Part III contains a selection of recent results on critical fractional equations. A careful balance is struck between rigorous mathematics and physical applications, allowing readers to see how these diverse topics relate to other important areas, including topology, functional analysis, mathematical physics, and potential theory.
613 citations