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January/February 2022 Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity
S. Rawat, K. Sreenadh
Differential Integral Equations 35(1/2): 89-121 (January/February 2022).

Abstract

In this paper, we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: \begin{align*} (-\Delta)^s u & = -\lambda|u|^{q-2}u + au + b \Big ( \int_{\Omega} \frac{(u^{+}(y))^{2^{*}_{\mu ,s}}} {|x-y|^ \mu}\, dy \Big ) (u^{+})^{2^{*}_{\mu ,s}-2}u \quad\text{in} \; \Omega,\\ u & = 0\quad \text{in} \; \mathbb{R}^{N}\backslash\Omega, \end{align*} where $\Omega$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$. Here, $(-\Delta)^s$ is the fractional Laplace operator, $\lambda > 0$ is a real parameter, $q \in (1, 2)$, $a > 0$ and $b > 0$ are given constants, $0 < \mu < \min\{N,4s\}$ and $2^{*}_{\mu ,s} = \frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and the notation $u^{+} = \max \{u, 0\}$. We prove that the above problem has at least three nontrivial solutions using the Mountain pass Lemma and Linking theorem.

Citation

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S. Rawat. K. Sreenadh. "Three solutions for a fractional elliptic problem with asymmetric critical Choquard nonlinearity." Differential Integral Equations 35 (1/2) 89 - 121, January/February 2022.

Information

Published: January/February 2022
First available in Project Euclid: 6 January 2022

Subjects:
Primary: 35A15 , 35J20 , 35J60

Rights: Copyright © 2022 Khayyam Publishing, Inc.

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Vol.35 • No. 1/2 • January/February 2022
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