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arXiv · 1907.02837

Variable order nonlocal Choquard problem with variable exponents

Reshmi Biswas·Sweta Tiwari
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Abstract

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents (-Δ)_{p(\cdot)}^{s(\cdot)}u(x)&=λ|u(x)|^{α(x)-2}u(x)+ \left(\DD\int_Ω\frac{F(y,u(y))}{|x-y|^{μ(x,y)}}dy\right)f(x,u(x)), x\in Ω, u(x)&=0, x\in \mathbb R^N\setminusΩ, where $Ω\subset\mathbb R^N$ is a smooth and bounded domain, $N\geq 2$, $p,s,μ$ and $α$ are continuous functions on $\mathbb R^N\times\mathbb R^N$ and $f(x,t)$ is Carathédory function. Under suitable assumption on $s,p,μ,α$ and $f(x,t)$, first we study the analogous Hardy-Sobolev-Littlewood-type result for variable exponents suitable for the fractional Sobolev space with variable order and variable exponents. Then we give the existence/multiplicity results for the above equation.

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Reshmi Biswas, Sweta Tiwari. 2019-07-05. Variable order nonlocal Choquard problem with variable exponents. https://arxiv.org/abs/1907.02837

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