Multiplicity and uniform estimate for a class of variable order fractional $p(x)$-Laplacian problems with concave-convex nonlinearities

Abstract

In this article, we study the existence/multiplicity results for the following variable order nonlocal Choquard problem with variable exponents \begin{equation*} \begin{array}{rl} (-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)),\\ &~\hspace{6cm} x\in Ω, \\ u(x)&=0 ,\hspace{20mm} x\in Ω^c:=\mathbb R^N\setminusΩ, \end{array} \end{equation*} where $\Om\subset\mathbb R^N$ is a smooth and bounded domain, $N\ge 2$, $p,s,\mu$ and $\alpha$ are continuous functions on ${\mathbb{R}}^N\times {\mathbb{R}}^N$ and $f(x,t)$ is continuous function with $F(x,t):={\int }_0^{t}f(x,s)ds$. Under suitable assumption on $s,p,\mu ,\alpha$ 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.

Modified the article with different tittle and abstract. Kindly see or cite "Variable order nonlocal Choquard problem with variable exponents" instead of this article. Link of it is given as: arXiv:1907.02837