Existence and multiplicity results for p(⋅)&q(⋅) fractional Choquard problems with variable order

Abstract

This paper is concerned with the existence and multiplicity of solutions for the fractional variable order Choquard type problem $\begin{array}{r}\begin{array}{l}(-\mathrm{\Delta }{)}_{p(\cdot )}^{s(\cdot )}u(x)+(-\mathrm{\Delta }{)}_{q(\cdot )}^{s(\cdot )}u\left(x\right)=\lambda |u\left(x\right){|}^{\text{β}\left(x\right)-2}u\left(x\right)\\ +\left({\int }_{\mathrm{\Omega }}\frac{F(y,u(y\left)\right)}{|x-y{|}^{\mu (x,y)}}\mathrm{d}y\right)f(x,u(x\left)\right)+k\left(x\right)& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u\left(x\right)=0& \mathrm{i}\mathrm{n}{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}\end{array}$ where $(-\mathrm{\Delta }{)}_{p(\cdot )}^{s(\cdot )}$ and $(-\mathrm{\Delta }{)}_{q(\cdot )}^{s(\cdot )}$ are two fractional Laplace operators with variable order $s(\cdot ):{\mathbb{R}}^{2N}\to (0,1)$ and with different variable exponents $p(\cdot ):{\mathbb{R}}^{2N}\to (1,\mathrm{\infty })$ and $q(\cdot ):{\mathbb{R}}^{2N}\to (1,\mathrm{\infty })$. Here $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded smooth domain with at least $N\ge 2$, λ is a real parameter, β, μ and k are continuous variable parameters, while F is the primitive function of a suitable f. Under some appropriate conditions on β and k, through variational methods, we prove existence and multiplicity of solutions for the above problem.

Keywords:

• $p(\cdot )\mathrm{\&}q(\cdot )$ fractional Laplacian
• variable exponent
• Choquard type nonlinearity
• variational methods

AMS Subject Classifcations:

• 35R11
• 47G20
• 35S15
• 35J60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work is supported by the Fundamental Research Funds for Central Universities [grant number 2019B44914] and the National Key Research and Development Program of China [grant number 2018YFC1508100]. This paper is also supported by China Scholarship Council [grant number 201906710004]. A. Fiscella is member of Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). A. Fiscella realized the manuscript within the auspices of the INdAM-GNAMPA project titled Equazioni alle derivate parziali: problemi e modelli [grant number Prot_20191219-143223-545], of the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Project titled Operators with non standard growth [grant number 2019/23917-3], of the FAPESP Thematic Project titled Systems and partial differential equations [grant number 2019/02512-5] and of the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Project titled Variational methods for singular fractional problems [grant number 3787749185990982].