Existence and multiplicity of solutions for critical nonlocal equations with variable exponents

ABSTRACT

In this paper, we are interested in a class of critical nonlocal problems with variable exponents of the form: $\{\begin{array}{l}M(T_{p(\cdot ,\cdot )}(u))[(-\mathrm{\Delta }{)}_{p(x,y)}^{s}u+|u{|}^{\tilde{p}(x)-2}u]=\lambda f(x,u)+|u{|}^{q(x)-2}u,\text{in }{\mathbb{R}}^N,\\ u\in W^{s,p(\cdot ,\cdot )}({\mathbb{R}}^N),\tilde{p}(x)=p(x,x),\end{array}$where $T_{p(\cdot ,\cdot )}(u):={\iint }_{{\mathbb{R}}^{2N}}\frac{|u(x)-u(y){|}^{p(x,y)}}{p(x,y)|x-y{|}^{N+sp(x,y)}}\mathrm{d}x\mathrm{d}y+{\int }_{{\mathbb{R}}^N}\frac{1}{\tilde{p}(x)}|u{|}^{\tilde{p}(x)}\mathrm{d}x,$M is the Kirchhoff function, λ is a real parameter and f is a continuous function. We also assume that $\{x\in {\mathbb{R}}^N:q(x)=p_s^{\ast }(x)\}\ne \mathrm{\varnothing }$, where $p_s^{\ast }(x)=N\tilde{p}(x)/(N-s\tilde{p}(x))$ is the critical Sobolev exponent for variable exponents. The strategy of the proof for these results is to approach the problem variationally by using the mountain pass theorem and the concentration-compactness principles for fractional Sobolev spaces with variable exponents. In addition, we obtain the existence and multiplicity of nontrivial solutions for the above problem in non-degenerate and degenerate cases.

KEYWORDS:

• Fractional p-Laplacian
• $p(\cdot )$-Laplacian
• concentration-compactness principles
• variational methods

2010 MSC:

• 35B33
• 35D30
• 35J20
• 46E35
• 49J35

Disclosure statement

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

Additional information

Funding

S. Liang was supported by the Foundation for China Postdoctoral Science Foundation [grant number 2019M662220], Natural Science Foundation of Jilin Province [grant number YDZJ202201ZYTS582], Natural Science Foundation of Changchun Normal University [grant number 2017-09]. P. Pucci is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and is partly supported by the INdAM – GNAMPA Project 2022 Equazioni differenziali alle derivate parziali in fenomeni non lineari [grant number CUP_E55F22000270001], [grant number U-UFMBAZ-2020-000761]. P. Pucci was also partly supported by the Fondo Ricerca di Base di Ateneo – Esercizio 2017–2019 of the University of Perugia, named PDEs and Nonlinear Analysis. B. Zhang was supported by the National Natural Science Foundation of China [grant number 11871199].