Abstract
We are concerned with the following elliptic equations with variable exponents:M([u]s,p(⋅,⋅))Lu(x)+V(x)|u|p(x)−2u=λρ(x)|u|r(x)−2u+h(x,u)in RN,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ M \\bigl([u]_{s,p(\\cdot,\\cdot)} \\bigr)\\mathcal{L}u(x) +\\mathcal {V}(x) \\vert u \\vert ^{p(x)-2}u =\\lambda\\rho(x) \\vert u \\vert ^{r(x)-2}u + h(x,u) \\quad \\text{in } \\mathbb {R}^{N}, $$\\end{document} where [u]_{s,p(cdot,cdot)}:=int_{mathbb {R}^{N}}int_{mathbb {R}^{N}} frac{|u(x)-u(y)|^{p(x,y)}}{p(x,y)|x-y|^{N+sp(x,y)}} ,dx ,dy, the operator mathcal{L} is the fractional p(cdot)-Laplacian, p, r: {mathbb {R}^{N}} to(1,infty) are continuous functions, M in C(mathbb {R}^{+}) is a Kirchhoff-type function, the potential function mathcal {V}:mathbb {R}^{N} to(0,infty) is continuous, and h:mathbb {R}^{N}timesmathbb {R} tomathbb {R} satisfies a Carathéodory condition. Under suitable assumptions on h, the purpose of this paper is to show the existence of at least two non-trivial distinct solutions for the problem above for the case of a combined effect of concave–convex nonlinearities. To do this, we use the mountain pass theorem and variant of the Ekeland variational principle as the main tools.
Highlights
In the last two decades an increasing deal of attention has been paid to the investigation on problems of differential equations and variational problems with nonstandard growth conditions because they can be corroborated as a model for many physical phenomena which arise in the research of elastic mechanics, electro-rheological fluid (“smart fluids”) and image processing, etc
Under suitable assumptions on h, the purpose of this paper is to show the existence of at least two non-trivial distinct solutions for the problem above for the case of a combined effect of concave–convex nonlinearities
In recent years the study of fractional Sobolev spaces and the corresponding nonlocal equations has received a great amount of attention because of their occurrence in many different applications such as optimization, fractional quantum mechanics, the thin obstacle problem, phase transition phenomena, image process, game theory and Lévy processes; see [14, 23, 28, 36, 39, 45] and the references therein for more details
Summary
In the last two decades an increasing deal of attention has been paid to the investigation on problems of differential equations and variational problems with nonstandard growth conditions because they can be corroborated as a model for many physical phenomena which arise in the research of elastic mechanics, electro-rheological fluid (“smart fluids”) and image processing, etc. In recent years the study of fractional Sobolev spaces and the corresponding nonlocal equations has received a great amount of attention because of their occurrence in many different applications such as optimization, fractional quantum mechanics, the thin obstacle problem, phase transition phenomena, image process, game theory and Lévy processes; see [14, 23, 28, 36, 39, 45] and the references therein for more details In this direction it is a natural question to see which results can be recovered when we replace the local p(·)-Laplacian, defined as – div(|∇u|p(x)–2∇u), with the nonlocal fractional p(·)-Laplacian. Let us first assume that a Kirchhoff function M : R+0 → R+ satisfies the following conditions:
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