Abstract
This paper is concerned with the existence of solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity: M∬R2N|u(x)-u(y)|N/s|x-y|2Ndxdy(-Δ)N/ssu=f(x,u)inΩ,u=0inRN\\Ω,\\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}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} M\\left( \\displaystyle \\iint _{{\\mathbb {R}}^{2N}}\\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy\\right) (-\\Delta )^{s}_{N/s}u=f(x,u)\\,\\, \\ &{}\\quad \\mathrm{in}\\ \\Omega ,\\\\ u=0\\ \\ \\ \\ &{}\\quad \\mathrm{in}\\ {\\mathbb {R}}^N{\\setminus } \\Omega , \\end{array}\\right. } \\end{aligned}$$\\end{document}where (-Delta )^{s}_{N/s} is the fractional N / s-Laplacian operator, Nge 1, sin (0,1), Omega subset {mathbb {R}}^N is a bounded domain with Lipschitz boundary, M:{mathbb {R}}^+_0rightarrow {mathbb {R}}^+_0 is a continuous function, and f:Omega times {mathbb {R}}rightarrow {mathbb {R}} is a continuous function behaving like exp (alpha t^{2}) as trightarrow infty for some alpha >0. We first obtain the existence of a ground state solution with positive energy by using minimax techniques combined with the fractional Trudinger–Moser inequality. Next, the existence of nonnegative solutions with negative energy is established by using Ekeland’s variational principle. The main feature of this paper consists in the presence of a (possibly degenerate) Kirchhoff model, combined with a critical Trudinger–Moser nonlinearity.
Highlights
Introduction and main resultsIn this paper, we study the following fractional Kirchhoff-type problem: in, in RN \, (1.1)where N ≥ 1, s ∈ (0, 1), ⊂ RN is a bounded domain with Lipschitz boundary, M :[0, ∞) → [0, ∞) is a continuous function, f : × R → R is a continuous functionN behaving like exp(α|t| N−s ) as t → ∞ for some α > 0, and (−)sN/s is the fractionalN /s-Laplacian operator which, up to a normalization constant, is defined as )sN/s φ(x ) lim ε→0+ RN \ Bε (x ) |φ (x )
Where N ≥ 1, s ∈ (0, 1), ⊂ RN is a bounded domain with Lipschitz boundary, M :
To the best of our knowledge, Theorems 1.1–1.3 are the first results for the Kirchhoff-type problems involving critical Trudinger–Moser nonlinearities in the fractional setting
Summary
We study the following fractional Kirchhoff-type problem: in , in RN \ ,. To study the existence of solutions for problem (1.1), let us recall some results related to the fractional Sobolev space W0s,p( ). When n = 2, W 1,2( ) → Lr ( ) for 1 ≤ r < ∞ but W 1,2( ) → L∞( ) To fill this gap, Trudinger [37] proved that that there exists τ > 0 such that W01,2( ) is embedded into the Orlicz space Lφτ ( ) determined by the Young function φτ = exp(τ t2 −1). In the setting of the fractional Laplacian, Iannizzotto and Squassina [17] investigated existence of solutions for the following Dirichlet problem. Perera and Squassina [32] studied the bifurcation results for the following problem with Trudinger–Moser nonlinearity (− )sN /s u = λ|u|(N −2s)/s exp(|u|N /(N −s)) u=0 in , in RN \ , where λ > 0 is a parameter
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