Fractional Kirchhoff problems with critical Trudinger–Moser nonlinearity

By Xiang Mingqi, Vicentiu D. Rădulescu, Binlin Zhang. Calculus of Variations and Partial Differential Equations, 2019

166 citations recorded by the source corpus.

This paper is concerned with the existence of solutions for a class of fractional Kirchhoff-type problems with Trudinger–Moser nonlinearity: $$\begin{aligned} {\left\{ \begin{array}{ll} M\left( \displaystyle \iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^{N/s}}{|x-y|^{2N}}dxdy\right) (-\Delta )^{s}_{N/s}u=f(x,u)\,\, \ &{}\quad \mathrm{in}\ \Omega ,\\ u=0\ \ \ \ &{}\quad \mathrm{in}\ {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. } \end{aligned}$$M∬R2N|u(x)-u(y)|N/s|x-y|2Ndxdy(-Δ)N/ssu=f(x,u)inΩ,u=0inRN\Ω,where $$(-\Delta )^{s}_{N/s}$$(-Δ)N/ss is the fractional N / s-Laplacian operator, $$N\ge 1$$N≥1, $$s\in (0,1)$$s∈(0,1), $$\Omega \subset {\mathbb {R}}^N$$Ω⊂RN is a bounded domain with Lipschitz boundary, $$M:{\mathbb {R}}^+_0\rightarrow {\mathbb {R}}^+_0$$M:R0+→R0+ is a continuous function, and $$f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}} $$f:Ω×R→R is a continuous function behaving like $$\exp (\alpha t^{2})$$exp(αt2) as $$t\rightarrow \infty $$t→∞ for some $$\alpha >0$$α>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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c7fd4359e207f162285b824d3217a2f1f003da11

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In Applied Mathematics

  1. R. Arora
  2. T. Mukherjee
  3. Patrick Winkert
  4. A. Fiscella
  5. Vicentiu D. Rădulescu
  6. Ky Ho
  7. Leszek Gasinski
  8. Patrizia Pucci

In Computational Theory and Mathematics

  1. R. Arora
  2. Patrick Winkert
  3. T. Mukherjee
  4. A. Fiscella
  5. Vicentiu D. Rădulescu
  6. Leszek Gasinski
  7. Ky Ho
  8. Patrizia Pucci

In Mathematical Physics

  1. Patrick Winkert
  2. T. Mukherjee
  3. R. Arora
  4. A. Fiscella
  5. Binlin Zhang
  6. Patrizia Pucci
  7. Mingqi Xiang
  8. Leszek Gasinski

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