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Bifurcation results for a fractional elliptic equation with critical exponent in \(\mathbb {R}^n\)

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Abstract

In this paper we study some nonlinear elliptic equations in \({\mathbb R}^n\) obtained as a perturbation of the problem with the fractional critical Sobolev exponent, that is

$$\begin{aligned} (-\Delta )^s u = \varepsilon \,h\,u_+^q + u_+^p \ { \text{ in } }{\mathbb R}^n, \end{aligned}$$

where \(s\in (0,1)\), \(n>4s\), \(\varepsilon >0\) is a small parameter, \(p=\frac{n+2s}{n-2s}\), \(0<q<p\) and h is a continuous and compactly supported function. To construct solutions to this equation, we use the Lyapunov–Schmidt reduction, that takes advantage of the variational structure of the problem. For this, the case \(0<q<1\) is particularly difficult, due to the lack of regularity of the associated energy functional, and we need to introduce a new functional setting and develop an appropriate fractional elliptic regularity theory.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Melbourne, 813 Swanston St, Parkville, VIC, 3010, Australia

    Serena Dipierro & Enrico Valdinoci

  2. School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Perth, WA, 6009, Australia

    Serena Dipierro & Enrico Valdinoci

  3. Weierstraß-Institut für Angewandte Analysis und Stochastik, Hausvogteiplatz 5/7, 10117, Berlin, Germany

    Serena Dipierro & Enrico Valdinoci

  4. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    María Medina

  5. Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile

    María Medina

  6. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Ireneo Peral

  7. Dipartimento di Matematica, Università degli studi di Milano, Via Saldini 50, 20133, Milan, Italy

    Enrico Valdinoci

  8. Istituto di Matematica Applicata e Tecnologie Informatiche, Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100, Pavia, Italy

    Enrico Valdinoci

Authors
  1. Serena Dipierro
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  2. María Medina
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  3. Ireneo Peral
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  4. Enrico Valdinoci
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Correspondence to Serena Dipierro.

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Dipierro, S., Medina, M., Peral, I. et al. Bifurcation results for a fractional elliptic equation with critical exponent in \(\mathbb {R}^n\) . manuscripta math. 153, 183–230 (2017). https://doi.org/10.1007/s00229-016-0878-3

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