On a Fractional \( p \& q\) Laplacian Problem with Critical Sobolev–Hardy Exponents

  • Vincenzo AmbrosioEmail author
  • Teresa Isernia


We consider the following fractional \( p \& q\) Laplacian problem with critical Sobolev–Hardy exponents
$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}_{p} u + (-\Delta )^{s}_{q} u = \frac{|u|^{p^{*}_{s}(\alpha )-2}u}{|x|^{\alpha }}+ \lambda f(x, u) &{} \text{ in } \Omega \\ u=0 &{} \text{ in } \mathbb {R}^{N}{\setminus } \Omega , \end{array} \right. \end{aligned}$$
where \(0<s<1\), \(1\le q<p<\frac{N}{s}\), \((-\Delta )^{s}_{r}\), with \(r\in \{p,q\}\), is the fractional r-Laplacian operator, \(\lambda \) is a positive parameter, \(\Omega \subset \mathbb {R}^{N}\) is an open bounded domain with smooth boundary, \(0\le \alpha <sp\), and \(p^{*}_{s}(\alpha )=\frac{p(N-\alpha )}{N-sp}\) is the so-called Hardy–Sobolev critical exponent. Using concentration-compactness principle and the mountain pass lemma due to Kajikiya [23], we show the existence of infinitely many solutions which tend to be zero provided that \(\lambda \) belongs to a suitable range.


Fractional \( p \& q\) Laplacians critical Sobolev–Hardy exponents symmetric mountain pass lemma 

Mathematics Subject Classification

47G20 35R11 35A15 58E05 



The authors would like to thank the anonymous referee for her/his useful comments and valuable suggestions which improved and clarified the paper.


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Authors and Affiliations

  1. 1.Dipartimento di Scienze Matematiche Informatiche e FisicheUniversità di UdineUdineItaly
  2. 2.Dipartimento di Ingegneria Industriale e Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly

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