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

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Abstract

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.

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Acknowledgements

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. Dipartimento di Scienze Matematiche Informatiche e Fisiche, Università di Udine, via delle Scienze, 206, 33100, Udine, Italy

    Vincenzo Ambrosio

  2. Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131, Ancona, Italy

    Teresa Isernia

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  1. Vincenzo Ambrosio
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  2. Teresa Isernia
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Correspondence to Vincenzo Ambrosio.

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Ambrosio, V., Isernia, T. On a Fractional \( p \& q\) Laplacian Problem with Critical Sobolev–Hardy Exponents. Mediterr. J. Math. 15, 219 (2018). https://doi.org/10.1007/s00009-018-1259-9

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  • DOI: https://doi.org/10.1007/s00009-018-1259-9

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