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Multiplicity of Solution for a Quasilinear Equation with Singular Nonlinearity

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Abstract

For an open, bounded domain \(\Omega \) in \(\mathbb {R}^N\) which is strictly convex with smooth boundary, we show that there exists a \(\Lambda >0\) such that for \(0<\lambda <{\Lambda } \), the quasilinear singular problem

$$\begin{aligned} \begin{aligned} -\Delta _pu&= \lambda u^{-\delta }+u^q\,\,\text { in }\,\,\Omega \\ u&= 0\,\,\text { on }\,\,\partial \Omega ;\, \,\,u>0\,\,\text { in }\,\,\Omega \end{aligned} \end{aligned}$$

admits at least two distinct solutions u and v in \(W^{1,p}_{loc}(\Omega )\cap L^{\infty }(\Omega )\) provided \(\delta \ge 1\), \(\frac{2N+2}{N+2}<p<N\) and \(p-1<q<\frac{Np}{N-p}-1\).

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Acknowledgements

The first author is supported by DST-Inspire Faculty Award MA-2013029. The second author is supported by NBHM Fellowship No. 2/39(2)/2014/NBHM/R&D-II/8020/June 26, 2014. The authors would also like to thank the anonymous referee for his/her careful reading of the manuscript and providing us with valuable insight about the literature.

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  1. Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, Uttar Pradesh, 208016, India

    Kaushik Bal & Prashanta Garain

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  1. Kaushik Bal
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  2. Prashanta Garain
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Correspondence to Kaushik Bal.

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Kaushik Bal is supported by DST-Inspire Faculty Award MA-2013029. Prashanta Garain is supported by NBHM Fellowship no: 2/39(2)/2014/NBHM/R&D-II/8020/June 26, 2014.

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Bal, K., Garain, P. Multiplicity of Solution for a Quasilinear Equation with Singular Nonlinearity. Mediterr. J. Math. 17, 91 (2020). https://doi.org/10.1007/s00009-020-01515-5

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