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Solutions with Sign Information for Noncoercive Double Phase Equations

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Abstract

We consider a nonautonomous (p, q)-equation with unbalanced growth and a reaction which exhibits the combined effects of a parametric “concave" (\((p-1)\)-sublinear) term and of a \((p-1)\)-linear perturbation, which asymptotically stays above the principal eigenvalue \({\widehat{\lambda }}_{1}^{a}>0\) of the Dirichlet \(-\Delta _{p}^{a}\) operator. Using variational tools, truncation and comparison techniques and critical groups, we show that for all small values of the parameter the problem has at least three nontrivial bounded solutions with sign information (positive, negative, nodal), which are ordered.

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Acknowledgements

The authors wish to thank the two knowledgeable referees for their helpful remarks. The research of Jian Zhang and Wen Zhang was supported by the National Natural Science Foundation of China (12271152), the Natural Science Foundation of Hunan Province (2021JJ30189, 2022JJ30200), the Key project of Scientific Research Project of Department of Education of Hunan Province (21A0387, 22A0461), and Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.

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Authors and Affiliations

  1. Department of Mathematics, National Technical University, Zografou Campus, 15780, Athens, Greece

    Nikolaos S. Papageorgiou

  2. College of Science, Hunan University of Technology and Business, Changsha, 410205, Hunan, China

    Jian Zhang & Wen Zhang

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  1. Nikolaos S. Papageorgiou
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  2. Jian Zhang
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  3. Wen Zhang
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Correspondence to Jian Zhang.

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Papageorgiou, N.S., Zhang, J. & Zhang, W. Solutions with Sign Information for Noncoercive Double Phase Equations. J Geom Anal 34, 14 (2024). https://doi.org/10.1007/s12220-023-01463-y

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