Abstract
We consider a nonlinear parametric Neumann problem driven by the anisotropic (p, q)-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive solutions. Using a combination of topological and variational tools together with suitable truncation and comparison techniques, we prove a bifurcation-type result describing the set of positive solutions as the positive parameter λ varies. We also show the existence of minimal positive solutions \(u_{\lambda }^{*}\) and determine the monotonicity and continuity properties of the map \(\lambda \mapsto u_{\lambda }^{*}\).
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The authors wish to thank the two anonymous reviewers for their remarks and constructive criticisms that helped us to improve the presentation.
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The authors have been supported by the Slovenian Research Agency grants P1-0292, J1-8131, N1-0064, N1-0083, and N1-0114. The work of Vicenţiu D. Rădulescu was supported by a grant of the Romanian Ministry of Education and Research, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2020-0068, within PNCDI III.
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All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by all authors. The first draft of the manuscript was written by Nikolaos S. Papageorgiou and all authors commented on previous versions of the manuscript. The authors read and approved the final manuscript.
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This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, N1-0064, N1-0083, and N1-0114. The work of Vicenţiu D. Rădulescu was supported by a grant of the Romanian Ministry of Education and Research, CNCS-UEFISCDI, project number PN-III-P4-ID-PCE-2020-0068, within PNCDI III.
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Papageorgiou, N.S., Rădulescu, V.D. & Repovš, D.D. Anisotropic Singular Neumann Equations with Unbalanced Growth. Potential Anal 57, 55–82 (2022). https://doi.org/10.1007/s11118-021-09905-4
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DOI: https://doi.org/10.1007/s11118-021-09905-4