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Multiple constant sign and nodal solutions for nonlinear nonhomogeneous elliptic equations depending on a parameter

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Abstract

We study a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which has z-dependent zeros of constant sign. For all big values of the parameter \(\lambda >0\), we prove two multiplicity theorems producing two positive solutions, two negative solutions, and two nodal solutions. In the first we do not impose any asymptotic conditions on the reaction \(f(z,\cdot )\) at zero. In the second we do not impose any asymptotic conditions on the reaction \(f(z,\cdot )\) at \(\pm \infty \). Then we produce a total of twelve nontrivial smooth solutions all with sign information. Our proofs use variational methods together with flow invariance arguments and suitable truncation techniques.

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Acknowledgements

The authors wish to thank two very knowledgeable referees very much for their valuable comments and helpful suggestions which improved the paper considerably.

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Authors and Affiliations

  1. School of Computation Science, Zhongkai University of Agriculture and Engineering, Guangzhou, 510225, China

    Tieshan He, Pengfei Guo & Li Liu

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  1. Tieshan He
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  2. Pengfei Guo
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  3. Li Liu
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Correspondence to Tieshan He.

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Communicated by Y. Giga.

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Supported by Guangdong Basic and Applied Basic Research Foundation (Nos. 2020A1515010459, 2020A1515110958).

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He, T., Guo, P. & Liu, L. Multiple constant sign and nodal solutions for nonlinear nonhomogeneous elliptic equations depending on a parameter. Calc. Var. 60, 82 (2021). https://doi.org/10.1007/s00526-021-01977-9

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