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Sign-Changing Solutions for Resonant and Superlinear Nonhomogeneous Elliptic Equations

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Abstract

We consider nonlinear Dirichlet problems driven by a nonlinear nonhomogeneous differential operator and with a \((p-1)\)-superlinear Carathéodory reaction term which doesn’t satisfy the usual Ambrosetti–Rabinowitz condition. Also, at the origin, the primitive of the reaction satisfies a nonuniform nonresonance condition with respect to the first eigenvalue of \((-\Delta _{p},W_{0}^{1,p}(\Omega ))\). We consider two distinct cases. In the first one, the reaction \(f(z,\cdot )\) is \((p-1)\)-superlinear growth near \(\pm \infty \). In the second case, the reaction \(f(z,\cdot )\) is \((p-1)\)-superlinear growth near \(+\infty \) (\(-\infty \) ) and it is \((p-1)\)-sublinear near \(-\infty \) (\(+\infty \) ) and resonant with respect to any nonprincipal eigenvalue. We prove several multiplicity results producing sign-changing solutions. Our approach uses variational methods together with flow invariance arguments.

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

    Tieshan He

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Supported by Guangdong Basic and Applied Basic Research Foundation (No. 2020A1515010459) and by Social science-Philosophy plan project of Guangzhou (No. 2019GZGJ125)

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He, T. Sign-Changing Solutions for Resonant and Superlinear Nonhomogeneous Elliptic Equations. Mediterr. J. Math. 18, 250 (2021). https://doi.org/10.1007/s00009-021-01914-2

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