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A Robin boundary-value problem with non-homogeneous differential operator, indefinite potential, and reaction defined only near zero is investigated. The existence of one or more nodal solutions is achieved by using truncation, perturbation, and comparison techniques, results from Morse theory, besides variational methods.
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Umberto Guarnotta, Salvatore A. Marano, Nikolaos S. Papageorgiou, Multiple nodal solutions to a Robin problem with sign-changing potential and locally defined reaction. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 2, pp. 269–294