Abstract
We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator with a growth of order \((p-1)\) near \(+\infty \) and with a reaction which has the competing effects of a parametric singular term and a \((p-1)\)-superlinear perturbation which does not satisfy the usual Ambrosetti–Rabinowitz condition. Using variational tools, together with suitable truncation and strong comparison techniques, we prove a “bifurcation-type” theorem that describes the set of positive solutions as the parameter \(\lambda \) moves on the positive semiaxis. We also show that for every \(\lambda >0\), the problem has a smallest positive solution \(u^*_\lambda \) and we demonstrate the monotonicity and continuity properties of the map \(\lambda \mapsto u^*_\lambda \).
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This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, N1-0114, N1-0064, and N1-0083.
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Papageorgiou, N.S., Rădulescu, V.D. & Repovš, D.D. Nonlinear nonhomogeneous singular problems. Calc. Var. 59, 9 (2020). https://doi.org/10.1007/s00526-019-1667-0
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DOI: https://doi.org/10.1007/s00526-019-1667-0