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arXiv · 2004.12583

Nonlinear nonhomogeneous singular problems

Nikolaos S. Papageorgiou·Vicenţiu D. Rădulescu·Dušan D. Repovš
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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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Nikolaos S. Papageorgiou, Vicenţiu D. Rădulescu, Dušan D. Repovš. 2020-04-27. Nonlinear nonhomogeneous singular problems. https://doi.org/10.1007/s00526-019-1667-0

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