Abstract
We consider a nonlinear parametric Neumann problem driven by a nonhomogeneous differential operator and a strictly \((p-1)\)-sublinear reaction term. We prove a bifurcation-type result establishing the existence of a critical parameter value \(\lambda _*>0\) such that for all \(\lambda >\lambda _*\) the problem has at least two positive solutions, for \(\lambda =\lambda _*\) it has at least one positive solution and for \(\lambda \in (0,\lambda _*)\) there are no positive solutions. Also, for \(\lambda \ge \lambda _*\) we show that the problem has a smallest positive solution \(\bar{u}_{\lambda }\) and we investigate the continuity and monotonicity properties of the map \(\lambda \rightarrow \bar{u}_{\lambda }\).
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Papageorgiou, N.S., Papalini, F. Existence, nonexistence and multiplicity of positive solutions for nonlinear, nonhomogeneous Neumann problems. manuscripta math. 154, 257–274 (2017). https://doi.org/10.1007/s00229-017-0919-6
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DOI: https://doi.org/10.1007/s00229-017-0919-6