Nonlinear, nonhomogeneous parametric Neumann problems

Sergiu Aizicovici, Nikolaos S. Papageorgiou, Vasile Staicu

DOI: http://dx.doi.org/10.12775/TMNA.2016.035

Abstract


We consider a parametric nonlinear Neumann problem driven by a nonlinear nonhomogeneous differential operator, with a Carathéodory reaction $f$ which is $p$-superlinear in the second variable, but not necessarily satisfying the usual in such cases Ambrosetti-Rabinowitz condition. We prove a bifurcation type result describing the dependence of positive solutions on the parameter $\lambda> 0$, show the existence of a smallest positive solution $\overline{u}_{\lambda}$ and investigate properties of the map $\lambda\mapsto\overline{u}_{\lambda}$. Finally, we show the existence of nodal solutions.

Keywords


Positive solutions; nonlinear nonhomogeneous differential operator; nonlinear regularity; nonlinear maximum principle; bifurcation type result; nodal solutions

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References


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