Some results on the p(u)-Laplacian problem
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The p-Laplacian problem with the exponent of nonlinearity p depending on the solution u itself is considered in this work. Both situations when p(u) is a local quantity or when p(u) is nonlocal are studied here. For the associated boundary-value local problem, we prove the existence of weak solutions by using a singular perturbation technique. We also prove the existence of weak solutions to the nonlocal version of the associated boundary-value problem. The issue of uniqueness for these problems is addressed in this work as well, in particular by working out the uniqueness for a one dimensional local problem and by showing that the uniqueness is easily lost in the nonlocal problem.
Mathematics Subject Classification35J60 35J05 35D30
We are very greatful to the referees for their constructive remarks. This work was performed when the first author was visiting the USTC in Hefei and during a part time employment at the S. M. Nikolskii Mathematical Institute of RUDN University, 6 Miklukho-Maklay St, Moscow, 117198, supported by the Ministry of Education and Science of the Russian Federation. He is greatful to these institutions for their support. Main part of this work was carried out also during the visit of the second author to the University of Zurich during the first quarter of 2018. Besides the Grant SFRH/BSAB/135242/2017 of the Portuguese Foundation for Science and Technology (FCT), Portugal, which made this visit possible, the second author also wishes to thank to Prof. Michel Chipot who kindly welcomed him at the University of Zurich.
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