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Positive solutions for some generalized $ p $–Laplacian type problems

  • * Corresponding author: Anna Maria Candela

    * Corresponding author: Anna Maria Candela 

Dedicated to Patrizia Pucci on the occasion of her 65th birthday

Partially supported by Fondi di Ricerca di Ateneo 2015/16 and Research Funds INdAM – GNAMPA Project 2018 "Problemi ellittici semilineari: alcune idee variazionali"

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  • In this paper, we prove the existence of nontrivial weak bounded solutions of the nonlinear elliptic problem

    $ \left\{ \begin{array}{ll} - {\rm div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) = f(x,u) &\hbox{in $\Omega$,}\\ u \ge 0 &\hbox{in $\Omega$,}\\ u\ = \ 0 & \hbox{on $\partial\Omega$,} \end{array} \right. $

    where $ \Omega \subset \mathbb {R}^N $ is an open bounded domain, $ N\ge 3 $, and $ A(x, t, \xi) $, $ f(x, t) $ are given functions, with $ A_t = \frac{\partial A}{\partial t} $, $ a = \nabla_\xi A $.

    To this aim, we use variational arguments which are adapted to our setting and exploit a weak version of the Cerami–Palais–Smale condition.

    Furthermore, if $ A(x, t, \xi) $ grows fast enough with respect to $ t $, then the nonlinear term related to $ f(x, t) $ may have also a supercritical growth.

    Mathematics Subject Classification: Primary: 35J20, 35J92; Secondary: 35J25, 58E05.


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