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

Anna Maria Candela; Addolorata Salvatore

doi:10.3934/dcdss.2020151

Article Contents

2020, Volume 13, Issue 7: 1935-1945. Doi: 10.3934/dcdss.2020151

This issue Previous Article Positive radial solutions for the Minkowski-curvature equation with Neumann boundary conditions Next Article Existence theorems for generalized nonlinear quadratic integral equations via a new fixed point result

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

Anna Maria Candela^, and
Addolorata Salvatore

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy

^* Corresponding author: Anna Maria Candela
^* Corresponding author: Anna Maria Candela

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

Received: June 30, 2018

Revised: November 30, 2018

Published: April 2020

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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Abstract

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.

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

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