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July  2020, 13(7): 1935-1945. doi: 10.3934/dcdss.2020151

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

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

* Corresponding author: Anna Maria Candela

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

Received  July 2018 Revised  December 2018 Published  November 2019

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

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.
Citation: Anna Maria Candela, Addolorata Salvatore. Positive solutions for some generalized $p$–Laplacian type problems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (7) : 1935-1945. doi: 10.3934/dcdss.2020151
##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar [2] D. Arcoya and L. Boccardo, Critical points for multiple integrals of the calculus of variations, Arch. Rational Mech. Anal., 134 (1996), 249-274.  doi: 10.1007/BF00379536.  Google Scholar [3] D. Arcoya and L. Boccardo, Some remarks on critical point theory for nondifferentiable functionals, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 79-100.  doi: 10.1007/s000300050066.  Google Scholar [4] G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 977-1009.  doi: 10.1007/s00030-012-0193-y.  Google Scholar [5] G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbb {R}^N$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016.  Google Scholar [6] A. M. Candela and G. Palmieri, Multiple solutions of some nonlinear variational problems, Adv. Nonlinear Stud., 6 (2006), 269-286.  doi: 10.1515/ans-2006-0209.  Google Scholar [7] A. M. Candela and G. Palmieri, Infinitely many solutions of some nonlinear variational equations, Calc. Var. Partial Differential Equations, 34 (2009), 495-530.  doi: 10.1007/s00526-008-0193-2.  Google Scholar [8] A. M. Candela and G. Palmieri, Some abstract critical point theorems and applications, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications, 7th AIMS Conference, Suppl., (2009), 133–142.  Google Scholar [9] A. M. Candela and G. Palmieri, Multiplicity results for some quasilinear equations in lack of symmetry, Adv. Nonlinear Anal., 1 (2012), 121-157.   Google Scholar [10] A. M. Candela and G. Palmieri, An abstract three critical points theorem and applications, in Proceedings of Dynamic Systems and Applications, Dynamic Publishers Inc., Atlanta, 6 (2012), 70–77. Google Scholar [11] A. M. Candela and G. Palmieri, Multiplicity results for some nonlinear elliptic problems with asymptotically $p$-linear terms, Calc. Var. Partial Differential Equations, 56 (2017), Art. 72, 39 pp. doi: 10.1007/s00526-017-1170-4.  Google Scholar [12] A. M. Candela, G. Palmieri and K. Perera, Multiple solutions for $p$-Laplacian type problems with asymptotically $p$-linear terms via a cohomological index theory, J. Differential Equations, 259 (2015), 235-263.  doi: 10.1016/j.jde.2015.02.007.  Google Scholar [13] A. M. Candela, G. Palmieri and A. Salvatore, Multiple solutions for some symmetric supercritical problems, Commun. Contemp. Math., (to appear). doi: 10.1142/S0219199719500755.  Google Scholar [14] A. M. Candela, G. Palmieri and A. Salvatore, Infinitely many solutions for quasilinear elliptic equations with lack of symmetry, Nonlinear Anal., 172 (2018), 141-162.  doi: 10.1016/j.na.2018.02.011.  Google Scholar [15] A. M. Candela and A. Salvatore, Infinitely many solutions for some nonlinear supercritical problems with break of symmetry, Opuscula Math., 39 (2019), 175-194.  doi: 10.7494/OpMath.2019.39.2.175.  Google Scholar [16] A. Canino, Multiplicity of solutions for quasilinear elliptic equations, Topol. Methods Nonlinear Anal., 6 (1995), 357-370.  doi: 10.12775/TMNA.1995.050.  Google Scholar [17] P. Lindqvist, On the equation div $(|\nabla u|^{p-2}\nabla u) + \lambda |u|^{p-2}u =0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.  doi: 10.1090/S0002-9939-1990-1007505-7.  Google Scholar [18] B. Pellacci and M. Squassina, Unbounded critical points for a class of lower semicontinuous functionals, J. Differential Equations, 201 (2004), 25-62.  doi: 10.1016/j.jde.2004.03.002.  Google Scholar [19] P. Pucci and V. Rădulescu, Combined effects in quasilinear elliptic problems with lack of compactness, Rend. Lincei Mat. Appl., 22 (2011), 189-205.  doi: 10.4171/RLM/595.  Google Scholar [20] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Commun. Pure Appl. Math., 20 (1967), 721-747.  doi: 10.1002/cpa.3160200406.  Google Scholar

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##### References:
 [1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.  doi: 10.1016/0022-1236(73)90051-7.  Google Scholar [2] D. Arcoya and L. Boccardo, Critical points for multiple integrals of the calculus of variations, Arch. Rational Mech. Anal., 134 (1996), 249-274.  doi: 10.1007/BF00379536.  Google Scholar [3] D. Arcoya and L. Boccardo, Some remarks on critical point theory for nondifferentiable functionals, NoDEA Nonlinear Differential Equations Appl., 6 (1999), 79-100.  doi: 10.1007/s000300050066.  Google Scholar [4] G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 977-1009.  doi: 10.1007/s00030-012-0193-y.  Google Scholar [5] G. Autuori and P. Pucci, Elliptic problems involving the fractional Laplacian in $\mathbb {R}^N$, J. Differential Equations, 255 (2013), 2340-2362.  doi: 10.1016/j.jde.2013.06.016.  Google Scholar [6] A. M. Candela and G. Palmieri, Multiple solutions of some nonlinear variational problems, Adv. Nonlinear Stud., 6 (2006), 269-286.  doi: 10.1515/ans-2006-0209.  Google Scholar [7] A. M. Candela and G. Palmieri, Infinitely many solutions of some nonlinear variational equations, Calc. Var. Partial Differential Equations, 34 (2009), 495-530.  doi: 10.1007/s00526-008-0193-2.  Google Scholar [8] A. M. Candela and G. Palmieri, Some abstract critical point theorems and applications, Discrete Contin. Dyn. Syst., Dynamical Systems, Differential Equations and Applications, 7th AIMS Conference, Suppl., (2009), 133–142.  Google Scholar [9] A. M. Candela and G. Palmieri, Multiplicity results for some quasilinear equations in lack of symmetry, Adv. Nonlinear Anal., 1 (2012), 121-157.   Google Scholar [10] A. M. Candela and G. Palmieri, An abstract three critical points theorem and applications, in Proceedings of Dynamic Systems and Applications, Dynamic Publishers Inc., Atlanta, 6 (2012), 70–77. Google Scholar [11] A. M. Candela and G. Palmieri, Multiplicity results for some nonlinear elliptic problems with asymptotically $p$-linear terms, Calc. Var. Partial Differential Equations, 56 (2017), Art. 72, 39 pp. doi: 10.1007/s00526-017-1170-4.  Google Scholar [12] A. M. Candela, G. Palmieri and K. Perera, Multiple solutions for $p$-Laplacian type problems with asymptotically $p$-linear terms via a cohomological index theory, J. Differential Equations, 259 (2015), 235-263.  doi: 10.1016/j.jde.2015.02.007.  Google Scholar [13] A. M. Candela, G. Palmieri and A. Salvatore, Multiple solutions for some symmetric supercritical problems, Commun. Contemp. Math., (to appear). doi: 10.1142/S0219199719500755.  Google Scholar [14] A. M. Candela, G. Palmieri and A. Salvatore, Infinitely many solutions for quasilinear elliptic equations with lack of symmetry, Nonlinear Anal., 172 (2018), 141-162.  doi: 10.1016/j.na.2018.02.011.  Google Scholar [15] A. M. Candela and A. Salvatore, Infinitely many solutions for some nonlinear supercritical problems with break of symmetry, Opuscula Math., 39 (2019), 175-194.  doi: 10.7494/OpMath.2019.39.2.175.  Google Scholar [16] A. Canino, Multiplicity of solutions for quasilinear elliptic equations, Topol. Methods Nonlinear Anal., 6 (1995), 357-370.  doi: 10.12775/TMNA.1995.050.  Google Scholar [17] P. Lindqvist, On the equation div $(|\nabla u|^{p-2}\nabla u) + \lambda |u|^{p-2}u =0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.  doi: 10.1090/S0002-9939-1990-1007505-7.  Google Scholar [18] B. Pellacci and M. Squassina, Unbounded critical points for a class of lower semicontinuous functionals, J. Differential Equations, 201 (2004), 25-62.  doi: 10.1016/j.jde.2004.03.002.  Google Scholar [19] P. Pucci and V. Rădulescu, Combined effects in quasilinear elliptic problems with lack of compactness, Rend. Lincei Mat. Appl., 22 (2011), 189-205.  doi: 10.4171/RLM/595.  Google Scholar [20] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Commun. Pure Appl. Math., 20 (1967), 721-747.  doi: 10.1002/cpa.3160200406.  Google Scholar
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