# American Institute of Mathematical Sciences

December  2020, 13(12): 3335-3345. doi: 10.3934/dcdss.2020241

## Existence of minimizers for some quasilinear elliptic 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 Gisèle Ruiz Goldstein on the occasion of her 60th birthday

Received  December 2018 Revised  June 2019 Published  January 2020

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"

The aim of this paper is investigating the existence of at least one weak bounded solution of the quasilinear 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$,}\\ u\ = \ 0 & \hbox{on$\partial\Omega$,} \end{array} \right.$
where
 $\Omega \subset \mathbb R^N$
is an open bounded domain and
 $A(x,t,\xi)$
,
 $f(x,t)$
are given real functions, with
 $A_t = \frac{\partial A}{\partial t}$
,
 $a = \nabla_\xi A$
.
We prove that, even if
 $A(x,t,\xi)$
makes the variational approach more difficult, the functional associated to such a problem is bounded from below and attains its infimum when the growth of the nonlinear term
 $f(x,t)$
is "controlled" by
 $A(x,t,\xi)$
. Moreover, stronger assumptions allow us to find the existence of at least one positive solution.
We use a suitable Minimum Principle based on a weak version of the Cerami–Palais–Smale condition.
Citation: Anna Maria Candela, Addolorata Salvatore. Existence of minimizers for some quasilinear elliptic problems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3335-3345. doi: 10.3934/dcdss.2020241
##### References:
 [1] 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 [2] D. Arcoya, L. Boccardo and L. Orsina, Critical points for functionals with quasilinear singular Euler–Lagrange equations, Calc. Var. Partial Differential Equations, 47 (2013), 159-180.  doi: 10.1007/s00526-012-0514-3.  Google Scholar [3] R. Bartolo, A. M. Candela and A. Salvatore, $p$–Laplacian problems with nonlinearities interacting with the spectrum, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1701-1721.  doi: 10.1007/s00030-013-0226-1.  Google Scholar [4] L. Boccardo and B. Pellacci, Critical points of non–regular integral functionals, Rev. Math. Iberoam., 34 (2018), 1001-1020.  doi: 10.4171/RMI/1013.  Google Scholar [5] 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 [6] A. M. Candela and G. Palmieri, Some abstract critical point theorems and applications, Discrete Contin. Dynam. Syst., 2009 (2009), 133-142.   Google Scholar [7] A. M. Candela, G. Palmieri and A. Salvatore, Some results on supercritical quasilinear elliptic problems, Commun. Contemp. Math., (2019) 1950075 (20 pages). doi: 10.1142/S0219199719500755.  Google Scholar [8] 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 [9] A. M. Candela and A. Salvatore, Positive solutions for a generalized $p$–Laplacian type problem, Discrete Contin. Dyn. Syst. Ser. S, (to appear). doi: 10.3934/dcdss.2020151.  Google Scholar [10] 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 [11] 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 [12] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar [13] G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$–Laplacian, Portugaliae Mathematica, 58 (2001), 339-378.   Google Scholar [14] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.   Google Scholar [15] P. Lindqvist, On the equation ${\rm {div}} (|\nabla u|^{p-2}\nabla u) + \lambda|u|^{p-2}u=0$, Proc. Am. Math. Soc., 109 (1990), 157-164.  doi: 10.1090/S0002-9939-1990-1007505-7.  Google Scholar [16] J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar [17] 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] 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 [2] D. Arcoya, L. Boccardo and L. Orsina, Critical points for functionals with quasilinear singular Euler–Lagrange equations, Calc. Var. Partial Differential Equations, 47 (2013), 159-180.  doi: 10.1007/s00526-012-0514-3.  Google Scholar [3] R. Bartolo, A. M. Candela and A. Salvatore, $p$–Laplacian problems with nonlinearities interacting with the spectrum, NoDEA Nonlinear Differential Equations Appl., 20 (2013), 1701-1721.  doi: 10.1007/s00030-013-0226-1.  Google Scholar [4] L. Boccardo and B. Pellacci, Critical points of non–regular integral functionals, Rev. Math. Iberoam., 34 (2018), 1001-1020.  doi: 10.4171/RMI/1013.  Google Scholar [5] 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 [6] A. M. Candela and G. Palmieri, Some abstract critical point theorems and applications, Discrete Contin. Dynam. Syst., 2009 (2009), 133-142.   Google Scholar [7] A. M. Candela, G. Palmieri and A. Salvatore, Some results on supercritical quasilinear elliptic problems, Commun. Contemp. Math., (2019) 1950075 (20 pages). doi: 10.1142/S0219199719500755.  Google Scholar [8] 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 [9] A. M. Candela and A. Salvatore, Positive solutions for a generalized $p$–Laplacian type problem, Discrete Contin. Dyn. Syst. Ser. S, (to appear). doi: 10.3934/dcdss.2020151.  Google Scholar [10] 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 [11] 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 [12] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear Anal. TMA., 56 (2004), 213-226.  doi: 10.1016/j.na.2003.09.008.  Google Scholar [13] G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$–Laplacian, Portugaliae Mathematica, 58 (2001), 339-378.   Google Scholar [14] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.   Google Scholar [15] P. Lindqvist, On the equation ${\rm {div}} (|\nabla u|^{p-2}\nabla u) + \lambda|u|^{p-2}u=0$, Proc. Am. Math. Soc., 109 (1990), 157-164.  doi: 10.1090/S0002-9939-1990-1007505-7.  Google Scholar [16] J. Q. Liu, Y. Q. Wang and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, Ⅱ, J. Differential Equations, 187 (2003), 473-493.  doi: 10.1016/S0022-0396(02)00064-5.  Google Scholar [17] 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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