# American Institute of Mathematical Sciences

doi: 10.3934/dcdss.2022013
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## Existence of minimizers for a quasilinear elliptic system of gradient type

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

* Corresponding author: Federica Mennuni

Received  August 2021 Early access January 2022

Fund Project: The research that led to the present paper was partially supported by MIUR-PRIN project "Qualitative and quantitative aspects of non linear PDEs"(2017 JPCAPN-005), Fondi di Ricerca di Ateneo 2017/18 "Problemi differenziali non lineari"

The aim of this paper is to investigate the existence of weak solutions for the coupled quasilinear elliptic system of gradient type
 $\left\{ \begin{array}{ll} - {\rm div} (a(x, u, \nabla u)) + A_t (x, u,\nabla u) = g_1(x, u, v) &{\rm{ in}} \; \Omega ,\\ - {\rm div} (B(x, v, \nabla v)) + B_t (x, v,\nabla v) = g_2(x, u, v) &{\rm{ in}}\; \Omega ,\\ \quad u = v = 0 &{\rm{ on}}\;\partial\Omega , \end{array} \right.$
where
 $\Omega \subset \mathbb R^N$
is an open bounded domain,
 $N \geq 2$
and
 $A(x,t,\xi)$
,
 $B(x,t, {\xi})$
are
 $\mathcal{C}^1$
–Carathéodory functions on
 $\Omega \times \mathbb R \times { \mathbb R}^{N}$
with partial derivatives
 $A_t = \frac{\partial A}{\partial t}$
,
 $a = {\nabla}_{\xi}A$
, respectively
 $B_t = \frac{\partial B}{\partial t}$
,
 $b = {\nabla}_{{\xi}}B$
, while
 $g_1(x,t,s)$
,
 $g_2(x,t,s)$
are given Carathéodory maps defined on
 $\Omega \times \mathbb R\times \mathbb R$
which are partial derivatives with respect to
 $t$
and
 $s$
of a function
 $G(x,t,s)$
.
We prove that, even if the general form of the terms
 $A$
and
 $B$
makes the variational approach more difficult, under suitable hypotheses, the functional related to the problem is bounded from below and attains its minimum in a "right" Banach space
 $X$
.
The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition and a suitable generalization of the Weierstrass Theorem.
Citation: Federica Mennuni, Addolorata Salvatore. Existence of minimizers for a quasilinear elliptic system of gradient type. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022013
##### 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. [2] L. Boccardo and D. G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 309-323.  doi: 10.1007/s00030-002-8130-0. [3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. [4] 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. [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. [6] A. M. Candela and G. Palmieri, Some abstract critical point theorems and applications, Discrete Contin. Dyn. Syst. Ser. S, (2009), 133-142. [7] 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), 39 pp. [8] A. M. Candela, G. Palmieri and A. Salvatore, Multiple solutions for some symmetric supercritical problems, Commun. Contemp. Math., 22 (2020), 20 pp. [9] A. M. Candela and A. Salvatore, Existence of minimizer for some quasilinear elliptic problems, Discrete Contin. Dynam. Syst. Ser. S, 13 (2020), 3335-3345.  doi: 10.3934/dcdss.2020241. [10] A. M. Candela, A. Salvatore and C. Sportelli, Existence and multiplicity results for a class of coupled quasilinear elliptic systems of gradient type, Adv. Nonlinear Stud., 21 (2021), 461-488.  doi: 10.1515/ans-2021-2121. [11] A. M. Candela and C. Sportelli, Nontrivial solutions for a class of gradient-type quasilinear elliptic systems, Topol. Methods Nonlinear Anal.. [12] A. Canino, Multiplicity of solutions for quasilinear elliptic equations, Topol. Methods Nonlinear Anal., 6 (1995), 357-370.  doi: 10.12775/TMNA.1995.050. [13] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, 1989. [14] L. F. O. Faria, O. H. Miyagaki, D. Motreanu and M. Tanaka, Existence results for nonlinear elliptic equations with Leray–Lions operator and dependence on the gradient, Nonlinear Anal., 96 (2014), 154-166.  doi: 10.1016/j.na.2013.11.006. [15] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. [16] 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. [17] M. Squassina, Existence, Multiplicity, Perturbation, and Concentration Results for a Class of Quasi-linear Elliptic Problems, Electron. J. Differ. Equ. Monogr., 7, Texas State University-San Marcos, San Marcos TX, 2006.

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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. [2] L. Boccardo and D. G. de Figueiredo, Some remarks on a system of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl., 9 (2002), 309-323.  doi: 10.1007/s00030-002-8130-0. [3] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. [4] 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. [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. [6] A. M. Candela and G. Palmieri, Some abstract critical point theorems and applications, Discrete Contin. Dyn. Syst. Ser. S, (2009), 133-142. [7] 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), 39 pp. [8] A. M. Candela, G. Palmieri and A. Salvatore, Multiple solutions for some symmetric supercritical problems, Commun. Contemp. Math., 22 (2020), 20 pp. [9] A. M. Candela and A. Salvatore, Existence of minimizer for some quasilinear elliptic problems, Discrete Contin. Dynam. Syst. Ser. S, 13 (2020), 3335-3345.  doi: 10.3934/dcdss.2020241. [10] A. M. Candela, A. Salvatore and C. Sportelli, Existence and multiplicity results for a class of coupled quasilinear elliptic systems of gradient type, Adv. Nonlinear Stud., 21 (2021), 461-488.  doi: 10.1515/ans-2021-2121. [11] A. M. Candela and C. Sportelli, Nontrivial solutions for a class of gradient-type quasilinear elliptic systems, Topol. Methods Nonlinear Anal.. [12] A. Canino, Multiplicity of solutions for quasilinear elliptic equations, Topol. Methods Nonlinear Anal., 6 (1995), 357-370.  doi: 10.12775/TMNA.1995.050. [13] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, 1989. [14] L. F. O. Faria, O. H. Miyagaki, D. Motreanu and M. Tanaka, Existence results for nonlinear elliptic equations with Leray–Lions operator and dependence on the gradient, Nonlinear Anal., 96 (2014), 154-166.  doi: 10.1016/j.na.2013.11.006. [15] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. [16] 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. [17] M. Squassina, Existence, Multiplicity, Perturbation, and Concentration Results for a Class of Quasi-linear Elliptic Problems, Electron. J. Differ. Equ. Monogr., 7, Texas State University-San Marcos, San Marcos TX, 2006.
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