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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 |
$ \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. $ |
$ \Omega \subset \mathbb R^N $ |
$ N \geq 2 $ |
$ A(x,t,\xi) $ |
$ B(x,t, {\xi}) $ |
$ \mathcal{C}^1 $ |
$ \Omega \times \mathbb R \times { \mathbb R}^{N} $ |
$ A_t = \frac{\partial A}{\partial t} $ |
$ a = {\nabla}_{\xi}A $ |
$ B_t = \frac{\partial B}{\partial t} $ |
$ b = {\nabla}_{{\xi}}B $ |
$ g_1(x,t,s) $ |
$ g_2(x,t,s) $ |
$ \Omega \times \mathbb R\times \mathbb R $ |
$ t $ |
$ s $ |
$ G(x,t,s) $ |
$ A $ |
$ B $ |
$ X $ |
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. |
show all references
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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