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Quasilinear nonlocal elliptic problems with variable singular exponent
Existence and concentration of nodal solutions for a subcritical p&q equation
Departamento de Matemática - Universidade de Brasília, 70.910-900, Brasília - DF, Brazil |
$ -\Delta_{p}u- \Delta_{q}u+V( x)(|u|^{p-2}u+|u|^{q-2}u) = f(u) \quad \mbox{in} \ \mathbb{R}^{N}. $ |
$ \mathbb{R}^{N} $ |
References:
[1] |
C. O. Alves and M. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153–1166.
doi: 10.1007/s00033-013-0376-3. |
[2] |
C. O. Alves and G. M. Figueiredo, Multiplicity of positive solutions for a quasilinear problem in $I\!\!R^{N}$ via penalization method, Adv. Nonlinear Stud., 5 (2005), 551-572.
doi: 10.1515/ans-2005-0405. |
[3] |
C. O. Alves and S. H. M. Soares,
On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations., J. Math. Anal. Appl., 296 (2004), 563-577.
doi: 10.1016/j.jmaa.2004.04.022. |
[4] |
C. O. Alves and S. H. M. Soares,
Nodal solutions for singularly perturbed equations with critical exponential growth, J. Differ. Equ., 234 (2007), 464-484.
doi: 10.1016/j.jde.2006.12.006. |
[5] |
C. O. Alves and G. M. .Figueiredo,
Multiplicity and Concentration of Positive Solutions for a Class of Quasilinear Problems., Adv. Nonlinear Stud., 11 (2011), 265-295.
doi: 10.1515/ans-2011-0203. |
[6] |
C. O. Alves and G. M. .Figueiredo,
Existence and concentration of nodal solutions to a class of quasilinear problems, Topol. Method. Nonl. Anal., 29 (2007), 279-293.
|
[7] |
S. Barile and G. M. Figueiredo,
Existence of least energy positive, negative and nodal solutions for a class of $p$ & $q$-problems, J. Math. Anal. Appl., 427 (2015), 1205-1233.
doi: 10.1016/j.jmaa.2015.02.086. |
[8] |
T. Bartsch, T. Weth and M. Willem,
Partial symmetry of least energy nodal solution to some variational problems, J. Anal. Math., 96 (2005), 1-18.
doi: 10.1007/BF02787822. |
[9] |
A. Castro, J. Cossio and J. Neuberger, A sign-changing solution for a super linear Dirichlet problem, Rocky Mt. J. Math., 27 (1997), 1041-1053.
doi: 10.1216/rmjm/1181071858. |
[10] |
M. Del Pino and P. Felmer,
Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. equ., 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[11] |
E. Di Benedetto E,
$C^{1, \alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1985), 827-850.
doi: 10.1016/0362-546X(83)90061-5. |
[12] |
Y. Deng and W. Shuai,
Existence and concentration behavior of sign-changing solutions for quasilinear Schrödinger equations, Sci. China Math., 59 (2016), 1095-1112.
|
[13] |
G. M. Figueiredo,
Existence of positive solutions for a class of p & q elliptic problems with critical growth on $\mathbb{R}^{N}$, J. Math. Anal. Appl., 378 (2011), 507-518.
doi: 10.1016/j.jmaa.2011.02.017. |
[14] |
G. M. Figueiredo and M. T. O. Pimenta, Nodal solutions of an NLS equation concentrating on lower dimensional spheres, Bound. Value Probl., 168 (2015), 19pp.
doi: 10.1186/s13661-015-0411-8. |
[15] |
N. S. Papageorgiou, C. Vetro and F. Vetro, Multiple solutions for parametric double phase Dirichlet problems, Commun. Contemp. Math., (2020), 18pp.
doi: 10.1142/S0219199720500066. |
[16] |
N. S. Papageorgiou, C. Vetro and F. Vetro,
Continuous spectrum for a two phase eigenvalue problem with an indefinite and unbounded potential, J. Differ. Equ., 268 (2020), 4102-4118.
doi: 10.1016/j.jde.2019.10.026. |
[17] |
P. H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[18] |
Y. Sato,
Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency, Commun. Pure. Appl. Anal., 7 (2008), 883-903.
doi: 10.3934/cpaa.2008.7.883. |
[19] |
C. Vetro, An elliptic equation on n-dimensional manifolds, Complex Var. Elliptic, (2020), 17pp.
doi: 10.1080/17476933.2020.1711745. |
[20] |
C. Vetro, Parametric and nonparametric A-Laplace problems: Existence of solutions and asymptotic analysis, Asymptot. Anal., (2020), 14pp. |
[21] |
C. Vetro,
Pairs of nontrivial smooth solutions for nonlinear Neumann problems, Appl. Math. Lett., 103 (2020), 1-7.
doi: 10.1016/j.aml.2019.106171. |
[22] |
C. Vetro,
Semilinear Robin problems driven by the Laplacian plus an indefinite potential, Complex Var. Elliptic, 65 (2020), 573-587.
doi: 10.1080/17476933.2019.1597066. |
[23] |
C. Vetro and F. Vetro,
On problems driven by the (p(·), q(·))-Laplace operator, Mediterr. J. Math, 17 (2019), 1-11.
|
[24] |
X. Wang,
On concentration of positive bound states of nonlinear Schr$\ddot{o}$dinger equations, Commun. Math. Phys., 53 (1993), 229-244.
|
[25] |
M. Willem, Minimax theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
show all references
References:
[1] |
C. O. Alves and M. Souto, Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains, Z. Angew. Math. Phys., 65 (2014), 1153–1166.
doi: 10.1007/s00033-013-0376-3. |
[2] |
C. O. Alves and G. M. Figueiredo, Multiplicity of positive solutions for a quasilinear problem in $I\!\!R^{N}$ via penalization method, Adv. Nonlinear Stud., 5 (2005), 551-572.
doi: 10.1515/ans-2005-0405. |
[3] |
C. O. Alves and S. H. M. Soares,
On the location and profile of spike-layer nodal solutions to nonlinear Schrödinger equations., J. Math. Anal. Appl., 296 (2004), 563-577.
doi: 10.1016/j.jmaa.2004.04.022. |
[4] |
C. O. Alves and S. H. M. Soares,
Nodal solutions for singularly perturbed equations with critical exponential growth, J. Differ. Equ., 234 (2007), 464-484.
doi: 10.1016/j.jde.2006.12.006. |
[5] |
C. O. Alves and G. M. .Figueiredo,
Multiplicity and Concentration of Positive Solutions for a Class of Quasilinear Problems., Adv. Nonlinear Stud., 11 (2011), 265-295.
doi: 10.1515/ans-2011-0203. |
[6] |
C. O. Alves and G. M. .Figueiredo,
Existence and concentration of nodal solutions to a class of quasilinear problems, Topol. Method. Nonl. Anal., 29 (2007), 279-293.
|
[7] |
S. Barile and G. M. Figueiredo,
Existence of least energy positive, negative and nodal solutions for a class of $p$ & $q$-problems, J. Math. Anal. Appl., 427 (2015), 1205-1233.
doi: 10.1016/j.jmaa.2015.02.086. |
[8] |
T. Bartsch, T. Weth and M. Willem,
Partial symmetry of least energy nodal solution to some variational problems, J. Anal. Math., 96 (2005), 1-18.
doi: 10.1007/BF02787822. |
[9] |
A. Castro, J. Cossio and J. Neuberger, A sign-changing solution for a super linear Dirichlet problem, Rocky Mt. J. Math., 27 (1997), 1041-1053.
doi: 10.1216/rmjm/1181071858. |
[10] |
M. Del Pino and P. Felmer,
Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differ. equ., 4 (1996), 121-137.
doi: 10.1007/BF01189950. |
[11] |
E. Di Benedetto E,
$C^{1, \alpha}$ local regularity of weak solutions of degenerate elliptic equations, Nonlinear Anal., 7 (1985), 827-850.
doi: 10.1016/0362-546X(83)90061-5. |
[12] |
Y. Deng and W. Shuai,
Existence and concentration behavior of sign-changing solutions for quasilinear Schrödinger equations, Sci. China Math., 59 (2016), 1095-1112.
|
[13] |
G. M. Figueiredo,
Existence of positive solutions for a class of p & q elliptic problems with critical growth on $\mathbb{R}^{N}$, J. Math. Anal. Appl., 378 (2011), 507-518.
doi: 10.1016/j.jmaa.2011.02.017. |
[14] |
G. M. Figueiredo and M. T. O. Pimenta, Nodal solutions of an NLS equation concentrating on lower dimensional spheres, Bound. Value Probl., 168 (2015), 19pp.
doi: 10.1186/s13661-015-0411-8. |
[15] |
N. S. Papageorgiou, C. Vetro and F. Vetro, Multiple solutions for parametric double phase Dirichlet problems, Commun. Contemp. Math., (2020), 18pp.
doi: 10.1142/S0219199720500066. |
[16] |
N. S. Papageorgiou, C. Vetro and F. Vetro,
Continuous spectrum for a two phase eigenvalue problem with an indefinite and unbounded potential, J. Differ. Equ., 268 (2020), 4102-4118.
doi: 10.1016/j.jde.2019.10.026. |
[17] |
P. H. Rabinowitz,
On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys., 43 (1992), 270-291.
doi: 10.1007/BF00946631. |
[18] |
Y. Sato,
Sign-changing multi-peak solutions for nonlinear Schrödinger equations with critical frequency, Commun. Pure. Appl. Anal., 7 (2008), 883-903.
doi: 10.3934/cpaa.2008.7.883. |
[19] |
C. Vetro, An elliptic equation on n-dimensional manifolds, Complex Var. Elliptic, (2020), 17pp.
doi: 10.1080/17476933.2020.1711745. |
[20] |
C. Vetro, Parametric and nonparametric A-Laplace problems: Existence of solutions and asymptotic analysis, Asymptot. Anal., (2020), 14pp. |
[21] |
C. Vetro,
Pairs of nontrivial smooth solutions for nonlinear Neumann problems, Appl. Math. Lett., 103 (2020), 1-7.
doi: 10.1016/j.aml.2019.106171. |
[22] |
C. Vetro,
Semilinear Robin problems driven by the Laplacian plus an indefinite potential, Complex Var. Elliptic, 65 (2020), 573-587.
doi: 10.1080/17476933.2019.1597066. |
[23] |
C. Vetro and F. Vetro,
On problems driven by the (p(·), q(·))-Laplace operator, Mediterr. J. Math, 17 (2019), 1-11.
|
[24] |
X. Wang,
On concentration of positive bound states of nonlinear Schr$\ddot{o}$dinger equations, Commun. Math. Phys., 53 (1993), 229-244.
|
[25] |
M. Willem, Minimax theorems, Birkhäuser, Boston, 1996.
doi: 10.1007/978-1-4612-4146-1. |
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