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November  2020, 19(11): 5077-5095. doi: 10.3934/cpaa.2020227

Existence and concentration of nodal solutions for a subcritical p&q equation

Gustavo S. Costa and  Giovany M. Figueiredo ,

Departamento de Matemática - Universidade de Brasília, 70.910-900, Brasília - DF, Brazil

* Corresponding author

Received  February 2020 Revised  May 2020 Published  November 2020 Early access  July 2020

Fund Project: Gustavo S. Costa and Giovany M. Figueiredo were partially supported by CNPq, Capes and Fapesp - Brazil

In this paper we prove existence and concentration results for a family of nodal solutions for a some quasilinear equation with subcritical growth, whose prototype is
$ -\Delta_{p}u- \Delta_{q}u+V( x)(|u|^{p-2}u+|u|^{q-2}u) = f(u) \quad \mbox{in} \ \mathbb{R}^{N}. $
Each nodal solution changes sign exactly once in
$ \mathbb{R}^{N} $
 and has an exponential decay at infinity. Here we use variational methods and Del Pino and Felmer's technique [10] in order to overcome the lack of compactness.
Keywords: p&q Laplacian operator, Del Pino and Felmer's penalization method.
Mathematics Subject Classification: Primary: 35J60; Secondary: 35J10, 35J20.
Citation: Gustavo S. Costa, Giovany M. Figueiredo. Existence and concentration of nodal solutions for a subcritical p&q equation. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5077-5095. doi: 10.3934/cpaa.2020227
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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