Rocky Mountain Journal of Mathematics

Existence of solutions for quasilinear Kirchhoff type problems with critical nonlinearity in $\mathbb {R}^N$

Jing Zhang and Alatancang Chen

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We study the existence of solutions for a class of Kirchhoff type problems with critical growth in $\mathbb {R}^N$:$$-\varepsilon ^2\biggl (a+b\int _{\mathbb {R}^N}|\nabla u|^2\,dx\biggr )\Delta u + V(x)u -\varepsilon ^2a\Delta (u^2)u = |u|^{22^\ast -2}u + h(x,u),$$ $(t, x) \in \mathbb {R} \times \mathbb {R}^N$. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We prove that it has at least one solution and for any $m \in \mathbb {N}$, it has at least $m$ pairs of solutions. The proofs are based on the variational methods and concentration-compactness principle.

Article information

Source
Rocky Mountain J. Math., Volume 49, Number 5 (2019), 1725-1753.

Dates
First available in Project Euclid: 19 September 2019

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1568880099

Digital Object Identifier
doi:10.1216/RMJ-2019-49-5-1725

Mathematical Reviews number (MathSciNet)
MR4010581

Subjects
Primary: 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)
Secondary: 58E50: Applications

Keywords
Kirchhoff type problems critical nonlinearity variational method critical point

Citation

Zhang, Jing; Chen, Alatancang. Existence of solutions for quasilinear Kirchhoff type problems with critical nonlinearity in $\mathbb {R}^N$. Rocky Mountain J. Math. 49 (2019), no. 5, 1725--1753. doi:10.1216/RMJ-2019-49-5-1725. https://projecteuclid.org/euclid.rmjm/1568880099


Export citation

References

  • A. Ambrosetti, M. Badiale and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal. 140 (1997), 285–300.
  • A. Ambrosetti, A. Malchiodi and D. Ruiz, Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Anal. Math. 98 (2006), 317–348.
  • A. Ambrosetti, A. Malchiodi and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal. 159 (2001), 253–271.
  • A. Ambrosetti and Z.-Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential Integral Equations 18 (2005), 1321–1332.
  • A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. Eur. Math. Soc. (JEMS) 7 (2005), 117–144.
  • A. Ambrosetti and A. Malchiodi, Perturbation methods and semilinear elliptic problems on ${\mathbb R}^n$, Progress in Mathematics 240, Birkhäuser, Basel (2006).
  • V. Benci, On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc. 274 (1982), 533–572.
  • V. Benci, C.R. Grisanti and A.M. Micheletti, Existence and non existence of the ground state solution for the nonlinear Schroedinger equations with $V(\infty)=0$, Topol. Methods Nonlinear Anal. 26 (2005), 203–219.
  • H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), 486–490.
  • J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. Ration. Mech. Anal. 165 (2002), 295–316.
  • J. Byeon and Z.-Q. Wang, Standing waves with a critical frequency for nonlinear Schrödinger equations, II, Calc. Var. Partial Differential Equations 18 (2003), 207–219.
  • D. Cao and S. Peng, Semi-classical bound states for Schrödinger equations with potentials vanishing or unbounded at infinity, Comm. Partial Differential Equations 34 (2009), 1566–1591.
  • G. Cerami, G. Devillanova and S. Solimini, Infinitely many bound states for some nonlinear scalar field equations, Calc. Var. Partial Differential Equations 23 (2005), 139–168.
  • J. Chabrowski, Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents, Calc. Var. Partial Differential Equations 3 (1995), 493–512.
  • S. Cingolani and M. Lazzo, Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. Differential Equations 160 (2000), 118–138.
  • M. Clapp and Y. Ding, Minimal nodal solutions of a Schrödinger equation with critical nonlinearity and symmetric potential, Differential Integral Equations 16 (2003), 981–992.
  • M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Anal. 56 (2004), 213–226.
  • F.J.S.A. Corrêa and G.M. Figueiredo, On an elliptic equation of $p$-Kirchhoff type via variational methods, Bull. Austral. Math. Soc. 74 (2006), 263–277.
  • F.J.S.A. Corrêa and R.G. Nascimento, On a nonlocal elliptic system of $p$-Kirchhoff-type under Neumann boundary condition, Math. Comput. Modelling 49 (2009), 598–604.
  • M. del Pino and P.L. Felmer, Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137.
  • M. del Pino and P.L. Felmer, Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal. 149 (1997), 245–265.
  • M. del Pino and P.L. Felmer, Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 127–149.
  • Y. Ding and F. Lin, Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations 30 (2007), 231–249.
  • Y. Ding and J. Wei, Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials, J. Funct. Anal. 251 (2007), 546–572.
  • J.M. do Ó and U. Severo, Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal. 8 (2009), 621–644.
  • M.J. Esteban and P.-L. Lions, Stationary solutions of nonlinear Schrödinger equations with an external magnetic field, pp. 401–449 in Partial differential equations and the calculus of variations, vol. I, Progr. Nonlinear Differential Equations Appl. 1, Birkhäuser, Boston (1989).
  • A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), 397–408.
  • X. He and W. Zou, Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. Engl. Ser. 26 (2010), 387–394.
  • X. He and W. Zou, Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal. 70 (2009), 1407–1414.
  • X. He and W. Zou, Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb R^3$, J. Differential Equations 252 (2012), 1813–1834.
  • G. Kirchhoff, Vorlesungen über Mechanik, Teubner, Leipzig (1883).
  • S. Liang and J. Zhang, Existence of solutions for Kirchhoff type problems with critical nonlinearity in $\mathbb{R}^3$, Nonlinear Anal. Real World Appl. 17 (2014), 126–136.
  • J.-L. Lions, On some questions in boundary value problems of mathematical physics, pp. 284–346 in Contemporary developments in continuum mechanics and partial differential equations (Rio de Janeiro, 1977), North-Holland Math. Stud. 30, North-Holland, Amsterdam (1978).
  • P.-L. Lions, The concentration-compactness principle in the calculus of variations: the locally compact case, II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223–283.
  • C. Liu, Z. Wang and H.-S. Zhou, Asymptotically linear Schrödinger equation with potential vanishing at infinity, J. Differential Equations 245 (2008), 201–222.
  • J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Soliton solutions for quasilinear Schrödinger equations, II, J. Differential Equations 187 (2003), 473–493.
  • J.-Q. Liu, Y.-Q. Wang and Z.-Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations 29 (2004), 879–901.
  • T.F. Ma and J.E. Muñoz Rivera, Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. Math. Lett. 16 (2003), 243–248.
  • A. Moameni, Existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in $\mathbb R^N$, J. Differential Equations 229 (2006), 570–587.
  • V. Moroz and J. Van Schaftingen, Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials, Calc. Var. Partial Differential Equations 37 (2010), 1–27.
  • Y.-G. Oh, Correction to “Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class $(V)_a$”, Comm. Partial Differential Equations 14 (1989), 833–834.
  • Y.-G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys. 131 (1990), 223–253.
  • K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006), 246–255.
  • P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics 65, Amer. Math. Soc., Providence, RI (1986).
  • P.H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (1992), 270–291.
  • M. Reed and B. Simon, Methods of modern mathematical physics, IV: Analysis of operators, Academic Press, New York (1978).
  • J. Su, Z.-Q. Wang and M. Willem, Nonlinear Schrödinger equations with unbounded and decaying radial potentials, Commun. Contemp. Math. 9 (2007), 571–583.
  • J. Su, Z.-Q. Wang and M. Willem, Weighted Sobolev embedding with unbounded and decaying radial potentials, J. Differential Equations 238 (2007), 201–219.
  • X. Wang and B. Zeng, On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions, SIAM J. Math. Anal. 28 (1997), 633–655.
  • M. Willem, Minimax theorems, Progress in Nonlinear Differential Equations and their Applications 24, Birkhäuser, Boston (1996).
  • X. Wu, Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in ${\mathbb R}^N$, Nonlinear Anal. Real World Appl. 12 (2011), 1278–1287.
  • M. Yang and Y. Ding, Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in $\mathbb{R}^N$, Ann. Mat. Pura Appl. (4) 192 (2013), 783–804.
  • H. Zou, Existence and non-existence for Schrödinger equations involving critical Sobolev exponents, J. Korean Math. Soc. 47 (2010), 547–572.