Skip to main content Skip to main navigation menu Skip to site footer
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Online First
  • Archives
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Privacy Statement
    • Contact
  • Login
  • Language:
  • English
  • Język Polski

Topological Methods in Nonlinear Analysis

Multiplicity of solutions of some quasilinear equations in ${\mathbb{R}^{N}}$ with variable exponents and concave-convex nonlinearities
  • Home
  • /
  • Multiplicity of solutions of some quasilinear equations in ${\mathbb{R}^{N}}$ with variable exponents and concave-convex nonlinearities
  1. Home /
  2. Archives /
  3. Vol 47, No 2 (June 2016) /
  4. Articles

Multiplicity of solutions of some quasilinear equations in ${\mathbb{R}^{N}}$ with variable exponents and concave-convex nonlinearities

Authors

  • Claudianor O. Alves
  • José L. P. Barreiroy
  • José Valdo Gonçalves

DOI:

https://doi.org/10.12775/TMNA.2016.015

Keywords

Variational methods, quasilinear problems, weak solutions

Abstract

We prove multiplicity of solutions for a class of quasilinear problems in $ \mathbb{R}^{N} $ involving variable exponents and nonlinearities of concave-convex type. The main tools used are variational methods, more precisely, Ekeland's variational principle and Nehari manifolds.

References

S. Adachi and K. Tanaka, Four positive solutions for the semilinear elliptic equation: −∆u + u = a(x)up + f (x) in R^N , Calc. Var. Partial Differential Equations 11 (2000), 63–95.

C.O. Alves, Existence and Multiplicity of solution for a class of quasilinear equations, Adv. Nonlinear Stud. 5 (2005), 73–87.

C.O. Alves, Existence of solution for a degenerate p(x)-Laplacian equation in R^N, J. Math. Anal. Appl. 345 (2008), 731–742.

C.O. Alves and J.L.P. Barreiro, Multiplicity of solutions for a class of quasilinear problem involving variable exponent, Asympt. Anal 96 (2016), 161–184.

C.O. Alves and M.C. Ferreira, Existence of solutions for a class of p(x)-laplacian equations involving a concave-convex nonlinearity with critical growth in R^N, Topol. Methods Nonlinear Anal. 45 (2015), 399–435.

C.O. Alves and S. Liu, On superlinear p(x)-Laplacian equations in R^N, Nonlinear Anal. 73 (2010), 2566–2579.

C.O. Alves and M.A.S. Souto, Existence of solutions for a class of problems in R^N involving p(x)-Laplacian, Progr. Nonlinear Differential Equations Appl. 66 (2005), 17–32.

S. Antontsev and S. Shmarev, Elliptic equations with anisotropic nonlinearity and nonstandard conditions, Handbook of Differential Equations (M. Chipot and P. Quittner, eds.), Stationary Partial Differential Equations vol. 3, Elsevier B.V., North-Holland, 2006, 1–100.

G. Autuori and P. Pucci, Existence of entire solutions for a class of quasilinear elliptic equations, NoDEA Nonlinear Differential Equations Appl. 20 (2013), 977–1009.

K.J. Brown and T.F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electron. J. Differential Equations 69 (2007), 1–9.

D.M. Cao, Multiple solutions of a semilinear elliptic equations in RN , Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1996), 593–604.

D.M. Cao and E.S. Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problem in RN , Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), no. 5, 567–588.

D.M. Cao and H.S. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in RN , Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 443–463.

J. Chabrowski and Y. Fu, Existence of solutions for p(x)-laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), 604–618.

L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka, Lebesgue and Sobolev Spaces with variable exponents, Lecture Notes in Math. Vol. 2017, Springer–Verlag, Heidelberg, 2011.

L. Diening, P. Hästö and A. Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, FSDONA04 Proceedings (P. Drábek and J. Rákosnı́k, eds.), Milovy, Czech Republic (2004), 38–58.

X. Fan, p(x)-Laplacian equations in RN with periodic data and nonperiodic perturbations, J. Math. Anal. Appl. 341 (2008), 103–119.

X. Fan and X. Han, Existence and multiplicity of solutions for p(x)-Laplacian equations in R^N , Nonlinear Anal. 59 (2004), 173–188.

X. Fan, S. Shen and D. Zhao, Sobolev embedding theorems for spaces W k,p(x) (Ω), J. Math. Anal. Appl. 262 (2001), no. 2, 749–760.

X. Fan and D. Zhao, On the spaces Lp(x) (Ω) and W m,p(x) (Ω), J. Math. Anal. Appl. 263 (2001), 424–446.

X. Fan, Y. Zhao and D. Zhao, Compact embedding theorems with symmetry of Strauss–Lions type for the space W 1,p(x) (R^N), J. Math. Anal. Appl. 255 (2001, 333–348.

Y. Fu, The principle of concentration compactness in Lp(x) spaces and its application, Nonlinear Anal. 71 (2009), 1876–1892.

Y. Fu and X. Zhang, Solutions of p(x)-Laplacian equations with critical exponent and perturbations in RN , Electron. J. Differential Equations 120 (2011), 1–14.

N. Hirano, Existence of entire positive solutions for nonhomogeneous elliptic equations, Nonlinear Anal. 29 (1997), 889–901.

N. Hirano and N. Shioji, A multiplicity result including sign-changing solutions for a nonlinear problem in RN , Adv. Nonlinear Stud. 7 (2007), 513–532.

T.S Hsu, H.L. Lin and C.C Hu, Multiple positive solutions of quasilinear elliptic equations in RN , J. Math. Anal. Appl. 388 (2012), 500–512.

K. Hu and C.L. Tang, Existence and multiplicity of positive solutions of semilinear elliptic equations in unbounded domains, J. Differential Equations 251 (2011), 609–629.

L. Jeanjean, Two positive solutions for nonhomogeneous elliptic equations, Differential Integral Equations 10 (1997), 609–624.

O. Kovăčik and J. Răkosnik, On spaces Lp(x) and W 1,p(x) , Czechoslovak Math. J. 41 (116) (1991), no. 4, 592–618.

A. Kristály, V. Radulescu and C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications, vol. 136, Cambridge University Press, Cambridge, 2010.

H.L. Lin, Multiple positive solutions for semilinear elliptic systems, J. Math. Anal. Appl. (2012), 107–118.

H.L. Lin, Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in RN , Bound. Value Probl. 2012 (24) (2012), 1–17.

R.A. Mashiyev, S. Ogras, Z. Yucedag and M. Avci, The Nehari manifold approach for Dirichlet problem involving the p(x)-Laplacian equation, J. Korean Math. Soc. 47 (2010), 1–16.

M. Mihăilescu and V. Rădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. Roy. Soc. Edinburgh Sect. A 462 (2006), 2625–2641.

M. Mihăilescu and V. Rădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc. 135 (2007), 2929–2937.

M. Mihăilescu, V. Rădulescu and D. Stancu-Dumitru, On a Caffarelli–Kohn–Nirenberg type inequality in bounded domains involving variable exponent growth conditions and applications to PDE’s, Complex Var. Elliptic Equ. 56 (2011), 659–669.

P. Pucci and V. Rădulescu, Combined effects in quasilinear elliptic problems with lack of compactness, Rend. Lincei Mat. Appl. 22 (2011), 189–205, special volume dedicated to the memory of Prof. G. Prodi.

P. Pucci and Q. Zhang, Existence of entire solutions for a class of variable exponent elliptic equations, J. Differential Equations 257 (2014), 1529–1566.

M. Růžička, Electrorheological fluids: Modeling and mathematical theory, Lecture Notes in Mathematics, vol. 1748, Springer–Verlag, Berlin, 2000.

V. Rădulescu and D. Repovš, Partial Differential Equations with Variable Exponent: Variational Methods and Qualitative Analysis, CRC Press, Taylor and Francis Group, Boca Raton, FL, 2015.

S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms Spec. Funct. 16 (2005), 461–482.

G. Tarantello, On nonhomogeneous elliptic involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), 281–304.

M. Willem, Minimax Theorems, Birkhäuser, Boston, 1996.

T.F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl. (2006), 253–270.

T.F. Wu, Multiplicity of positive solutions for semilinear elliptic equations in R^N, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 647–670.

T.F. Wu, The existence of multiple positive solutions for a semilinear elliptic equations in R^N , Nonlinear Anal. 72 (2010), 3412–3421.

Downloads

  • PREVIEW
  • FULL TEXT

Published

2016-06-01

How to Cite

1.
ALVES, Claudianor O., BARREIROY, José L. P. and GONÇALVES, José Valdo. Multiplicity of solutions of some quasilinear equations in ${\mathbb{R}^{N}}$ with variable exponents and concave-convex nonlinearities. Topological Methods in Nonlinear Analysis. Online. 1 June 2016. Vol. 47, no. 2, pp. 529 - 559. [Accessed 7 July 2025]. DOI 10.12775/TMNA.2016.015.
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol 47, No 2 (June 2016)

Section

Articles

Stats

Number of views and downloads: 0
Number of citations: 1

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Newsletter

Subscribe Unsubscribe
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop