Sage Journals: Discover world-class research

Abstract

In this paper we prove the multiplicity of solutions for a class of quasilinear problems in $R^{N}$ involving variable exponents. The main tool used in the proof are the variational methods, Ekeland’s variational principle and some properties related to Nehari manifold.

Keywords

existence multiplicity variable exponents variational methods

Get full access to this article

View all access options for this article.

References

[1]

Adachi and

Tanaka, Four positive solutions for the semilinear elliptic equation:

- Δ u + u = a (x) u^{p} + f (x)

R^{N}

, Calc. Var. Partial Differential Equations 11 (2000), 63–95.

[2]

C.O.

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

[3]

C.O.

Alves, Existence of solution for a degenerate

p (x)

-Laplacian equation in

R^{N}

, J. Math. Anal. Appl. 345(2) (2008), 731–742.

[4]

C.O.

Alves and

M.C.

Ferreira, Nonlinear perturbations of a

p (x)

-Laplacian equation with critical growth in

R^{N}

, Math. Nachr. 287 (2014), 849–868.

[5]

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), 399–422.

[6]

C.O.

Alves and

M.A.S.

Souto, Existence of solutions for a class of problems in

R^{N}

involving

p (x)

-Laplacian, Prog. Nonlinear Differ. Eq. Appl. 66 (2005), 17–32.

[7]

Antontsev and

Shmarev, Elliptic equations with anisotropic nonlinearity and nonstandard conditions, in: Handbook of Differential Equations,

Chipot and

Quittner, eds, Stationary Partial Differential Equations, Vol. 3, Elsevier B.V., North-Holland, 2006, pp. 1–100.

[8]

S.N.

Antontsev and

J.F.

Rodrigues, On stationary thermo-rheological viscous flows, Ann. Univ. Ferrara Sez. Sci. Mat. 52 (2006), 19–36.

[9]

D.M.

Cao and

E.S.

Noussair, Multiplicity of positive and nodal solutions for nonlinear elliptic problem in

R^{N}

, Ann. Inst. H. Poincaré 13(5) (1996), 567–588.

10.

[10]

D.M.

Cao and

H.S.

Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in

R^{N}

, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 443–463.

11.

[11]

Chabrowski and

Fu, Existence of solutions for

p (x)

-Laplacian problems on a bounded domain, J. Math. Anal. Appl. 306 (2005), 604–618.

12.

[12]

Chen,

Levine and

Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66(4) (2006), 1383–1406.

13.

[13]

Diening,

Harjulehto,

Hästö and

Ruzicka, Lebesgue an Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Vol. 2017, Springer, Heidelberg, 2011.

14.

[14]

Diening,

Hästö and

Nekvinda, Open problems in variable exponent Lebesgue and Sobolev spaces, in: FSDONA04 Proceedings, Milovy, Czech Republic,

Drábek and

Rákosník, eds, 2004, pp. 38–58.

15.

[15]

Fan,

p (x)

-Laplacian equations in

R^{N}

with periodic data and nonperiodic perturbations, J. Math. Anal. Appl. 341 (2008), 103–119.

16.

[16]

Fan and

X.Y.

Han, Existence and multiplicity of solutions for

p (x)

-Laplacian equations in

R^{N}

, Nonlinear Anal. 59 (2004), 173–188.

17.

[17]

Fan,

Shen and

Zhao, Sobolev imbedding theorems for spaces

W^{k, p (x)} (Ω)

, J. Math. Anal. Appl. 262 (2001), 749–760.

18.

[18]

Fan and

Zhao, On the spaces

L^{p (x)} (Ω)

and

W^{m, p (x)} (Ω)

, J. Math. Anal. Appl. 263 (2001), 424–446.

19.

[19]

Fan,

Zhao and

Zhao, Compact imbedding theorems with symmetry of Strauss–Lions type for the space

W^{1, p (x)} (R^{N})

, J. Math. Anal. Appl. 255 (2001), 333–348.

20.

[20]

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

21.

[21]

Hirano and

Shioji, A multiplicity result including sign-changing solutions for a nonlinear problem in

R^{N}

, Adv. Nonlinear Stud. 7 (2007), 513–532.

22.

[22]

T.S.

Hsu,

H.L.

Lin and

C.C.

Hu, Multiple positive solutions of quasilinear elliptic equations in

R^{N}

, J. Math. Anal. Appl. 388 (2012), 500–512.

23.

[23]

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.

24.

[24]

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

25.

[25]

Kovăčik and

Răkosnik, On spaces

L^{p (x)}

and

W^{1, p (x)}

, Czechoslovak Math. J. 41(116)(4) (1991), 592–618.

26.

[26]

Kristály,

Rădulescu and

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 Univ. Press, Cambridge, 2010.

27.

[27]

H.L.

Lin, Multiple positive solutions for semilinear elliptic systems, J. Math. Anal. Appl. 391 (2012), 107–118.

28.

[28]

Mihailescu and

Radulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. A 462 (2006), 2625–2641.

29.

[29]

Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Vol. 1748, Springer, Berlin, 2000.

30.

[30]

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.

31.

[31]

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

32.

[32]

Willem, Minimax Theorems, Birkhäuser, Boston, 1996.

33.

[33]

T.F.

Wu, Multiplicity of positive solutions for semilinear elliptic equations in

R^{N}

, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 647–670.

34.

[34]

T.F.

Wu, The existence of multiple positive solutions for a semilinear elliptic equations in

R^{N}

, Nonlinear Anal. 72 (2010), 3412–3421.

Multiple solutions for a class of quasilinear problems involving variable exponents

Abstract

Keywords

Get full access to this article

References