## Topological Methods in Nonlinear Analysis

### Existence of solutions for $p(x)$-Laplacian problem on an unbounded domain

Fu Yongqiang

#### Abstract

In this paper we study the following $p(x)$-Laplacian problem: \begin{alignat}{2} -{\rm div}(a(x)|\nabla u|^{p(x)-2}\nabla u)+b(x)|u|^{p(x)-2}u&=f(x,u) &\quad& x\in \Omega,\\ u&=0 &\quad&\text{on }\partial\Omega, \end{alignat} where $1< p_{1}\le p(x)\le p_{2}< n$, $\Omega\subset {\mathbb R}^{n}$ is an exterior domain. Applying Mountain Pass Theorem we obtain the existence of solutions in $W_{0}^{1,p(x)}(\Omega)$ for the $p(x)$-Laplacian problem in the superlinear case.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 30, Number 2 (2007), 235-249.

Dates
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1463150092

Mathematical Reviews number (MathSciNet)
MR2387827

Zentralblatt MATH identifier
1159.35322

#### Citation

Yongqiang, Fu. Existence of solutions for $p(x)$-Laplacian problem on an unbounded domain. Topol. Methods Nonlinear Anal. 30 (2007), no. 2, 235--249. https://projecteuclid.org/euclid.tmna/1463150092

#### References

• \ref E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth , Arch. Rational Mech. Anal., 156 (2001), 121–140
• \ref E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids , Arch. Rational Mech. Anal., 164 (2002), 213–259
• \ref Y. Alkhutov, The Harnack inequality and the Hölder property of solutions of nonlinear elliptic equations with a nonstandard growth conditions , Differential Equations, 33 (1997), 1653–1662
• \ref T. Bartsch and Z. Liu, On a superlinear elliptic $p$-Laplacian equation , J. Differential Equations, 198 (2004), 149–175
• \ref K. Chaib, Necessary and sufficient conditions of existence for a system involving the $p$-Laplacian $(1<p<N)$ , J. Differential Equations, 189 (2003), 513–525
• \ref V. Chiad Piat and A. Coscia, Hölder continuity of minimizers of functionals with variable growth exponent , Manuscripta Math., 93 (1997), 283–299
• \ref A. Coscia and G. Mingione, Hölder continuity of the gradient of $p(x)$–harmonic mappings , C.R. Acad. Sci. Paris Ser., 328 (1999), 363–368
• \ref D. Costa and O. Miyagaki, Nontrivial solutions for perturbations of the $p$-Laplacian on unbounded domain , J. Math. Anal. Appl., 193 (1995), 737–755
• \ref P. De Napoli and M. Mariani, Mountain pass solutions to equations of $p$-Laplacian type , Nonlinear Anal., 54 (2003), 1205–1219
• \ref G. Dinca, P. Jebelean and J. Mawhin, Variational and topological methods for Dirichlet problems with $p$-Laplacian , Portugal Math., 58 (2001), 339–378
• \ref D. Edmunds, J. Lang and A. Nekvinda, On $L^{p(x)}$ Norms , R. Soc. London Proc. Sec. A, 1981 (1999), 219–225
• \ref D. Edmunds and J. Rakosnik, Sobolev embending with variable exponent , Studia Math., 143 (2000), 267–293
• \ref D. Edmunds and J. Rakosnik, Sobolev embending with variable exponent \romII, Math. Nachr., 246–247 (2002), 53–67
• \ref H. Egnell, Existence and nonexistence results for $m$-Laplace equations involving critical Sobolev exponents , Arch. Rational Mech. Anal., 104 (1988), 57–77
• \ref X. Fan and X. Han, Existence and multiplicity of solutions for $p(x)$-Laplacian equations in ${\Bbb R}^{N}$ , Nonlinear Anal., 59 (2004), 173–188
• \ref X. Fan, J. Shen and D. Zhao, Sobolev Embedding Theorems for Spaces $W^{k,p(x)}(\Omega)$ , J. Math. Anal. Appl., 262 (2001), 749–760
• \ref X. Fan and Q. Zhang, Existence of solutions for $p(x)$-Laplacian Dirichlet problem , Nonlinear Anal., 52 (2003), 1843–1852
• \ref X. Fan and D. Zhao, The quasi-minimizer of integal functionals with $m(x)$ growth conditions , Nonlinear Anal., 39 (2000), 807–816
• \ref Y. Huang, Existence of positive solutions for a class of the $p$-Laplace equations , J. Austral. Math. Soc. Sec. B, 36 (1994), 249–264
• \ref O. Kovacik and J. Rakosnik, On spaces $L^{p(x)}$ and $W_{0}^{k,p(x)}$ , Czechoslovak Math. J., 41 (1991), 592–618
• \ref W. Ni and J. Serrin, Existence and nonexistence theorems for ground states of quasilinear partial differential equations. The anomalous case , Accad. Naz. Lincei, 77 (1986), 231–257
• \ref M. Ruzicka, Electro-Rheological Fluids: Modeling and Mathematical Theory, Springer–Verlag (2000)
• \ref S. Samko, Convolution and potential type operators in $L^{p(x)}(\Bbb{R}^{n})$ , Integral Transform. Spec. Funct., 7 (1998), 261–284
• \ref A. Wu, Existence of multiple nontrivial solutions for nonlinear $p$-Laplacian problems on $\Bbb{R}^{N}$ , Proc. Roy. Soc. Edinburgh Sec. A, 129 (1999), 855–883
• \ref L. Yu, Nonlinear $p$-Laplacian problems on unbounded domains , Proc. Amer. Math. Soc., 115 (1992), 1037–1045