## Taiwanese Journal of Mathematics

### INFINITELY MANY SOLUTIONS FOR A CLASS OF DEGENERATE ANISOTROPIC ELLIPTIC PROBLEMS WITH VARIABLE EXPONENT

Maria-Magdalena Boureanu

#### Abstract

We study the nonlinear degenerate problem $-\sum_{i=1}^N \partial_{x_i} a_i \left(x,\partial_{x_i}u \right) = f(x,u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) is a bounded domain with smooth boundary, $\sum_{i=1}^N \partial_{x_i} a_i \left(x,\partial_{x_i}u \right)$ is a $\overset{\rightarrow} p(\cdot)$ - Laplace type operator and the nonlinearity $f$ is $(P_+^+-1)$ - superlinear at infinity (with $\overset{\rightarrow} p(x) = (p_1(x), p_2(x), ... p_N(x))$ and $P_+^+ = \max_{i \in \{1,...,N\}} \left\{\sup_{x \in \Omega} p_i(x) \right\}$). By means of the symmetric mountain-pass theorem of Ambrosetti and Rabinowitz, we establish the existence of a sequence of weak solutions in appropriate anisotropic variable exponent Sobolev spaces.

#### Article information

Source
Taiwanese J. Math., Volume 15, Number 5 (2011), 2291-2310.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406435

Digital Object Identifier
doi:10.11650/twjm/1500406435

Mathematical Reviews number (MathSciNet)
MR2880405

Zentralblatt MATH identifier
1237.35039

#### Citation

Boureanu, Maria-Magdalena. INFINITELY MANY SOLUTIONS FOR A CLASS OF DEGENERATE ANISOTROPIC ELLIPTIC PROBLEMS WITH VARIABLE EXPONENT. Taiwanese J. Math. 15 (2011), no. 5, 2291--2310. doi:10.11650/twjm/1500406435. https://projecteuclid.org/euclid.twjm/1500406435

#### References

• A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory, J. Funct. Anal., 14 (1973), 349-381.
• S. Antontsev and S. Shmarev, Vanishing solutions of anisotropic parabolic equations with variable nonlinearity, J. Math. Anal. Appl., 361 (2010), 371-391.
• M.-M. Boureanu, Existence of solutions for anisotropic quasilinear elliptic equations with variable exponent, Advances in Pure and Applied Mathematics, 1 (2010), 387-411.
• M.-M. Boureanu, P. Pucci and V. R${\rm\breve{a}}$dulescu, Multiplicity of solutions for a class of anisotropic elliptic equations with variable exponent, Complex Variables and Elliptic Equations, 2010, 1-13, i First.
• H. Brézis and F. Browder, Partial Differential Equations in the 20th Century, Advances in Mathematics, 135 (1998), 76-144.
• Y. Chen, S. Levine and R. Rao, Functionals with $p(x)$-growth in image processing, Duquesne University, Department of Mathematics and Computer Science Technical Report 2004-01.
• L. Diening, Theoretical and Numerical Results for Electrorheological Fluids, Ph.D. thesis, University of Frieburg, Germany, 2002.
• D. E. Edmunds, J. Lang and A. Nekvinda, On $L^{p(x)}$ norms, Proc. Roy. Soc. London Ser. A, 455 (1999), 219-225. \def\ipzz\' i
• D. E. Edmunds and J. Rákosn\ipzzk, Density of smooth functions in $W^{k,p(x)}(\Omega)$, Proc. Roy. Soc. London Ser. A, 437 (1992), 229-236. \def\ipzz\' i
• D. E. Edmunds and J. Rákosn\ipzzk, Sobolev embedding with variable exponent, Studia Math., 143 (2000), 267-293.
• X. Fan, Solutions for $p(x)$-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl., 312 (2005), 464-477.
• X. Fan, Q. Zhang and D. Zhao, Eigenvalues of $p(x)$-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306-317.
• I. Fragalà, F. Gazzola and B. Kawohl, Existence and nonexistence results foranisotropic quasilinear equations, Ann. Inst. H. Poincaré, Analyse Non Linéaire, 21 (2004), 715-734.
• M. Ghergu and V. R${\rm\breve{a}}$dulescu, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications, 37, Oxford University Press, 2008.
• T. C. Halsey, Electrorheological fluids, Science, 258 (1992), 761-766.
• P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values, Math. Bohem., 132(2) (2007), 125-136.
• P. Hästö, On the density of continuous functions in variable exponent Sobolev spaces, Rev. Mat. Iberoamericana, \bf23 (2007), 74-82.
• Y. Jabri, The Mountain Pass Theorem. Variants, Generalizations and some Applications, Cambridge University Press, 2003.
• C. Ji, An eigenvalue of an anisotropic quasilinear elliptic equation with variable exponent and Neumann boundary condition, Nonlinear Analysis T.M.A., 71 (2009), 4507-4514.
• B. Kone, S. Ouaro and S. Traore, Weak solutions for anisotropic nonlinear elliptic equations with variable exponents, Electronic Journal of Differential Equations, 2009 (2009), No. 144, 1-11.
• O. Kováčc ik and J. Rákosn\ipzzk, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.
• A. Kristály, H. Lisei and C. Varga, Multiple solutions for $p$-Laplacian type equations, Nonlinear Anal. TMA, 68 (2008), 1375-1381.
• A. Kristály, V. R${\rm\breve{a}}$dulescu 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, 136, Cambridge University Press, Cambridge, 2010.
• V. K. Le, On a sub-supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal., 71 (2009), 3305-3321.
• M. Mih${\rm\breve{a}}$ilescu and G. Moroşanu, Existence and multiplicity of solutions for an anisotropic elliptic problem involving variable exponent growth conditions, Applicable Analysis, 89 (2010), 257-271.
• M. Mih${\rm\breve{a}}$ailescu, P. Pucci and V. R${\rm\breve{a}}$dulescu, Nonhomogeneous boundary value problems in anisotropic Sobolev spaces, C. R. Acad. Sci. Paris, Ser. I, 345 (2007), 561-566.
• M. Mih${\rm\breve{a}}$ilescu, P. Pucci and V. R${\rm\breve{a}}$dulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl., 340 (2008), 687-698.
• M. Mih${\rm\breve{a}}$ilescu and V. R${\rm\breve{a}}$dulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. Roy. Soc. London Ser. A, 462 (2006), 2625-2641.
• M. Mih${\rm\breve{a}}$ilescu and V. R${\rm\breve{a}}$dulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., 135 (2007), 2929-2937.
• J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics, Vol. 1034, Springer, Berlin, 1983.
• S. M. Nikol'skii, On imbedding, continuation and approximation theorems for differentiable functions of several variables, Russian Math. Surv., 16 (1961), 55-104.
• C. Pfeiffer, C. Mavroidis, Y. Bar-Cohen and B. Dolgin, Electrorheological fluid based force feedback device, Proc. 1999 SPIE Telemanipulator and Telepresence Technologies VI Conf. (Boston, MA), \bf3840 (1999), 88-99.
• P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math., 65, Amer. Math. Soc., 1986.
• J. Rákosn\ipzzk, Some remarks to anisotropic Sobolev spaces I, Beiträge zur Analysis, 13 (1979), 55-68.
• J. Rákosn\ipzzk, Some remarks to anisotropic Sobolev spaces II, Beiträge zur Analysis, 15 (1981), 127-140.
• K. R. Rajagopal and M. R${\rm\dot{u}}$žz ičc ka, Mathematical modelling of electrorheological fluids, Continuum Mech. Thermodyn., 13 (2001), 59-78.
• M. R${\rm\dot{u}}$žz ičc ka, Electrorheological Fluids: Modeling and Mathematical Theory,Springer-Verlag, Berlin, 2002.
• S. Samko and B. Vakulov, Weighted Sobolev theorem with variable exponent for spatial and spherical potential operators, J. Math. Anal. Appl., 310 (2005), 229-246.
• M. Seed, G. S. Hobson, R. C. Tozer and A. J. Simmonds, Voltage-controlled Electrorheological brake, Proc. IASTED Int. Symp. Measurement, Sig. Proc. and Control: Paper No. 105-092-1, Taormina, Italy: ACTA Press, 1986.
• A. J. Simmonds, Electro-rheological valves in a hydraulic circuit, IEE Proceedings-D, 138(4) (1991), 400-404.
• R. Stanway, J. L. Sproston and A. K. El-Wahed, Applications of electro-rheological fluids in vibration control: a survey, Smart Mater. Struct., 5 (1996), 464-482.
• M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18 (1969), 3-24.
• L. Ven'-tuan, On embedding theorems for spaces of functions with partial derivatives of various degree of summability, Vestnik Leningrad Univ., 16 (1961), 23-37.
• W. Winslow, Induced fibration of suspensions, J. Appl. Phys., 20 (1949), 1137-1140.
• V. V. Zhikov, Averaging of functionals in the calculus of variations and elasticity, Math. USSR Izv., 29 (1987), 33-66.