Taiwanese Journal of Mathematics

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR DEGENERATE $p(x)$-LAPLACE EQUATIONS INVOLVING CONCAVE-CONVEX TYPE NONLINEARITIES WITH TWO PARAMETERS

Ky Ho and Inbo Sim

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Abstract

We show the existence of two nontrivial nonnegative solutions and infinitely many solutions for degenerate $p(x)$-Laplace equations involving concave-convex type nonlinearities with two parameters. By investigating the order of concave and convex terms and using a variational method,  we determine the existence according to the range of each parameter. Some Caffarelli-Kohn-Nirenberg type problems with variable exponents are also discussed.

Article information

Source
Taiwanese J. Math. Volume 19, Number 5 (2015), 1469-1493.

Dates
First available in Project Euclid: 4 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499133719

Digital Object Identifier
doi:10.11650/tjm.19.2015.5187

Subjects
Primary: 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations 35J70: Degenerate elliptic equations 47J10: Nonlinear spectral theory, nonlinear eigenvalue problems [See also 49R05] 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Keywords
$p(x)$-Laplacian weighted variable exponent Lebesgue-Sobolev spaces concave-convex nonlinearities nonnegative solutions multiplicity

Citation

Ho, Ky; Sim, Inbo. EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR DEGENERATE $p(x)$-LAPLACE EQUATIONS INVOLVING CONCAVE-CONVEX TYPE NONLINEARITIES WITH TWO PARAMETERS. Taiwanese J. Math. 19 (2015), no. 5, 1469--1493. doi:10.11650/tjm.19.2015.5187. https://projecteuclid.org/euclid.twjm/1499133719


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References

  • B. Abdellaoui, E. Colorado and I. Peral, Some critical quasilinear elliptic problems with mixed Dirichlet-Neumann boundary conditions: relation with Sobolev and Hardy-Sobolev optimal constants, J. Math. Anal. Appl., 332 (2007), 1165-1188.
  • A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal., 14 (1973), 349-381.
  • A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122(2) (1994), 519-543.
  • T. Bartsch, S. Peng and Z. Zhang, Existence and non-existence of solutions to elliptic equations related to the Caffarelli-Kohn-Nirenberg inequalities, Calc. Var. Partial Differential Equations, 30(1) (2007), 113-136.
  • H. Brezis and L. Nirenberg, $H^1$ versus $C^1$ local minimizers, C. R. Acad. Sci. Paris Sér. I Math., 317(5) (1993), 465-472.
  • L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.
  • Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66(4) (2006), 1383-1406.
  • R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 1, Physical Origins and Classical Methods, Springer-Verlag, Berlin, 1985.
  • D. de Figueiredo, J.-P. Gossez and P. Ubilla, Local “superlinearity” and “sublinearity” for the $p$-Laplacian, J. Funct. Anal., 257 (2009), 721-752.
  • L. Diening, P. Harjulehto, P. Hästö and M. R\ptmrs \ru\ptmrs ži\ptmrs čka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics 2017, Springer-Verlag, Heidelberg, 2011.
  • P. Drábek, A. Kufner and F. Nicolosi, Quasilinear elliptic equations with degenerations and singularities. de Gruyter Series in Nonlinear Analysis and Applications 5, Walter de Gruyter and Co., Berlin, 1997.
  • I. Ekeland and N. Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc., 39 (2002), 207-265.
  • X. Fan, J. S. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)} (\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760.
  • X. Fan and D. Zhao, On the spaces $L^{p(x) }(\Omega ) $ and $W^{m,p(x) }(\Omega ) $, J. Math. Anal. Appl., 263 (2001), 424-446.
  • 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, Solutions for p(x)-Laplacian Dirichlet problems with singular coefficients, J. Math. Anal. Appl., 312 (2005), 464-477.
  • X. L. Fan, On the sub-supersolution method for p(x)-Laplacian equations, J. Math. Anal. Appl., 330 (2007), 665-682.
  • J. Garc\ptmrs ía, I. Peral and J. Manfredi, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Commun. Contemp. Math., 2(3) (2000), 385-404.
  • N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc., 12 (2000), 5703-5743.
  • D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer-Verlag, Heidelberg, 2001.
  • Z. Guo and Z. Zhang, $W^{1,p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Appl., 286 (2003), 32-50.
  • K. Ho and I. Sim, Existence and some properties of solutions for degenerate elliptic equations with exponent variable, Nonlinear Anal., 98 (2014), 146-164.
  • K. Ho and I. Sim, Existence results for degenerate $p(x)$-Laplace equations with Leray-Lions type operators, submitted.
  • Y. H. Kim and I. Sim, Existence of solutions and positivity of the infimum eigenvalue for degenerate elliptic equations with variable exponents, Discrete Contin. Dyn. Syst. Supplement, (2013), 695-707.
  • Y. H. Kim, L. Wang and C. Zhang, Global bifurcation of a class of degenerate elliptic equations with variable exponents, J. Math. Anal. Appl., 371 (2010), 624-637.
  • O. Kov\ptmrs ăčik and J. R\ptmrs ăkosnik, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41(116) (1991), 592-618.
  • V. K. Le, On a sub-supersolutionmethod for variational inequalities with Leray-Lions operators in variable exponent spaces, Nonlinear Anal., 71 (2009), 3305-3321.
  • M. Mih\ptmrs ăilescu and V. R\ptmrs ădulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. A, 462 (2006), 2625-2641.
  • M. Mih\ptmrs ăilescu and V. R\ptmrs ădulescu, On a nonhomogeneous quasilinear eigenvalue problem in Sobolev spaces with variable exponent, Proc. Amer. Math. Soc., 135(9) (2007), 2929-2937 (electronic).
  • M. Mih\ptmrs ăilescu, V. R\ptmrs ădulescu and D. Stancu-Dumitru, A Caffarelli-Kohn-Nirenberg-type inequality with variable exponent and applications to PDEs, Complex Var. Elliptic Equ., 56(7-9) (2011), 659-669.
  • V. Murthy and G. Stampacchia, Boundary value problems for some degenerate-elliptic operators, Ann. Mat. Pura Appl., 80 (1968), 1-122
  • M. R\ptmrs \ru\ptmrs ži\ptmrs čka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, 1748, Springer-Verlag, Berlin, 2000.
  • M. Willem, Minimax theorems, Birkhäuser, Boston, 1996.
  • V. V. Zhikov, On some variational problems, Russ. J. Math. Phys., 5 (1997), 105-116.
  • J. F. Zhao, Structure Theory of Banach Spaces, Wuhan University Press, Wuhan, 1991 (in Chinese).