본문 바로가기
KERD
논문KCI 등재

A GLOBAL BIFURCATION FOR NONLINEAR ELLIPTIC EQUATIONS INVOLVING NONHOMOGENEOUS OPERATORS OF p(x)-LAPLACE TYPE

김윤호
장전수학회 (Jangjeon Mathematical Society)|2018.01|Advanced Studies in Contemporary Mathematics|제28권 제1호|pp. 27-39|DOI http://dx.doi.org/10.17777/ascm2017.28.1.27

2

조회

0

다운로드

0

인용

0

참고문헌

키워드

Variable exponent Lebesgue-Sobolev spacesWeak solutionsEigenvalueBifurcationp(x)-Laplacian
주제분류수학

초록

We are concerned with the following nonlinear problem -div( Ψ(x, ∇u)) = μ|u|p(x)-2u + f(λ, x, u, ∇u) in Ω subject to Dirichlet boundary condition, provided that μ is not an eigenvalue of the p(x)-Laplacian. The aim of this paper is to study the structure of the set of weak solutions of nonlinear equations of p(x)-Laplace type, by applying a bifurcation result for nonlinear operator equations.

Abstract

We are concerned with the following nonlinear problem -div( Ψ(x, ∇u)) = μ|u|p(x)-2u + f(λ, x, u, ∇u) in Ω subject to Dirichlet boundary condition, provided that μ is not an eigenvalue of the p(x)-Laplacian. The aim of this paper is to study the structure of the set of weak solutions of nonlinear equations of p(x)-Laplace type, by applying a bifurcation result for nonlinear operator equations.

참고문헌 (30)

  1. Y. Chen, "Variable exponent, linear growth functionals in image restoration", SIAM J. Appl. Math., 66, pp.1383~1406 (2006)[DOI]
  2. M. A. Del Pino, "Global bifurcation from the eigenvalues of the p-Laplacian", J. Differential Equations, 92, pp.226~251 (1991)[DOI]
  3. L. Diening, "Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces <TEX>$L^{P({\cdot})}$<", TEX> and <TEX>$W^{k,P({\cdot})}$<, Math. Nachr., pp.2004, 268[DOI]
  4. L. Diening, "Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Math., vol. 2017" (2011), Springer-Verlag
  5. P. Drabek, "On the global bifurcation for a class of degenerate equations", Ann. Mat. Pura Appl., 159, pp.1~16 (1991)[DOI]
  6. P. Draek, "Quasilinear Elliptic Equations with Degenerations and Singularities" (1997), de Gruyter
  7. D. E. Edmunds, "Sobolev embeddings with variable exponent", Studia Math., 143, pp.267~293 (2000)[DOI]
  8. X. Fan, "Sobolev embedding theorems for spaces <TEX>$W^{k,P({x})}({\Omega})$<", TEX>, 2001, pp.262, 749~760
  9. X. Fan, "Eigenvalues of p(x)-Laplacian Dirichlet problem", J. Math. Anal. Appl., 302, pp.306~317 (2005)[DOI]
  10. X. Fan, "On the spaces <TEX>$L^{P(x)}{(\Omega)}$<", TEX> and <TEX>$W^{m,p(x)} (Q) $<, J. Math. Anal. Appl., pp.2001, 263[DOI]
  11. N. Fukagai, "A bifurcation problem of some nonlinear degenerate elliptic equations", Adv. Differential Equations, 2, pp.895~926 (1997)
  12. E. Giorgieri, "Nonlinear spectral theory for homogeneous operators", Nonlinear Funct. Anal. Appl., 7, pp.589~618 (2002)
  13. B. -H. Hwang, "Existence of an unbounded branch of the set of solutions for Neumann problems involving the p(x)-Laplacian", Bound. Value Probl., 92, pp.1~16 (2014)
  14. I. H. Kim, "Mountain pass type solutions and positivity of the infimum eigenvalue for quasilinear elliptic equations with variable exponents", Maunscripta Math., 147, pp.169~191 (2015)[DOI]
  15. I. -S. Kim, "A global bifurcation result for generalized Laplace operators in <TEX>${\mathbb{R}}^N$<", TEX>, 2012, pp.13, 667~680
  16. Y. -H. Kim, "Existence of an unbounded branch of the set of solutions for equations of p(x)-Laplace type", Bound. Value Probl., 27, pp.1~20 (2014)
  17. Y. -H. Kim, "Global solution branches for equations involving nonhomogeneous operators of p-Laplace type", Nonlinear Anal., 74, pp.1878~1891 (2011)[DOI]
  18. Y. -H. Kim, "Global bifurcation for a class of degenerate elliptic equations with variable exponents", J. Math. Anal. Appl., 371, pp.624~637 (2010)[DOI]
  19. O. Kovacik, "On spaces <TEX>$L^{P(x)}$<", TEX> and <TEX>$W^{k,P({x})}$<, Czechoslovak Math. J., pp.1991, 41[DOI]
  20. M. A. Krasnoselskii, "Topological Methods in the Theory of Nonlinear Integral Equations" (1965), Pergamon
  21. V. K. Le, "On a sub-Supersolution method for variational inequalities with Leray-Lions operators in variable exponent spaces", Nonlinear Anal., 71, pp.3305~3321 (2009)[DOI]
  22. V. K. Le, "Global Bifurcation in Variational Inequalities: Applications to Obstacle and Unilateral Problems" (1997), Springer
  23. J. Musielak, "Orlicz spaces and modular spaces, Lecture Notes in Math., vol. 1034" (1983), Springer-Verlag
  24. P. H. Rabinowitz, "Some global results for non-linear eigenvalue problems", J. Funct. Anal., 7, pp.487~513 (1971)[DOI]
  25. M. Ruzicka, "Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol 1748" (2000), Springer
  26. K. Schmitt, "Bifurcation problems associated with generalized Laplacians", Adv. Differential Equations, 9, pp.797~828 (2004)
  27. C. A. Stuart, "Some bifurcation theory for k-set contractions", Proc. London Math. Soc., 27, pp.531~550 (1973)
  28. M. Vath, "Global bifurcation of the p-Laplacian and related operators", J. Differential Equations, 213, pp.389~409 (2005)[DOI]
  29. B, "Nonlinear Functional Analysis and its Applications II", pp.1990, E. Zeidler
  30. V. V. Zhikov, "On the density of smooth functions in Sobolev-Orlicz spaces", Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov., 226, pp.67~81 (2004)

관련 논문AI추천

EXISTENCE OF THREE SOLUTIONS FOR EQUATIONS OF p(x)-LAPLACE TYPE김연호, 이승대장전수학회 · Proceedings of the Jangjeon Mathematical Society(장전수학회 논문집)2017 · 제20권 제2호Global bifurcation for generalized Laplacian operatorsIn-Sook Kim대한수학회 · 대한수학회지2009 · 제46권 제1호ON SUPERLINEAR p(x)-LAPLACIAN-LIKE PROBLEM WITHOUT AMBROSETTI AND RABINOWITZ CONDITIONGe Bin대한수학회 · 대한수학회보2014 · 제51권 제2호EXISTENCE OF WEAK NON-NEGATIVE SOLUTIONS FOR A CLASS OF NONUNIFORMLY BOUNDARY VALUE PROBLEMHoang Quoc Toan, Trinh Thi Minh Hang대한수학회 · 대한수학회보2012 · 제49권 제4호Multiple solutions to a Neumann problem with p(x)-LaplacianAmina Bounouala, N. Benouhiba장전수학회 · Advanced Studies in Contemporary Mathematics2014 · 제24권 제2호Multiple solutions for equations of $p(x)$-Laplace type with nonlinear Neumann boundary condition김윤호, 박기섭대한수학회 · 대한수학회보2016 · 제53권 제6호

이 간행물의 다른 논문 (12건)

FOURIER SERIES OF r-DERANGEMENT AND HIGHER-ORDER DERANGEMENT FUNCTIONS김대산, 권혁인, 김태균pp. 1-11A New Combined Homotopy-Laplace Decomposition Method for Solving DDEs of Order (1, 2)R. Rangarajan, Ananth Kumar S. R.pp. 13-25OBSERVATIONS ON IDENTITIES AND RELATIONS FOR INTERPOLATION FUNCTIONS AND SPECIAL NUMBERSIrem Kucukoglu, Yilmaz Simsekpp. 41-56SOME THETA-FUNCTION IDENTITIES OF LEVEL SIX AND ITS APPLICATIONS TO PARTITIONSANUSHA KUMARI, B. R. SRIVATSA KUMARpp. 57-71SOME IDENTITIES OF DERANGEMENT NUMBERS ARISING FROM DIFFERENTIAL EQUATIONS김태균, 장관우, 권혁인pp. 73-82
원문 보기