Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Bifurcation problems for the $p$-Laplacian in $R^N$


Authors: Pavel Drábek and Yin Xi Huang
Journal: Trans. Amer. Math. Soc. 349 (1997), 171-188
MSC (1991): Primary 35B32, 35J70, 35P30
DOI: https://doi.org/10.1090/S0002-9947-97-01788-1
MathSciNet review: 1390979
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the bifurcation problem

\begin{equation*}-\text {div\ } (|{\nabla } u|^{p-2}{\nabla } u)={\lambda } g(x)|u|^{p-2}u+f({\lambda } , x, u), \end{equation*}

in ${R^N} $ with $p>1$. We show that a continuum of positive solutions bifurcates out from the principal eigenvalue ${\lambda } _{1}$ of the problem

\begin{equation*}-\text {div\ } (|{\nabla } u|^{p-2}{\nabla } u)={\lambda } g(x)|u|^{p-2}u. \end{equation*}

Here both functions $f$ and $g$ may change sign.


References [Enhancements On Off] (What's this?)

  • [AD] R.A. Adams, Sobolev Spaces, Academic Press, 1975. MR 56:9247
  • [AH] W. Allegretto and Y.X. Huang, Eigenvalues of the indefinite weight $p$-Laplacian in weighted ${R^N} $ spaces, Funkc. Ekvac. 38 (1995), 233-242.
  • [A] A. Anane, Simplicité et isolation de la première valeur propre du $p$-laplacien avec poids, C.R. Acad. Sci. Paris 305 I (1987), 725-728. MR 89e:35124
  • [BH] P.A. Binding and Y.X. Huang, Bifurcation from eigencurves of the $p$-Laplacian, Diff. Int. Equa. 8 (1995), 415-428. MR 95j:35165
  • [BP] F.E. Browder and W.F. Petryshyn, Approximation methods and the generalized topological degree for nonlinear mappings in Banach spces, J. Funct. Anal. 3 (1969), 217-245. MR 39:6126
  • [DA] E.N. Dancer, On the structure of the solutions of nonlinear eigenvalue problems, Indiana Univ. Math. J. 23 (1974), 1069-1076. MR 50:1065
  • [DM] M.A. Del Pino and R. Manásevich, Global bifurcation from the eigenvalues of the $p$-Laplacian, J, Diff. Equa. 92 (1991), 226-251. MR 92g:35028
  • [D1] P. Drábek, On the global bifurcation for a class of degenerate equations, Ann. Mat. Pura Appl. 159 (1991), 1-16. MR 93d:47112
  • [D2] P. Drábek, Solvability and Bifurcations of Nonlinear Equations, Pitman Research Notes in Math. 264, Longman, Harlow, 1992. MR 94e:47084
  • [D3] P. Drábek, Nonlinear eigenvalue problem for the $p$-Laplacian in ${R^N} $, Math. Nach. 173 (1995), 131-139. MR 96b:35064
  • [ER] A. Edelson and A. Rumbos, Linear and semilinear eigenvalue problems in ${R^N} $, Comm. Part. Diff. Equa. 18 (1993), 215-240. MR 94b:35101
  • [FK] S. Fu\v{c}ik and A. Kufner, Nonlinear Differential Equations, Elsevier, Holland, 1980. MR 81e:35001
  • [HS] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag, New York, 1975. MR 51:3363
  • [KA1] Y. Kabeya, Existence theorems for quasilinear elliptic problems in ${R^N} $, Funkc. Ekvac. 35 (1992), 603-616. MR 94b:35103
  • [KA2] Y. Kabeya, On some quasilinear elliptic problems involving critical Sobolev exponents, Funkc. Ekvac. 36 (1993), 385-404. MR 94k:35106
  • [KU] I.A. Kuzin, On multiple solvability of some elliptic problems in ${R^N} $, Soviet Math. Dokl. 44 (1992), 700-704.
  • [LY] G. Li and S. Yan, Eigenvalue problem for quasilinear elliptic equations in ${R^N} $, Comm. Part. Diff. Equa. 14 (1989), 1291-1314. MR 91a:35072
  • [LQ] P. Lindqvist, On the equation $\operatorname {div}(|{\nabla } u|^{p-2}{\nabla } u)+{\lambda } |u|^{p-2}u=0$, Proc. Amer. Math. Soc. 109 (1990), 157-164. MR 90h:35088
  • [R] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513. MR 46:745
  • [RE] A. Rumbos and A. Edelson, Bifurcation properites of semilinear elliptic equations in ${R^N} $, Diff. Int. Equa. 7 (1994), 399-410. MR 94m:35028
  • [SC] I. Schindler, Quasilinear elliptic boundary-value problems on unbounded cylinders and a related mountain-pass lemma, Arch. Rat. Mech. Anal. 120 (1992), 363-374. MR 93k:35093
  • [SE] J. Serrin, Local behavior of solutions of quasilinear equations, Acta Math. 111 (1964), 247-302. MR 30:3327
  • [SK] I.V. Skrypnik, Nonlinear Elliptic Equations of Higher Order (in Russian), Naukovaja Dumka, Kyjev, 1973. MR 55:8549 Nonlinear Elliptic Boundary Value Problems (in English), Teubner Texte fur Math. 91, Teubner-Verlag, Leipzig, 1986. MR 89g:35040
  • [TO] P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Diff. Equa. 51 (1984), 126-150. MR 85g:35047
  • [TR] N.S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721-747. MR 37:1788
  • [Y] L.S. Yu, Nonlinear $p$-Laplacian problems on unbounded domains, Proc. Amer. Math. Soc. 115 (1992), 1037-1045. MR 93e:35027

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 35B32, 35J70, 35P30

Retrieve articles in all journals with MSC (1991): 35B32, 35J70, 35P30


Additional Information

Pavel Drábek
Affiliation: Department of Mathematics, University of West Bohemia, P.O. Box 314, 30614 Pilsen, Czech Republic

Yin Xi Huang
Affiliation: Department of Mathematical Sciences, University of Memphis, Memphis, Tennessee 38152
Email: huangy@mathsci.msci.memphis.edu

DOI: https://doi.org/10.1090/S0002-9947-97-01788-1
Keywords: $p$-Laplacian, global positive solutions, weighted spaces
Received by editor(s): November 18, 1994
Received by editor(s) in revised form: March 10, 1995
Additional Notes: The first author was partially supported by the Grant Agency of the Czech Republic under the Grant No. 201/94/0008
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society