Some applications of minimax and topological degree to the study of the Dirichlet problem for elliptic partial differential equations

Leszek Gęba; Tadeusz Pruszko

Annales Polonici Mathematici (1991)

  • Volume: 56, Issue: 1, page 49-61
  • ISSN: 0066-2216

Abstract

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This paper treats nonlinear elliptic boundary value problems of the form (1) L[u] = p(x,u) in Ω ⊂ ℝ n , u = D u = . . . = D m - 1 u on ∂Ω in the Sobolev space W 0 m , 2 ( Ω ) , where L is any selfadjoint strongly elliptic linear differential operator of order 2m. Using both topological degree arguments and minimax methods we obtain existence and multiplicity results for the above problem.

How to cite

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Leszek Gęba, and Tadeusz Pruszko. "Some applications of minimax and topological degree to the study of the Dirichlet problem for elliptic partial differential equations." Annales Polonici Mathematici 56.1 (1991): 49-61. <http://eudml.org/doc/262472>.

@article{LeszekGęba1991,
abstract = {This paper treats nonlinear elliptic boundary value problems of the form (1) L[u] = p(x,u) in $Ω ⊂ ℝ^n$, $u = Du = ... = D^\{m-1\}u$ on ∂Ω in the Sobolev space $W_0^\{m,2\}(Ω)$, where L is any selfadjoint strongly elliptic linear differential operator of order 2m. Using both topological degree arguments and minimax methods we obtain existence and multiplicity results for the above problem.},
author = {Leszek Gęba, Tadeusz Pruszko},
journal = {Annales Polonici Mathematici},
keywords = {multiplicity result; mountain-pass lemma},
language = {eng},
number = {1},
pages = {49-61},
title = {Some applications of minimax and topological degree to the study of the Dirichlet problem for elliptic partial differential equations},
url = {http://eudml.org/doc/262472},
volume = {56},
year = {1991},
}

TY - JOUR
AU - Leszek Gęba
AU - Tadeusz Pruszko
TI - Some applications of minimax and topological degree to the study of the Dirichlet problem for elliptic partial differential equations
JO - Annales Polonici Mathematici
PY - 1991
VL - 56
IS - 1
SP - 49
EP - 61
AB - This paper treats nonlinear elliptic boundary value problems of the form (1) L[u] = p(x,u) in $Ω ⊂ ℝ^n$, $u = Du = ... = D^{m-1}u$ on ∂Ω in the Sobolev space $W_0^{m,2}(Ω)$, where L is any selfadjoint strongly elliptic linear differential operator of order 2m. Using both topological degree arguments and minimax methods we obtain existence and multiplicity results for the above problem.
LA - eng
KW - multiplicity result; mountain-pass lemma
UR - http://eudml.org/doc/262472
ER -

References

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  1. [1] H. Amann and S. A. Weiss, On the uniqueness of the topological degree, Math. Z. 130 (1973), 39-54. Zbl0249.55004
  2. [2] A. Ambrosetti and G. Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal. 3 (1979), 635-645. Zbl0433.35025
  3. [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381. Zbl0273.49063
  4. [4] K.-C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102-129. Zbl0487.49027
  5. [5] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York 1969. Zbl0224.35002
  6. [6] L. Gå rding, Dirichlet's problem for linear elliptic partial differential equations, Math. Scand. 1 (1953), 55-72. 
  7. [7] E. M. Landesman and A. C. Lazer, Linear eigenvalues and a nonlinear boundary value problem, Pacific J. Math. 33 (1970), 311-328. Zbl0204.12002
  8. [8] A. C. Lazer and P. J. McKenna, On the number of solutions of a nonlinear Dirichlet problem, J. Math. Anal. Appl. 84 (1981), 282-294. Zbl0496.35039
  9. [9] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University, 1974. Zbl0286.47037
  10. [10] R. S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165-171. Zbl0119.09201
  11. [11] W. V. Petryshyn, Variational solvability of quasilinear elliptic boundary value problems at resonance, Nonlinear Anal. 5 (1981), 1095-1108. Zbl0477.35041
  12. [12] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. in Math. 65, Amer. Math. Soc., Providence, R.I., 1986. 
  13. [13] M. Struve, A note on a result of Ambrosetti and Mancini, Ann. Mat. Pura Appl. 131 (1982), 107-115. 
  14. [14] M. Vaĭnberg, On the continuity of some operators of special type, Dokl. Akad Nauk SSSR 73 (1950), 253-255 (in Russian) Zbl0039.33702

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