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

• Autorzy: Leszek Gęba , Tadeusz Pruszko
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1991-1992
• Tom: 56
• Numer: 1
• Strony: 49-61

Daty

wydano

1991

otrzymano

1990-04-02

poprawiono

1991-03-02

Twórcy

autor

Leszek Gęba

• Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland

autor

Tadeusz Pruszko

• Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland

Bibliografia

• [1] H. Amann and S. A. Weiss, On the uniqueness of the topological degree, Math. Z. 130 (1973), 39-54.
• [2] A. Ambrosetti and G. Mancini, Sharp nonuniqueness results for some nonlinear problems, Nonlinear Anal. 3 (1979), 635-645.
• [3] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349-381.
• [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.
• [5] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York 1969.
• [6] L. Gå rding, Dirichlet's problem for linear elliptic partial differential equations, Math. Scand. 1 (1953), 55-72.
• [7] E. M. Landesman and A. C. Lazer, Linear eigenvalues and a nonlinear boundary value problem, Pacific J. Math. 33 (1970), 311-328.
• [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.
• [9] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute of Mathematical Sciences, New York University, 1974.
• [10] R. S. Palais and S. Smale, A generalized Morse theory, Bull. Amer. Math. Soc. 70 (1964), 165-171.
• [11] W. V. Petryshyn, Variational solvability of quasilinear elliptic boundary value problems at resonance, Nonlinear Anal. 5 (1981), 1095-1108.
• [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] M. Struve, A note on a result of Ambrosetti and Mancini, Ann. Mat. Pura Appl. 131 (1982), 107-115.
• [14] M. Vaĭnberg, On the continuity of some operators of special type, Dokl. Akad Nauk SSSR 73 (1950), 253-255 (in Russian)