Bifurcation theorems of Rabinowitz type for certain differential operators of the fourth order

• Autorzy: Jolanta Przybycin
• Języki publikacji: EN

Abstrakty

EN

This paper was inspired by the works of P. H. Rabinowitz. We study nonlinear eigenvalue problems for some fourth order elliptic partial differential equations with nonlinear perturbation of Rabinowitz type. We show the existence of an unbounded continuum of nontrivial positive solutions bifurcating from (μ₁,0), where μ₁ is the first eigenvalue of the linearization about 0 of the considered problem. We also prove the related theorem for bifurcation from infinity. The results obtained are similar to those proved by Rabinowitz for second order elliptic partial differential equations ([5]-[7]). The methods used are based, in principle, on the results of [1], [5], [6].

Słowa kluczowe

EN

nonlinear partial differential equations; bifurcation in partial differential equations

Kategorie tematyczne

• 35J65: Nonlinear boundary value problems for linear elliptic equations
• 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 1992
• Tom: 57
• Numer: 1
• Strony: 21-28

Daty

wydano

1992

otrzymano

1990-10-20

poprawiono

1991-07-02

Twórcy

autor

Jolanta Przybycin

• Institute of Mathematics, Academy of Mining And Metallurgy, Mickiewicza 30, 30-059 Kraków, Poland

Bibliografia

• [1] J. Bochenek, On some properties of eigenvalues and eigenfunctions of certain differential equations of the fourth order, Ann. Polon. Math. 24 (1971), 113-119.
• [2] J. Bochenek, Positive solution of asymptotically linear elliptic eigenvalue problems for certain differential equations of the fourth order, Ann. Polon. Math. 45 (1985), 231-236.
• [3] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute, New York 1974.
• [4] J. Przybycin, Some applications of bifurcation theory to ordinary differential equations of the fourth order, Ann. Polon. Math. 53 (1991), 153-160.
• [5] P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161-202.
• [6] P. H. Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1973), 462-475.
• [7] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513.