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].
@article{bwmeta1.element.bwnjournal-article-apmv57z1p21bwm,
author = {Jolanta Przybycin},
title = {Bifurcation theorems of Rabinowitz type for certain differential operators of the fourth order},
journal = {Annales Polonici Mathematici},
volume = {57},
year = {1992},
pages = {21-28},
zbl = {0781.35004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv57z1p21bwm}
}
Jolanta Przybycin. Bifurcation theorems of Rabinowitz type for certain differential operators of the fourth order. Annales Polonici Mathematici, Tome 57 (1992) pp. 21-28. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv57z1p21bwm/
[000] [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. | Zbl 0212.45102
[001] [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. | Zbl 0596.35044
[002] [3] L. Nirenberg, Topics in Nonlinear Functional Analysis, Courant Institute, New York 1974.
[003] [4] J. Przybycin, Some applications of bifurcation theory to ordinary differential equations of the fourth order, Ann. Polon. Math. 53 (1991), 153-160. | Zbl 0729.34022
[004] [5] P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161-202. | Zbl 0255.47069
[005] [6] P. H. Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1973), 462-475. | Zbl 0272.35017
[006] [7] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513. | Zbl 0212.16504