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

Annales Polonici Mathematici (1992)

- Volume: 57, Issue: 1, page 21-28
- ISSN: 0066-2216

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topJolanta Przybycin. "Bifurcation theorems of Rabinowitz type for certain differential operators of the fourth order." Annales Polonici Mathematici 57.1 (1992): 21-28. <http://eudml.org/doc/262460>.

@article{JolantaPrzybycin1992,

abstract = {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].},

author = {Jolanta Przybycin},

journal = {Annales Polonici Mathematici},

keywords = {nonlinear partial differential equations; bifurcation in partial differential equations; nonlinear perturbation; unbounded continuum of nontrivial positive solutions; bifurcation from infinity},

language = {eng},

number = {1},

pages = {21-28},

title = {Bifurcation theorems of Rabinowitz type for certain differential operators of the fourth order},

url = {http://eudml.org/doc/262460},

volume = {57},

year = {1992},

}

TY - JOUR

AU - Jolanta Przybycin

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

JO - Annales Polonici Mathematici

PY - 1992

VL - 57

IS - 1

SP - 21

EP - 28

AB - 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].

LA - eng

KW - nonlinear partial differential equations; bifurcation in partial differential equations; nonlinear perturbation; unbounded continuum of nontrivial positive solutions; bifurcation from infinity

UR - http://eudml.org/doc/262460

ER -

## References

top- [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. Zbl0212.45102
- [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. Zbl0596.35044
- [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. Zbl0729.34022
- [5] P. H. Rabinowitz, Some aspects of nonlinear eigenvalue problems, Rocky Mountain J. Math. 3 (1973), 161-202. Zbl0255.47069
- [6] P. H. Rabinowitz, On bifurcation from infinity, J. Differential Equations 14 (1973), 462-475. Zbl0272.35017
- [7] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487-513. Zbl0212.16504