Some global results for nonlinear fourth order eigenvalue problems

Abstract

Abstract:In this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems. MSC:

Global bifurcation results for some fourth‐order nonlinear eigenvalue problem with a spectral parameter in the boundary condition

“…If 𝜆 1 = 0, then using the above ideas and plus the approximation argument given on pp. 468-469 of Aliyev, 18 we can justify the validity of the statements of Theorem 4.2.…”

“…Global bifurcation from zero in nonlinear eigenvalue problems for ordinary differential equations of the second and fourth orders, elliptic partial differential equations of second order, and the one‐dimensional Dirac equation was studied in previous studies 4,5,7,10,12–24 . These papers prove the existence of unbounded continua of nontrivial solutions bifurcating from the points and intervals of the line of trivial solutions and contained in the classes with fixed oscillation count (these classes consist of functions that have oscillatory properties of linear problems obtained from nonlinear problems by equating nonlinear terms to zero).…”

“…Global bifurcation from zero in nonlinear eigenvalue problems for ordinary differential equations of the second and fourth orders, elliptic partial differential equations of second order, and the one-dimensional Dirac equation was studied in previous studies. 4,5,7,10,[12][13][14][15][16][17][18][19][20][21][22][23][24] These papers prove the existence of unbounded continua of nontrivial solutions bifurcating from the points and intervals of the line of trivial solutions and contained in the classes with fixed oscillation count (these classes consist of functions that have oscillatory properties of linear problems obtained from nonlinear problems by equating nonlinear terms to zero). Global bifurcation from infinity in the above nonlinear eigenvalue problems was studied in other works, 15,23,[25][26][27][28][29][30] where the authors show the existence an unbounded continua of nontrivial solutions bifurcating from points and intervals of R × {∞} and contained in the classes with fixed oscillation count in some neighborhoods of these points and intervals.…”

Bifurcation of positive and negative solutions of nonlinearizable Sturm-Liouville problems with indefinite weight

“…) corresponding to the usual nodal properties and emanating from bifurcation points and intervals surrounding the eigenvalues of the linear problem (1.1)-(1.3) with f ≡ 0. Similar problems for the nonlinear eigenvalue problems of ordinary differential equations of fourth order have been considered in [1,2].…”

Global Bifurcation of Fourth-Order Nonlinear Eigenvalue Problems’ Solution

“…Problems ( 1)-(2) were considered in [4] (see also [5]) in the case when r(x) is strictly positive on [0, 1], h � f + g, and f satisfies the condition…”