In this article, by employing an oscillatory condition on the nonlinear term, a result is proved for the existence of connected component of solutions set of a nonlocal boundary value problem, which bifurcates from infinity and asymptotically oscillates over an interval of parameter values. An interesting and immediate consequence of such oscillation property of the connected component is the existence of infinitely many solutions of the nonlinear problem for all parameter values in that interval.
Xian XU
,
Baoxia QIN
,
Zhen WANG
. ASYMPTOTIC BEHAVIOR OF SOLUTION BRANCHES OF NONLOCAL BOUNDARY VALUE PROBLEMS[J]. Acta mathematica scientia, Series B, 2020
, 40(2)
: 341
-354
.
DOI: 10.1007/s10473-020-0203-9
[1] Sun Jingxian, Xu Xian, O'regan D. Nodal solutions for m-point boundary value problems using bifurcation methods. Nonlinear Analysis, 2006, 68:3034-3046
[2] Ma R, O'Regan D. Nodal solutions for second-order m-point boundary value problems with nonlinearities across several eigenvalues. Nonlinear Analysis, 2006, 64:1562-1577
[3] Rynne B P. Spectral properties and nodal solutions for second-order, m-point boundary value problems. Nonlinear Analysis, 2007, 67:3318-3327
[4] Schaaf R, Schmitt K. A class of nonlinear Sturm-Liouville problems with infinitely many solutions. Trans Amer Math Soc, 1988, 306:853-859
[5] Davidson F A, Rynne B P. Asymptotic oscillations of continua of positive solutions of a semilinear SturmLiouville problem. J Math Anal Appl, 2000, 252:617-630
[6] Gong Lin, Li Xiang, Qin Baoxia, Xu Xian. Solutions Branches for Nonlinear Problems with Asymptotic Oscillation Property. Electronic J Diff Equas, 2015, 205(269):1-15
[7] Kielhofer H, Maier S. Infinitely many positive solutions of semilinear elliptic problems via sub- and supersolutions. Comm Partial Differential Equations, 1993, 18:1219-1229
[8] Maier-Paape S, Schmitt K. Asymptotic behaviour of solution continua for semilinear elliptic problems. Can Appl Math Quart, 1996, 4:211-228
[9] Rynne B P. Oscillating global continua of positive solutions of nonlinear elliptic problems. Proc Amer Math Soc, 2000, 128:229-236
[10] Schaaf R, Schmitt K. Periodic perturbations of linear problems at resonance on convex domains. Rocky Mountain J Math, 1990, 20:1119-1131
[11] Schaaf R, Schmitt K. Asymptotic behaviour of positive solution branches of elliptic problems with linear part at resonance. Z Angew Math Phys, 1992, 43:645-676
[12] Kong Lingju, Kong Qingkai, Kwong Man K, Wong J S W. Linear Sturm-Liouville problems wit multi-point boundary conditions. Math Nachr, 2013, 286(11/12):1167-1179
[13] Rabinowitz P H. On bifurcation from infinity. J Differential Equations, 1973, 14:462-475
[14] Berestycki H. On Some nonlinear Sturm-Liovuille problems. J Differential Equations, 1977, 26:375-390
[15] Schmitt K, Smith H L. On eigenvalue problems for nondifferentiable mappings. J Differential Equations, 1979, 33(3):294-319
[16] Zettl A. Sturm-Liouville Theory//Mathematical Surveys and Monographs. Vol. 121. American Mathematical Society, 2005
[17] Xu Xian. Multiple sign-changing solutions for some m-point boundary-value problems. Electron J Differential Equations, 2004, 89:14
