Bifurcation from infinity and nodal solutions of quasilinear problems without the signum condition

• Published in: Journal of mathematical analysis and applications Vol. 397; no. 1; pp. 119 - 123
• Main Authors: Dai, Guowei, Ma, Ruyun, Lu, Yanqiong
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2013
• Subjects: Bifurcation; Nodal solutions; Nodal solutions; Bifurcation
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2012.07.056

In this paper, we shall establish a unilateral global bifurcation theorem from infinity for a class of p-Laplacian problems. As an application of the above result, we shall study the global behavior of the components of nodal solutions of the following problem {(φp(u′))′+λa(t)f(u)=0,t∈(0,1),u(0)=u(1)=0, where φp(s)=|s|p−2s, a∈C([0,1],[0,+∞)) with a≢0 on any subinterval of [0,1]; f:R→R is continuous, and there exist two constants s2<0<s1 such that f(s2)=f(s1)=f(0)=0, f(s)s>0 for s∈R∖{s2,0,s1}. Moreover, we give the intervals for the parameter λ which ensure the existence of multiple nodal solutions for the problem if f0∈(0,+∞) and f∞∈(0,+∞), where f(s)/φp(s) approaches f0 and f∞ as s approaches 0 and ∞, respectively. We use topological methods and nonlinear analysis techniques to prove our main results.