Unilateral global bifurcation and nodal solutions for the p-Laplacian with sign-changing weight

2012/03/15 by Dai, Guowei, Ma, Ruyun
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1203.3262

Abstract

In this paper, we shall establish a Dancer-type unilateral global bifurcation result for a class of quasilinear elliptic problems with sign-changing weight. Under some natural hypotheses on perturbation function, we show that (μ_k^ν(p),0) is a bifurcation point of the above problems and there are two distinct unbounded continua, (C_k^ν)⁺ and (C_k^ν)^-, consisting of the bifurcation branch C_k^ν from (μ_k^ν(p), 0), where μ_k^ν(p) is the k-th positive or negative eigenvalue of the linear problem corresponding to the above problems, ν∈\+,-\. As the applications of the above unilateral global bifurcation result, we study the existence of nodal solutions for a class of quasilinear elliptic problems with sign-changing weight. Moreover, based on the bifurcation result of Drábek and Huang (1997) [\refDH], we study the existence of one-sign solutions for a class of high dimensional quasilinear elliptic problems with sign-changing weight.

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