An exact bifurcation diagram for ap-qLaplacian boundary value problem

Abstract

We study positive solutions to thep–qLaplacian two-point boundary value problem:{−μ[(u′)p−1]′−[(u′)q−1]′=λu(1−u)on (0,1)u(0)=0=u(1)whenp=4andq=2. Hereλ>0is a parameter andμ≥0is a weight parameter influencing the higher-order diffusion term. Whenμ=0(the Laplacian case) the exact bifurcation diagram for a positive solution is well-known, namely, whenλ≤π2there are no positive solutions, and forλ>π2there exists a unique positive solutionuλ,μsuch that‖uλ,μ‖∞→0asλ→π2and‖uλ,μ‖∞→1asλ→∞. Here, we will prove that for allμ>0similar bifurcation diagrams preserve, and they all bifurcate from(λ,u)=(π2,0). Our results are established via the method of sub-super solutions and a quadrature method. We also present computational evaluations of these bifurcation diagrams for various values ofμand illustrate how they evolve whenμvaries.