An exact bifurcation diagram for a $p$–$q$ Laplacian boundary value problem

Acharya, Ananta and Munoz, Victor and Nichols, Dustin and Shivaji, Ratnasingham (2023) An exact bifurcation diagram for a $p$–$q$ Laplacian boundary value problem. ELECTRONIC JOURNAL OF QUALITATIVE THEORY OF DIFFERENTIAL EQUATIONS, 2023 (7). pp. 1-10. ISSN 1417-3875

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Abstract

We study positive solutions to the p–q Laplacian two-point boundary value problem: −µ[(u′)p−1]′ − [(u′)q−1]′ = λu(1 − u) on (0,1) u(0) = 0 = u(1) when p = 4 and q = 2. Here λ > 0 is a parameter and µ ≥ 0 is a weight parameter influencing the higher-order diffusion term. When µ = 0 (the Laplacian case) the exact bifurcation diagram for a positive solution is well-known, namely, when λ ≤ π2 there are no positive solutions, and for λ > π2 there exists a unique positive solution uλ,µ such that ∥uλ,µ∥∞ → 0 as λ → π2 and ∥uλ,µ∥∞ → 1 as λ → ∞. Here, we will prove that for all µ > 0 similar 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 of µ and illustrate how they evolve when µ varies.

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