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

Abstract

We study positive solutions to the $p$–$q$ Laplacian two-point boundary value problem: \begin{align*} \begin{cases} -\mu[(u')^{p-1}]' - [(u')^{q-1}]' = \lambda u(1-u) \quad \text{on }(0,1) \\ u(0) = 0 = u(1) \end{cases} \end{align*} when $p = 4$ and $q=2$. Here $\lambda>0$ is a parameter and $\mu \geq 0$ is a weight parameter influencing the higher-order diffusion term. When $\mu = 0$ (the Laplacian case) the exact bifurcation diagram for a positive solution is well-known, namely, when $\lambda \leq \pi^2$ there are no positive solutions, and for $\lambda > \pi^2$ there exists a unique positive solution $u_{\lambda,\mu}$ such that $\|u_{\lambda,\mu}\|_{\infty} \rightarrow 0$ as $\lambda \rightarrow \pi^2$ and $\|u_{\lambda,\mu}\|_{\infty} \rightarrow 1$ as $\lambda \rightarrow \infty$. Here, we will prove that for all $\mu > 0$ similar bifurcation diagrams preserve, and they all bifurcate from $(\lambda,u) = (\pi^2,0)$. 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 $\mu$ and illustrate how they evolve when $\mu$ varies.

Keywords

• positive solutions
• $p$–$q$ laplacian
• dirichlet boundary conditions
• exact bifurcation diagram