Global bifurcation of positive solutions for a superlinear $p$-Laplacian system

Abstract

We are concerned with the principal eigenvalue of \begin{equation*} \begin{cases} -\Delta_p u= \lambda\theta_1\varphi_p(v), &x\in \Omega,\\ -\Delta_p v= \lambda\theta_2\varphi_p(u), &x\in \Omega,\\ u=0=v,\ &x\in\partial\Omega \end{cases} \tag{P} \end{equation*} and the global structure of positive solutions for the system \begin{equation*} \begin{cases} -\Delta_p u= \lambda f(v), &x\in \Omega,\\ -\Delta_p v= \lambda g(u), &x\in \Omega,\\ u=0=v,\ &x\in\partial\Omega, \end{cases} \tag{Q} \end{equation*} where $\varphi_p(s)=|s|^{p-2}s$, $\Delta_p s=\text{div}(|\nabla s|^{p-2}\nabla s)$, $\lambda>0$ is a parameter, $\Omega\subset\mathbb{R}^N$, $N> 2$, is a bounded domain with smooth boundary $\partial\Omega$, $f,g:\mathbb{R}\to(0,\infty)$ are continuous functions with $p$-superlinear growth at infinity. We obtain the principal eigenvalue of $(P)$ by using a nonlinear Krein–Rutman theorem and the unbounded branch of positive solutions for $(Q)$ via bifurcation technology.

Keywords

• $p$-laplacian
• principal eigenvalue
• positive solutions
• bifurcation