Existence and uniqueness of positive eigenvalues for certain eigenvalue systems (Q2721870)

This paper is devoted to the eigenvalue problem of the form NEWLINE\[NEWLINE\begin{cases} -\Delta_p u+a\psi_p(u)= \lambda^{p-1} \sum^k_{i=1} a_i(x) u^{ \alpha_i} v^{p-1-\alpha_i} \quad & \text{in }\Omega\\ -\Delta_q v+b \psi_q (v) =\lambda^{q-1} \sum^N_{j=1} b_j(x)u^{q-1- \beta_j} v^{\beta_j} \quad & \text{in }\Omega\\ u=v=0\quad & \text{on }\partial \Omega \end{cases}.\tag{1}NEWLINE\]NEWLINE Under some natural assumptions the authors prove that (1) has a unique eigenvalue with positive eigenfunctions, and that the eigenfunction is unique up to a scalar multiple.