Estimate of the first eigenvalue of a selfadjoint elliptic operator

Let $L$ be a symmetric positive elliptic operator o! order m with smooth coefficients in a bounded domain. The problem considered is that of estimating a minimum number $\lambda$ for which the homogeneous Dirichlet problem for the equation $Lu-\lambda Qu=0$ has a non-trivial solution. It is assumed that $Q(x)\ge0$, $\int Q^\alpha(x)\,dx=1$, where $\alpha$ is a real number, $\alpha\ne0$.