We prove the existence of at least two distinct solutions of problem (1), (2) under some restrictions on $Q$, $f$ and $\lambda$. One solution has been obtained by a local minimization and the second one by applying the mountain pass theorem.
J. Chabrowski. "On multiple solutions for the nonhomogeneous $p$-Laplacian with a critical Sobolev exponent." Differential Integral Equations 8 (4) 705 - 716, 1995. https://doi.org/10.57262/die/1369055606