Let Ω ⊂ ℝn be a bounded starshaped domain and consider the (p; q)-Laplacian problem -∆pu - ∆pu = λ(x)|u|p*-2u + μ|u|r-2u where μ is a positive parameter, 1 < q ≤ p < n, r ≥ p* and is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the (p; q)-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

• 35B33 - Critical exponents in context of PDEs
• 35J35 - Variational methods for higher-order elliptic equations
• 35J92 - Quasilinear elliptic equations with \(p\)-Laplacian
• 58E05 - Abstract critical point theory (Morse theory, Lyusternik-Shnirel'man theory, etc.) in infinite-dimensional spaces