Resumen de Liouville Results for Double Phase Problems in RN

Phuong Le

• We prove Liouville theorems for the double phase problem −div(|∇u| p−2∇u + w(x)|∇u| q−2∇u) = f (x)|u| r−1u in RN , where q ≥ p ≥ 2,r > q−1 and w, f ∈ L1 loc(RN ) are two nonnegative functions such that w(x) ≤ C1|x| a and f (x) ≥ C2|x| b for all |x| > R0, where R0,C1,C2 > 0 and a, b ∈ R. Our Liouville results hold for stable solutions in dimension N < N, where N is explicitly computed. We also prove Liouville theorems for finite energy solutions as well as solutions stable outside a compact set when N+b r+1 > max N−p p , N−q+a q .

  Methods of integral estimates and a Pohožaev type identity for double phase problems are exploited in our proofs.