Liouville properties for differential inequalities with (p,q) Laplacian operator

Source abstract

Abstract In this paper, we establish several Liouville‐type theorems for a class of nonhomogenenous quasilinear inequalities. In the first part, we prove various Liouville results associated with nonnegative solutions to where , and is any exterior domain of . In particular, we prove that for , inequality does not admit any positive solution when and admits a positive solution if , where is the Serrin exponent for the ‐Laplacian. Further, we show that when and , the only nonnegative solution to is the trivial solution. On the other hand, for we prove that is the only nonnegative solution for for any . In the second part, we consider the inequality where , and . We prove that, for , the only positive solution to is constant, provided . The Liouville property continues to hold in the critical case if . This, in particular, proves that if , then any nonnegative solution to with and is the trivial solution. To prove Liouville in the range , we first prove an almost optimal lower estimate of any nonnegative supersolution of and then leveraging this estimate we prove Liouville result. To the best of our knowledge, this technique is completely new and provides an alternative approach to the capacity method of Mitidieri–Pohozaev provided higher regularity is available.