Liouville properties for differential inequalities with Laplacian operator

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 (Formula presented.) (Formula presented.) where (Formula presented.), (Formula presented.) and (Formula presented.) is any exterior domain of (Formula presented.). In particular, we prove that for (Formula presented.), inequality (Formula presented.) does not admit any positive solution when (Formula presented.) and (Formula presented.) admits a positive solution if (Formula presented.), where (Formula presented.) is the Serrin exponent for the (Formula presented.) -Laplacian. Further, we show that when (Formula presented.) and (Formula presented.), the only nonnegative solution to (Formula presented.) is the trivial solution. On the other hand, for (Formula presented.) we prove that (Formula presented.) is the only nonnegative solution for (Formula presented.) for any (Formula presented.). In the second part, we consider the inequality (Formula presented.) (Formula presented.) where (Formula presented.), (Formula presented.) and (Formula presented.). We prove that, for (Formula presented.), the only positive solution to (Formula presented.) is constant, provided (Formula presented.). The Liouville property continues to hold in the critical case if (Formula presented.). This, in particular, proves that if (Formula presented.), then any nonnegative solution to (Formula presented.) with (Formula presented.) and (Formula presented.) is the trivial solution. To prove Liouville in the range (Formula presented.), we first prove an almost optimal lower estimate of any nonnegative supersolution of (Formula presented.) 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.