Monotonicity and Symmetry of Positive Solutions to Logarithmic Laplacian Equations and Systems with Gradient Terms

Abstract

This paper is concerned with logarithmic Laplacian equations and systems with gradient terms in bounded convex domains. Under some assumptions on the nonlinearities, the authors prove the strict monotonicity and symmetry of positive solutions. In particular, the authors establish one new estimate of the logarithmic Laplacian operator that plays an important role in applying the direct method of moving planes. This paper seems to be the first one to deal with the monotonicity and symmetry of positive solutions to logarithmic Laplacian equations and systems with gradient terms.

Keywords

Logarithmic Laplacian / Monotonicity / Positive solution / Convex domain / 35J67 / 35R11

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