The Direct Method Of Moving Planes In Several Kinds Of The Logarithmic Laplacian Equations And Applications

The Logarithmic Laplacian operator,which is a non-local pseudo-differential operator,is developed from the fractional Laplacian operator.Its properties and characteristics are significantly different from the fractional Laplacian operator.In this paper,the symmetry and monotonicity of solutions of equations and systems involving the Logarithmic Laplacian are studied by the direct method of moving planes.The first part develops the Maximum principle,Narrow region principle and Decay at infinity of the equations involving the Logarithmic Laplacian firstly.Then the radial symmetry and monotonicity of the positive solutions of nonlinear logarithmic Laplacian equations in bounded or unbounded regions are proved by the direct method of moving planes.The second part studies the system involving the Logarithmic Laplacian.At first,we prove the Narrow region principle and Decay at infinity of the system.Then we obtain the radial symmetry of the positive solutions of the logarithmic Laplacian system in RN by the direct method of moving planes.The third part,at first,prove the Hopf lemma with the Logarithmic Laplacian in the ball.Then we extend the Hopf lemma to the general region on its basis.Finally,we show how to use the Hopf’s lemma in the process of moving planes to derive radial symmetry of positive solutions.The fourth part investigates the nonlinear Schr(?)dinger equation involving the Logarithmic Laplacian and obtains the symmetry and monotonicity properties for positive solutions.