Research On The Properties Of Solutions To A-harmonic Equation Under Natural Growth Conditions

Partial differential equations theory has a strong practical background.The elliptic equation is an important part of PDEs.In the PDE equation,the A-harmonic equation is a very important category,and has great significance to study the harmonic equation.However,most of the studies on the properties of the solutions of non-homogeneous harmonic equations are concentrated in controlled growth conditions.The research on the solutions of A-harmonic equations under natural growth conditions is just starting.The article studies the A-harmonic equation(?) under the condition of natural growth,and hopes to contribute to the theoretical study of PDEs.The article first introduces the research status of the A-harmonic equation and expounds the uniqueness,existence,regularity and other properties of the solution.Then,the article mainly studies two aspects: the singularity of the weak solution of the harmonic equation under the natural growth condition,and the singularity of the weak solution of its weighted form under the natural growth condition.On the one hand,I study the removable singularities of the weak solutions of the A-harmonic equation under the condition of natural growth.The Caccioppoli inequality for the weak solution of the A-harmonic equation under the condition of natural growth was obtained.Then by the method of the peak function and capacity,its removable singular was obtained.On the other hand,I study the properties of the A-harmonic equation under the condition of natural growth to obtain its weighted.By choosing the appropriate test function,the Caccioppoli inequality for the weak solution of the A-harmonic equation under the condition of natural growth is obtained.Then by using the method of peak function and capacity,its removable singularities to the weighted A-harmonic equation under the condition of natural growth was obtained.