On The Geometric Properties Of The Level Sets Of Solutions To Elliptic Partial Differential Equations

The main goal of this paper is devoted to study the geometric properties of the solutions to certain elliptic partial differential equations, namely, we are interested in the following equation Here N≥2,Ωis a smooth region of RN.This kind of equations can be viewed as a natural generalization of some equa-tions which has been studied by B.Gidas,WeiMing. Ni, L. Nirenberg, M.Grossi and R. Molle.By using the modification of the method of M. Grossi and R.Molle's, we prove that the starshapedness of the level sets of the above solutions whenΩis starshaped and under some other conditions of f. Then we apply our result to the p-Laplacian equation.