The Symmetry Breaking For Hénon-type P-Laplace Equations

This thesis concerns the study of the symmetry breaking for Henon-type p_ Laplace equations. we can obtain radial and nonradial solutions, and we analyze their asymptotic behaviours.We first consider the equation -Δpu+up-1=|x|αuq-1,u>0 inΩ, (?)=0, on (?)Ω, whereα> 0, p 0 inΩ*,u=0,on (?)Ω*, where a> 0, p< q,Ω*={x∈(?)n|1<|x|<3}, we present some estimates for functional in the global space and radial space. It will lead us to a first symmetry breaking result, which means the global least energy solution uq is nonradial. Then we prove that uq concentrates as q→p* at precisely one point of the boundary (?)Ω*, which has two connected components. So a second nonradial solution can then be found. We get a third nonradial solution by Mountain Pass lemma.