Existence Of Multiple Solutions For Quasilinear P-Laplace Partial Differential Equations

Because of the extensive background in physics and chemistry, nonlinear elliptic equations with p-Laplace operator, especially the quasilinear ones have attracted the attention of the academic world both at home and abroad during the past few decades. We study the existence of multiple solutions of two kinds of quasilinear p-Laplace equations by Mountain Pass Theorem and Ekeland’s variational principle in this thesis.(1) Consider the following quasilinear elliptic equationWhen f satisfies the growth condition and g satisfies certain sublinear condition, the corresponding energy function Φ satisfies the (PS) condition. So a critical point which is also a solution for the problem can be get through applying the Mountain Pass Theorem. Then a local minimum of Φ near zero leads to another solution for the problem. Finally, the non-negativity of the two solutions is illustrated and a concrete example is given.(2) Consider the elliptic boundary value problem-△pu=λ1,b(x)|u|P-2u+f(x,u), x∈Ω,|u=0, x∈(?)Ω. When b(x) is bounded and f(x,u) satisfies the strong resonance condition, thecorresponding energy function Ⅰsatisfies the (Ce) condition, so at least one solution for the equation can be got by using Ekeland’s variational principle. If f(x,0)=0, then u=0is a trivial solution of this equation. But when f(x, u) also satisfies the following two conditions,(H1) limsup f(x,t)/|t|p-2t<0a.e.inΩ. (H2) There exist real numbers t-<0<t+such that the equation has at least three nontrival solutions u0±, u1∈W01,p(Ω).