本文選題：薛定諤泊松方程 + p-拉普拉斯算子　； 參考：《中北大學(xué)》2017年碩士論文

【摘要】：變分法是以臨界點(diǎn)理論為理論基礎(chǔ)的,將微分方程邊值問(wèn)題化為變分問(wèn)題,來(lái)證明解的存在性,多重性,及求近似解的方法.本文主要運(yùn)用變分方法研究?jī)深愑袕?qiáng)大物理背景的橢圓微分方程,即帶有臨界指數(shù)的薛定諤泊松方程和帶有P-拉普拉斯算子的基爾霍夫方程.第二章研究以下帶有臨界指數(shù)的薛定諤泊松方程Ω是R3中具有光滑邊界的有界區(qū)域,r ∈(0,1),f,g為C(Ω)上的非負(fù)函數(shù).通過(guò)引入Nehari流形,運(yùn)用Ekeland變分原理和集中緊性原理得到常數(shù)M4,說(shuō)明對(duì)于0λM4,此方程至少有一個(gè)正解uλ,uλ ∈Nλ+.第三章研究以下帶有P-拉普拉斯算子的基爾霍夫方程Ω是R3中有光滑邊界的有界區(qū)域.△p=div(|%絬|p-2%絬)是p-拉普拉斯算子,1pN,且a,b0,a,+ b0,λ ≥ 0,0s1,0r ≤ p*-1.對(duì)任意∈Ω,f Lp*+s-1/p*(Ω)且(x)0.另外p*=N-p/Np 通過(guò)變分法,勒貝格控制收斂定理以及算子的第一特征值說(shuō)明此方程具有唯一解.
[Abstract]:The variational method is based on the critical point theory. The boundary value problem of differential equation is transformed into a variational problem to prove the existence, multiplicity and approximate solution of the solution.In this paper, the variational method is used to study two kinds of elliptic differential equations with strong physical background, namely, Schrodinger Poisson equation with critical exponent and Kirchhoff equation with P- Laplacian operator.In chapter 2, we study that the Schrodinger Poisson equation with critical exponent 惟 is a bounded domain with smooth boundary in R3 and that r 鈭，

本文編號(hào)：1736551