【摘要】：本文主要研究了幾類擬線性橢圓型問題解的相關(guān)性質(zhì),具體包括解的存在性、非存在性以及多解性等.第一章研究擬線性橢圓型方程組正解的存在性與非存在性,其中Ω (?) RN為有界光滑區(qū)域或者Ω=RN(當(dāng)上下解方法,我們得到當(dāng)Ω為有界光滑區(qū)域或者Q=RN時,該問題至少存在一個正解,且當(dāng)Ω=RN時,該問題不存在徑向?qū)ΨQ的有界正解.第二章研究一類含凹凸項(xiàng)和臨界項(xiàng)的p-q-拉普拉斯方程組數(shù)人*0使得當(dāng)λp/(p-r)+p/(μp-r)∈(0,∧*)時,該問題至少存在cat(Ω)+1個不同的正解,從而建立了方程組解的個數(shù)與區(qū)域拓?fù)渲g的關(guān)系.第三章研究一類帶有非線性邊界條件的非齊次擬線性橢圓型方程組路引理和Ekeland變分準(zhǔn)則,我們得到存在常數(shù)λ*,L0使得當(dāng)入∈(0,λ*)時,該問題至少存在兩個非平凡解,其中第四章研究一類含變號位勢的p-Kirchhoff型方程組V∈C(RN,R)可變號,且函數(shù)F∈C1(RN×R2,R)滿足一定的增長性條件.利用對稱的山路引理,我們證得該問題存在無窮多個高能量的非平凡解.第五章在RN中研究一類具有臨界指數(shù)增長的N-Kirchhoff型問題解的存在性與多解性,其中且函數(shù)f滿足臨界指數(shù)增長條件.結(jié)合變分法和RN中的Trudinger-Moser不等式,我們得到存在常數(shù)A0使得當(dāng)λ∈(0,A)時,該問題至少存在兩個正解.
[Abstract]:In this paper, we mainly study the related properties of solutions for quasilinear elliptic problems, including the existence, nonexistence and multiplicity of solutions. In chapter 1, we study the existence and nonexistence of positive solutions for quasilinear elliptic equations, where 惟 (?) RN is a bounded smooth region or 惟 -RN (when the upper and lower solution method, we obtain that if 惟 is a bounded smooth region or Q=RN, there is at least one positive solution to the problem, and if 惟 RN, the problem does not have a radial symmetric bounded positive solution. In chapter 2, we study a class of p-q-Laplacian equations with concave and convex terms and critical terms such that when 位 p / (p-r) p / (渭 p-r) 鈭，

本文編號：2187887