Qualitative Analysis Of Solutions For The Quasilinear Elliptic Problems

In this dissertation, we investigate some characteristics of solutions to quasi-linear elliptic problems, including existence, non-existence and multiplicity, etc.In Chapter 1, we study the existence and non-existence of positive solutions for the quasilinear elliptic systems where Ω is a bounded domain in RN with a smooth boundary or Ω= RN (when Ω= RN, the condition u= v= 0 on (?)Ω should be understood as u(x)â†' 0, v(x) â†' 0 as |x|â†'∞),p, q> 1. Each ai(x) (i= 1,2) is a positive C0,α(/Ω) (α∈(0,1)) function, Fi:Ω× (0, ∞) × (0, ∞) â†' (0, ∞) is continuously differentiable on its domain, and we suppose Fi is singular in one of variables and non-singular in the others. By a sub-super solution argument, we obtain the existence of positive solutions for this problem when Ω (?) RN is bounded or Ω= RN. Moreover, we also prove that the problem admits no radial positive bounded solutions when Ω= RN.In Chapter 2, we study the existence and multiplicity of positive solutions for a p-q-Laplacian system where Ω(?) RN is a bounded domain with smooth boundary, λ,μ>0, α,β>1 satisfy p<p*. By using the Nehari manifold, fibering maps and the Lusternik-Schnirelman category, we prove that there exists A*> 0 such that if λP/P-r＋μP/P-r∈(0, Λ*), the problem admits at least cat(Ω)+1 distinct positive solutions, and then we establish the relationship between the number of positive solutions for the problem and the topological property of Q.In Chapter 3, we investigate the multiplicity of nontrivial solutions for a class of quasilinear elliptic systems involving nonhomogeneous nonlinearities and nonlinear boundary conditions degree α,β, respectively. By using the Mountain Pass theorem and the Ekeland’s variational principle, we prove that there exist A*, L> 0 such that the problem admitsIn Chapter 4,we consider the multiplicity of the following p-Kirchhoff type sys-tem V ∈ C(RN,R) is allowed to be sign-changing, and the function F ∈ C1(RN× R2,R) satisfies some more general conditions. By using the Symmetric Mountain Pass theorem, we obtain the existence of infinitely many high energy solutions for this problem.In Chapter 5, we concerns with the existence and multiplicity of positive solu-tions for the following class of N-Kirchhoff type problem in RN A is a positive function in Lσ(RN) with σ=N/N-P, and f is a continuous function having critical exponential growth. By applying variational methods together with Trudinger-Moser inequality in RN, we obtain that there exists a constant A> 0 such that if λ∈ (0, Λ), the problem admits at least two positive solutions.