Study on Some Elliptic and Parabolic Equations

论文摘要

The work of this thesis deals with existence and nonexistence of solutions for semilinear elliptic and parabolic equations on bounded domains in RN. This is an interesting and a widely investigated field.（Ⅰ） The general form of the elliptic equations under Dirichlet boundary con-dition that we study is whereΩis a bounded domain with smooth boundary in RN（N≥3）, x = （y, z）∈Ω（?） Rk×RN-k = RN-, 2≤k<N,λ∈R, t∈（0, 2） and 2*（t）:=2（N-t）/N-2 is the critical Sobolev-Hardy exponent for the Sobolev embedding H01（Ω）→L2*（t）（Ω,|y|-t）.Throughout the thesis the approach is variational, as （0.0.1） is the Euler-Lagrange equation of the functionalTo state the main results, it is convenient to introduce the "limiting problem" （see [36]） of （0.0.1） as Let F0∞: D1,2（RN）→R given by denote the energy functional corresponding to the limiting problem （P0∞）.Our results mainly focus on existence and nonexistence of nontrivial solutions to problem （0.0.1） in a bounded domain in RN. The notion of solution we refer to is in the sense of critical points for the Euler functional related to the equation. The first result is the following global compactness theorem Theorem 0.0.1.If N≥3,λ∈R,{um} （?） H01（Ω） such that Jλ（um）≤c, DJλ（um）→0 strongly in H-1（Ω） as m→∞.Then （i） um can be decomposed as where wm→0 in H01（Ω） and u0 is a critical point of Jλ（u） and l∈N.For 1≤j≤l,Rmj→∞and{（0,zmj）} converge to （0,z0j）∈Ωas m→∞,v0j are solutions of （P0∞）.The usual proof of this theorem is based on rescaling arguments. Such meth-ods have been repeatedly used to extract convergent subsequences from families of solutions or minimizing sequences to nonlinear variational problems.One of the original results of this thesis the following existence theorems Theorem 0.0.2. Assume N≥4, t=1 andλ∈（0,λ1）. Then problem （0.0.1） has a positive solution in H01 （Ω）.Actuary, to prove the existence result in Theorem 0.0.2, we just need com-pactness for"low-energy" （P.S）-sequences. Such a property is proven in Lemma 4.2.2. Since the cases N=3 and N≥4 are quite different, we prove the following theorem by arguments similar to those used by Jannelli in [26]. Theorem 0.0.3. If N = 3 and t = 1, problem （0.0.1） has at least one solu-tion u∈H01 （Ω） whenλ*<λ<λ1, whereλ* is a suitable positive number.Based on the result of Theorem 0.0.2 and the linking （dual） theory respec-tively, we prove the sign-changing solutions of problem （0.0.1）.In a bounded domain the problem （0.0.1） does not have a solution in general due to the critical exponent. The nonexistence phenomenon is due to the lack of compactness of Jλ. We prove the following nonexistence theorem Theorem 0.0.4. Letλ≤0 andΩ（?）RN be an open set with smooth boundary and is strictly star-shaped with respect to some point（0,z0）.Supposc in addi-tion （?）Ωis orthogonal to the singular set, then problem （0.0.1）, has a nontrivial solution only ifΩ=RN.（Ⅱ） The general form of the parabolic equations with Cauchy boundary con-dition that we study is in Lq（RN）, q=N（γ-1）/2＞1,and fi（u）∈C1,（i=0,1…,n）, where C0, Ci are some positive constants andβ,γ>1 are fixed parameters.The various functions are essentially pure power functions. The function f0（u） behaves lik, |u|γand the other functions fi（u） all behave like |u|β.The result without the fi is well-known and first proved in [41]. Our treat-ment differs from [17], in that we use a different space.Here, the proof of existence result is based on a contraction mapping argu-ment in an appropriate space of functions which yields global in time solutions automatically.

论文目录

• Acknowledgements
• Abstract
• 1 Introduetion
• 2 Preliminaries
• 2.1 Introduction
• 2.2 Functional setting and weak solutions
• 2.3 Hardy-Sobolev inequality
• 2.4 Regularity results
• 3 A global compactness result
• 4 Existence and nonexistence results
• 4.1 Introduction
• 4.2 Existence of positive solutions
• 4.2.1 The case N≥4
• 4.2.2 The case N=3
• 4.3 Nonexistence of nontrivial solutions
• 5 Existence of sign-changing solutions
• 6 Global solutions of a semilinear parabolic equation
• References
• List of Published PaPer