The Application Of Moving Plane Method To The Study Of Quasilinear Elliptic Equations

This paper introduces the application of moving plane method to the of quasilinear elliptic equations. Firstly, we introduce the moving plane method in proving the symmetry property of the solution. Then we present the application of the moving plane method in the nonexistence and the property of the solution. Finally, we briefly introduce the improvement of the moving plane method which is about the partial application of the method of moving spheres. The structure of this dissertation is the following:Chapter one is the introduction of this paper. Firstly, we present the origin of the moving plane method, which was first created by Alexandrov, then improved by Serrin and has become an important tool to prove the symmetry of solutions of quasilinear elliptic equation. Then we introduce some specific application of the moving plane method, including the application in proving the symmetry property of the solution, the existence and nonexistence of the solution, and the local property of the solution. In studying the symmetry property of the solution, we introduce the results about Laplace equation, p-Laplace equation and different nonlinear elliptic equations. In studying the existence and nonexistence of the solution, we mainly introduce the application to quasilinear equation system. And in studying the local property of the solution, we introduce the application to singular perturbation of p-Laplace equation primarily. Finally, we present the deformation of the moving plane method, namely the moving sphere method and its applications.Chapter two presents the application of the moving spheres method in proving the symmetry of solutions of the quasilinear elliptic equation. And we give three specific examples, they respectively are the symmetry properties of smooth solutions of a kind of the Laplace equation in R², the symmetry properties of positive solutions of a kind of Laplace equation in R^N (N > 2) space and the symmetry properties of positive solutions of the p-Laplace equation in R^N (N≥2).The first problem studied is the following,The main result is:Theorem 1. The solution of equation (1) is point-symmetry about a point in R².The next problem studied iswhere TV≥2 and f is sublinear at the origin.We also assume that(F): Function f(t) is continuous in (0, +∞), and is locally Lipschitz continuous in (0, +∞). There exists aδ> 0, such that if t∈(0, 8), f(t) > 0, then f'(t) is nonincreasing and integral from n=0 toδ((dt)/(f(t)) < +∞.(A). Function a(r) is locally bounded and nonincreasing for r∈[0,∞).The result isTheorem 2. We assume that function f(t) satisfies (F), and a(|x|) satisfies (A), u∈C²(R^N) is a solution of equation(2) such that f(t)≥0.in [0,|u|_L∞(R^N)] then the solution u is radial symmetric about one point x₀ in R^N, and u_r < 0, r = |x - x₀| > 0.The 3rd problem isfor 1 < p < 2, withâ–³_pu = div(|Du|^p-2Du).To be more precise, we assume that(H1) f is locally Lipschitz continuous in (0.∞).(H2) There exists a s₀ > 0 such that / is nonincreasing on (0, s₀).Then we haveTheorem 3. Under the assumptions of (H1), (H2). if u∈C¹(R^N)∩W^1,p(R^N) is a solution of (3), then the solution u is radial symmetric about a point x₀ in R^N with u = u(r), r = |x - x₀|, u^r < 0. for r > 0.Chapter three introduces the applications of moving plane method to proving the existence of the solutions and the property of the solutions. The specific applications include the existence of the solution and the property of the solutions of a quasilinear elliptic equation on convex domain, and the problem of nonexistence of the positive solution of a kind of semilinear elliptic systems. The first example mainly presents the change of the local property of the solutions if the conditions change via moving plane method. The second example shows the nonexistence of the positive solution to some semilinear elliptic systems. To get the results, we firstly used the Kelvin transforms, then use the method of moving plane to prove the nonexistence of the positive solution to some semilinear ellipticsystems. The main results are as follows.The first problem considered is -∈△_pu(x) = f(u(x)), x∈Ω; u = 0, x∈(?)Ω, (4)with nonlinear termf(u) = u^q -u^p-1,u≥0,where∈> 0 is a small parameter, p > 2.â–³_pu = div(|Du|^p-2Du),Du =(D₁u,…,D_Nu), D_iu =(((?)u)/((?)x_i)),Ω(?)R^N (N≥2) is a bounded smooth domain. qsatisfies that p - 1 < q < ((Np)/(N-p))- 1. p < N; p - 1 < q <∞, p≥N.The result isTheorem 4. LetΩbe a convex domain, p > 2. Then, as∈→0, (4) has a minimum energic solution u_∈. which has only one local (thus global) maximum point x_∈∈Ω, dist(x_∈, (?)Ω)≥σ> 0; u_∈tends to 0 outside of any neighborhood of x_∈, and u_∈(x_∈)→ω(0). whereωis the only positive radial solution of the problemâ–³_pω(x) + f(ω(x)) = 0,x∈R^N;ω(x)→0,|x|→∞, whereω(0) >β, with positiveβsatisfying in integral n=0 toβf(s)ds = 0. Also. u_∈(∈^1/p ? +x_∈) converges uniformly to toω(·), in C_loc¹(Ω_∈) withΩ_∈= {y;∈^1/py + x_∈∈Ω}.The next problem studied is the following system of equations.where x∈R^N, N≥3. The main result isTheorem 5. (A) Forα,β,γ> 0. withα,β,γ≤((N+2)/(N-2)), but they are not equal ((N+2)/(N-2)) at the same time. Then (5) only has the trivial solution u = 0, v = 0,ω= 0.(B) Ifα=β=γ= ((N+2)/(N-2)), then u, v, w are spherically symmetric about a point in R^N (N≥2). Chapter four presents the application of the moving spheres method to quasilinear elliptic systems. As an example, we study the existence and nonexis-tence of positive solutions of a semilinear elliptic systems via the moving spheres method. In the process of the proving, we firstly use the Kelvin transforms, then moving the sphere to find the needed results step by stcp. The main results are as follows.The problem studied iswith x∈R^N (N≥3).The main results areTheorem 6. For every u, v∈[0,∞). f(u. v)=a₁u^k+a₂v^p, g(u, v)a₃u^q+ a₄v^t, with a_i>0, k, t≥0, p, q > 0. If max{k, p}≤(N+2)/(N-2) and max{q, t}≤(N+2)/(N-2), but k, p, q, t are not (N+2)/(N-2) at the same time. Then Problem (6) has no positive solution in C\²(R^N).Theorem 7. For any u, v∈[0,∞). f(u, v) = a₁u^k+a₂v^p, g(u,v)a₃u^q+ a₄v^t, with a_i>0. If k = p = q = t = (N+2)/(N-2), then Problem, (6) has only such a positive solution in C²(R^N) that for any constant d > 0, (x)∈R^N,where c₁,c₂>0 and satisfy...