Positive Solutions To Some Operator Equations And Its Applications

Nonlinear problems usually arise in mathematics, physics and other natural sciences. They can well describe the various natural phenomenon in nature. Then it has been received extensive attention of the scholars. Kirchhoff equations is a kind of very important nonlinear differential equations. In addition, the existence and multiplicity of solutions for nonlinear operator equations is a very interesting topic. In this paper, we study the existence and multiplicity for the equation with a homogeneous operator by using the fixed point index theory. As an application, by using our main theorem we can obtain a symmetrical positive solution to the one dimensional Kirchhoff equation. Moreover, by using the Leggett-Williams three solutions theorem skillfully, we obtain two positive solutions to Kirchhoff equations under some appropriate conditions. As a direct corollary, we also obtain the same result to elliptic equations. In addition, we discover and prove a new inequality in the article. In the end of the paper, we present an example which makes the elliptic equation has infinite many positive radial solutions and give the approximate images of the nonlinear term.The thesis consists of four chapters.Chapter 1 is the preface.In Chapter 2, we introduce some basic concepts and theorems.In Chapter 3, we consider the following equation with an a homogeneous operator φ(u)u=λaAu+Bu+U0, (3.1.1) where λ≥0 is a parameter, A:Eâ†'E is a positive linear completely continuous operator, B:Pâ†'P is a completely continuous a homogeneous operator,φ(u)=a+b‖u‖β and u0 is a given element of P.By using the fixed point index theory, we obtain the following two main theorems to the operator equation (3.1.1).Theorem 3.2.2 Let α-γβ>1 where γ=sgnb. Assume that A:Eâ†'E is a positive linear completely continuous operator, B:Pâ†'P is an a homogeneous completely continuous operator satisfying If λa‖A‖a+(α-β-1)Mβ-1((bβ)/(M(α-1)))(α-1)/α-β-1), where M = supu∈p,‖u‖=1 ‖Bu‖> 0, then, for any u0 ∈P with the equation (3.1.1) has at least one positive solution when u0 = 0, and has at least two positive solutions when u0≠ 0.Theorem 3.2.5 Let a>0. Assume that A : Eâ†'E is a positive linear completely continuous operator and A‖A‖ < 1. Suppose that B : Pâ†' P is an α homogeneous com-pletely continuous operator and (3.2.1) holds. If α-γβ> 1, or α =β + 1 but 0 < b < μ, then, for any u0 € P with ‖u0‖<(α-1)M((a(1-λ‖A‖))/(αM)/(α-1), where M = supu∈P,‖u‖=1‖Bu‖ > 0, the equation (3.1.1) has at least one positive solution when u0 = 0, and has at least two positive solutions when U0≠0.As an application, we consider the existence of symmetrical positive solutions to the following one dimensional Kirchhoff problem where the constants a, b≥0 with a + b > 0 and f∈ C(R+,R+),R+ := [0,∞).Let the function f satisfies(f1) the following limits exist(f2) f is increasing and there is c>0 such that f(x)≥kc4 for x ∈6 |c, 8c], where 3kc3=28a+216bc2.By using the Theorem 3.2.2, we obtain the multiple solutions result to the one dimen-sional Kirchhoff problem (3.1.4).Theorem 3.3.5 Assume that f satisfies (f1) and (f2). Then the problem (3.1.4) has at least two positive solutions.This chapter has been published in the journal of mathematical analysis and applica-tions, see J. Math. Anal. Appl. 422 (2015) 544-558.In Chapter 4, we study the multiplicity of positive radial solutions to the following Kirchhoff equation where B1 = {x ∈RN :|x| <1},(?)B1 = {x ∈RN :|x|=1}, the constants a > 0, b≥0 and f∈C(R+,R+).Taking a = 1 and b = 0 in the Kirchhoff equation (4.1.1), we get the elliptic equationIn order to make the argument simple, we first consider the following operator equation [a+bΦ(x)]x = Ax, (4.1.3) where A : P â†'P is a completely continuous operator and a > 0, b ≥0 are two constants. The functional φ : Pâ†'R+ is continuous and satisfies φ(x)≤k(‖x‖) for all x ∈ P, where the function k : R+â†'R+ is continuous and nondecreasing. By using the Leggett-Williams three solutions theorem, we obtain a new simple three solutions theorem to the equation (4.1.3) under some appropriate conditions.Theorem 4.2.1 Let a≥1. Assume that A : Pâ†'P is a completely continuous operator and α : Pâ†' R+ is a continuous concave functional which satisfies α(x)≤‖x‖ for all x E P. Suppose that there exist 0 < d0 <a0 < c0 such that(â…°) ‖Ax‖≤ ac0 for all x ∈ Pc0;(â…±) ‖Ax‖< ad0 for all x ∈ Pd0;(â…²) {x∈ P(α,a0,c0) : a(x) > a0}≠(?), and α{Ax) > a0[a + bk(c0)] for all x ∈ P(α,a0,c0). Then the equation (4.1.3) has at least three solutions x1,x2 and x3 in Pc0, which satisfy that ‖x1‖ < d0,‖x2‖> d0 and α(x2) < a0, a0 < α(x3).Theorem 4.2.3 Let a > 0. Assume that A : Pâ†'P is a completely continuous operator and α : Pâ†' R+ is a continuous concave functional which satisfies α(x)≤‖x‖ for all x ∈ P, and α(tx) = tα(x) for all x ∈ P and t ∈ R+. Suppose that there exist 0 < d0 <a0 <c0 such that(â…°) ‖Ax‖≤ac0 for all x ∈Pc0;(â…±)‖Ax‖< ad0 for all x ∈Pd0;(â…²) {x ∈ P(α,a0, c0) : α(x) a0} ≠(?), and α(Ax) > a0[a +bk(c0)] for all x ∈ P(α,a0,c0). Then the equation (4.1.3) has at least three fixed points x1,x2 and x3 in Pc0 which satisfy that ||x1|| <d0, ||x2|| > do and α{x2) <a0,a0 <α(x3).By using the new three solutions theorem, we obtain two positive radial solutions to the Kirchhoff equation (4.1.1) ant the elliptic equation (4.1.2) under adding some appropriate conditions to nonlinearity /.Theorem 4.3.1 Let N≠2 and a > 0. Assume that there exist 0 < d0 < a0 < C0 with a0(a + bNωNc02)≤σ Naco such that the nonlinear term / satisfies(f1) f{x) ≤N ac0 for all x ∈ [0, c0];(f2) f(x) < N ad0 for all x ∈ [0, d0];(f3) f(x)≥ a0(a + bNωNc02)/σ for all x ∈ [a0,c0]. Then the equation (4.1.1) has at least three nonnegative radial solutions, where the constant σ=(?)0εNG(εN,s)ds=1/(N2)(2/N)2/(N-2)=((εn)/N/)2.Theorem 4.3.4 Let N = 2 and a > 0. Assume that there exist 0 < d0 <a0 <c0 with 2ea0 (a + 2Ï€bc02)≤ac0 such that the nonlinear term / satisfies(f1) f{x)≤ 2ac0 for all x ∈ [0,c0];(f2) f(x) < 2ad0 for all x ∈ [0,d0];(f3) f(x) > 4ca0 (a + 2Ï€bc02) x ∈ [a0, c0]. Then the equation (4.1.1) has at least three nonnegative radial solutions.Taking a =1 and b = 0 in Theorem 4.3.1 and Theorem 4.3.4, then we obtain two positive radial solutions to the elliptic equation (4.1.2).