Existence And Multiplicity To Solutions For A Kind Of P-Kirchhoff Type Equations

Nonlinear problems are the general problems in the natural science and engineering ares. Because they can well explain the various natural phenomenon, a large number of domestic and foreign researchers have paid attention to the problems for a long time. Moreover for p-Kirchhoff equations as the most fundamental nonlinear equations, its origin is described in micro vibration on elastic string. In the process of researching the p-Kirchhoff equation with initial boundary value problem, the existence, multiplicity and nonexistence of solutions have also been a hot spot for the foreign and domestic scholars.In this paper, we use variational methods, such as the nehari manifold, the fountain theorem, the clarke theorem, to discuss the existence and multiplicity of solutions for a p-Kirchhoff equation.The thesis consists of three sections.Chapter 1 is the preface.In Chapter 2, we consider the multiplicity of positive solutions to the following p-Kirchhoff equation: where Ω is a bounded smooth domain in RN, △pu= div(|▽u|p-2▽u) is the p-Laplace oper-ator with p ∈ (1,N) and γ> 0 is a parameter. For M, f and g, we assume:(H1) for k≥ 0, M(t)=tk,t∈E [0, ∞);(H2) 1<q<m<r<p*, where m= p(k+1) and p*= Np/(N - p);(H3)f, g ∈ C(Ω) and f+g+≠ 0, where f+= max{f,0}.We obtain the following result via the decomposition of the Nehari manifold.Theorem 2.1.1 Suppose that (H1), (H2) and (H3) hold. Then there exists λ*> 0 such that for λ ∈ (0, λ), the equation (2.1) has at least two positive solutions.In Chapter 3, we consider the existence of infinitely many solutions to the following p-Kirchhoff equation: where Ω. is a bounded smooth domain in RN, △pu= div(|▽u|p-2▽u) is the p-Laplace oper-ator with p ∈ (1, N). For M and h, we assume:(A1) for k≥ 0, M(t)= tk,t∈ [0, ∞);(A2) h(x,z)= λf(x)｜z｜q-2z+g(x)｜z｜r-2z, where λ> 0,1< q< m< r< p*,m= p(k+1),p*= Np/(N-p),f∈ L∞(Ω),g∈= C(Ω),g> m0> 0;(A3) h(x,-z)=-h(x, z), (x,z) ∈ Ω ×R;(A4) 1<p≤r<m<p*, there exists d1,d2> 0 such thatFor the equation, we first use two important inequalities in RN to prove that the energy functional satisfy (PS)c condition, and then use Fountain theorem and Clarke theorem to obtain the following two main results.Theorem 3.1.1 Suppose that (A{) and (A2) hold. Then the equation (3.1) has a sequence of positive-energy solutions.Theorem 3.1.2 Suppose that (A1), (A3) and (A4) hold. Then the equation (3.1) has a sequence of negative-energy solutions.