Existence And Multiplicity Of Solutions For Some Boundary Value Problems Of Differential Equations

Shortly after the birth of calculus, mathematicians and physicists have started to use differential equations to describe a variety of problems which arise from mechanics and physics. Since then, the differential equations has always played an important role in mathematics. In recent years, since the application areas of differential equations were becoming more and more comprehensive, the research of differential equations aroused wide interest in mathematics, physics, chemistry, engineering, biology, eco-nomics, and many other sciences. At the same time, many applied problems come from interrelated subjects, which give a challenge to the research of general theory for differential equations. many people were devoted to the developments of the general theory for differential equations. However, the people find that it is very difficult to deal with differential equations, even in the case of linear equations, sometimes it is very complex. In the case of nonlinear equation, we have to establish different methods for different equations. In this process, many work were completed, which significantly advanced the developments of differential equations.From the seventies of last century so far, using variational method to study the existence and multiplicity for elliptic boundary value problems has been widespread concerned. The idea of this approach is to translates the solutions of equations to the critical point of corresponding energy functional. Many scholars applied Minimax Principle to obtain the existence and multiplicity in the framework of variational.Coming from physics and engineering and having profound backgrounds, many models of elliptic equations aroused wide interest in recent years. Among these prob-lems, the fourth-order elliptic equation and the p-Laplacian equation received extensive attention.Fourth-order elliptic equation with biharmonic operator have a wide applications in physics, engineering, and many other science. We give a application here which is about the problem of oscillation in a suspension bridge. In suspension bridge, the stays connecting the cable to the desk of the bridge are fundamentally nonlinear, in that if you pull on a rope, it resists, whereas if you push, it does not. We shall treat the stays as one-sided springs, obeying Hooke's law, with a restoring force proportional to the displacement from the unstretched state if stretched, and with no restoring force if compressed. The roadbed will be treated as a one-dimensional vibrating beam This gives rise to the following equation where the suspension bridge is seen as a beam of length L, the downward deflection is measured by u(x,t), the stays are seen as nonlinear springs with springs constent k, the weight per unit length of the bridge W(x) pushing it down, andεf(x, t) is the external forcing term.If we takeεf(x,t)=0 and look for the travelling wave solution with the form then y satisfies Generalize the equation to N-dimension, we get In addition, Similar results can be applied to the naval architecture and oceanics to Compute the hydroelastic response of floating structure in waves.The N-dimensional equations also arise in such as communication satellites, space shuttles, and space stations, which are equipped with large antennas mounted on long flexible masts (beams), see [1,2].p-Laplacian has become a hot topics in the nonlinearity science. Since it has very abundant applications, p-Laplacian has been widespread concern. In [31], Drabek called the p-Laplacican is the mascot of nonlinear analysis.In [52], it is pointed out p-Laplacian operator appears in rheology, glacelogy, ra-diation of heat, and plastic moulding. Some recent advances indicate that even the Brownian motion has its counterpart and a mathematical game " Tug of War" leads to the case p=∞.Let us present mathematical model of the behavior of compressible fluid in a homogeneous isotropic rigid porous medium. This model is come from [31].Letρ(x, t) denote the density,φbe a volumetric moisture content and (?)(x, t) be a seepage velocity. Then the continuity equation reads as follows: In the laminar regime through the porous medium the momentum velocity pV and the pressure P= P(x, t) are connected by the Darcy law In turbulent regimes, however, the flow rate is different and several authors proposed a nonlinear relation. Namely, the nonlinear Darcy law of the following form is often considered whereα＞1 is a suitable real constant. Taking into account the equation of state for the polytropic gas with some constant of proportionality c＞0, we get After the change of variables and notations this equation becomes where p＞1. the operator is called p-Laplacian.In this paper we focus on the stationary case and discuss the equations of the typeThe main content of this paper is divided into two aspects. On one hand, we study fourth-order elliptic equations with Navier boundary condition. On the other hand, we study some p-Laplacian problems, we first consider some p-Laplacian equations with Dirichlet boundary value condition, then we study some 1-dimensional p-Laplacian equations with periodic boundary value condition. we use variational method, upper-lower solution method, critical point theory and many other non-linear analysis meth-ods to study two types of equations. some existence and multiplicity results are given.In Chapter 3, we first study biharmonic fourth-order elliptic equations with Navier-type boundary value condition where△2 is the biharmonic operator, c is a constant, c＜λ1, whereλ1 is the first eigenvalue of (-△, H01(Ω)).Ω(?)RN (N＞2) is a bounded smooth domain and f(x, s) is a continuous function on (?)×R. When the nonlinearity is superlinear, we obtain the multiplicity of solutions of the problems, without assuming the famous Ambrosetti-Rabinowitz condition. Then we will study the combined nonlinearity problems whereλ≥0 is a parameter, h∈L∞(Ω), h(x)≥0, h(x)(?)0, and f(x, s) is a continuous function on (?)×R.We discuss how different nonlinearities and differentλeffect the existence and multiplicity of solutions. Assuming nonlinearity is asymptotically linear at∞, we give the existence theorem about five solutions.We first deals with the caseλ= 0. Then we consider the caseλ＞0. Forλ＞0 small enough, we give the existence of five solutions:Two of the solutions are of mountain pass type, two extra solutions are found as local minima of the energy functional, and there exists other different solution, which can change sign. Our method to obtain the fifth solution follows the ideas developed in [4] for Laplacian operator. However, while in [4] the authors assumed that nonlinearity is super linear at∞, we discuss here a local asymptotically linear problem.The result is local, sinceλis small enough. A global result of Ambrosetti-Brezis-Cerami type also be considered.Besides, when h(x) is negative, we discuss the case of h(x)≡-1. consider the following problem whereλ＞0,1＜q＜2.Chapter 4 is devoted to the p-Laplacian equations with Dirichlet boundary value condition. We first consider the super p-linear problem where△p=div(|(?)u|p-2(?)u),1＜p＜+∞,Ω(?)RN, f(x, s) is continuous on (?)×R. We discuss how combined nonlinearities effect the existence and multiplicity of solutions. Consider where△p=div(|(?)u|p-2(?)u),1＜p＜+∞,Ω(?)RN, fλ(x, u) is continuous on (?)×R.Assuming nonlinearity is super p-linear at∞, we give the result about the existence and multiplicity of solutions.In Chapter 5, we study some p-Laplacian equations with periodic boundary value conditions. By using continuation theorem, the existence of periodic solutions is ob-tained. We first consider Duffing type p-Laplace equations where p＞1 andφp:Rn→Rn is given byφp(s)=|s|p-2s for s≠0 andφp(0)=0, F:Rn→R is a C1 function, g:R×Rn→Rn is continuous with g(t,·)= g(t+T,·), and e:R→Rn is continuous with e(t)= e(t+T).Then, we consider two class of equations with periodic boundary value condition, which are generalized from p-Laplacian equations. One of them isφ-Laplacian type equations where the operatorφis defined as p＞2, F:RN→R is a C1 function, g:R×RN→RN continuous, g(t,·)= g(t+T,·), e:R→RN continuous, such that e(t)= e(t+T).Besides, we discuss a class of generalize equation, which can describe forced pen-dulum with relativistic effects: where the operatorφis defined by where p＞1. When p= 2, this model described forced pendulum with relativistic effects.We give the results about the existence of periodic solutions. Our approach is continuation theorem and degree theory.