Class Of Nonlinear Elliptic Partial Differential Equations Are Solutions

All sorts of nonlinear problems have resulted from electricity, hydrodynamics, heat, magnetism, optics, elasticity, chemistry, biology, medicine, economics, engineering, other applied disciplines and so on. During the development of solving such problems, since the 18th century, nonlinear partial differential equations has been one of the most important and active research fields in Modern mathematics. Variational theory (see [1] [40] [41] [42] [43] [44]) is one of the most important methods and effective of partial differential equations. In the early 1960's, variational theory had already won enormous developments especially mountain pass lemma and Morse theory were put into the framework of differential topology on infinite dimensional manifolds by R. Palais and S. Smale [P-S]. Therefore, it has driven the enormous development of partial differential equation. Also it provides a much effect theoretical tool for solving many nonlinear problems in the fields of the science and technology.We mainly consider nonlinear elliptic boundary value problem(BVP) in this thesis. This kind of nonlinear elliptic boundary value problem(BVP) has all been concerns of the mathematics worker all the time. In recent years, semilinear elliptic boundary value problem with Sublinear (Singular) and concave nonlinearities have obtained extensive research [3]-[38]. This kind of question stems from applied mathematics and physics Extremely each dynamic branch of population. Therefore, it is one of the most active fields in mathematics at present. (Such as seeing [2]-[25]And subsequent references). A lot of scholars have already done a large amount of researchabout semilinear elliptic boundary value problem with Sublinear (Singular) and Superlinear nonlinearities. At the same time, semilinear elliptic boundary value problem with Singular and Superlinear nonlinearities has relatively little research at present. Ambrosetti. A., Brezis,H. and Cerami. G. had researched about the kind of problem. Tan zhong and Yao zhengan had research the semilinear elliptic boundary value problem with convex and concave nonlinearities. They studied the existence of solutions of the kind boundary value problem(BVP) by means of lower and supper solutions and variational method. In this test, we consider the more general kind of the semilinear elliptic boundary value problem with convex and concave nonlinearitiesovally. we obtain the two solve of the equation by means of lower and'supper solutions and variational method.Since later stage of the seventies, A lot of scholars have already done a large amount of research about semilinear elliptic boundary value problem with Singular nonlinearities because of the need of application of science.However, because of the Singular nonlinearities, a large amount of work was leaning towards nonlinear partial differential equations around bound or nonbound with negative exponent until 1996. Sun yijing and Wu shaoping, Long yingmin[34] and yanghaitao had considered the existence of solutions of the Singular nonlinearities elliptic boundary value problem(BVP) with negative exponent and positive exponent by means of lower and supper solutions and variational method. In this test, we consider the more general kind of the the Singular nonlinearities elliptic boundary value problem(BVP) with negative exponent and positive exponent, we obtain the the more general consequence of the kind of equation by meansof Ekeland principle and lower and supper solutions method. This thesis is divided into three chapter. In the first chapter and the second chapter, we study the existence of solutions of nonlinear elliptic boundary value problem (BVP). In the third chapter, we study the existence of solutions of singular nonlinear elliptic boundary value problem(BVP).In the first chapter, we consider the existence of positive solutions and multiple solutions of the semilinear elliptic boundary value problem withconvex and concave nonlinearities.where is a real parameter and satisfying some appropriate conditions.In [5], Tan...