| Nonlinear elliptic problem has been receiving extensive concern for a long time. It is because a lot of mathematical and physical problem, such as nonlinear diffusion theory which resource is nonlinear resource, the gas burning theory of thermodynamics, quantum field theory, statistical mechanics and the gravity balance theory of galaxy have stemmed from it. Moreover, some branches of mathematics, for example, Yamabe problem and equal-length inequality of geometry, Hardy-Littlewood-Sobolev inequality of Harmony Analysis, the existence of non-minimal solution of Yang-Mills function and population dynamics have closely connected with nonlinear elliptic equation. We study the existence of nonlinear elliptic equation by variational method and critical point theory, combined with sub- and sup-solution method. The thesis is composed of five chapters.Chapter 1 is the introduction of this paper, which introduces the problem and the background of the problem we studied.In the second chapter, we consider the existence of solutions for problemand p-Laplace equation with local p-convex and p-concave nonlinearitieswhere A is a positive parameter, is an bounded domain insatisfy some assumptions.In [1], A. Ambrosetti, H. Brezis, and G. Cerami studied the special cases of equation (1.1), namely, the special cases of a(x) = b(x) = 1. In [2], the authors studied the special cases of b(x) = 1, and the result is that there exists , such that equation (1.1) has one solution when equation (1.1) has no solution when . By means of sub- and sup-solution method and variational method, we obtain the main results as follows:Theorem 2.1.1 Let a(x),b(x) satisfy the assumption e different positive constants,1, tnen tnere exists a con-1) possesses a weak positive solutionTheorem 2.1.2 Let a(x), b(x) satisfy the assumption (H2) :a(x), b(x) , where c0 is a positive constant, and 0 < p < then there exists a constant A 6 R, A >0, such that(i) (1.2) possesses a minimal solution u such that I(u) < 0 for all (0, A), and A is increasing with respect to A if A e (0, A);(ii) (1.2) possesses at least one weak solution (iii) there are no solutions of (1.2) for A > A.Enlightened by [11], we study the p-Laplace equation (1.2), via the theory of critical point, we prove that equation (1.2) at least has two solutions, which generalizes the results of [1-3,11].The third chapter is concerned with the existence of solution for the singular elliptic problemwhere are positive parameters, is a bounded domain with smooth boundary,a(x),b(x) satisfy the assumption where co,Ciare two different positive constants.When a(x) = - 1, equation (2.1) are studied respectively in [18] and [19] (especially, [19] required that b(x) = 1.) Our goal is study equation (2.1) when 1 by means of sub- and supersolution method and variational method. We obtain the following results:Theorem 3.1.1 Let a(x),b(x) satisfy the assumption (H1), and c1> . Then there exists > 0, such that equation (2.1) has unique positive solution Singular elliptic equationis considered in [14], where p(x) > 0, possesses singularity at 0, and requires that f'(s) < 0, s >0, namely, f(s) is decreasing with respect to s.We devote to the singular elliptic equation, which the nonlinear function f(x, s). does not decreasing with respect to s. Via variational method, we get the existence and uniqueness of positive solution for equation (2.2).In the fourth chapter, we study the existence of solution for elliptic problems with critical exponents and Hardy potentialswhere is critical Sobolev exponents, is a smooth and bounded domain which contains the origin. Under appropriate assumption on f(x,u), via Hardy inequality and variational method, we prove that , there exists such that , problem (3.1) has a solution. Our results contain the results of [24,25], and generalize the results of [27-29]. Moveover, the method we used is different from that of [27-29].The last chapter is concerned with the existence of positive solutions for the p(x)-Laplace problemwhere...
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