The Existence Of Multiple Solutions For Some Weighted Semilinear Elliptic Equations

In this thesis, we are devoted to study of the existence of a nontrivial solution for a semilinear problem: where is an unbounded domain with smooth boundary, 2*＝2N/(N-2) (N＞3), satisfies the following conditions:(A1) k(x)＝k+(x)-k-(x), k±(x)＝max｛±k(x),0}≠0(A2)0＜r＜R,And a class of weighted semilinear elliptic equations:The thesis consists of five chapters.In chapter one, we introduce some results on above mentioned weighted semi-linear problcms.In chapter two, we introduce some basic knowledge of Sobolev spaces and some basic lemmas. In addition, we give some notations.In chapter three, we adopt the linking theorem and delicate estimates for prob-lem (1.1) to obtain a nontrivial solution.In chapter four, under the condition of 0≤μ≤μ-(1 + a)2, we obtain two positive solutions of problem (1.2) by use of the (PS) sequence, the translation method and the Mountain Pass Lemma. In chapter five, under the condition of 0≤μ＜μ-(2 + 2a)2, (N +μ-μ)/(μ+μ-μ)＜q＜2, we obtain a sign changing solution of prob-lem (1.2). To attain this goal, we separate the Nehari-type set of the considered problem, the study minimization problems on its proper subset; And we obtain infinitely many critical values with negative energy by use of the Dual Fountain Theorem.