| This paper mainly studies the existence of multiple solutions for three types of nonlinear elliptic equations. The thesis consists of four chapters.In chapter one, we introduce some research background and our new results.In chapter two, we study the existence of multi-bump solutions for the semilinear Schr(o|¨)dinger equationwhere N≥1,2 < p < 2*,2* is the critical Sobolev exponent defined by 2* =(?) if N≥3 and 2* =∞if N = 1 or N = 2, andε> 0 is a parameter. The function a is assumed to satisfy the following conditions: a∈C(RN),a(x) > 0 in RN,a(x) =o(1) and ln(a(x)) =o(|x|) as |x|→∞.For any positive integer n, we prove that there existsε(n) > 0 such that, for 0 <ε<ε(n), the equation has an n-bump positive solution. Therefore, the equation has more and more multi-bump positive solutions asε→0.In chapter three, letε> 0 be a small parameter. We study existence of multiple multi-bump positive solutions for the semilinear Schr(o|¨)dinger equationwhere N≥1,2 *,a∈C(RN),a(x) > 0 for x∈RN,and lim|x|→∞ a(x) = 0.We also study existence of multiple multi-tower positive solutions for the prescribed scalar curvature equationwhere N≥3, K∈C([0,∞)), K(r) > 0 for r > 0, limr→0 K(r) = 0, and limr→∞ K(r) = 0.We also consider the more general equationand another related equation iswhere N≥3 and q > 1.We would like to emphasize that we do not impose any upper bound on q.In chapter four, the goal of this paper is to establish the existence of multiplicity of multi-bubble positive solutions for the following critical elliptic equations of Caffarelli-Kohn-Nirenberg type Where N≥3, a,b,p,v satisfy some conditions, K∈C(RN),K(x) > 0 for x∈RN, lim|x|→∞ K(x) = 0,lim|x|→∞K(x) = 0 andε> 0 is a small parameter.
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