Existence Of Solutions For Nonlinear Elliptic Equations On Compact Riemannian Manifolds

This thesis is concerned with the existence of multiple solutions for semilinear elliptic equa-tions, and the existence of positive solutions for quasilinear elliptic equations on compact Rieman-nian manifolds.In chapter one, we introduce some background and results for semilinear elliptic equationsand quasilinear elliptic equations. We state our main results.In chapter two, we give some basic knowledge, including the definition and properties ofcategory, and some basic knowledge about manifolds.In chapter three, we consider the existence of multiple solutions for the following semilinearelliptic equation(?)where M is a compact, connected, orientable, boundaryless Riemannian manifold of class C∞,dim M = n≥3, and g denotes the metric tensor, ?g is the Beltrami-Laplace operator, 2 < p <2? = n2?n2. Assume V (x) and K(x) are positive continuous functions, we show that the shape ofV (x) and K(x) a?ects the number of solutions for equation (3) , and prove problem (3) possessesmultiple solutions.In chapter four, we consider the existence of positive solutions for the following quasilinearelliptic equation(?)where M is a compact, connected, orientable, boundaryless Riemannian manifold of class C∞,dim M = n≥3, and g denotes the metric tensor. (?p)gu = divg(| gu|p?2 gu), f∈C1(M×R×Rn). Since problem (4) does not have a variational structure in general, we can not get theexistence of solutions by usual critical point theory. In order to overcome this di?culty, we combinethe blow-up method with a Liouville type theorem to obtain a priori estimates of the solutionsfor problem (4), then we use topological method to prove that problem (4) possesses a positivesolution u∈C1,α(M), where 0 <α< 1.