On positive solutions of semilinear equation delta u + lambda u - hu(p) = 0 on compact manifolds

In this thesis, we study the existence of positive solutions of semi-linear equation {dollar}Delta u + lambda u - husp{lcub}p{rcub} = 0{dollar} on compact Riemannian manifolds as well as on bounded smooth domains in {dollar}Rsp{lcub}n{rcub}{dollar} with homogeneous Dirichlet or Neumann boundary conditions.; Let (M,g) be a compact Riemannian manifold without boundary of dimension {dollar}geq{dollar}3, g be a metric on M. Consider the following semilinear problem:(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}eqalign{lcub}Delta u + lambda u - husp{lcub}p{rcub}&= 0quad {lcub}rm on{rcub} M,crcr u&> 0quad {lcub}rm on{rcub} M,cr{rcub}leqno(1.1){dollar}{dollar}(TABLE/EQUATION ENDS)where {dollar}lambda > 0, p > 1{dollar} are constants and {dollar}h(x) geq 0{dollar} is a {dollar}Csp{lcub}1{rcub}{dollar} {dollar}-{dollar} function on M. Let(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}eqalign{lcub}Msb+&= {lcub}x in M mid h(x) > 0{rcub}quadrm andcr Msb0&= Mbar Msb+.cr{rcub}{dollar}{dollar}(TABLE/EQUATION ENDS); Our main result may be stated as follows.; Theorem 1. Assume that {dollar}h geq 0(notequiv 0){dollar} is a smooth function on M. (i) If {dollar}Msb0 = emptyset{dollar}, then for every {dollar}lambda > 0{dollar}, there exists a unique solution {dollar}u(lambda){dollar} of problem (1). (ii) If {dollar}Msb{lcub}0{rcub} not= emptyset{dollar}, then there is a positive {dollar}barlambda in (0,infty){dollar} such that for any {dollar}lambda < barlambda{dollar} there exists a unique solution {dollar}u(lambda){dollar} of (1), and for {dollar}lambda geq barlambda{dollar} there is no solution of (1). Moreover{dollar}{dollar}{lcub}limlimitssb{lcub}lambda to barlambda{rcub}{rcub}Vert u(lambda) Vertsb{lcub}Lsp2(M){rcub} = infty.{dollar}{dollar}; Furthermore, suppose {dollar}lambdasb{lcub}1{rcub} > 0{dollar} is the first eigenvalue of {dollar}Delta{dollar} on {dollar}Msb0{dollar} with zero Dirichlet boundary condition. i.e.(UNFORMATTED TABLE OR EQUATION FOLLOWS){dollar}{dollar}eqalign{lcub}Deltavarphi + lambdasb1varphi&= 0quad {lcub}rm in{rcub} Msb0,crvarphi&> 0quad {lcub}rm on{rcub} Msb0,crvarphi& = 0quad {lcub}rm on{rcub} partial Msb0,crcrbarlambda&= lambdasb1.cr{rcub}{dollar}{dollar}(TABLE/EQUATION ENDS)...