| In this paper we consider an existence of a solution for a nonlinear nonmonotone wave equation in (0,{dollar}pi{dollar}) {dollar}times{dollar} {dollar}IR{dollar} and an existence of a positive solution for a non-positone Dirichlet problem in a bounded subset of {dollar}IRsp{lcub}rm n{rcub}{dollar}.; For a wave equation u{dollar}sb{lcub}rm tt{dollar} {dollar}-{dollar} u{dollar}sb{lcub}rm xx{rcub}{dollar} + {dollar}lambda{dollar}u = f(x,t) + g(u(x,t)), u(0,t) = u({dollar}pi{dollar},t) = 0, u(x,t) = u(x,t + 2{dollar}pi{dollar}), we show that if f = cq + r, where q and r {dollar}in{dollar} L{dollar}sp2{dollar} and {dollar}lambda in IR - {lcub}{dollar}j{dollar}sp2 -{dollar} k{dollar}sp2{dollar}:k = 1,2, dots, j = 0,1, dots{dollar}{rcub}{dollar}, then there exists c{dollar}sb0{dollar} such that if {dollar}vert{dollar}c{dollar}vert{dollar} {dollar}>{dollar} c{dollar}sb0{dollar} the wave equation has a weak solution. The solution is obtained using an iteration argument.; For the elliptic Dirichlet problem {dollar}-lambdaDelta{dollar}u = f(u) in {dollar}Omega{dollar}, u = 0 on {dollar}partialOmega{dollar}, where {dollar}Delta{dollar} is the Laplace operator, and {dollar}Omega{dollar} is a smooth bounded domain in {dollar}IRsp{lcub}rm n{rcub}{dollar}, we show that if {dollar}lambda{dollar} {dollar}>{dollar} 0, f is a nondecreasing function such that f(x) = f(0) {dollar}<{dollar} 0 for x {dollar}leq{dollar} 0, and f satisfies some additional conditions, then there exists {dollar}lambdasb0{dollar} {dollar}>{dollar} 0 such that if 0 {dollar}<{dollar} {dollar}lambda{dollar} {dollar}<{dollar} {dollar}lambdasb0{dollar} the Dirichlet problem has a positive solution. The solution is obtained using the so called "mountain pass lemma."... |
