We study the existence of sign changing solutions to following problem(0.1){Δu+|u|p−1u=0inΩε;u=0on∂Ωε, where p=n+2n−2 is critical Sobolev exponent and Ωε a bounded smooth domain in Rn, n≥3, form Ωε=Ω\B(0,ε). Here Ω containing origin 0 B(0,ε) denotes ball centered at with radius ε>0. construct new type sign-changing high energy problem (0.1), when parameter ε small enough.

We describe the qualitative properties of the solutions of the second order scalar equation ẍ + q(t)g(x) = 0, where q is a changing sign function, and consider the problem of existence and multiplicity of solutions which satisfy various different boundary conditions. In particular we outline some difficulties which arise in the use of the shooting approach.

This article concerns the third-order three-point boundary-value problem u′′′(t) = f(t, u(t)), t ∈ [0, 1], u′(0) = u(1) = u′′(η) = 0. Although the corresponding Green’s function is sign-changing, we still obtain the existence of at least 2m−1 positive solutions for arbitrary positive integer m under suitable conditions on f .