Nonlinear Perturbations at Resonance

Abstract

This chapter describes the nonlinear perturbations at resonance. It presents nonlinear differential equations of the type Lu + Xu = Nu, where L is a linear self-adjoint differential operator over a real Hubert space S with preassigned linear homogeneous boundary conditions, λ an eigenvalue of the associated linear problem Lu + Xu = 0, and N a nonlinear operator over S. It also presents conditions for the existence of solutions of the abstract semilinear problem Lu + Xu = Nu under suitable hypotheses on N. The chapter also focuses on nonlinear problems where the nonlinearity is not defined over the entire Hubert space S. It describes differential operators L that together with the boundary conditions admit of a natural decomposition in the form TT*. The chapter also presents an assumption in which L is a linear self-adjoint differential operator with preassigned homogeneous boundary conditions over a smooth bounded domain in Rn and S is the real Hilbert space L2(Ω) with norm and inner product. It also presents another assumption in which L is such that the associated eigenvalue problem Lu + Xu = 0 has a countable system of real eigenvalues.