Global solution curves in harmonic parameters, and multiplicity of solutions

Abstract

For the semilinear Dirichlet problem Δ u + g ( u ) = f ( x ) for x ∈ Ω , u = 0 on ∂ Ω decompose f ( x ) = μ 1 φ 1 + e ( x ) , where φ 1 is the principal eigenfunction of the Laplacian with zero boundary conditions, and e ( x ) ⊥ φ 1 in L 2 ( Ω ) , and similarly write u ( x ) = ξ 1 φ i + U ( x ) , with U ⊥ φ 1 in L 2 ( Ω ) . We study properties of the solution curve ( u ( x ) , μ 1 ) ( ξ 1 ) , and in particular its section μ 1 = μ 1 ( ξ 1 ) , which governs the multiplicity of solutions. We consider both general nonlinearities, and some important classes of equations, and obtain detailed description of solution curves under the assumption g ′ ( u ) < λ 2 . We obtain particularly detailed results in case of one dimension. This approach is well suited for numerical computations, which we perform to illustrate our results.