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The multiplicity of solutions and geometry of a nonlinear elliptic equation

Q. ChoiSungki ChunTacksun Jung — 1996

Studia Mathematica

Let Ω be a bounded domain in n with smooth boundary ∂Ω and let L denote a second order linear elliptic differential operator and a mapping from L 2 ( Ω ) into itself with compact inverse, with eigenvalues - λ i , each repeated according to its multiplicity, 0 < λ1 < λ2 < λ3 ≤ ... ≤ λi ≤ ... → ∞. We consider a semilinear elliptic Dirichlet problem L u + b u + - a u - = f ( x ) in Ω, u=0 on ∂ Ω. We assume that a < λ 1 , λ 2 < b < λ 3 and f is generated by ϕ 1 and ϕ 2 . We show a relation between the multiplicity of solutions and source terms in the equation....

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