The multiplicity of solutions and geometry of a nonlinear elliptic equation

Let Ω be a bounded domain in ${\displaystyle {ℝ}^n}$ with smooth boundary ∂Ω and let L denote a second order linear elliptic differential operator and a mapping from ${\displaystyle L^2(\Omega )}$ into itself with compact inverse, with eigenvalues ${\displaystyle -{\lambda }_i}$, each repeated according to its multiplicity, 0 < λ_{1} < λ_{2} < λ_{3} ≤ ... ≤ λ_{i} ≤ ... → ∞. We consider a semilinear elliptic Dirichlet problem ${\displaystyle Lu+b{u}^{\pm }a{u}^{\equiv }f\left(x\right)}$ in Ω, u=0 on ∂ Ω. We assume that ${\displaystyle a<{\lambda }_1}$, ${\displaystyle {\lambda }_2<b<{\lambda }_3}$ and f is generated by ${\displaystyle {\varphi }_1}$ and ${\displaystyle {\varphi }_2}$. We show a relation between the multiplicity of solutions and source terms in the equation.