We consider the infinite multiplicity of entire solutions for the elliptic equation Δu + K(x)eu + μf(x) = 0 in ℝn, n ⩾ 3. Under suitable conditions on K and f, the equation with small μ ⩾ 0 possesses a continuum of entire solutions with a specific asymptotic behaviour. Typically, K behaves like |x|ℓ at ∞ for some ℓ > −2 and the entire solutions behave asymptotically like − (2 + ℓ)log |x| near ∞. Main tools of the analysis are comparison principle for separation structure, asymptotic expansion of solutions near ∞, barrier method and strong maximum principle. The linearized operator for the equation has two characteristic behaviours related with the stability and the weak asymptotic stability of the solutions as steady states for the corresponding parabolic equation.

Triangular algebras, and maximal triangular algebras in particular, have been objects of interest for over 50 years. Rich families of examples have been studied in the context of many w*- and C*-algebras, but there remains a dearth of concrete examples in $B({\cal H})$. In previous work, we described a family of maximal triangular algebras of finite multiplicity. Here, we investigate a related family of maximal triangular algebras with infinite multiplicity, and unearth a new asymptotic structure exhibited by these algebras.