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.

We establish that, for $n\geq3$, the elliptic equation

$$ -\Delta u=\lambda|x|^\mu|u|^{q-2}u+|x|^\nu|u|^{p-2}u $$

on a ball with zero Dirichlet data possesses a pair of nodal radial solutions for all $\lambda>0$ provided that

$$ \mu,\nu>-2,\quad\max\bigg\{2,\frac{n+2\mu+2}{n-2}\bigg\}<q<\frac{2(n+\mu)}{n-2}\quad\text{and}\quad p=\frac{2(n+\nu)}{n-2}. $$

When $q=2$ and $n>2\mu+6$, the same result holds for $\lambda>0$ small. Canonical transformations convert the equation into a quasi-linear elliptic equation and an equation with Hardy term. Then the results correspond to the results for the transformed equations. For example, the equation

$$ -\Delta w-\frac{\chi}{|y|^2}w=\tilde{\lambda}|y|^a|w|^{q-2}w +|y|^\nu|w|^{p-2}w, $$

on a ball with zero Dirichlet data, possesses a pair of nodal radial solutions for all $\tilde\lambda>0$ provided that $a,\nu>-2$ and

$$ \max\bigg\{2,\frac{n+a-\sqrt{\bar{\chi}-\chi}}{\sqrt{\bar{\chi}}}\bigg\} <q<\frac{n+a}{\sqrt{\bar{\chi}}}\quad\text{with~}\bar{\chi}=\bigg(\frac{n-2}{2}\bigg)^2. $$

When $q=2$, $n>2a+6$ and $0<\chi<\bar\chi-(a+2)^2$, the same result holds for $\tilde\lambda>0$ small.