The aim of this paper is twofold. The first aim is to describe the entire solutions of the partial differential equation (PDE) $u_{z_1}^2+2Bu_{z_1}u_{z_2}+u_{z_2}^2=e^g$, where B is a constant and g is a polynomial or an entire function in $\mathbb {C}^2$. The second aim is to consider the entire solutions of another PDE, which is a generalization of the well-known PDE of tubular surfaces.

In this paper, we study the entire solutions of the Fisher–KPP (Kolmogorov–Petrovsky–Piskunov) equation ut = uxx + f(u) on the half line [0, ∞) with Dirichlet boundary condition at x = 0. (1) For any $c \ge 2\sqrt {f'(0)} $, we show the existence of an entire solution ${{\cal U}^c}(x,t)$ which connects the traveling wave solution φc(x + ct) at t = −∞ and the unique positive stationary solution V(x) at t = +∞; (2) We also construct an entire solution ${{\cal U}}(x,t)$ which connects the solution of ηt = f(η) at t = −∞ and V(x) at t = +∞.

We examine the regularity of the extremal solution of the nonlinear eigenvalue problem

on a general bounded domain Ω in ℝN, with Navier boundary condition u = Δu on ∂Ω. Firstly, we prove the extremal solution is smooth for any p > 1 and N ⩽ 4, which improves the result of Guo and Wei (Discrete Contin. Dynam. Syst. A 34 (2014), 2561–2580). Secondly, if p = 3, N = 3, we prove that any radial weak solution of this nonlinear eigenvalue problem is smooth in the case Ω = 𝔹, which completes the result of Dávila et al. (Math. Annalen348 (2009), 143–193). Finally, we also consider the stability of the entire solution of Δ2u = –l/up in ℝN with u > 0.

We are interested in entire solutions for the semilinear biharmonic equation Δ2 u = f(u) in ℝN , where f(u) = eu or –u –p (p > 0). For the exponential case, we prove that for the polyharmonic problem Δ2m u = eu with positive integer m, any classical entire solution verifies Δ2m–1 u < 0; this completes the results of Dupaigne et al. (Arch. Ration. Mech. Analysis208 (2013), 725–752) and Wei and Xu (Math. Annalen313 (1999), 207–228). We also obtain a refined asymptotic expansion of the radial separatrix solution to Δ2 u = eu in ℝ3, which answers a question posed by Berchio et al. (J. Diff. Eqns252 (2012), 2569–2616). For the negative power case, we show the non-existence of the classical entire solution for any 0 < p ⩽ 1.

This paper is concerned with entire solutions of a class of bistable delayed latticedifferential equations with nonlocal interaction. Here an entire solution is meant by asolution defined for all (n,t) ∈ ℤ × ℝ. Assuming that the equation has anincreasing traveling wave front with nonzero wave speed and using a comparison argument,we obtain a two-dimensional manifold of entire solutions. In particular,it is shown that the traveling wave fronts are on the boundary of the manifold.Furthermore, uniqueness and stability of such entire solutions are studied.

The paper describes entire solutions to the eiconal type non-linear partial differential equations, which include the eiconal equations ${{({{X}_{1}}(u))}^{2}}\,+\,{{({{X}_{2}}(u))}^{2}}\,=\,1$ as special cases, where ${{X}_{1}}\,=\,{{p}_{1}}\partial /\partial {{z}_{1}}\,+\,{{p}_{2}}\partial /\partial {{z}_{2}},\,{{X}_{2}}\,=\,{{p}_{3}}\partial /\partial {{z}_{1}}\,+\,{{p}_{4}}\partial /\partial {{z}_{2}}$ are linearly independent operators with ${{p}_{j}}$ being arbitrary polynomials in ${{\mathbf{C}}^{2}}$.