We prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.

A version of the centre manifold theorem is established which is suitable for quasilinear hyperbolic equations. As an application, the Benard problem for a viscoelastic fluid is discussed.

In this paper we discuss the problem

We show that for c > 0 there exists a positive constant μ* such that (*)μ possesses at least one solution if μ ∈ (0, μ*) and no solutions if μ > μ*. Furthermore, there exists a positive constant μ**≦μ* such that (*)μ possesses at least two solutions if μ∈(0, μ**), 2<N<6. For N≧6, μ∈(0,μ**), we show that problem (*)μ possesses a unique solution if f(x) is radial with f′(r) < 0(r = |x|).

In this paper, an Lp version of the “div-curl lemma” is generalised in a very general framework. Another form of the Lp -theorem of compensated compactness is also exploited.