Let ⊂ R n be a bounded Lipschitz domain and consider the Dirichlet energy functionalover the space of measure preserving mapsIn this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler-Lagrange equations associated with F over A( ). The main result here is that in even dimensions the latter equations admit infinitely many solutions, modulo isometries, amongst such maps. We investigate various qualitative properties of these solutions in view of a remarkably interesting previously unknown explicit formula.

Let Ω ⊂ R n be a bounded Lipschitz domain and consider the energy functionalIn this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler-Lagrange equations associated with F p over A p (Ω). The main result is a surprising discrepancy between even and odd dimensions. Here we show that in even dimensions the latter system of equations admit infinitely many smooth solutions, modulo isometries, amongst such maps. In odd dimensions this number reduces to one. The result relies on a careful analysis of the full versus the restricted Euler-Lagrange equations where a key ingredient is a necessary and sufficient condition for an associated vector field to be a gradient.

Let R n be a bounded Lipschitz domain and consider the energy functionalover the space of admissible mapsIn this paper we introduce a class of maps referred to as generalised twists and examine them in connection with the Euler-Lagrange equations associated with F over A./. The main result is a complete characterisation of all twist solutions and this points at a surprising discrepancy between even and odd dimensions. Indeed we show that in even dimensions the latter system of equations admit infinitely many smooth solutions, modulo isometries, amongst such maps. In odd dimensions this number reduces to one. The result relies on a careful analysis of the full versus the restricted Euler-Lagrange equations.

No abstract

Let Ω ⊂ R n be a bounded starshaped domain and consider the (p, q)-Laplacian problemwhere µ is a positive parameter, 1 < q ≤ p < n, r ≥ p and p := np n−p is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the (p, q)-Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.