A multiplicity result for the scalar field equation

We prove the existence of $N - 1$ distinct pairs of nontrivial solutions of the scalar field equation in ${\mathbb R}^N$ under a slow decay condition on the potential near infinity, without any symmetry assumptions. Our result gives more solutions than the existing results in the literature when $N \ge 6$. When the ground state is the only positive solution, we also obtain the stronger result that at least $N - 1$ of the first $N$ minimax levels are critical, i.e., we locate our solutions on particular energy levels with variational characterizations. Finally we prove a symmetry breaking result when the potential is radial. To overcome the difficulties arising from the lack of compactness we use the concentration compactness principle of Lions, expressed as a suitable profile decomposition for critical sequences

Reinventing the public mission of the research university in the Asian century: a gateway approach

The recently released White Paper on Australia in the Asian Century reflected a consensus that higher education is at the cutting edge of our Asian engagement. To this end the White Paper prescribes an important role for public universities in the unfolding Asian Century. It suggests that universities – like other public and private institutions – should deepen our engagement with Asia.But what does this deep internationalisation mean for our public research universities? The argument in this Policy Brief is that varied forms of internationalisation will have different forms of balance between private and public purposes and benefits pursued by our research universities. Internationalisation or the ‘deep internationalisation’ proposed by the White Paper challenges us to consider the public purposes and benefits beyond the box of ‘national state’, and yet achieve this without letting the market model dominate the ‘public’ enterprise of the research university. As Simon Marginson (2012) – an astute observer of higher education – has maintained: how do we redefine the ‘public’ as universities operate on global and regional scales

Existence results for double-phase problems via Morse theory

We obtain nontrivial solutions for a class of double-phase problems using Morse theory. In the absence of a direct sum decomposition, we use a cohomological local splitting to get an estimate of the critical groups at zero.Comment: 11 page

Asymptotic behavior of the eigenvalues of the p(x)-Laplacian

We obtain asymptotic estimates for the eigenvalues of the p(x)-Laplacian defined consistently with a homogeneous notion of first eigenvalue recently introduced in the literature.Comment: 10 pages, revised versio