1+1 spectral problems arising from the Manakov-Santini system

This paper deals with the spectral problem of the Manakov Santini system. The point Lie symmetries of the Lax pair have been identified. Several similarity reductions arise from these symmetries. An important benefit of our procedure is that the study of the Lax pair instead of the partial differential equations yields the reductions of the eigenfunctions and also the spectral parameter. Therefore, we have obtained five interesting spectral problems in 1+1 dimensions

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

Nonautonomous fractional problems with exponential growth

We study a class of nonlinear non-autonomous nonlocal equations with subcritical and critical exponential nonlinearity. The involved potential can vanish at infinity.Comment: 12 page