The generalized Kadomtsev-Petviashvili system of equations in three space dimensions, $$ \begin{cases} u_t + u^p u_x + u_{xxx} - v_y - w_z = 0, \\ v_x = u_y, \\ w_x = u_z, \end{cases} \tag*{(*)} $$ has been shown by de Bouard and Saut to possess solitary-wave solutions if and only if $ 1 \le < 4/3. $ It is demonstrated here that these localized traveling-waves, when considered as solutions of the initial-value problem for $ (*) $, are dynamically unstable to perturbations.

In this paper we prescribe a fourth-order conformal invariant on the standard $n$-sphere, with $ n\geq5 $, and study the related fourth-order elliptic equation. We first find some existence results in the perturbative case. After some blow-up analysis we build a homotopy to pass from the perturbative case to the nonperturbative one under some flatness condition. Finally we state some existence results under the assumption of symmetry.

In this paper we will obtain the asymptotic profile of solutions of the equation $u_t=\Delta$ log $u$, $u>0$, in $\Omega\times (0,\infty)$, $u=c_1$ on $\partial\Omega\times (0,\infty)$, $u(x,0)=u_0(x)\ge 0$ on $\Omega\subset R^n$ for all $n\in\mathcal {Z}^+$ and $0 <c_1 <\infty$ where $\Omega$ is a smooth, convex, bounded domain.

A result of Amann-Zehnder type for quasilinear elliptic equations is proved when the origin is a degenerate critical point. For this purpose, the critical groups of the associated continuous functionals are studied. An adaptation of the Generalized Morse Lemma is proved.