We prove the existence of non-negative non-trivial solutions of the quasilinear equation $\Delta_mu+f(u)=0$ in $\mathbb R^n$ and of its associated free boundary problem, where $\Delta_m$ denotes the $m$-Laplace operator. The nonlinearity $f(u)$, defined for $u>0$, is required to be Lipschitz continuous on $(0,\infty)$, and in $L^1$ on $(0,1)$ with $\int_0^uf(s)\ ds <0$ for small $u>0$; the usual condition $f(0)=0$ is thus completely removed. When $n>m$, existence is established essentially for all subcritical behavior of $f$ as $u\to\infty$, and, with some further restrictions, even for critical and supercritical behavior. When $n=m$ we treat various exponential growth conditions for $f$ as $u\to\infty$, while when $n <m$ no growth conditions of any kind are required for $f$. The proof of the main results moreover yield as a byproduct an a priori estimate for the supremum of a ground state in terms of $n$, $m$ and elementary parameters of the nonlinearity. Our results are thus new and unexpected even for the semilinear equation $\Delta u+f(u)=0$. The proofs use only straightforward and simple techniques from the theory of ordinary differential equations; unlike well known earlier demonstrations of the existence of ground states for the semilinear case, we rely neither on critical point theory [6] nor on the Emden-Fowler inversion technique [2, 3].

The Kuramoto-Sivashinsky equation is a dissipative evolution equation in one space dimension which, despite its apparent simplicity, gives rise to a very rich dynamical behavior, as evidenced for instance by the study in [16], of its complicated set of stationary solutions and stationary and Hopf bifurcations. The large time behavior of the solutions is usually embodied by the attractor and the inertial manifolds which have been the object of many studies. In the present article, explicit expressions which can be completely evaluated are obtained for the dimension of an inertial manifold for the Kuramoto-Sivashinsky equation. This involves reworking the analysis in [1] to estimate the radius of the absorbing ball. From there, the choice of phase space, spectral gap condition, and preparation of the equation outside the absorbing ball are varied and the results compared over a moderate domain length. A new preparation of the equation is introduced which leads to the smallest dimension of those compared. The dimension is also obtained for the equation prepared using radii smaller than that of the known absorbing ball, down to the radius of the global attractor suggested by computations.

We study the continuous as well as discontinuous viscosity solutions of a certain Hamilton-Jacobi equation, $u_t + H(u, D u)=0$ in $\mathbb R^{\,n} \times \mathbb R_+$ with $u(x,0)=u_0(x)$. We obtain explicit formulas for continuous as well as for the sub- and supersolutions. In the latter case, furthermore, if $ H(u,p) > 0 $ for $|p| \neq 0 $ then the supersolution becomes a solution.

Here we study the behaviour near a punctual singularity of the positive solutions of semilinear elliptic systems in ${\mathbb R}^{N}(N\geq 3)$ given by \[ \left\{ \begin{array}{c} \Delta u+\left| x\right| ^{a}u^{s}v^{p}=0, \\ \Delta v+\left| x\right| ^{b}u^{q}v^{t}=0, \end{array} \right. \] (where $a,b,p,q,s,t\in $ ${\mathbb R}$ , $p,q>0,s,t\geq 0$). We describe the first undercritical case, and the sublinear and linear cases. The proofs do not use any variational methods, but lie essentially upon comparison properties between the two solutions $u$ and $v$, and the properties of the subsolutions and supersolutions of the scalar equation \[ \Delta f+\left| x\right| ^{\sigma }f^{\eta }=0 \] ($\sigma ,\eta \in $ ${\mathbb R}$ , $\eta >0$). This extends the classical study of the scalar equation when $0 <\eta <\max $ $(N,(N+\sigma ))/(N-2)$.

We prove that the problem $$ \begin{cases} -\Delta u=u^p\quad & \text{in $\Omega$}\\ u>0\quad & \text{in $\Omega$ } \\ u=0\quad & \text{on $\partial\Omega$} \end{cases} $$ has only one solution if $\Omega$ is a convex symmetric domain of $\Bbb R^N $, $N\ge3$ and $p <{{N+2}\over{N-2}}$ is close to ${{N+2}\over{N-2}}$. Moreover, we show that this solution is nondegenerate.

We investigate the evolution of compact hypersurfaces of $\mathbb R^N$ depending, not only on terms of curvature of the surface, but also on non local terms such as the measure of the set enclosed by the surface. We present some existence and convexity results for such motions, even for evolutions which do not preserve the inclusion of the initial surfaces.

We consider the prescribed Levi curvature equation, a second order quasilinear equation whose associated operator can be represented as a sum of squares of nonlinear vector fields. For this equation we introduce a notion of derivatives modeled on the geometry of the associated operator and prove an a priori $L^2$ estimate for these second order intrinsic derivatives of a viscosity solution. We then show that viscosity solutions are strong solutions in a natural sense and satisfy the equation almost everywhere.

For $1 <p,q <+\infty$, we show the maximal $L^q$ regularity for the inhomogeneous evolution equation in $L^p(\Omega)$ associated with the infinitesimal generator of a semigroup whose kernel satisfies suitable upper bounds, with $\Omega$ a subset of a space of homogeneous type.

We prove a result of existence and multiplicity for local minima of a functional defined on maps from $\mathbb R^3$ to a compact Riemannian manifold $\mathcal M$. The interest in such a minimization problem lies in possible applications to field theory. Namely, the solutions to our variational problem are related to the existence of topological solitons.