We give a complete classification over the range of parameters for existence or nonexistence both of global and of local solutions to a class of semilinear elliptic problems with a first order term and singular coefficients, as well as of solutions to the related Dirichlet problem in bounded domains containing the origin. We also investigate symmetry properties of ground state solutions, proving that symmetry breaking occurs in a suitable range of parameters.

We study the existence and stability of travelling-front solutions for parabolic systems of the form \begin{equation} u_{t} = A u_{xx} + f(u, u_{x}),~~x \in \mathbb R,~ t>0,~ u(x,t) \in \mathbb R^N, \tag*{(0.1)} \end{equation} where $A$ is a positive-definite diagonal matrix. The nonlinearity $f$ is a ``monostable'' function with equilibria $E^- < E^+ , ~E^-$ ``stable'' and $E^+$ ``unstable'', that depends not only on $u$ but also on $u_{x}$ and satisfies both conditions sufficient for a comparison principle to hold for (0.1) (so is ``weakly-coupled''), and also either a subquadratic growth condition in $u_x$ or alternatively a mild monotonicity property in $u_x$. It is shown that there exist ${c^{*}}$ and $c_{0}$ with $c^{*} \geq c_{0}$ such that there is a ``stable-monotone direction'' at ${E^{+}}$ for all $c \geq c_{0}$, and solutions $w_{c} (x-ct)$ of (0.1) that connect $E^-$ to ${E^{+}}$ exist for all $c \geq c^{*}$ (${c^{*}} \neq c_{0}$ in general). We then discuss the stability of fronts of speed $c > {c_{0}}$ in the framework of exponentially weighted spaces, using Sattinger's approach [36] that was also exploited by the Volperts in [41]. The stability of the travelling front $w_{c}$ depends on whether $w_{c}'$ belongs to a $c-$dependent space, $X_{c}$ say. If $w_{c}' \in X_{c}$, the solution $u^{{E^{+}}hi}$ of (0.1) with initial data $\phi \in {BUC}^{1}$ converges in $X_{c}$ to a {\em translate} of $w_{c}$ if $\phi(x) - w_{c}(x)$ is small when $|x|$ is large and $\phi$ behaves like $w_{c}$ in a neighbourhood of the ``unstable'' equilibrium ${E^{+}}$. If $w_{c}' \not\in X_{c}$, the theorem is local --- $u^{\phi}$ converges in $X_{c}$ to $w_{c}$ if $\phi$ is close to $w_{c}$ in $X_{c}$. We also show that, as for scalar equations, at most one front solution of the parabolic system (0.1) can have ``fast-decay'', and if such a front exists, its velocity must be $c=c^{*}$.

We prove certain bifurcation results for the quasilinear elliptic system \begin{align*} & -\Delta_{p}u = \lambda\, a(x)\, |u|^{p-2}u+\lambda\, b(x)\, |u|^{\alpha}\, |v|^{\beta}\, v +f(x,\lambda,u,v), \\ & -\Delta_{q}v = \lambda\, d(x)\, |v|^{q-2}v+\lambda\, b(x)\, |u|^{\alpha}\, |v|^{\beta}\, u +g(x,\lambda,u,v), \end{align*} defined on an arbitrary domain (bounded or unbounded) of $\mathbb{R}^N$, where the functions $a$, $d$, $f$ and $g$ may change sign. To this end we establish the isolation of the principal eigenvalue of the corresponding unperturbed system and apply topological degree theory.

We extend to the vector $p-$Laplace operator $(\phi_p(u'))'=(|u'|^{p-2}u')',$ $p>1,$ ($|\cdot|$ denotes the Euclidean norm in $\mathbb R^N$) a method that uses a suitable homotopy along $p$ to evaluate at the level $p=2$ a Leray-Schauder degree for an associated operator depending on $p.$ We apply this result to prove existence of nontrivial solutions to the system $$(\phi_p(u'))'=f(t,u)\quad\text{a.e.}\quad \mbox{in}\quad (0,T),\quad u(0)=0,\quad u(T)=0, $$ where $f:[0,T]\times\mathbb R^N\mapsto \mathbb R^N$ is a Carathéodory function, with $f(t,0)=0.$

The aim of this paper is to present a new joint approach to the Helmholtz decomposition in infinite cylinders and in infinite layers $\Omega=\Sigma {\times} {\mathbb R}^k$ in the function space $L^q({\mathbb R}^k;L^r({\Sigma}))$ using even arbitrary Muckenhoupt weights with respect to $x'\in \Sigma\subset {\mathbb R}^{n-k}$ and, if possible, exponential weights with respect to $x''\in {\mathbb R}^k, \;1\leq k\leq n-1,\;n\geq 2.$ For $n=2$, we get the Helmholtz decomposition for a strip, for $n=3$ in an infinite cylinder or an infinite layer and for $n>3$ in some (non--physical) unbounded domains of cylinder or layer type. The proof based on a weak Neumann problem uses a partial Fourier transform and operator--valued multiplier functions, the ${\mathcal R}$--boundedness of the family of multiplier operators and an extrapolation property in weighted $L^q$--spaces.