We prove the existence of at least two distinct solutions of problem (1), (2) under some restrictions on $Q$, $f$ and $\lambda$. One solution has been obtained by a local minimization and the second one by applying the mountain pass theorem.

The paper deals with an equation $\dot x=f(t,x)$, $(t,x)\in I\times \Bbb R$, where $[0,1]\subset I$. The notion characteristic exponent for a {\it closed} solution is extended to an {\it arbitrary} solution $\varphi(t)$ defined on $[0,1]$---in the obvious way. If such solutions exist, the characteristic exponent as a function $\mu=\mu(\xi)$, $\xi=\varphi(0)$ is defined. This function is investigated by means of two formulae. The first one is a formula for the sum $\mu(\xi_1)+\mu(\xi_2)$, the second one is a formula for the value $\mu(\xi_2)$ in terms of arbitrary values $\xi_1$ and $\xi_3$ such that $\xi_1<\xi_2<\xi_3$. Besides these formulae a main result is that $\mu(\xi)$ is weakly convex if $f_x'(t,x)$ is weakly convex in $x$ for all fixed $t$. Moreover, the Riccati equation (in several ways) is characterized inside the class of equations where $f'_x(t,x)$ has the mentioned convexity property. Finally, an earlier result on closed solutions is completed by a stability discussion.

A necessary and sufficient condition for the existence of a bounded solution of the equation of a forced oscillator is obtained. The proof is based on the method of guiding functions.

We study a singularly perturbed reaction-diffusion equation with a small parameter $\epsilon>0$. This problem is viewed as an approximation of the evolution of an interface by its mean curvature with a forcing term. We derive a quasi-optimal error estimate of order $\mathcal{O}(\epsilon^2| \rm{log}\, \epsilon|^2)$ for the interfaces, which is valid before the onset of singularities, by constructing suitable sub and super solutions. The proof relies on the behavior at infinity of functions appearing in the truncated asymptotic expansion, and by using a modified distance function combined with a vertical shift.

We consider strongly coupled quasilinear reaction-diffusion systems subject to nonlinear boundary conditions. Our aim is to develop a geometric theory for these types of equations. Such a theory is necessary in order to describe the dynamical behavior of solutions. In our main result we establish the existence and attractivity of center manifolds under suitable technical assumptions. The technical ingredients we need consist of the theory of strongly continuous analytic semigroups, maximal regularity, interpolation theory and evolution equations in extrapolation spaces.

We present a novel approach to the Kadomtsev-Petviashvili (KP) hierarchy and its modified counterpart, the mKP hierarchy based on factorizations of formal pseudo-differential operators and a matrix-valued Lax operator for the mKP hierarchy. As a result of this framework we obtain new B\"acklund transformations for the KP hierarchy and the possibility of transferring classes of KP solutions into those of mKP solutions, and vice versa. As an application of our techniques we provide a new derivation of soliton solutions of the KP and mKP equation.

We combine results from nonlinear functional analysis relating nonlinear eigenvalues of the type $ g'(u) = \rho u $ to the derivatives of the critical value function $ \gamma (t) := \sup_{\|u\|^2=t} g(u) $ with concentration compactness techniques to study the Dirichlet boundary value problem on $\Omega$, $$ -\Delta u + u = \lambda f(x,u), \tag 0.1 $$ where $\Omega$ is an unbounded cylindrical domain and the dependence on $x$ in the unbounded direction is periodic. We give sufficient conditions on $f$ to obtain an interval in which the $\lambda$'s for which (0.1) has a weak solution are dense.

This article is concerned with semilinear elliptic equations of the form $$ \Delta u+\lambda g(x)f(u)=0\quad\text{in }\Bbb R^n. $$ In particular, it addresses the questions of nonexistence, uniqueness and nonuniqueness of positive solutions $u$ such that $$ \begin{align} &\lim_{|x|\to\infty}u(x)=0\quad\text{when }n\ge3,\qquad \text{no condition at infinity when }n=1,2. \end{align} $$ A typical example for $f$ is the function $f(u)=u-u^{1+p}$, $p>0$, arising in population genetics models; moreover, the function $g$ may change sign. Our results show the importance of the behavior of $g$ at infinity. We derive the nonexistence and uniqueness results by establishing a series of identities involving solutions of the problem, as well as solutions of a linear problem. For nonuniqueness, we use sub and super solution techniques in connection with monotonicity arguments.

We prove a Wiener energy estimate for relaxed Dirichlet problems $Lu + \mu u =\nu$ in $\Omega$, with $L$ a uniformly elliptic operator with bounded coefficients, $\mu$ a measure of $\mathcal {M}_0(\Omega)$, $\nu$ a Kato measure and $\Omega$ a bounded open set of $\mathbb{R}^N$, $N \geq 2$. Choosing a particular $\mu$, we obtain an energy estimate also for classical variational Dirichlet problems.

Using the notion of viscosity solutions for Hamilton-Jacobi equation and the theory of differential games, a Trotter formula for Hamilton-Jacobi equations with max-min Hamiltonians is given.

In this paper, the $L^p$ behavior of systems of linear, hyperbolic partial differential equations is examined by means of the theory of integrated semigroups. We show in particular how the degree of integration and therefore the regularity of the solution depends on the multiplicity of the eigenvalues of the symbol.

In this paper we study the qualitative properties (existence, uniqueness and regularity) of solutions to a class of hyperbolic systems which occur in integrated circuits modeling.

We derive an inequality of Carleman's type for solutions to Schroedinger's equation, with singular weight function. As a consequence, we obtain a new unique continuation result for solutions in a bounded domain.

Within the class of quadratic perturbations we show analytically or numerically how many limit cycles can be bifurcated at first order out of the periodic orbits surrounding the center of the quadratic system $$ \dot x=x(1-x-ay)\quad\text{and}\quad \dot y=y(-1+ax+y),\quad\text{where }1<a<\infty. $$

Sufficient conditions for the asymptotic stability of invariant manifolds of nonlinear systems of ordinary differential equations, $\dot x=f(t,x)$, are expressed directly in terms of $f$. A special case of these conditions is the well-known Poincaré criterion for the asymptotic orbital stability of a limit cycle.

We study convergence of value-functions associated to control systems with a singular perturbation. In the nonlinear case, we prove new convergence results: the limit of optimal costs of the perturbed system is an optimal cost for the reduced system. We furthermore provide an estimation of the rate of convergence when the solutions of the reduced are regular enough