We consider the Dirichlet problem (*)$ -\bigtriangleup u=\mu u+f$ in $\Omega,u=0$ on $\partial\Omega$. Let $\widehat{\lambda}$ be an eigenvalue, with $\widehat{\varphi}$ an associated eigenfunction. Under suitable assumptions on $f$ and on the nodal domains of $\widehat{\varphi}$, we show that, if $\mu$ is sufficiently close to $\widehat{\lambda}$, then the solution $u$ of (*) has the same number of nodal domains as $\widehat{\varphi}$, and moreover the nodal domains of $u$ appear as small deformations of those of $\widehat{\varphi}$.

In 1981, Peter Hess established a multiplicity result for solutions of boundary-value problems for nonlinear perturbations of the Laplace operator. The sufficient conditions given were later shown to be also necessary by Dancer and the second author. In this paper, we show that similar (and slightly more general) results hold when the Laplace operator is replaced by the $p-$Laplacian. Some applications to singular problems are given, as well.

We discuss existence and multiplicity of bounded variation solutions of the mixed problem for the prescribed mean curvature equation \begin{equation*} -{\rm div } \Big({\nabla u}/{ \sqrt{1+{|\nabla u|}^2}}\Big) = f(x,u) \hbox{\, in $\Omega$}, \quad u=0 \hbox{\, on $\Gamma_{D}$}, \quad \partial u / \partial \nu =0 \hbox{\, on $ \Gamma_{N}$}, \end{equation*} where $\Gamma_{D} $ is an open subset of $\partial \Omega$ and $\Gamma_{N}=\partial \Omega\setminus \Gamma_{D}$. Our approach is based on variational techniques and a lower and upper solutions method specially developed for this problem.

For every $2 < p < 3$, we show that $u \in W^{1,p}(B^3;S^2)$ can be strongly approximated by maps in $C^\infty(\overline B \,\!^3;S^2)$ if, and only if, the distributional Jacobian of $u$ vanishes identically. This result was originally proved by Bethuel-Coron-Demengel-H\'elein, but we present a different strategy which is motivated by the $W^{2,p}$-case.

We prove the existence of a sequence of solutions approaching zero for an elliptic system of the form $-\Delta u=|v|^{q-2}v+g(x,v)$, $-\Delta v=|u|^{p-2}u+f(x,u)$ in a bounded domain, under Dirichlet homogeneous boundary conditions. We assume that $1 <p,q <2$ and that both $f(x,u)$ and $g(x,v)$ are small enough near the origin.

This work is devoted to the existence and qualitative properties of ground state solutions of the Dirchlet problem for the semilinear equation $-\Delta u-\lambda u=\vert u\vert^{2^*-2}u$ in a bounded domain. Here, $2^*$ is the critical Sobolev exponent, and the term ground state refers to minimizers of the corresponding energy within the set of nontrivial solutions. We focus on the indefinite case where $\lambda$ is larger than the first Dirichlet eigenvalue of the Laplacian, and we present a particularly simple approach to the study of ground states.

We prove the existence of chaotic dynamics in a simple Hamiltonian system of the form $\ddot{x} + q(t) f(x) = 0,$ where $q(t)$ is a periodic function of constant sign. Applications are given to a pendulum equation with variable length.

We consider perturbations, depending on a small parameter $\lambda$, of a non-invertible differential operator having a nonnegative spectrum. Given a pair of lower and upper solutions, belonging to the kernel of the differential operator, without any prescribed order, we prove the existence of a solution, when $\lambda$ is sufficiently small. Our method of proof has the advantage of permitting a uniform choice of $\lambda$ for a whole class of functions. Applications are given in a variety of situations, ranging from ODE problems to equations of parabolic type, or involving the $p\,$-Laplacian operator.

We first prove the existence of periodic solutions to systems of the form \[ (\phi_{p}(u^{\prime}))^{\prime}+\frac{d}{dt}(\nabla F(u))+\nabla G(u)=e(t). \] We then study the asymptotic behavior of all solutions to such systems, and give sufficient conditions for uniform ultimate boundedness of solutions.

We investigate new conditions for the existence of heteroclinics connecting $\pm 1$ for a non-autonomous equation of the form \begin{equation} \label{hetero} \ddot u=a(t)f(u) \end{equation} where $a(t)$ is a bounded positive function and $f(\pm 1)=0$. Here $f=F'$, where $F$ is a $C^1$ non-negative function such that $F(-1)=F(1)=0$. We are interested mainly in the case where $a(t)$ approaches its positive limit, as $|t|\to\infty$, from above, but we allow also the (``asymptotically asymmetric") case where $|\lim _{t\to-\infty}a(t)-\lim _{t\to+\infty}a(t)|$ is a sufficiently small positive number. Variational methods are used in the proofs.

We prove the existence of quasi-periodic solutions and of Aubry-Mather sets for a resonant reversible equation of the form $x^{\prime\prime} + ax^+ - bx^- +\varphi(x) +f(x,x',t)= p(t)$; the functions $p$ and $f$ are $2\pi$-periodic in $t$ and the perturbation $\varphi$ is bounded.

We study the problems $$ \begin{array}{lcr} -\Delta u= f_{\theta}(u)\text{ in }\Omega, & & u=0\text{ on }\partial\Omega,\\ -\Delta u + u= f_\theta(u)\text{ in }\Omega, & & \partial_{\nu} u=0\text{ on }\partial\Omega, \end{array} $$ where $f_{\theta}$ is a slowly superlinearly growing nonlinearity, and $\Omega$ is a bounded domain. Namely, we are interested in generalizing the results obtained in [4], where the model nonlinearity $f_\theta(u)= | {u} | ^{\theta-2} u$ was considered in the case of Dirichlet boundary conditions. We derive the asymptotic behaviour of ground state and least energy nodal solutions when $\theta\to 2$, leading to symmetry results for $\theta$ small. Our assumptions permit us to study some typical nonlinearities such as a superlinear perturbation of a small pure power or the sum of small powers and slowly exponentialy growing nonlinearities in dimension $2$.