This paper characterizes the pointwise behavior of smooth supersolutions of a general class of elliptic linear boundary value problems of mixed type. Our main result is a substantial extension of W.~Walter [9, Theorem 2] to cover the case of general mixed boundary conditions. As an application of this result, the theory of [6] is adapted to obtain a weak version---within the context of Schauder's theory---of the characterization of the strong maximum principle of H. Amann and J. López-Gómez [3].

We show the existence of bounded and periodic solutions for certain weakly damped linear beam equations with slowly periodic perturbations resting on weakly nonlinear elastic bearings. The homoclinicity of the bounded solutions to periodic ones are also derived.

We study conditions on $f$ which ensure the existence of nonnegative, nontrivial radial solutions vanishing at infinity of the quasilinear elliptic equation $-\Delta _{p}u=f(u)$ in $\mathbb{R}^{n},$ with $n>p.$ Both the behaviors of $f$ at the origin and at infinity are important. We discuss several different subcritical growth conditions at infinity, and we show that it is possible to obtain existence of solutions also in some supercritical cases. We also show that, after an arbitrarily small $L^{q}$ perturbation ($1\le q <\infty$) on $f $, solutions can be obtained without any restrictions on the behavior at infinity. In our proofs we use techniques from calculus of variations and arguments from the theory of ordinary differential equations such as shooting methods and the Emden-Fowler inversion.

We are looking for infinitely many weak solutions for a semilinear elliptic equation with indefinite nonlinearity. The presence of an $L^2$ function perturbs the symmetry of the problem. The result is obtained using the approach introduced by Rabinowitz for positive nonlinearities.

We classify special solutions $u>0$ for the diffusion equation $$ u_t=\left(u^{m-1} u_x\right)_{x}, $$ in the very singular range of parameters, $m\le 0$ (very fast diffusion). We investigate the existence and properties of traveling waves and of self-similar solutions of three types: forward in time, backward in time and exponential type. The study is a necessary preliminary for the construction of a general existence and uniqueness theory of very fast diffusion. We find interesting differences with respect to the standard range $m>0$. In particular, there exist three options in the behaviour at infinity or near a singularity (extending the standard choice between slow and fast rates). The novelty is the existence of solutions with very fast decay as $|x|\to \infty$. There are other ways in which this range differs from usual nonlinear diffusion. Thus, we construct very singular solutions and show that there exists not only one as in the case $m>0$, but infinitely many; moreover, they are classical solution.