In this paper we investigate in a variational setting, the elliptic boundary value problem $-\Delta u={\rm sign}u/|u|^{\alpha+1}$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is an open connected bounded subset of ${\mathbb R}^N$, and $\alpha> 0$. For the positive solution, which is checked as a minimum point of the formally associated functional $$ E(u)=\frac 12\int_\Omega|\nabla u|^2+\frac{1}{\alpha} \int_\Omega \frac1{|u|^\alpha}, $$ we prove dependence on the domain $\Omega$. Moreover, an approximative functional $E_\varepsilon$ is introduced, and an upper bound for the sequence of mountain pass points $u_\varepsilon$ of $E_\varepsilon$, as $\varepsilon\to 0$, is given. For the one-dimensional case, all sign-changing solutions of $-u''={\rm sign}u/|u|^{\alpha+1}$ are characterized by their nodal set as the mountain pass point and $n$-saddle points ($n> 1$) of the functional $E$.

We study the impact of symmetries on the occurrence of periodic solutions in systems of van der Pol equations. We apply the equivariant degree theory to establish existence results for multiple nonconstant periodic solutions and classify their symmetries. The computations of the algebraic invariants in the case of dihedral, tetrahedral, octahedral and icosahedral symmetries for a van der Pol system of equations are included.

New criteria are established for the existence of multiple positive solutions of a Hammerstein integral equation of the form $$ u(t)= \int_{0}^1 k(t,s)g(s)f(s,u(s))\,ds \equiv Au(t) $$ where $k$ can have discontinuities in its second variable and $g \in L^{1}$.

These criteria are determined by the relationship between the behaviour of $f(t,u)/u$ as $u$ tends to $0^+$ or $\infty$ and the principal (positive) eigenvalue of the linear Hammerstein integral operator $$ Lu(t)=\int_{0}^1 k(t,s)g(s)u(s)\,ds. $$ We obtain new results on the existence of multiple positive solutions of a second order differential equation of the form $$ u''(t)+g(t)f(t,u(t))=0 \quad\text{a.e. on } [0,1], $$ subject to general separated boundary conditions and also to nonlocal $m$-point boundary conditions. Our results are optimal in some cases. This work contains several new ideas, and gives a unified approach applicable to many BVPs.

The aim of this paper is to prove the existence of infinitely many radial solutions of a superlinear elliptic problem with rotational symmetry and non-homogeneous boundary data.

We study the properties and the large time asymptotics of radially symmetric solutions of a chemotaxis system in a disc of ${\mathbb R}^2$ when the parameter is either critical and equal to $8\pi$ or subcritical.

We consider the following system of Schrödinger-Maxwell equations in the unit ball $B_1$ of ${\mathbb R}^3$ $$ -\frac{\hbar^2}{2m}\Delta v+ e\phi v=\omega v, \quad -\Delta\phi=4\pi e v^2 $$ with the boundary conditions $ u=0$, $ \phi=g$ on $\partial B_1$, where $\hbar, m, e, \omega > 0$, $v$, $\phi\colon B_1\rightarrow {\mathbb R}$, $g\colon \partial B_1\to {\mathbb R}$. Such system describes the interaction of a particle constrained to move in $B_1$ with its own electrostatic field. We exhibit a family of positive solutions $(v_\hbar, \phi_\hbar)$ corresponding to eigenvalues $\omega_\hbar$ such that $v_\hbar$ concentrates around some points of the boundary $\partial B_1$ which are minima for $g$ when $\hbar\rightarrow 0$.

The existence of bounded solutions of the Dirichlet problem for a ceratin class of elliptic partial differential equations is discussed here. We use variational methods based on the subdifferential theory and the comparison principle for difergence form operators. We present duality and variational principles for this problem. As a consequences of the duality we obtain also the variational principle for minimizing sequences of $J$ which gives a measure of a duality gap between primal and dual functional for approximate solutions.