For a proper, Gromov-hyperbolic metric space and a discrete, non-elementary, group of isometries, we define a natural subset of the limit set at infinity of the group called the ergodic limit set. The name is motivated by the fact that every ergodic measure which is invariant for the geodesic flow on the quotient metric space is concentrated on geodesics with endpoints belonging to the ergodic limit set. We refine the classical Bishop–Jones theorem proving that the packing dimension of the ergodic limit set coincides with the critical exponent of the group.

Using Patterson–Sullivan measures, we investigate growth problems for groups acting on a metric space with a strongly contracting element.

We consider the non-linear Schrödinger equation(Pμ)

\begin{equation*}\begin{array}{lc}-\Delta u + V(x) u = \mu f(u) + |u|^{2^*-2}u, &\end{array}\end{equation*}

$\mathbb{R}^N$

$N\geq3$

$f(s)/s$

Using a dual variational approach, we obtain nontrivial real-valued solutions of the critical nonlinear Helmholtz equation

$$-\Delta u-k^2u = Q(x) \vert u \vert ^{2^*-2}u,\quad u\in W^{2,2^*}({\open R}^{N})$$

$Q \in L^{\infty }({\open R}^{N}){\setminus }\{0\}$

In this paper, we study negative classical solutions and stable solutions of the following k-Hessian equation

$$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$

In this paper, we consider the following critical Kirchhoff type equation:

$$\left\{ _{u\,=\,0,\,\,\,\,\,\,\text{on}\,\partial \Omega \text{,}}^{-(a\,+\,b{{\int }_{\Omega }}|\nabla u{{|}^{2}})\Delta u\,=\,\text{Q(}x)|u{{|}^{4}}u\,+\,\lambda |u{{|}^{q-1}}u,\,\,\,\text{in}\,\Omega \text{,}} \right.$$

By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when $3\,<\,q\,<\,5$. The relation between the number of maxima of $\text{Q}$ and the number of positive solutions for the problem is also investigated.

In this paper, we study the positive solutions for a semilinear equation in hyperbolic space. Using the heat semigroup and by constructing subsolutions and supersolutions, a Fujita-type result is established.

We study the following polyharmonic Hénon equation:

where (m)* = 2N/(N – 2m) is the critical exponent, B1(0) is the unit ball in ℝN, N ⩾ 2m + 2 and K(|y|) is a bounded function. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large.

In this paper, we prove the existence, uniqueness and multiplicity of positive solutions of a nonlinear perturbed fourth-order problem related to the $Q$ curvature.

For an $N$-step self-avoiding walk on the hypercubic lattice ${{Z}^{d}}$, we prove that the meansquare end-to-end distance is at least ${{N}^{4/(3d)}}$ times a constant. This implies that the associated critical exponent $v$ is at least $2/(3d)$, assuming that $v$ exists.

Power-law distributions and other skew distributions, observed in various models and real systems, are considered. A model, describing evolving systems with increasing number of elements, is considered to study the distribution over element sizes. Stationary power-law distributions are found. Certain non-stationary skew distributions are obtained and analyzed, based on exact solutions and numerical simulations.

Let w be an infinite fixed point of a binary k-uniform morphism f, and let Ew be the critical exponent of w. We give necessary and sufficient conditions for Ew to be bounded, and an explicit formula to compute it when it is. In particular, we show that Ew is always rational. We also sketch an extension of our method to non-uniform morphisms over general alphabets.