We study the field equation $$ -\Delta u+V(x)u+\varepsilon^r(-\Delta_pu+W'(u))=\mu u $$ on $\mathbb R^n$, with $\varepsilon$ positive parameter. The function $W$ is singular in a point and so the configurations are characterized by a topological invariant: the topological charge. By a min-max method, for $\varepsilon$ sufficiently small, there exists a finite number of solutions $(\mu(\varepsilon),u(\varepsilon))$ of the eigenvalue problem for any given charge $q\in{\mathbb Z}\setminus\{0\}$.

We study some variational principles which imply the existence of multiple critical points for a functional $f$, using the properties of both $f$ and $\nabla f$ on some suitable sets. We derive some multiplicity theorems for a certain class of strongly indefinite functionals and we apply these results for finding multiple solutions of an elliptic system of reaction-diffusion type.

In this paper we are concerned with the problem of finding solutions for the following nonlinear field equation $$ -\Delta u + V(hx)u-\Delta_{p}u+ W'(u)=0, $$ where $u:\mathbb R^{N}\rightarrow \mathbb R^{N+1}$, $N\geq3$, $p> N$ and $h> 0$. We assume that the potential $V$ is positive and $W$ is an appropriate singular function. In particular we deal with the existence of solutions obtained as critical (not minimum) points for the associated energy functional when $h$ is small enough. Such solutions will eventually exhibit some notable behaviour as $h\rightarrow 0^{+}$. The proof of our results is variational and consists in the introduction of a modified (penalized) energy functional for which mountain pass solutions are studied and soon after are proved to solve our equation for $h$ sufficiently small. This idea is in the spirit of that used in M. Del Pino and P. Felmer [Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. Var. Partial Differential Equations 4 (1996), 121–137], [Semi-classical states for nonlinear Schrödinger equations, J. Funct. Anal. 149 (1997), 245–265] and [Multi-peak bound states for nonlinear Schrödinger equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 127–149], where "local mountain passes" are found in certain nonlinear Schrödinger equations.

The aim of this paper is to consider Hardy's inequality with weight on unbounded domains. In particular, using a decomposition lemma, we study the existence of a minimizer for $$ S_\varepsilon(\Omega):= \inf_{u \in D_{\varepsilon}^{1,2}(\Omega)} \frac {\int_{\Omega}{\vert\nabla u\vert}^2{\delta^{\varepsilon}}dx} {\int_{\Omega}{\vert u\vert}^2\delta^{\varepsilon - 2}dx}. $$

In this paper we prove the existence of critical points of non differentiable functionals of the kind $$ J(v)=\frac12\int_\Omega A(x,v)\nabla v\cdot\nabla v-\frac1{p+1}\int_\Omega (v^+)^{p+1}, $$ where $1< p< (N+2)/(N-2)$ if $N> 2$, $p> 1$ if $N\leq 2$ and $v^+$ stands for the positive part of the function $v$. The coefficient $A(x,s)=(a_{ij}(x,s))$ is a Carathéodory matrix derivable with respect to the variable $s$. Even if both $A(x,s)$ and $A'_s(x,s)$ are uniformly bounded by positive constants, the functional $J$ fails to be differentiable on $H^1_0(\Omega)$. Indeed, $J$ is only derivable along directions of $H^1_0(\Omega)\cap L^{\infty}(\Omega)$ so that the classical critical point theory cannot be applied.

We will prove the existence of a critical point of $J$ by assuming that there exist positive continuous functions $\alpha(s),\beta(s)$ and a positive constants $\alpha_0$ and $M$ satisfying $\alpha_0|\xi|^2\leq \alpha(s)|\xi|^2 \leq A(x,s)\xi\cdot \xi$, $A(x,0)\leq M$, $|A'_s(x,s)|\leq \beta(s)$, with $\beta(s)$ in $L^1(\mathbb R)$.

We consider a nonlinear model for age-dependent population dynamics subject to a density dependent factor which regulates the selection of newborn at age zero. The initial-boundary value problem is studied using a vanishing viscosity method (in the age direction) together with the fixed point theory. Existence and uniqueness are obtained, and also the positivity of the solution to the problem.

We are concerned with a class of penalized semilinear elliptic problems depending on a parameter. We study some multiplicity results and the limit problem obtained when the parameter goes to $\infty$. We obtain a "reversed" variational inequality, which is deeply investigated in low dimension.

By using the Conley connection index theory we prove the existence of travelling wave solutions for a class of reaction-diffusion systems. The results are applied to equations describing laser sustained plasma.