Volume 3, Issue S1 - Special Issue: Recent advances in the qualitative analysis of nonlinear partial differential equations

In this paper we review recent results on the existence of non-topological solitons in classical relativistic nonlinear field theories. We follow the Coleman approach, which is based on the existence of two conservation laws, energy and charge. In particular we show that under mild assumptions on the nonlinear term it is possible to prove the existence of solitons for a set of admissible charges. This set has been studied for the nonlinear Klein–Gordon equation, and in this paper we state new results in this direction for the Klein–Gordon–Maxwell system.

In this article we present recent results on optimal embeddings, and associated PDEs, of the space of functions whose distributional Laplacian belongs to L 1 . We discuss sharp embedding inequalities which allow to improve the optimal summability results for solutions of Poisson equations with L 1 -data by Maz'ya ( N ≥ 3) and Brezis–Merle ( N = 2). Then, we consider optimal embeddings of the mentioned space into L 1 , for the simply supported and the clamped case, which yield corresponding eigenvalue problems for the 1- biharmonic operator (a higher order analogue of the 1-Laplacian). We derive some properties of the corresponding eigenfunctions, and prove some Faber–Krahn type inequalities.

This paper deals with the Klein–Gordon–Maxwell system in a bounded spatial domain with a nonuniform coupling. We discuss the existence of standing waves in equilibrium with a purely electrostatic field, assuming homogeneous Dirichlet boundary conditions on the matter field and nonhomogeneous Neumann boundary conditions on the electric potential. Under suitable conditions we prove existence and nonexistence results. Since the system is variational, we use Ljusternik–Schnirelmann theory.

We prove the existence of N - 1 distinct pairs of nontrivial solutions of the scalar field equation in ℝ N under a slow decay condition on the potential near infinity, without any symmetry assumptions. Our result gives more solutions than the existing results in the literature when N ≥ 6. When the ground state is the only positive solution, we also obtain the stronger result that at least N - 1 of the first N minimax levels are critical, i.e., we locate our solutions on particular energy levels with variational characterizations. Finally we prove a symmetry breaking result when the potential is radial. To overcome the difficulties arising from the lack of compactness we use the concentration compactness principle of Lions, expressed as a suitable profile decomposition for critical sequences.

We study the following nonlinear Schrödinger equations: -Δu+V(x)u=f(u)inℝN.$ - \Delta u + V(x) u = f(u) \quad \text{in } {\mathbb {R}^N}. $ The purpose of this paper is to establish the existence of a positive solution under general conditions which are weaker than the Ambrosetti–Rabinowitz condition.

Given a bounded open regular set Ω of ℝ2$\mathbb {R}^2$, q1,...,qK∈Ω${q_1, \ldots , q_K \hspace*{-0.85358pt}\in \hspace*{-0.85358pt} \Omega }$, a regular bounded function ϱ:Ω→[0,+∞)${\varrho \hspace*{-0.56905pt}:\hspace*{-0.56905pt} \Omega \hspace*{-0.85358pt}\rightarrow \hspace*{-0.85358pt} [0,+\infty )}$ and a bounded function V:Ω→[0,+∞)${V: \Omega \rightarrow [0,+\infty )}$, we give a sufficient condition for the model problem -Δu-λϱ(x)|∇u|q=ε2V(x)eu$ -\Delta u -\lambda \varrho (x)\vert \nabla u\vert ^q = \varepsilon ^2 V(x) e^u $ to have a positive weak solution in Ω with u = 0 on ∂Ω${\partial \Omega }$, which is singular at each q i as the parameters ε and λ tend to 0, without considering any relation between them, essentially when the set of concentration points q i and the set of zeros of V are not necessarily disjoint and q∈[1,2)$q\in [1,2)$ is a real number.

In this paper we are interested to ensure the existence of multiple nontrivial solutions for some classes of problems under Dirichlet boundary conditions with impulsive effects. More precisely, by using a suitable analytical setting, the existence of at least three solutions is proved exploiting a recent three-critical points result for smooth functionals defined in a reflexive Banach space. Our approach generalizes some well-known results in the classical framework.