We obtain the variant of maximum principle for radial solutions of $p$-harmonic equation $-a\Delta_p(w)=\phi(w)$. As a consequence of this result we prove monotonicity of constant sign solutions, analyze the support of the solutions and study their oscillations. The results are applied to various type nonlinear eigenvalue problems and nonexistence theorems.

We prove a Liouville-type theorem for entire solutions of the elliptic system $-\Delta u = |v|^{q-2}v$, $-\Delta v=|u|^{p-2}u$ having finite relative Morse index in the sense of Abbondandolo. Here, $p,q > 2$ and $1/p+1/q> (N-2)/N$. In particular, this yields a result on a priori bounds in $L^{\infty}\times L^{\infty}$ for solutions of superlinear elliptic systems obtained by means of min-max theorems, for both Dirichlet and Neumann boundary conditions.

We obtain multiple nontrivial solutions of Neumann $p$-Laplacian systems via Morse theory.

Let $X$ be a real reflexive and separable Banach space having the Kadeč-Klee property, compactly imbedded in the real Banach space $V$ and let $G\colon V\rightarrow\mathbb{R}$ be a differentiable functional.

By using ``fountain theorem'' and ``dual fountain theorem'' (Bartsch [Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal. 20 (1993), 1205–1216] and Bartsch-Willem [On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc. 123 (1995), 3555–3561], respectively), we will study the multiplicity of solutions for operator equation $$ J_{\varphi}u=G'(u), $$ where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge function $\varphi$.

Equations having the above form with $J_{\varphi}$ a duality mapping on Orlicz-Sobolev spaces are considered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.

In this paper, we study multiplicity of weak solutions for the following class of quasilinear elliptic problems of the form $$ -\Delta_p u -\Delta u = g(u)-\lambda |u|^{q-2}u \quad \text{in } \Omega \text{ with } u=0 \text{ on } \partial\Omega, $$ where $ \Omega $ is a bounded domain in ${\mathbb{R}}^n $ with smooth boundary $\partial\Omega$, $ 1< q< 2< p\leq n$, $\lambda$ is a real parameter, $\Delta_p u = {\rm div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian and the nonlinearity $g(u)$ has subcritical growth. The proofs of our results rely on some linking theorems and critical groups estimates.

We investigate the existence of bounded solutions on the whole real line of the following strongly non-linear non-autonomous differential equation \begin{equation} (a(x(t))x'(t))'= f(t,x(t),x'(t)) \quad \text{a.e. } t\in \mathbb R \tag{${\rm E}$} \end{equation} where $a(x)$ is a generic continuous positive function, $f$ is a Carathéodory right-hand side.

We get existence results by combining the upper and lower-solutions method to fixed-point techniques. We also provide operative comparison criteria ensuring the well-ordering of pairs of upper and lower-solutions.

We consider a nonlinear Neumann problem, driven by the $p$-Laplacian, and with a nonlinearity which exhibits a $p$-superlinear growth near infinity, but does not necessarily satisfy the Ambrosetti-Rabinowitz condition. Using variational methods based on critical point theory, together with suitable truncation techniques and Morse theory, we show that the problem has at least three nontrivial solutions, of which two have a fixed sign (one positive and the other negative).

This paper gives some sufficient conditions for every solution of delay differential equation \begin{align*} \overset{\ldots}{x}(t) +f(t,x(t),x(t-r),\dot{x}(t),\dot{x}(t-r),\ddot{x}(t),\ddot{x}(t-r)) &+b(t)g(x(t-r),\dot{x}(t-r)) +c(t)h(x(t)) \\& =p(t,x(t),x(t-r),\dot{x}(t),\dot{x}(t-r),\ddot{x}(t)) \end{align*} to be bounded.

In the present paper we consider some extension properties for continuous maps in metric spaces and, using them, we develop in metric spaces some aspects of Granas theory of topological essentiality.

We define intrinsic, natural and metrizable topologies ${\mathcal T}_{\Omega}, {\mathcal T}, {\mathcal T}_{s,\Omega}$ and ${\mathcal T}_s$ in ${\mathcal G}(\Omega), \overline{\mathbb K}, {\mathcal G}_s(\Omega)$ and $\overline{\mathbb K}_s$, respectively. The topology ${\mathcal T}_{\Omega}$ induces ${\mathcal T}, {\mathcal T}_{s,\Omega}$ and ${\mathcal T}_s$. The topologies ${\mathcal T}_{s,\Omega}$ and ${\mathcal T}_s$ coincide with the Scarpalezos sharp topologies.

An inequality between James constants of Banach spaces $X_s$ and the James constant of their direct sum is obtained. This gives a characterization of uniform nonsquareness of sums of Banach spaces.

We introduce detailed connection matrices. We prove that the spectral sequence can be reconstructed from a detailed connection matrix in the category of filtered differential vector spaces.