In this paper we study a class of quasilinear elliptic systems of the type $$ \begin{cases} -{\rm div}(a_1(x,\nabla u_1,\nabla u_2))=f_1(x,u_1,u_2) & \text{in } \Omega,\\ -{\rm div}(a_2(x,\nabla u_1,\nabla u_2))=f_2(x,u_1,u_2) & \text{in } \Omega,\\ u_1 = u_2 = 0 & \text{on } \partial \Omega, \end{cases} $$ with $\Omega$ bounded domain in $\mathbb R^N$. We assume that $A\colon \Omega \times {\mathbb{R}}^N\times{\mathbb{R}}^N\rightarrow{\mathbb{R}}$, $F\colon \Omega \times {\mathbb{R}} \times {\mathbb{R}} \rightarrow {\mathbb{R}}$ exist such that $a=(a_1,a_2)=\nabla A$ satisfies the so called Leray-Lions conditions and $f_1={\partial F}/{\partial u_1}$, $f_2={\partial F}/{\partial u_2}$ are Carathéodory functions with subcritical growth.

The approach relies on variational methods and, in particular, on a cohomological local splitting which allows one to prove the existence of a nontrivial solution.

We will prove the existence of a nontrivial homoclinic solution for an autonomous second order Hamiltonian system $\ddot{q}+\nabla{V}(q)=0$, where $q\in\mathbb{R}^n$, a potential $V\colon\mathbb{R}^n\to\mathbb{R}$ is of the form $V(q)=-K(q)+W(q)$, $K$ and $W$ are $C^{1}$-maps, $K$ satisfies the pinching condition, $W$ grows at a superquadratic rate, as $|q|\to\infty$ and $W(q)=o(|q|^2)$, as $|q|\to 0$. A homoclinic solution will be obtained as a weak limit in the Sobolev space $W^{1,2}(\mathbb{R},\mathbb{R}^n)$ of a sequence of almost critical points of the corresponding action functional. Before passing to a weak limit with a sequence of almost critical points each element of this sequence has to be appropriately shifted.

Our aim is to prove a multiplicity result for periodic solutions of Hamiltonian systems in the plane, by the use of the Poincaré-Birkhoff Fixed Point Theorem. Our main theorem generalizes previous results obtained for scalar second order equations by Lazer and McKenna [Large scale oscillatory behaviour in loaded asymmetric systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987), 243–274] and Del Pino, Manasevich and Murua [On the number of $2\pi$-periodic solutions for $u''+g(u) =s(1+h(t))$ using the Poincaré–Birkhoff Theorem, J. Differential Equations 95 (1992), 240–258].

We consider a nonlinear nonvariational Neumann problem with a nonsmooth potential. Using the spectrum of the assymptotic (as $|x| \rightarrow\infty)$ differential operator and degree theoretic techniques based on the degree map of certain multivalued perturbations of (S)$_{+}$-operators, we establish the existence of at least one nontrivial smooth solution.

In this paper, we establish a sequence $\{\nu_k^\infty\}$ of eigenvalues for the following eigenvalue problem $$ \begin{cases} \varphi_p (u'(t))' + \nu h(t) \varphi_p(u(t)) = 0 &\text{for } t \in (0,1), \\ u(0) = 0=u(1), \end{cases} $$ where $\varphi_p(x)=|x|^{p-2}x$, $ 1< p< 2$, $\nu$ a real parameter. In particular, $h \in C((0,1),(0,\infty))$ is singular at the boundaries which may not be of $L^1(0,1)$. Employing global bifurcation theory and approximation technique, we prove several existence results of sign-changing solutions for problems of the form \begin{equation} \begin{cases} \varphi_p (u'(t))' + \lambda h(t) f (u(t)) = 0 &\text{for } t \in (0,1), \\ u(0) = 0= u(1), \end{cases} \tag{${\rm QP}_\lambda$} \end{equation} when $f \in C({\mathbb{R}}, {\mathbb{R}})$ and $uf(u) > 0$, for all $u \neq 0$ and is odd with various combinations of growth conditions at $0$ and $\infty$.

We study the problem of persistence of $T$-periodic solutions of the celebrated symmetric Euler top when subjected to a small $T$-periodic stimulus. All solutions of the unperturbed system are periodic (of different periods, including continua of equilibria). In the case that the perturbation depends also on the three components of the angular momentum (the unknowns of the system) we provide bifurcation functions whose simple zeros correspond to $T$-periodic solutions of the perturbed system.

The nonlocal initial value problem for an infinite system of parabolic semilinear functional-differential equations is studied. General operators of parabolic type of second order with variable coefficients are considered and the system is weakly coupled. We prove a theorem on existence of a classical solution in the class of continuous bounded functions and in the class of continuous functions satisfying a certain growth condition. Partial uniqueness result is obtained as well.

The existence of at least one solution to a nonlinear second order differential equation on the half-line with the boundary conditions $x'(0)=0$ and with the first derivative vanishing at infinity is proved.

We investigate a class of anisotropic elliptic equations in the whole $\mathbb R^N$. By a variational approach, we obtain existence and regularity of nontrivial solutions in the framework of anisotropic Sobolev spaces. In addition, when the data is assumed to be merely locally integrable, the existence of solutions is established for a subclass of equations.

This paper deals with the initial-boundary value problem for the viscoelastic equation with strong damping and nonlinear source. Firstly, we prove the local existence of solutions by using the Faedo-Galerkin approximation method and Contraction Mapping Theorem. By virtue of the potential well theory and convexity technique, we then prove that if the initial data enter into the stable set, then the solution globally exists and decays to zero with a polynomial rate, and if the initial data enter into the unstable set, then the solution blows up in a finite time. Moreover, we show that the solution decays to zero with an exponential or polynomial rate depending on the decay rate of the relaxation function.

We provide an alternative approach to the equivariant Nielsen fixed point theory developed by P. Wong in [Equivariant Nielsen numbers, Pacific J. Math. 159 (1993), 153–175] by associating an abstract simplicial complex to any $G$-map and defining two $G$-homotopy invariants that are lower bounds for the number of fixed points and orbits in the $G$-homotopy class of a given $G$-map in terms of this complex. We develop a relative equivariant Nielsen fixed point theory along the lines above and prove a minimality result for the Nielsen-type numbers introduced in this setting.