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In this paper, we study the quasilinear Schrödinger equations $$-\Delta u+V(x)u+\Delta(u^2)u = f(x, u),\qquad\forall x\in\mathbb{R}^N,$$ where $V\in C(\mathbb{R}^N;\mathbb{R})$ may change sign and $f$ is only locally defined for $|u|$ small. Under some new assumptions on $V$ and $f$, we show that the above equation has a sequence of solutions converging to zero. Some recent results in the literature are generalized and significantly improved and some examples are also given to illustrate our main theoretical results.

In this paper, we consider the multiplicity of homoclinic solutions for the following damped vibration problems $$ \ddot{x}(t)+B\dot{x}(t)-A(t)x(t)+H_{x}(t,x(t))=0,$$ where $A(t)\in (\mathbb{R},\mathbb{R}^{N})$ is a symmetric matrix for all $t\in \mathbb{R}$, $B=[b_{ij}]$ is an antisymmetric $N \times N$ constant matrix, and $H(t,x)\in C^{1}(\mathbb{R}\times B_{\delta},\mathbb{R})$ is only locally defined near the origin in $x$ for some $\delta>0$. With the nonlinearity $H(t,x)$ being partially sub-quadratic at zero, we obtain infinitely many homoclinic solutions near the origin by using a Clark's theorem.

Infinitely many solutions for a nonlinear sixth-order differential equation are obtained. The variational methods are adopted and an oscillating behaviour on the nonlinear term is required, avoiding any symmetry assumption.

The existence and the multiplicity of periodic solutions for a parameter dependent second order Hamiltonian system are established via linking theorems. A monotonicity trick is adopted in order to prove the existence of an open interval of parameters for which the problem under consideration admits at least two non trivial qualified solutions.

The aim of this paper is to study the existence of at least two non-trivial solutions to a boundary value problem for fourth-order elastic beam equations given by \begin{align*} u^{(4)}+Au''+Bu = \lambda f(x,u) \quad \text{in } [0,1],\\ u(0) =u(1) = 0,\quad u''(0)=u''(1) = 0, \end{align*} under suitable conditions on the nonlinear term on the right hand side. Our approach is based on variational methods, and in particular, on an abstract two critical points theorem given for differentiable functionals defined on a real Banach space.

In this paper, we are focusing to the following Schrödinger–Maxwell system: \begin{equation} \begin{cases} -\Delta_{g}u+\beta(x)u+eu\phi=\Psi(\lambda,x)f(u) & \mbox{in} \ M,\\ -\Delta_{g}\phi+\phi=qu^{2} & \mbox{in} \ M, \end{cases}\tag{$\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e}$} \end{equation} where $(M,g)$ is a 3-dimensional compact Riemannian manifold without boundary, $e,q>0$ are positive numbers, $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, $\beta\in C^{\infty}(M)$ and $\Psi\in C^{\infty}(\mathbb{R}_{+}\times M)$ are positive functions. By various variational approaches, existence of multiple solutions of the problem $(\mathcal{SM}_{\Psi(\lambda,\cdot)}^{e})$ is established.

The existence and the multiplicity of periodic solutions for a parameter dependent second order Hamiltonian system are established via linking theorems. A monotonicity trick is adopted in order to prove the existence of an open interval of parameters for which the problem under consideration admits at least two non trivial qualified solutions.

In this paper we deal with semilinear problems at resonance. We present a sufficient condition for the existence of a weak solution in terms of asymptotic properties of nonlinearity. Our condition generalizes the classical Landesman--Lazer condition and it also covers the cases of vanishing and oscillating nonlinearities.

In this paper, we investigate the existence of periodic solutions for the second order systems at resonance: \begin{equation} \begin{cases} \ddot u(t)+m^2\omega^2u(t)+\nabla F(t,u(t))=0\qquad \mbox{a.e. }t\in [0,T],\\ u(0)-u(T)=\dot u(0)-\dot u(T)=0, \end{cases} \end{equation} where $m>0$, the potential $F(t,x)$ is convex in $x$ and satisfies some general subquadratic conditions. The main results generalize and improve Theorem 3.7 in J. Mawhin and M. Willem [Critical point theory and Hamiltonian systems, Springer-Verlag, New York, 1989].

We study the existence of non-zero solutions for a fourth-order differential equation with nonlinear boundary conditions which models beams on elastic foundations. The approach is based on variational methods. Some applications are illustrated.

The aim of this paper is to prove the existence of infinitely many weak solutions for a mixed boundary value system with $(p_1,\dots,p_m)$-Laplacian. The approach is based on variational methods.