We investigate the existence of infinitely many periodic solutions for the $p\left(t\right)$-Laplacian Hamiltonian systems. By virtue of several auxiliary functions, we obtain a series of new super-$p^{+}$ growth and asymptotic-$p^{+}$ growth conditions. Using the minimax methods in critical point theory, some multiplicity theorems are established, which unify and generalize some known results in the literature. Meanwhile, we also present an example to illustrate our main results are new even in the case $p\left(t\right)\equiv p=2$.

We shall prove an existence result for a class of quasilinear elliptic equations with natural growth. The model problem is $$\left\{\begin{array}{cc}-\text{div}\left(\left(1+{\left|u\right|}^r\right)\nabla u\right)+{\left|u\right|}^{m-2}u{\left|\nabla u\right|}^2=f\hfill & \text{in}\mathrm{\Omega }\hfill \\ u=0\hfill & \text{su}\partial \mathrm{\Omega }.\hfill \end{array}\right.$$

In this paper we propose a new concept of quasi-uniform monotonicity weaker than the uniform monotonicity which has been developed in the study of nonlinear operator equation $Au=b$. We prove that if $A$ is a quasi-uniformly monotone and hemi-continuous operator, then $A^{-1}$ is strictly monotone, bounded and continuous, and thus the Galerkin approximations converge. Also we show an application of a quasi-uniformly monotone and hemi-continuous operator to the proof of the well-posedness and convergence...