Asymptotic behavior of solutions of some parabolic equation associated with the $p$-Laplacian as $p\to +\infty$ is studied for the periodic problem as well as the initial-boundary value problem by pointing out the variational structure of the $p$-Laplacian, that is, $\partial {\varphi }_p\left(u\right)=-{\Delta }_{p}u$, where ${\varphi }_p:L^2\left(\Omega \right)\to \left[0,+\infty \right]$. To this end, the notion of Mosco convergence is employed and it is proved that ${\varphi }_p$ converges to the indicator function over some closed convex set on $L^2\left(\Omega \right)$ in the sense of Mosco as $p\to +\infty$; moreover, an abstract theory relative to Mosco convergence and evolution equations governed by time-dependent subdifferentials is developed until the periodic problem falls within its scope. Further application of this approach to the limiting problem of porous-medium-type equations, such as $u_t=\Delta {\left|u\right|}^{m-2}u$ as $m\to +\infty$, is also given.

We prove the global existence and study decay properties of the solutions to the wave equation with a weak nonlinear dissipative term by constructing a stable set in $H^1\left({ℝ}^n\right)$.

We prove existence and multiplicity of solutions, with prescribed nodal properties, to a boundary value problem of the form $u^{″}+f\left(t,u\right)=0$, $u\left(0\right)=u\left(T\right)=0$. The nonlinearity is supposed to satisfy asymmetric, asymptotically linear assumptions involving indefinite weights. We first study some auxiliary half-linear, two-weighted problems for which an eigenvalue theory holds. Multiplicity is ensured by assumptions expressed in terms of weighted eigenvalues. The proof is developed in the framework of topological methods and is based on some relations between rotation numbers and weighted eigenvalues.

The topological degree for (S)${}_{+}$-mappings concerning a nonlinear eigenvalue problem associated with one-dimensional $p$-Laplacian is evaluated. The result is applied to a variational inequality, where the multiple existence of solutions is discussed.