In this paper we examine nonlinear periodic systems driven by the vectorial $p$-Laplacian and with a nondifferentiable, locally Lipschitz nonlinearity. Our approach is based on the nonsmooth critical point theory and uses the subdifferential theory for locally Lipschitz functions. We prove existence and multiplicity results for the “sublinear” problem. For the semilinear problem (i.e. $p=2$) using a nonsmooth multidimensional version of the Ambrosetti-Rabinowitz condition, we prove an existence theorem...

The purpose of this paper is to study the existence and multiplicity of a periodic solution for the non-autonomous second-order system $$\begin{array}{c}\frac{\mathrm{d}}{\mathrm{d}t}\left(\right|\dot{u}\left(t\right){|}^{p-2}\dot{u}\left(t\right))=\nabla F(t,u\left(t\right)),\text{a.e.}t\in [0,T],\\ u\left(0\right)-u\left(T\right)=\dot{u}\left(0\right)-\dot{u}\left(T\right)=0,\\ \Delta {\dot{u}}^i\left(t_j\right)={\dot{u}}^i\left(t_j^{+}\right)-{\dot{u}}^i\left(t_j^{-}\right)=I_{ij}\left(u^i\left(t_j\right)\right),i=1,2,\cdots ,N;j=1,2,\cdots ,m.\end{array}$$ By using the least action principle and the saddle point theorem, some new existence theorems are obtained for second-order $p$-Laplacian systems with or without impulse under weak sublinear growth conditions, improving some existing results in the literature.

Applying two three critical points theorems, we prove the existence of at least three anti-periodic solutions for a second-order impulsive differential inclusion with a perturbed nonlinearity and two parameters.

Dorsey, Di Bartolo and Dolgert (Di Bartolo et al., 1996; 1997) have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model, in the weak-$\kappa$ limit. These authors deduced a formal expansion for the superheating field in powers of ${\kappa }^{\frac{1}{2}}$ up to order four, extending the formula by De Gennes (De Gennes, 1966) and the two terms in Parr’s formula (Parr, 1976). In this paper, we construct asymptotic matched solutions at all orders leading to a complete expansion in powers...

Dorsey, Di Bartolo and Dolgert (Di Bartolo et al., 1996; 1997) have constructed asymptotic matched solutions at order two for the half-space Ginzburg-Landau model, in the weak-κ limit. These authors deduced a formal expansion for the superheating field in powers of ${\kappa }^{\frac{1}{2}}$ up to order four, extending the formula by De Gennes (De Gennes, 1966) and the two terms in Parr's formula (Parr, 1976). In this paper, we construct asymptotic matched solutions at all orders leading to a complete expansion...