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In this paper, we discuss several results regarding existence, non-existence and asymptotic properties of solutions to u ⁗ + q u ″ + f ( u ) = 0 , under various hypotheses on the parameter q and on the potential F ( t ) = ∫ 0 t f ( s ) d s , generally assumed to be bounded from below. We prove a non-existence result in the case q ≤ 0 and an existence result of periodic solution for: 1) almost every suitably small (depending on F ), positive values of q ; 2) all suitably large (depending on F ) values of q . Finally, we describe some conditions on F which ensure that some (or all) solutions u q to the equation satisfy ‖ u q ‖ ∞ → 0 , as q ↓ 0 . [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn–Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution – non-negativity, conservation of the total mass and dissipation of the energy – are automatically guaranteed by the construction from minimizing movements in the energy landscape. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the existence of positive solution for the following class of elliptic equation where f has a subcritical growth and V is a nonnegative potential, which can vanish at infinity, that is, as , or shortly, . [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of positive solutions and sign-changing solutions for the following boundary value problem in the half-space where λ is a positive parameter and the nonlinear term f is superlinear at zero and asymptotically linear at infinity. [ABSTRACT FROM AUTHOR]