Blow-up phenomena and asymptotic profiles passing from h1-critical to super-critical quasilinear Schr\uf6dinger equations

We study the asymptotic profile, as h \u2192 0, of positive solutions to where \u3b3 650 is a parameter with relevant physical interpretations, V and K are given potentials and the dimension N is greater than or equal to 5, as we look for finite L2-energy solutions. We investigate the concentrating behavior of solutions when \u3b3>0 and, differently from the case \u3b3=0 where the leading potential is V, the concentration is here localized by the source potential K. Moreover, surprisingly for \u3b3>0 we find a different concentration behavior of solutions in the case p=2NN-2 and when 2NN-24 NN-2. This phenomenon does not occur when \u3b3=0

Global vs Blow-Up Solutions and Optimal Threshold for Hyperbolic ODEs with Possibly Singular Nonlinearities

We consider a hyperbolic ordinary differential equation perturbed by a nonlinearity which can be singular at a point and in particular this includes MEMS type equations. We first study qualitative properties of the solution to the stationary problem. Then, for small value of the perturbation parameter as well as initial value, we establish the existence of a global solution by means of the Lyapunov function and we show that the omega limit set consists of a solution to the stationary problem. For strong perturbations or large initial values, we show that the solution blows up. Finally, we discuss the relationship between upper bounds of the perturbation parameter for the existence of time-dependent and stationary solutions, for which we establish an optimal threshold

Nonlocal Singular Problems and Applications to MEMS

We consider fourth order nonlinear problems which describe electrostatic actuation in MicroElectroMechanicalSystems (MEMS) both in the stationary case and in the evolution case; we prove existence, uniqueness and regularity theorems by exploiting the Near Operators Theory

Optimal Sobolev type inequalities in Lorentz spaces

It is well known that the classical Sobolev embeddings may be improved within the framework of Lorentz spaces L p,q : the space D 1,p (R n ) , 1\u2009<\u2009p\u2009<\u2009n, embeds into L p 17 ,q (R n ) , p\u2009 64\u2009q\u2009 64\u2009 1e. However, the value of the best possible embedding constants in the corresponding inequalities is known just in the case L p 17 ,p (R n ) . Here, we determine optimal constants for the embedding of the space D 1,p (R n ) , 1\u2009<\u2009p\u2009<\u2009n, into the whole Lorentz space scale L p 17 ,q (R n ) , p\u2009 64\u2009q\u2009 64\u2009 1e, including the limiting case q\u2009=\u2009p of which we give a new proof. We also exhibit extremal functions for these embedding inequalities by solving related elliptic problems