1,721,109 research outputs found
A sharp Trudinger-Moser type inequality for unbounded domains in R^n
Full text linkThe Trudinger-Moser inequality states that for functions u in the Sobolev space H^1,n
over a bounded domain with a certain exponential integral is bounded by a constant depending on the domain, but not on u. Recently, the second author has shown that for n = 2 the bound the constant is independent of the domain if the Dirichlet norm is replaced by the full Sobolev norm,
We extend here this result to arbitrary dimensions n > 2. Also, we prove that on all of R^n the corresponding supremum is attained. The proof is based on a blow-up procedure
Some identification problems related to thermal materials with loss of memory
No full textIl contributo è un articolo scientifico originale, non pubblicato altrove in nessuna forma.
The author considers a new class of identification problems for integro-differential parabolic equations describing the evolution of the temperature in a material with memory. The memory effects are taken into account by convolution kernels that are in general unknown. There is a large body of literature for identification problems consisting in determining both the temperature and the convolution memory kernel with the assumption that additional measurements on the temperature are taken into account. The new problem the author considers is the determination of a suitable additional function (together with the temperature and the convolution memory kernel) that appears in one of the extremes of integration, that is, in the convolution integral of the evolution equation of the temperature. This term takes into account the loss of memory of the material. In the references of the paper we find the motivation for the study of such new problems
A sharp Trudinger–Moser type inequality for unbounded domains in R2
No full textAbstractThe classical Trudinger–Moser inequality says that for functions with Dirichlet norm smaller or equal to 1 in the Sobolev space H01(Ω) (with Ω⊂R2 a bounded domain), the integral ∫Ωe4πu2dx is uniformly bounded by a constant depending only on Ω. If the volume |Ω| becomes unbounded then this bound tends to infinity, and hence the Trudinger–Moser inequality is not available for such domains (and in particular for R2).In this paper, we show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of ∫Ωe4πu2dx over all such functions is uniformly bounded, independently of the domain Ω. Furthermore, a sharp upper bound for the limits of Sobolev normalized concentrating sequences is proved for Ω=BR, the ball or radius R, and for Ω=R2. Finally, the explicit construction of optimal concentrating sequences allows to prove that the above supremum is attained on balls BR⊂R2 and on R2
Solutions of a nonlinear Dirac equation with external fields
Full text linkWe study the stationary Dirac equation with a matrix potential describing the external field, and a asymptotically quadratic nonlinearity modelling various types of interaction without any periodicity assumption. Our discussion includes the Coulomb potential as a special case, and for the semiclassical situation we handle the scalar fields. We obtain existence and multiplicity results of stationary solutions via critical point theory
Trudinger–Moser type inequalities with logarithmic weights in dimension N
No full textWe consider borderline embeddings of Trudinger–Moser type for weighted Sobolev spaces in bounded domains in RN . The embeddings go into Orlicz spaces with exponential growth functions. It turns out that the most interesting weights are powers of the logarithm, for which an explicit dependence of the maximal growth functions can be established. Corresponding Moser type results are also proved, with explicit sharp exponents. In the particular case of a logarithmic weight with the limiting exponent N − 1, a maximal growth of double exponential type is obtained, while for any larger exponent the embedding goes into L∞
On the number of closed solutions for polynomial ODE's and a special case of Hilbert's 16th problem
No full textOn a Liouville-type equation with sign-changing weight
Full text linkIn this paper we study the existence, non-existence and multiplicity of non-negative solutions for a family of elliptic problems with exponential nonlienarity and sign-changing weights. The techniques used in the proofs are a combination of upper and lower solutions, the Trudinger-Moser inequality and variational methods
Superlinear elliptic equations and systems
Full text linkIn this article we survey some recent results on superlinear elliptic equations and systems. A particular focus will be the borderline situations of so-called critical growth. In the existence theorems, we will use mostly variational methods, that is we look for critical points of functionals associated to the equations
and systems
Bifurcation beyond the principal eigenvalues for Neumann problems with indefinite weights
Full text linkThis paper is devoted to the study of the effects of indefinite weights on some following nonlinear Neumann problems. Our results establish a relation between the position of a parameter and the number of nontrivial classical solutions of these problems. The proof combines spectral analysis tools, variational methods and the Clark multiplicity theorem
Existence and Concentration of Semiclassical Solutions for Dirac Equations with Critical Nonlinearities
Full text linkWe study the semiclassical ground states of the Dirac equation with critical nonlinearity in R^3. The Dirac operator is unbounded from below and above, and so the associate energy functional is strongly indefinite. We develop
an argument to establish the existence of least energy solutions for small parameters. We also describe the concentration phenomena of the solutions as the parameter goes to zero
- …
