88 research outputs found
Existence and concentration of ground state solutions for a critical nonlocal Schr\"odinger equation in
We study the following singularly perturbed nonlocal Schr\"{o}dinger equation
-\vr^2\Delta u +V(x)u =\vr^{\mu-2}\Big[\frac{1}{|x|^{\mu}}\ast F(u)\Big]f(u)
\quad \mbox{in} \quad \R^2, where is a continuous real function on
, is the primitive of , and \vr is a positive
parameter. Assuming that the nonlinearity has critical exponential
growth in the sense of Trudinger-Moser, we establish the existence and
concentration of solutions by variational methods.Comment: 3
Group invariance and Pohozaev identity in Moser type inequalities
We study the so-called limiting Sobolev cases for embeddings of the spaces , where is a bounded domain. Differently from J. Moser, we consider optimal embeddings into Zygmund spaces: we derive related Euler-Lagrange equations, and show that Moser's concentrating sequences are the solutions of these equations and thus realize the best constants of the corresponding embedding inequalities. Furthermore, we exhibit a group invariance, and show that Moser's sequence is generated by this group invariance and that the solutions of the limiting equation are unique up to this invariance. Finally, we derive a Pohozaev-type identity, and use it to prove that equations related to perturbed optimal embeddings do not have solutions
Nonlocal planar Schr\"odinger-Poisson systems in the fractional Sobolev limiting case
We study the nonlinear Schr\"odinger equation for the fractional
Laplacian strongly coupled with the Poisson equation in dimension two and
with , which is the limiting case for the embedding of the
fractional Sobolev space . We prove existence of
solutions by means of a variational approximating procedure for an auxiliary
Choquard equation in which the uniformly approximated sign-changing logarithmic
kernel competes with the exponential nonlinearity. Qualitative properties of
solutions such as symmetry and decay are also established by exploiting a
suitable moving planes technique
A Fast Alternating Minimization Algorithm for Total Variation Deblurring Without Boundary Artifacts
Recently, a fast alternating minimization algorithm for total variation image
deblurring (FTVd) has been presented by Wang, Yang, Yin, and Zhang [{\em SIAM
J. Imaging Sci.}, 1 (2008), pp. 248--272]. The method in a nutshell consists of
a discrete Fourier transform-based alternating minimization algorithm with
periodic boundary conditions and in which two fast Fourier transforms (FFTs)
are required per iteration. In this paper, we propose an alternating
minimization algorithm for the continuous version of the total variation image
deblurring problem. We establish convergence of the proposed continuous
alternating minimization algorithm. The continuous setting is very useful to
have a unifying representation of the algorithm, independently of the discrete
approximation of the deconvolution problem, in particular concerning the
strategies for dealing with boundary artifacts. Indeed, an accurate restoration
of blurred and noisy images requires a proper treatment of the boundary. A
discrete version of our continuous alternating minimization algorithm is
obtained following two different strategies: the imposition of appropriate
boundary conditions and the enlargement of the domain. The first one is
computationally useful in the case of a symmetric blur, while the second one
can be efficiently applied for a nonsymmetric blur. Numerical tests show that
our algorithm generates higher quality images in comparable running times with
respect to the Fast Total Variation deconvolution algorithm
A Unified Approach to Singularly Perturbed Quasilinear Schrödinger Equations
AbstractIn this paper we present a unified approach to investigate existence and concentration of positive solutions for the following class of quasilinear Schrödinger equations,-ε2Δu+V(x)u∓ε2+γuΔu2=h(u),x∈RN,whereN⩾3,ε>0,V(x)is a positive external potential,his a real function with subcritical or critical growth. The problem is quite sensitive to the sign changing of the quasilinear term as well as to the presence of the parameterγ>0. Nevertheless, by means of perturbation type techniques, we establish the existence of a positive solutionuε,γconcentrating, asε→0, around minima points of the potential
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