1,720,969 research outputs found
Trudinger–Moser‐type inequality with logarithmic convolution potentials
Full text linkWe establish Moser-Trudinger type inequalities in presence of a logarithmic convolution potential when the domain is a ball or the entire space . Moreover, we characterize critical nonlinear growth rates for these inequalities to hold and for the existence of corresponding extremal functions. In addition, we show that extremal functions satisfy corresponding Euler-Lagrange equations, and we derive general symmetry and uniqueness results for solutions of these equations
On the planar Schrodinger-Poisson system
No full textWe develop a variational framework to detect high energy solutions of the planar Schrodinger-Poisson system
{-Delta u + a(x)u + gamma wu = 0,
{Delta w = u(2) in R-2
with a positive function a is an element of L-infinity(R-2) and gamma > 0. In particular, we deal with the periodic setting where the corresponding functional is invariant under Z(2)-translations and therefore fails to satisfy a global Palais-Smale condition. The key tool is a surprisingly strong compactness condition for Cerami sequences which is not available for the corresponding problem in higher space dimensions. In the case where the external potential a is a positive constant, we also derive, as a special case of a more general result, the existence of nonradial solutions (u, w) such that u has arbitrarily many nodal domains. Finally, in the case where a is constant, we also show that solutions of the above problem with u > 0 in R-2 and w(x) -> -infinity as vertical bar x vertical bar -> infinity are radially symmetric up to translation. Our results are also valid for a variant of the above system containing a local nonlinear term in u in the first equation
A Connection Between Symmetry Breaking for Sobolev Minimizers and Stationary Navier–Stokes Flows Past a Circular Obstacle
Full text linkFluid flows around a symmetric obstacle generate vortices which may lead to symmetry breaking of the streamlines. We study this phenomenon for planar viscous flows governed by the stationary Navier–Stokes equations with constant inhomogeneous Dirichlet boundary data in a rectangular channel containing a circular obstacle. In such (symmetric) framework, symmetry breaking is strictly related to the appearance of multiple solutions. Symmetry breaking properties of some Sobolev minimizers are studied and explicit bounds on the boundary velocity (in terms of the length and height of the channel) ensuring uniqueness are obtained after estimating some Sobolev embedding constants and constructing a suitable solenoidal extension of the boundary data. We show that, regardless of the solenoidal extension employed, such bounds converge to zero at an optimal rate as the length of the channel tends to infinity
Radial symmetry of positive solutions in nonlinear polyharmonic Dirichlet problems
No full textWe extend the symmetry result of Gidas-Ni-Nirenberg to semilinear polyharmonic Dirichlet problems in the unit ball. In the proof we develop a new variant of the method of moving planes relying on fine estimates for the Green function of the polyharmonic operator. We also consider minimizers for subcritical higher order Sobolev embeddings. For embeddings into weighted spaces with a radially symmetric weight function, we show that the minimizers are at least axially symmetric. This result is sharp since we exhibit examples of minimizers which do not have full radial symmetr
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Critical growth biharmonic elliptic problems under Steklov-type boundary conditions
No full textWe study the fourth-order nonlinear critical problem in a smooth, bounded domain , , subject to the boundary conditions on . We provide estimates for the range of parameters for which this problem admits a positive solution. If the domain is the unit ball, we obtain an almost complete description
On a fourth order Steklov eigenvalue problem
No full textWe study a biharmonic Stekloff eigenvalue problem. We prove some new results and we collect and refine a number of known
results. Moreover, we highlight the main open problems still to be solved
Positivity, symmetry and uniqueness for minimizers of second order Sobolev inequalities
No full textWe prove that minimizers for subcritical second-order Sobolev embeddings
in the unit ball are unique, positive and radially symmetric. Since the proofs of
the corresponding first-order results cannot be extended to the present situation, we
apply new and recently developed techniques
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