1,721,011 research outputs found
Yamabe systems and optimal partitions on manifolds with symmetries
No full textWe prove the existence of regular optimal G-invariant partitions, with an arbitrary number l ≥ 2 of components, for the Yamabe equation on a closed Riemannian manifold (M, g) when G is a compact group of isometries of M with infinite orbits. To this aim, we study a weakly coupled competitive elliptic system of l equations, related to the Yamabe equation. We show that this system has a least energy G-invariant solution with nontrivial components and we show that the limit profiles of its components separate spatially as the competition parameter goes to −∞, giving rise to an optimal partition. For l = 2 the optimal partition obtained yields a least energy sign-changing G-invariant solution to the Yamabe equation with precisely two nodal domains
Fully nontrivial solutions to elliptic systems with mixed couplings
No full textWe study the existence of fully nontrivial solutions to the system −Δui+λiui=∑j=1lβij|uj|p|ui|p−2uiinΩ,i=1,...,l,in a bounded or unbounded domain Ω in RN,N≥3. The λi’s are real numbers, and the nonlinear term may have subcritical (1<[Formula presented]), critical (p=[Formula presented]), or supercritical growth (p>[Formula presented]). The matrix (βij) is symmetric and admits a block decomposition such that the diagonal entries βii are positive, the interaction forces within each block are attractive (i.e., all entries βij in each block are non-negative) and the interaction forces between different blocks are repulsive (i.e., all other entries are non-positive). We obtain new existence and multiplicity results of fully nontrivial solutions, i.e., solutions where every component ui is nontrivial. We also find fully synchronized solutions (i.e., ui=ciu1 for all i=2,...,l) in the purely cooperative case whenever p∈(1,2)
Positive and nodal single-layered solutions to supercritical elliptic problems above the higher critical exponents
Full text linkWe study the problem
−Δv + \lambda v = |v|^{p−2} v in Ω, v = 0 on \partial Ω,
for \lambda\in R and supercritical exponents p, in domains of the form
Ω := {(y, z)\in R^{N−m−1} × R^{m+1} : (y, |z|) \in Θ},
where m \ge 1, N − m \ge 3, and Θ is a bounded domain in RN−m whose
closure is contained in \R^{N−m−1} ×(0,1). Under some symmetry assumptions
on Θ, we show that this problem has infinitely many solutions for every \lambda
in an interval which contains [0,1) and p > 2 up to some number which is
larger than the (m+ 1)st critical exponent 2^*_
{N,m} := \frac{2(N−m)}{N−m−2} . We also exhibit
domains with a shrinking hole, in which there are a positive and a nodal
solution which concentrate on a sphere, developing a single layer that blows
up at an m-dimensional sphere contained in the boundary of Ω, as the hole
shrinks and p \to 2^*_{N,m} from above. The limit profile of the positive solution,
in the transversal direction to the sphere of concentration, is a rescaling of
the standard bubble, whereas that of the nodal solution is a rescaling of a
nonradial sign-changing solution to the problem
−Δu = |u|2^*_{n−2} u, u\in D^{1,2}(\R^n),
where 2^*_n := \frac{2n}{n−2} is the critical exponent in dimension n
Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory
No full textWe consider the magnetic NLS equation (εi∇ + A(x)) 2 u + V (x)u = K(x) |u| p-2 u, x ∈ R N, where N ≥ 3, 2 < p < 2* := 2N/(N - 2), A : R N → R N is a magnetic potential and V : R N → R, K : R N → R are bounded positive potentials. We consider a group G of orthogonal transformations of R N and we assume that A is G-equivariant and V, K are G-invariant. Given a group homomorphism τ : G → S 1 into the unit complex numbers we look for semiclassical solutions u ε: R N → C to the above equation which satisfy u ε (gx) = τ(g)u ε (x) for all g ∈ G, x ∈ R N. Using equivariant Morse theory we obtain a lower bound for the number of solutions of this type
Intertwining semiclassical bound states to a nonlinear magnetic Schrödinger equation
No full textWe consider the magnetic NLS equation where N ≥ 3, 2 < p < 2 * 2N/(N - 2), is a magnetic potential and is a bounded electric potential. We consider a group G of orthogonal transformations of , and we assume that A(gx) = gA(x) and V(gx) = V(x) for any g ∈ G, . Given a group homomorphism into the unit complex numbers, we show the existence of semiclassical solutions to problem (0.1), which satisfy for all g ∈ G, . Moreover, we show that there is a combined effect of the symmetries and the electric potential V on the number of solutions of this type
Asymptotic profile of 2-nodal solutions to a semilinear elliptic problem on a Riemannian manifold
No full textSymmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory
No full textWe consider the magnetic NLS equation (εi∇ + A(x)) 2 u + V (x)u = K(x) |u| p-2 u, x ∈ R N, where N ≥ 3, 2 < p < 2* := 2N/(N - 2), A : R N → R N is a magnetic potential and V : R N → R, K : R N → R are bounded positive potentials. We consider a group G of orthogonal transformations of R N and we assume that A is G-equivariant and V, K are G-invariant. Given a group homomorphism τ : G → S 1 into the unit complex numbers we look for semiclassical solutions u ε: R N → C to the above equation which satisfy u ε (gx) = τ(g)u ε (x) for all g ∈ G, x ∈ R N. Using equivariant Morse theory we obtain a lower bound for the number of solutions of this type
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
Intertwining semiclassical solutions to a Schrödinger-Newton system
No full textWe study the problem
{(-epsilon i del + A(x))(2) u + V(x)u = epsilon(-2) (1/vertical bar x vertical bar * vertical bar u vertical bar(2)) u, u is an element of L-2(R-3, C), epsilon del u + iAu is an element of L-2 (R-3, C-3),
where A: R-3 -> R-3 is an exterior magnetic potential, V: R-3 -> R is an exterior electric potential, and epsilon is a small positive number. If A = 0 and epsilon = h is Planck's constant this problem is equivalent to the Schrodinger-Newton equations proposed by Penrose in [23] to describe his view that quantum state reduction occurs due to some gravitational effect. We assume that A and V are compatible with the action of a group G of linear isometrics of R-3. Then, for any given homomorphism T: G -> S-1 into the unit complex numbers, we show that there is a combined effect of the symmetries and the potential V on the number of semiclassical solutions u : R-3 -> C which satisfy u(gx) = T(g) u(x) for all g is an element of G, x is an element of R-3. We also study the concentration behavior of these solutions as epsilon -> 0
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