Nonexistence and multiplicity of solutions to elliptic problems with supercritical exponents

We consider the supercritical problem $$\begin{array}{l}{-}\Delta u=\left\vert u\right\vert ^{p-2}u \; {\rm in} \; \Omega,\quad u=0 \; {\rm on} \; \partial \Omega, \end{array}$$ where Ω is a bounded smooth domain in ${\mathbb{R}^{N}}$ , N ≥ 3, and ${p\geq2^{\ast}:=\frac{2N}{N-2}}$ . Bahri and Coron showed that if Ω has nontrivial homology this problem has a positive solution for p = 2^*. However, this is not enough to guarantee existence in the supercritical case. For ${p\geq\frac{2(N-1)}{N-3}}$ Passaseo exhibited domains carrying one nontrivial homology class in which no nontrivial solution exists. Here we give examples of domains whose homology becomes richer as p increases. More precisely, we show that for ${p>\frac{2(N-k)}{N-k-2}}$ with 1 ≤ k ≤ N?3 there are bounded smooth domains in ${\mathbb{R}^{N}}$ whose cup-length is k + 1 in which this problem does not have a nontrivial solution. For N = 4,8,16 we show that there are many domains, arising from the Hopf fibrations, in which the problem has a prescribed number of solutions for some particular supercritical exponents.     

Multiple solutions to nonlinear Schrödinger equations with singular electromagnetic potential

We consider the semilinear electromagnetic Schrödinger equation ${(-i{\nabla} + \mathcal{A}(x))^{2}u + V (x)u = |u|^{{2}^{\ast}-2}u, u\, {\in}\, D_{\mathcal{A},0}^{1,2}{(\Omega,\mathbb{C})}}$ , where ${\Omega = (\mathbb{R}^{m}\;{\backslash}\;\{0\}) {\times} {\mathbb{R}^{N-m}}}$ with 2 ≤ m ≤  N, N ≥ 3, 2* : = 2N/(N – 2) is the critical Sobolev exponent, V is a Hardy term and ${\mathcal{A}}$ is a singular magnetic potential of a particular form which includes the Aharonov– Bohm potentials. Under some symmetry assumptions on ${\mathcal{A}}$ we obtain multiplicity of solutions satisfying certain symmetry properties.     

On squares in polynomial products

Let ${f(X) \in \mathbb {Z}[X]}$ be an irreducible polynomial of degree D ≥ 2 and let N be a sufficiently large positive integer. We estimate the number of positive integers n ≤ N such that the product $$F(n) = \prod\limits_{k =1}^n f(k)$$ is a perfect square. We also consider more general questions and give a lower bound on the number of distinct quadratic fields of the form ${\mathbb {Q}(\sqrt{F(n)})}$ , n  = M + 1, ... , M + N.     

Amenable Operators of the Form Normal Plus Compact

An open question, raised independently by several authors, asks if a closed amenable subalgebra of ${\mathfrak{B}(\mathfrak{H})}$ must be similar to an C ^*-algebra. Recently, Choi, Farah and Ozawa have found a counter-example to this question, but their example is neither separable nor commutative, which leaves the question open for singly-generated algebras. In this paper we continue this line of investigation for special singly-generated algebras. It is shown that if an amenable operator T = N + K, where N is a normal operator, K is a compact operator and σ _e (N) has only finite accumulation points, then T is similar to a normal operator; if an amenable operator T = N + K, where N is a normal operator, ${K\in\mathcal{C}_p}$ for some p > 1 and ${\sigma(T)\cup\sigma(N)}$ is included in a smooth Jordan curve, then T is similar to a normal operator; if an amenable operator T = N + Q, where N is a normal operator, Q is a polynomial compact operator and NQ = QN, then T is similar to a normal operator; if there exists p, 1 < p < ∞, such that an amenable operator T satisfies one of the following conditions, then T is similar to a normal operator: (i) ${T-T^*\in\mathcal{C}_p}$ ; (ii) ${I-TT^*\in\mathcal{C}_p}$ ; (iii) ${I-T^*T\in\mathcal{C}_p}$ .     

Existence of semiclassical states for a quasilinear Schrödinger equation with critical exponent in {\mathbb{R}^N}

We consider the following perturbed version of quasilinear Schrödinger equation $$\begin{array}{lll}-\varepsilon^2\Delta u +V(x)u-\varepsilon^2\Delta (u^2)u=h(x,u)u+K(x)|u|^{22^*-2}u\end{array}$$ in ${\mathbb{R}^N}$ , where N ≥ 3, 22* = 4N/(N ? 2), V(x) is a nonnegative potential, and K(x) is a bounded positive function. Using minimax methods, we show that this equation has at least one positive solution provided that ${\varepsilon \leq \mathcal{E}}$ ; for any ${m\in\mathbb{N}}$ , it has m pairs of solutions if ${\varepsilon \leq \mathcal{E}_m}$ , where ${\mathcal{E}}$ and ${\mathcal{E}_m}$ are sufficiently small positive numbers. Moreover, these solutions ${u_\varepsilon \to 0}$ in ${H^1(\mathbb{R}^N)}$ as ${\varepsilon \to 0}$ .     

Random and algebraic permutations’ statistics

An algebraic permutation $\hat{A}\in S(N=n^{m})$ is the permutation of the N points of the finite torus ? _n ^m , realized by a linear operator A∈SL(m,? _n ). The statistical properties of algebraic permutations are quite different from those of random permutations of N points. For instance, the period length T(A) grows superexponentially with N for some (random) permutations A of N elements, whereas $T(\hat{A})$ is bounded by a power of N for algebraic permutations  $\hat{A}$ . The paper also contains a strange mean asymptotics formula for the number of points of the finite projective line P¹(? _n ) in terms of the zeta function.     

On multiplicative congruences

Let ${\varepsilon}$ be a fixed positive quantity, m be a large integer, x _j denote integer variables. We prove that for any positive integers N ₁, N ₂, N ₃ with ${N_1N_2N_3 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ contains almost all the residue classes modulo m (i.e., its cardinality is equal to m + o(m)). We further show that if m is cubefree, then for any positive integers N ₁, N ₂, N ₃, N ₄ with ${ N_1N_2N_3N_4 > m^{1+\varepsilon}, }$ the set $$\{x_1x_2x_3x_4 \quad ({\rm mod}\,m): \quad x_j\in [1,N_j]\}$$ also contains almost all the residue classes modulo m. Let p be a large prime parameter and let ${p > N > p^{63/76+\varepsilon}.}$ We prove that for any nonzero integer constant k and any integer ${\lambda\not\equiv 0 \,\, ({\rm mod}\,p)}$ the congruence $$p_1p_2(p_3+k)\equiv \lambda \quad ({\rm mod}\, p) $$ admits (1 + o(1))π(N)³/p solutions in prime numbers p ₁, p ₂, p ₃ ≤ N.     

Decay of mass for nonlinear equation with fractional Laplacian

The large time behavior of non-negative solutions to the reaction–diffusion equation ${\partial_t u=-(-\Delta)^{\alpha/2}u - u^p}$ , ${(\alpha\in(0,2], \;p > 1)}$ posed on ${\mathbb{R}^N}$ and supplemented with an integrable initial condition is studied. We show that the anomalous diffusion term determines the large time asymptotics for p > 1 + α/N, while nonlinear effects win if p ≤ 1 + α/N.     

Positive solution to semilinear parabolic equation associated with critical Sobolev exponent

We study the semilinear parabolic equation ${u_{t}- \Delta u = u^{p}, u \geq 0}$ on the whole space R ^N , ${N \geq 3}$ associated with the critical Sobolev exponent p = (N + 2)/(N ? 2). Similarly to the bounded domain case, there is threshold blowup modulus concerning the blowup in finite time. Furthermore, global in time behavior of the threshold solution is prescribed in connection with the energy level, blowup rate, and symmetry.     

Critical points for functionals with quasilinear singular Euler–Lagrange equations

For a bounded, open subset Ω of ${\mathbb{R}^{N}}$ with N > 2, and a measurable function a(x) satisfying 0 < α ≤ a(x) ≤ β, a.e. ${x \in \Omega}$ , we study the existence of positive solutions of the Euler–Lagrange equation associated to the non-differentiable functional $$\begin{array}{ll}J(v) = \frac{1}{2} \int \limits_{\Omega} [a(x)+|v|^{\gamma}]| \nabla v|^{2}- \frac{1}{p} \int \limits_{\Omega}(v_{+})^p,\end{array}$$ if γ > 0 and p > 1. Special emphasis is placed on the case ${2^{*} < p < \frac{2^{*}}{2} ( \gamma +2 )}$ .     

Sign-definiteness of q-eigenfunctions for a super-linear p-Laplacian eigenvalue problem

We consider the following q-eigenvalue problem for the p-Laplacian $$\left\{\begin{array}{ll}-{\rm div}\big( |\nabla u|^{p-2}\nabla u\big) = \lambda \|u\|_{L^{q}(\Omega)}^{p-q}|u|^{q-2}u \quad \quad\, {\rm in} \,\,\,\, \Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,\,{\rm on } \,\,\,\, \partial\Omega,\end{array}\right.$$ where \({\lambda\in\mathbb{R},}\) p > 1, Ω is a bounded and smooth domain of \({\mathbb{R}^{N},}\) N > 1, \({1\leq q < p^{\star}}\) , \({p^{\star}=\frac{Np}{N-p}}\) if p < N and \({p^{\star}=\infty}\) if \({p\geq N.}\) Let λ _q denote the first q-eigenvalue. We prove that in the super-linear case, \({p < q < p^{\star},}\) there exists \({\epsilon_{q}>0}\) such that if \({\lambda\in(\lambda_{q},\lambda _{q}+\epsilon_{q})}\) is a q-eigenvalue, then any corresponding q-eigenfunction does not change sign in Ω. As a consequence of this result we obtain, in the super-linear case, the isolatedness of λ _q for those Ω such that the Lane–Emden problem $$\left\{\begin{array}{ll}-{\rm div}\big(|\nabla u|^{p-2}\nabla u\big) = |u|^{q-2}u \qquad\quad\quad\quad \,\,{\rm in}\,\,\,\Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,{\rm on } \,\,\, \partial\Omega,\end{array}\right.$$ has exactly one positive solution.     

Radial and non radial solutions for Hardy–Hénon type elliptic systems

We discuss existence and non-existence of positive solutions for the following system of Hardy and Hénon type: $$\left\{\begin{array}{ll} {-\Delta v=|x|^{\alpha}u^{p},\,-\Delta u=|x|^{\beta}v^{q} \,\,{\rm in}\, \Omega,}\\ {u=v=0 \quad\quad\quad\quad\quad\quad\quad\quad\quad{\rm on}\, \partial \Omega}, \end{array}\right.$$ where ${\Omega\ni 0}$ is a bounded domain in ${\mathbb{R}^{N}}$ , N ≥ 3, p, q > 1, and α, β > ?N. We also study symmetry breaking for ground states when Ω is the unit ball in ${\mathbb{R}^{N}}$ .     

On strictly positive solutions for some semilinear elliptic problems

Let B be a ball in ${\mathbb{R}^{N}}$ , N ≥ 1, let m be a possibly discontinuous and unbounded function that changes sign in B and let 0 < p < 1. We study existence and nonexistence of strictly positive solutions for semilinear elliptic problems of the form ${-\Delta u=m(x) u^{p}}$ in B, u = 0 on ?B.     

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

The overlap, \({\mathcal{D}_N}\) , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\) . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\) . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\) , and \({\tilde{\gamma}}\) . In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.     

Semiclassical solutions of Schrödinger equations with magnetic fields and critical nonlinearities

We study the following nonlinear Schrödinger equations $$\begin{array}{lll}(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)g(|w|)w; \quad \quad \quad \quad \quad \quad \quad \quad \quad (0.1)\\(-i\varepsilon\nabla+A(x))^2 w + V(x)w = W(x)\left(g(|w|)+|w|^{2^*-2}\right)w,\quad \quad \quad\,\,(0.2)\end{array}$$ for ${w \in H^1\left( \mathbb{R}^N, \mathbb{C} \right)}$ , where g(|w|)w is super linear and subcritical, 2* = 2N/(N ? 2) if N > 2 and = ∞ if N = 2, min V > 0 and inf W > 0. Under proper assumptions we explore the existence and concentration phenomena of semiclassical solutions of (0.1). The most interesting result obtained here refers to the critical case. We establish the existence and describe the concentration of semiclassical ground states of (0.2) provided either min V <  τ ₀ for some τ₀ > 0, or ${\max W > \kappa_{0}}$ for some ${\kappa_0 > 0}$ .     

The influence of sources terms on the boundary behavior of the large solutions of quasilinear elliptic equations: the power like case

We study the explosive expansion near the boundary of the large solutions of the equation $$-\Delta_{p}u+u^{m}=f \quad{\rm in} \Omega$$ where ${\Omega}$ is an open bounded set of ${\mathbb{R}^{N}}$ , N > 1, with adequately smooth boundary, m > p?1 > 0, and f is a continuous nonnegative function in ${\Omega}$ . Roughly speaking, we show that the number of explosive terms in the asymptotic boundary expansion of the solution is finite, but it goes to infinity as m goes to p?1. For illustrative choices of the sources, we prove that the expansion consists of two possible geometrical and nongeometrical parts. For low explosive sources, the nongeometrical part does not exist, and all coefficients depend on the diffusion and the geometry of the domain. For high explosive sources, there are coefficients, relative to the nongeometrical part, independent on ${\Omega}$ and the diffusion. In this case, the geometrical part cannot exist, and we say then that the source is very high explosive. We emphasize that low or high explosive sources can cause different geometrical properties in the expansion for a given interior structure of the differential operator. This paper is strongly motivated by the applications, in particular by the non-Newtonian fluid theory where p ≠ 2 involves rheological properties of the medium.     

Compact Hankel Operators on Generalized Bergman Spaces of the Polydisc

Let ${\vartheta}$ be a measure on the polydisc ${\mathbb{D}^n}$ which is the product of n regular Borel probability measures so that ${\vartheta([r,1)^n\times\mathbb{T}^n) >0 }$ for all 0 < r < 1. The Bergman space ${A^2_{\vartheta}}$ consists of all holomorphic functions that are square integrable with respect to ${\vartheta}$ . In one dimension, it is well known that if f is continuous on the closed disc ${\overline{\mathbb{D}}}$ , then the Hankel operator H _f is compact on ${A^2_\vartheta}$ . In this paper we show that for n ≥ 2 and f a continuous function on ${{\overline{\mathbb{D}}}^n}$ , H _f is compact on ${A^2_\vartheta}$ if and only if there is a decomposition f = h + g, where h belongs to ${A^2_\vartheta}$ and ${\lim_{z\to\partial\mathbb{D}^n}g(z)=0}$ .     

Limiting Behavior of High Order Correlations for Simple Random Sampling

For ${N = 1, 2,\ldots,}$ let S _N be a simple random sample of size n = n _N from a population A _N of size N, where ${0 \leq n \leq N}$ . Then with f _N =  n/N, the sampling fraction, and 1 _A the inclusion indicator that ${A \in S_N}$ , for any ${H \subset A_N}$ of size ${k \geq 0}$ , the high order correlations $${\rm Corr}(k) = E \big(\mathop{\Pi}\limits_{A \in H} ({\bf 1}_A - f_N )\big)$$ depend only on k, and if the sampling fraction ${f_N \rightarrow f}$ as ${N \rightarrow \infty}$ , then $$N^{k/2}{\rm Corr}(k) \rightarrow [f (f - 1)]^{k/2}EZ^{k}, k \,{\rm even}$$ and $$N^{(k+1)/2}{\rm Corr}(k) \rightarrow [f (f - 1)]^{(k-1)/2}(2f - 1)\frac{1}{3}(k - 1)EZ^{k+1}, k \,{\rm odd}$$ where Z is a standard normal random variable. This proves a conjecture given in [2].