A Neumann problem of Ambrosetti–Prodi type

The aim of this paper is to establish an Ambrosetti–Proditype result for the problem

$$\left\{ \begin{array}{ll}-\Delta{u} = g(x, u,\nabla{u}) + t\varphi \quad {\rm in}\, \Omega,\\ \frac{\partial{u}}{\partial\eta} = 0 \qquad\qquad\qquad\quad {\rm on}\, \partial\Omega ;\end{array} \right.$$

i.e., under appropriate conditions, we will show that there exists a constant t ₀ such that the problem above has no solution if t > t ₀, at least a solution if t =  t ₀ and at least two solutions if t < t ₀. The proof is based on a combination of upper and lower solutions method and the Leray–Schauder degree.

     

Multiple solutions for a superlinear and periodic elliptic system on $${\mathbb{R}^N}$$

This paper is concerned with the following periodic Hamiltonian elliptic system

$$\left \{\begin{array}{l}-\Delta u+V(x)u=g(x,v)\, {\rm in }\,\mathbb{R}^N,\\-\Delta v+V(x)v=f(x,u)\, {\rm in }\, \mathbb{R}^N,\\ u(x)\to 0\, {\rm and}\,v(x)\to0\, {\rm as }\,|x|\to\infty,\end{array}\right.$$

where the potential V is periodic and 0 lies in a gap of the spectrum of ?Δ + V, f(x, t) and g(x, t) depend periodically on x and are superlinear but subcritical in t at infinity. By establishing a variational setting, existence of a ground state solution and multiple solution for odd f and g are obtained.

     

Infinitely many sign-changing solutions of an elliptic problem involving critical Sobolev and Hardy–Sobolev exponent

We study the existence and multiplicity of sign-changing solutions of the following equation

$$\begin{array}{@{}rcl@{}} \left\{\begin{array}{lllllllll} -{\Delta} u = \mu |u|^{2^{\star}-2}u+\frac{|u|^{2^{*}(t)-2}u}{|x|^{t}}+a(x)u \quad\text{in}\, {\Omega}, \\ u=0 \quad\text{on}\quad\partial{\Omega}, \end{array}\right. \end{array} $$

where Ω is a bounded domain in \(\mathbb {R}^{N}\), 0∈?Ω, all the principal curvatures of ?Ω at 0 are negative and μ≥0, a>0, N≥7, 0<t<2, \(2^{\star }=\frac {2N}{N-2}\) and \(2^{\star }(t)=\frac {2(N-t)}{N-2}\).

     

A threshold phenomenon for embeddings of $${H^m_0}$$ into Orlicz spaces

Given an open bounded domain \({\Omega\subset\mathbb {R}^{2m}}\) with smooth boundary, we consider a sequence \({(u_k)_{k\in\mathbb{N}}}\) of positive smooth solutions to

$\left\{\begin{array}{ll} (-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2} \quad\quad\quad\quad\quad {\rm in}\,\Omega\\ u_k=\partial_\nu u_k=\cdots =\partial_\nu^{m-1} u_k=0 \quad {\rm on }\, \partial \Omega, \end{array}\right.$

where λ _k → 0⁺. Assuming that the sequence is bounded in \({H^m_0(\Omega)}\) , we study its blow-up behavior. We show that if the sequence is not precompact, then

$\liminf_{k\to\infty}\|u_k\|^2_{H^m_0}:=\liminf_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx\geq \Lambda_1,$

where Λ₁ = (2m ? 1)!vol(S ^2m ) is the total Q-curvature of S ^2m .

     

On the equation $${{\rm det}\,\nabla{u}=f}$$ with no sign hypothesis

We prove existence of \({u\in C^{k}(\overline{\Omega};\mathbb{R}^{n})}\) satisfying

$\left\{\begin{array}{ll} det\nabla u(x) =f(x) \, x\in \Omega\\ u(x) =x \quad\quad\quad\quad x\in\partial\Omega\end{array}\right.$

where k ≥ 1 is an integer, \({\Omega}\) is a bounded smooth domain and \({f\in C^{k}(\overline{\Omega}) }\) satisfies

$\int\limits_{\Omega}f(x) dx={\rm meas} \Omega$

with no sign hypothesis on f.

     

Quasilinear parabolic problem with <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">x</Emphasis>)-laplacian: existence,uniqueness of weak solutions and stabilization

We discuss the existence and uniqueness of the weak solution of the following quasilinear parabolic equation

$$\left\{\begin{array}{ll}u_t-\Delta _{p(x)}u = f(x,u)&\quad \text{in }\quad Q_T \stackrel{{\rm{def}}}{=} (0,T)\times\Omega,\\u = 0 & \quad\text{on}\quad \Sigma_T\stackrel{{\rm{def}}}{=} (0,T)\times\partial\Omega,\\u(0,x)=u_0(x)& \quad \text{in}\quad \Omega \end{array}\right.\quad\quad (P_{T})$$

involving the p(x)-laplacian operator. Next, we discuss the global behaviour of solutions and in particular some stabilization properties.

     

$${\sigma_2}$$-Energy as a polyconvex functional on a space of self-maps of annuli in the multi-dimensional calculus of variations

Let \({\mathbb{X} \subset \mathbb {R}^n}\) be a bounded Lipschitz domain and consider the energy functional

$${{\mathbb F}_{\sigma_2}}[u; \mathbb{X}] := \int_\mathbb{X} {\mathbf F}(\nabla u) \, dx,$$

over the space of admissible maps

$${{\mathcal {A}_\varphi}(\mathbb{X}) :=\{u \in W^{1,4}(\mathbb{X}, {\mathbb{R}^n}) : {\rm det}\, \nabla u > 0\, {\rm for}\, {\mathcal {L}^n}{\rm -a.e. in}\, \mathbb{X}, u|_{\partial \mathbb{X}} =\varphi \}},$$

where the integrand \({{\mathbf F}\colon \mathbb M_{n\times n}\to \mathbb{R}}\) is quasiconvex and sufficiently regular. Here our attention is paid to the prototypical case when \({{\mathbf F}(\xi):=\frac{1}{2}\sigma_2(\xi)+\Phi(\det\xi)}\). The aim of this paper is to discuss the question of multiplicity versus uniqueness for extremals and strong local minimizers of \({\mathbb F_{\sigma_2}}\) and the relation it bares to the domain topology. In contrast, for constructing explicitly and directly solutions to the system of Euler–Lagrange equations associated to \({{\mathbb F}_{\sigma_2}}\), we use a topological class of maps referred to as generalised twists and relate the problem to extremising an associated energy on the compact Lie group \({\mathbf {SO}(n)}\). The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions amongst such maps whereas in odd dimensions this number reduces to one.

     

Some Generic Properties of Non Degeneracy for Critical Points of Functionals and Applications

We give some generic properties of non degeneracy for critical points of functionals. We apply these results, obtaining some theorems of multiplicity of solutions for the equation

$ \left\{\begin{array}{ll} -\varepsilon^2\Delta_g u + u = |u|^{p-2}u \quad {\rm in}\ M \\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad, \\ u \in H_g^1(M) \end{array}\right. $

where M is a compact Riemannian manifold of dimension n and \({2 < p < \frac{2n}{n\,-\,2}}\).

     

Multiple solutions to a magnetic nonlinear Choquard equation

We consider the stationary nonlinear magnetic Choquard equation

$(- {\rm i}\nabla+ A(x))^{2}u + V (x)u = \left(\frac{1}{|x|^{\alpha}}\ast |u|^{p}\right) |u|^{p-2}u,\quad x\in\mathbb{R}^{N}$

where A is a real-valued vector potential, V is a real-valued scalar potential, N ≥ 3, \({\alpha \in (0, N)}\) and 2 ? (α/N) < p < (2N ? α)/(N?2). We assume that both A and V are compatible with the action of some group G of linear isometries of \({\mathbb{R}^{N}}\) . We establish the existence of multiple complex valued solutions to this equation which satisfy the symmetry condition

$u(gx) = \tau(g)u(x)\quad{\rm for\, all }\ g \in G,\;x \in \mathbb{R}^{N},$

where \({\tau : G \rightarrow \mathbb{S}^{1}}\) is a given group homomorphism into the unit complex numbers.

     

Three solutions for a perturbed Navier problem

In this paper we prove the existence of at least three distinct solutions to the following perturbed Navier problem:

$$\left\{\begin{array}{ll}\Delta (|{\Delta u}|^{p-2}\Delta u) = f(x,u) + \lambda g(x,u) \quad{\rm in}\,\,\,\Omega \\ u=\Delta u = 0 \qquad\qquad\qquad\qquad\qquad\quad{\rm on}\,\,\, \partial \Omega,\end{array}\right.$$

where \({{\Omega \subset \mathbb {R}^N}}\) is an open bounded set with smooth boundary \({\partial \Omega}\) and \({\lambda \in \mathbb {R}}\) . Under very mild conditions on g and some assumptions on the behaviour of the potential of f at 0 and +∞, our result assures the existence of at least three distinct solutions to the above problem for λ small enough. Moreover such solutions belong to a ball of the space \({W^{2,p}(\Omega)\cap W_0^{1,p}(\Omega)}\) centered in the origin and with radius not dependent on λ.

     

Exact multiplicity results for a singularly perturbed Neumann problem

In this paper we study the number of the boundary single peak solutions of the problem $$\left\{\begin{array}{ll} -\varepsilon^{2} \Delta u + u = u^{p}, \quad {\rm in}\, \Omega \\ u > 0, \quad\quad\quad\quad\quad\quad {\rm in}\, \Omega \\ \frac{\partial u}{\partial {\nu}} = 0, \quad\quad\quad\quad\quad\,\,\, {\rm on}\, \partial {\Omega}\end{array}\right.$$ for ${\varepsilon}$ small and p subcritical. Under some suitable assumptions on the shape of the boundary near a critical point of the mean curvature, we are able to prove exact multiplicity results. Note that the degeneracy of the critical point is allowed.     

Waist and trunk of knots

Let K be a knot in the 3-sphere S ³. We define the waist of K as

$waist (K) = \mathop{\rm max}\limits_{F\in\mathcal{F}} \mathop{\rm min}\limits_{D\in\mathcal{D}_{F}} |D \cap K|,$

where \({\mathcal{F}}\) is the set of all closed incompressible surfaces in S ³?K and \({\mathcal{D}_F}\) is the set of all compressing disks for F in S ³. We define the trunk of K as

$trunk(K) = \mathop{\rm min}\limits_{h\in\mathcal{H}} \mathop{\rm max}\limits_{t\in\mathbb{R}} |h^{-1}(t) \cap K|,$

where \({\mathcal{H}}\) is the set of all Morse function \({h : S^3 \to \mathbb{R}}\) with two critical points. We show that

$waist (K) \le \frac{trunk(K)}{3}$

.

     

Stability of some versions of the Prékopa–Leindler inequality

We study the behavior of positive solutions of the following Dirichlet problem

$$\left \{ \begin{array}{ll} -\Delta_{p}u=\lambda u^{s-1}+u^{q-1} &\quad {\rm in}\enspace \Omega \\ u_{\mid\partial \Omega}=0 \end{array}\right. $$

when s → p ^?. Here \({p >1 , s\,{\in}\,]1,p]}\) and q > p with \({q\leq\frac{Np}{N-p}}\) if N > p.

     

Some borderline cases of nonlinear parabolic equations with irregular data

In this paper, we prove existence of solutions for nonlinear parabolic equations whose model is

$$u' - {\rm div} \, (|\nabla u|^{p-2}\nabla u) = f \quad {\rm on} \, \Omega \times (0,T),$$

with homogeneous Cauchy–Dirichlet boundary conditions, where \({1 < p < 2}\). Here f belongs to L ¹ or to L ^m , with m “small.”

     

Combinatorial Properties of Rogers-Ramanujan Type Identities Arising from Hall-Littlewood Polynomials

Here we consider the q-series coming from the Hall-Littlewood polynomials,

$$\begin{array}{l}{R_{v} (a, b; q) = {\sum_{\mathop {\lambda}\limits_{\lambda_1 \leq a}}} q^{c | \lambda |} P_{2\lambda}(1, q, q^{2}, \ldots ; q^{2b+d}).}\end{array}$$

These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence

$$p(5n+4) \equiv 0\quad ({\rm mod}\, 5).$$

     

The range of approximate unitary equivalence classes of homomorphisms from AH-algebras

Let C be a unital AH-algebra and A be a unital simple C*-algebras with tracial rank zero. It has been shown that two unital monomorphisms \({\phi, \psi: C\to A}\) are approximately unitarily equivalent if and only if

$ [\phi]=[\psi]\quad {\rm in}\quad KL(C,A)\quad {\rm and}\quad \tau\circ \phi=\tau\circ \psi \quad{\rm for\, all}\tau\in T(A),$

where T(A) is the tracial state space of A. In this paper we prove the following: Given \({\kappa\in KL(C,A)}\) with \({\kappa(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}\) and with κ([1 _C ]) = [1 _A ] and a continuous affine map \({\lambda: T(A)\to T_{\mathfrak f}(C)}\) which is compatible with κ, where \({T_{\mathfrak f}(C)}\) is the convex set of all faithful tracial states, there exists a unital monomorphism \({\phi: C\to A}\) such that

$[\phi]=\kappa\quad{\rm and}\quad \tau\circ \phi(c)=\lambda(\tau)(c)$

for all \({c\in C_{s.a.}}\) and \({\tau\in T(A).}\) Denote by \({{\rm Mon}_{au}^e(C,A)}\) the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map

$\Lambda: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++},$

where KLT(C, A)⁺⁺ is the set of compatible pairs of elements in KL(C, A)⁺⁺ and continuous affine maps from T(A) to \({T_{\mathfrak f}(C).}\) Moreover, we found that there are compact metric spaces X, unital simple AF-algebras A and \({\kappa\in KL(C(X), A)}\) with \({\kappa(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}}\) for which there is no homomorphism h: C(X) → A so that [h] = κ.

     

A note on existence of antisymmetric solutions for a class of nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation

$$-\triangle u + V(x)u= f(u)\quad {\rm in}\quad \mathbb{R}^N.$$

We assume that V is invariant under an orthogonal involution and show the existence of a particular type of sign changing solution. The basic tool employed here is the Concentration–Compactness Principle.

     

On an elliptic Kirchhoff-type problem depending on two parameters

In this paper, on a bounded domain \({\Omega\subset {\bf R}^n}\), we consider a non-local problem of the type

$\left\{\begin{array}{l}-K\left(\int_{\Omega}|\nabla u(x)|^2dx\right)\Delta u =\lambda f(x,u)+\mu g(x,u) \quad {\rm in}\,\,\Omega\\ u=0 \quad {\rm on}\,\,\partial\Omega.\end{array}\right.$

Under rather general assumptions on K and f, we prove, in particular, that there exists λ* > 0 such that, for each λ > λ* and each Carathéodory function g with a sub-critical growth, the above problem has at least three weak solutions for every μ ≥ 0 small enough.

     

Congruences for 5-regular partitions modulo powers of 5

Let \(b_{5}(n)\) denote the number of 5-regular partitions of n. We find the generating functions of \(b_{5}(An+B)\) for some special pairs of integers (A, B). Moreover, we obtain infinite families of congruences for \(b_{5}(n)\) modulo powers of 5. For example, for any integers \(k\ge 1\) and \(n\ge 0\), we prove that

$$\begin{aligned} b_{5}\left( 5^{2k-1}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}) \end{aligned}$$

and

$$\begin{aligned} b_{5}\left( 5^{2k}n+\frac{5^{2k}-1}{6}\right) \equiv 0 \quad (\mathrm{mod}\, 5^{k}). \end{aligned}$$

     

Thick clusters for the radially symmetric nonlinear Schrödinger equation

This article is devoted to the study of radially symmetric solutions to the nonlinear Schrödinger equation

$\varepsilon^2 \Delta u - V(r)u + |u|^{p-1}u = 0\, {\rm in} B,\quad \frac{\partial u}{\partial n} = 0\, {\rm on}\,{\partial}B,$

where B is a ball in \({\mathbb{R}}^N\) , 1 <  p <  (N +  2)/(N ? 2), N ≥ 3 and the potential V is radially symmetric. We construct positive clustering solutions in an annulus having O(1/?) critical points, as well as sign changing solutions with O(1/?) zeroes concentrating near zero.