The effect of the domain topology on the number of solutions of fractional Laplace problems

In this paper we are concerned with the multiplicity of solutions for the following fractional Laplace problem

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u= \mu |u|^{q-2}u + |u|^{2^*_s-2}u &{}\quad \text{ in } \Omega \\ u=0 &{}\quad \text{ in } {\mathbb {R}}^n{\setminus } \Omega , \end{array}\right. \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^n\) is an open bounded set with continuous boundary, \(n>2s\) with \(s\in (0,1),(-\Delta )^{s}\) is the fractional Laplacian operator, \(\mu \) is a positive real parameter, \(q\in [2, 2^*_s)\) and \(2^*_s=2n/(n-2s)\) is the fractional critical Sobolev exponent. Using the Lusternik–Schnirelman theory, we relate the number of nontrivial solutions of the problem under consideration with the topology of \(\Omega \). Precisely, we show that the problem has at least \(cat_{\Omega }(\Omega )\) nontrivial solutions, provided that \(q=2\) and \(n\geqslant 4s\) or \(q\in (2, 2^*_s)\) and \(n>2s(q+2)/q\), extending the validity of well-known results for the classical Laplace equation to the fractional nonlocal setting.

     

Rational Approximation of Functions in Hardy Spaces

Subsequent to our recent work on Fourier spectrum characterization of Hardy spaces \(H^p({\mathbb {R}})\) for the index range \(1\le p\le \infty ,\) in this paper we prove further results on rational Approximation, integral representation and Fourier spectrum characterization of functions for the Hardy spaces \(H^p({\mathbb {R}}), 0 < p\le \infty ,\) with particular interest in the index range \( 0< p \le 1.\) We show that the set of rational functions in \( H^p({\mathbb {C}}_{+1}) \) with the single pole \(-i\) is dense in \( H^p({\mathbb {C}}_{+1}) \) for \(0<p<\infty .\) Secondly, for \(0<p<1\), through rational function approximation we show that any function f in \(L^p({\mathbb {R}})\) can be decomposed into a sum \(g+h\), where g and h are, in the \(L^p({\mathbb {R}})\) convergence sense, the non-tangential boundary limits of functions in, respectively, \( H^p({\mathbb {C}}_{+1})\) and \(H^{p}({\mathbb {C}}_{-1}),\) where \(H^p({\mathbb {C}}_k)\ (k=\pm 1) \) are the Hardy spaces in the half plane \( {\mathbb {C}}_k=\{z=x+iy: ky>0\}\). We give Laplace integral representation formulas for functions in the Hardy spaces \(H^p,\) \(0<p\le 2.\) Besides one in the integral representation formula we give an alternative version of Fourier spectrum characterization for functions in the boundary Hardy spaces \(H^p\) for \(0<p\le 1\).     

On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts \(\zeta (s+i\tau )\), \(\tau \in \mathbb {R}\), of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts \(\zeta (s+ix_kh)\), \(k\in \mathbb {N}\), \(h>0\), where \(\{x_k\}\subset \mathbb {R}\) is such that the sequence \(\{ax_k\}\) with every real \(a\ne 0\) is uniformly distributed modulo 1, \(1\le x_k\le k\) for all \(k\in \mathbb {N}\) and, for \(1\le k\), \(m\le N\), \(k\ne m\), the inequality \(|x_k-x_m| \ge y^{-1}_N\) holds with \(y_N> 0\) satisfying \(y_Nx_N\ll N\).     

On generalized Ramanujan primes

Let \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) be the reduction mod 2 of the \(\Delta \) series. A modular form of level 1, \(f=\sum _{n\geqslant 0} c(n) \,q^n\), with integer coefficients, is congruent modulo \(2\) to a polynomial in \(\Delta \). Let us set \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { odd }} 1\), the number of odd Fourier coefficients of \(f\) of index \(\leqslant x\). The order of magnitude of \(W_f(x)\) (for \(x\rightarrow \infty \)) has been determined by Serre in the seventies. Here, we give an asymptotic equivalent for \(W_f(x)\). Let \(p(n)\) be the partition function and \(A_0(x)\) (resp. \(A_1(x)\)) be the number of \(n\leqslant x\) such that \(p(n)\) is even (resp. odd). In the preceding papers, the second-named author has shown that \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) for \(x\geqslant 3\) and \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) for \(x\geqslant 7\). Here, it is proved that \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) holds for \(x>1\) and that \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) holds for \(x\geqslant 2\). The main tools used to prove these results are the determination of the order of nilpotence of a modular form of level-\(1\) modulo \(2\), and of the structure of the space of those modular forms as a module over the Hecke algebra, which have been given in a recent work of Serre and the second-named author.     

Classifying First Extremal Points for a Fractional Boundary Value Problem with a Fractional Boundary Condition

Let \(n\in \mathbb {N}\), \(n\ge 2\), \(\beta >0\) fixed, and \(0<b\le \beta \). For \(n-1<\alpha \le n\), we look to classify extremal points for the fractional differential equation \(D_{0^+}^{\alpha }u+p(t) u=0\), satisfying the boundary conditions \(u^{(i)}(0)=0\), \(i=0,\ldots ,n-2\), \(D_{0^+}^\gamma u(b)=0\), where p(t) is a continuous nonnegative function on \([0,\beta ]\) which does not vanish identically on any nondegenerate compact subinterval of \([0,\beta ]\). Using the theory of Krein and Rutman, first extremal points of this boundary value problem are classified. As an application, the results are applied, along with a fixed-point theorem, to show the existence of a solution of a nonlinear fractional boundary value problem.     

The Hardy–Schrödinger operator with interior singularity: the remaining cases

We consider the remaining unsettled cases in the problem of existence of energy minimizing solutions for the Dirichlet value problem \(L_\gamma u-\lambda u=\frac{u^{2^*(s)-1}}{|x|^s}\) on a smooth bounded domain \(\Omega \) in \({\mathbb {R}}^n\) (\(n\ge 3\)) having the singularity 0 in its interior. Here \(\gamma <\frac{(n-2)^2}{4}\), \(0\le s <2\), \(2^*(s):=\frac{2(n-s)}{n-2}\) and \(0\le \lambda <\lambda _1(L_\gamma )\), the latter being the first eigenvalue of the Hardy–Schrödinger operator \(L_\gamma :=-\Delta -\frac{\gamma }{|x|^2}\). There is a threshold \(\lambda ^*(\gamma , \Omega ) \ge 0\) beyond which the minimal energy is achieved, but below which, it is not. It is well known that \(\lambda ^*(\Omega )=0\) in higher dimensions, for example if \(0\le \gamma \le \frac{(n-2)^2}{4}-1\). Our main objective in this paper is to show that this threshold is strictly positive in “lower dimensions” such as when \( \frac{(n-2)^2}{4}-1<\gamma <\frac{(n-2)^2}{4}\), to identify the critical dimensions (i.e., when the situation changes), and to characterize it in terms of \(\Omega \) and \(\gamma \). If either \(s>0\) or if \(\gamma > 0\), i.e., in the truly singular case, we show that in low dimensions, a solution is guaranteed by the positivity of the “Hardy-singular internal mass” of \(\Omega \), a notion that we introduce herein. On the other hand, and just like the case when \(\gamma =s=0\) studied by Brezis and Nirenberg (Commun Pure Appl Math 36:437–477, 1983) and completed by Druet (Ann Inst H Poincaré Anal Non Linéaire 19(2):125–142, 2002), \(n=3\) is the critical dimension, and the classical positive mass theorem is sufficient for the merely singular case, that is when \(s=0\), \(\gamma \le 0\).     

The solution gap of the Brezis–Nirenberg problem on the hyperbolic space

We consider the positive solutions of the nonlinear eigenvalue problem \(-\Delta _{\mathbb {H}^n} u = \lambda u + u^p, \) with \(p=\frac{n+2}{n-2}\) and \(u \in H_0^1(\Omega ),\) where \(\Omega \) is a geodesic ball of radius \(\theta _1\) on \(\mathbb {H}^n.\) For radial solutions, this equation can be written as an ordinary differential equation having n as a parameter. In this setting, the problem can be extended to consider real values of n. We show that if \(2<n<4\) this problem has a unique positive solution if and only if \(\lambda \in \left( n(n-2)/4 +L^*\,,\, \lambda _1\right) .\) Here \(L^*\) is the first positive value of \(L = -\ell (\ell +1)\) for which a suitably defined associated Legendre function \(P_{\ell }^{-\alpha }(\cosh \theta ) >0\) if \(0 < \theta <\theta _1\) and \(P_{\ell }^{-\alpha }(\cosh \theta _1)=0,\) with \(\alpha = (2-n)/2\).     

An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process

Let \(\{X(t):t\in \mathbb R_+\}\) be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, \(\mathbb E X(t) = 0, \mathbb E X^2(t) = 1\) and correlation function satisfying (i) \(r(t) = 1 - C|t|^{\alpha } + o(|t|^{\alpha })\) as \(t\rightarrow 0\) for some \(0\le \alpha \le 2\) and \(C>0\); (ii) \(\sup _{t\ge s}|r(t)|<1\) for each \(s>0\) and (iii) \(r(t) = O(t^{-\lambda })\) as \(t\rightarrow \infty \) for some \(\lambda >0\). For any \(n\ge 1\), consider n mutually independent copies of X and denote by \(\{X_{r:n}(t):t\ge 0\}\) the rth smallest order statistics process, \(1\le r\le n\). We provide a tractable criterion for assessing whether, for any positive, non-decreasing function \(f, \mathbb P(\mathscr {E}_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text { i.o.})\) equals 0 or 1. Using this criterion we find, for a family of functions \(f_p(t)\) such that \(z_p(t)=\mathbb P(\sup _{s\in [0,1]}X_{r:n}(s)>f_p(t))=O((t\log ^{1-p} t)^{-1})\), that \(\mathbb P(\mathscr {E}_{f_p})= 1_{\{p\ge 0\}}\). Consequently, with \(\xi _p (t) = \sup \{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}\), for \(p\ge 0\) we have \(\lim _{t\rightarrow \infty }\xi _p(t)=\infty \) and \(\limsup _{t\rightarrow \infty }(\xi _p(t)-t)=0\) a.s. Complementarily, we prove an Erdös–Révész type law of the iterated logarithm lower bound on \(\xi _p(t)\), namely, that \(\liminf _{t\rightarrow \infty }(\xi _p(t)-t)/h_p(t) = -1\) a.s. for \(p>1\) and \(\liminf _{t\rightarrow \infty }\log (\xi _p(t)/t)/(h_p(t)/t) = -1\) a.s. for \(p\in (0,1]\), where \(h_p(t)=(1/z_p(t))p\log \log t\).     

Contributions to a conjecture of Mueller and Schmidt on Thue inequalities

Let \(F(X,Y)=\sum \nolimits _{i=0}^sa_iX^{r_i}Y^{r-r_i}\in {\mathbb {Z}}[X,Y]\) be a form of degree \(r=r_s\ge 3\), irreducible over \({\mathbb {Q}}\) and having at most \(s+1\) non-zero coefficients. Mueller and Schmidt showed that the number of solutions of the Thue inequality

$$\begin{aligned} |F(X,Y)|\le h \end{aligned}$$

is \(\ll s^2h^{2/r}(1+\log h^{1/r})\). They conjectured that \(s^2\) may be replaced by s. Let

$$\begin{aligned} \Psi = \max _{0\le i\le s} \max \left( \sum _{w=0}^{i-1} \frac{1}{r_i-r_w},\sum _{w= i+1}^{s}\frac{1}{r_w-r_i}\right) . \end{aligned}$$

Then we show that \(s^2\) may be replaced by \(\max (s\log ^3s, se^{\Psi })\). We also show that if \(|a_0|=|a_s|\) and \(|a_i|\le |a_0|\) for \(1\le i\le s-1\), then \(s^2\) may be replaced by \(s\log ^{3/2}s\). In particular, this is true if \(a_i\in \{-1,1\}\).

     

Quantum MDS codes with relatively large minimum distance from Hermitian self-orthogonal codes

It has become common knowledge that constructing q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\) is significantly more difficult than constructing those with minimum distance less than or equal to \(q/2+1\). Despite of various constructions of q-ary quantum MDS codes, all known q-ary quantum MDS codes have minimum distance bounded by \(q/2+1\) except for some lengths. The purpose of the current paper is to provide some new q-ary quantum MDS codes with minimum distance bigger than \(q/2+1\). In this paper, we provide several classes of quantum MDS codes with minimum distance bigger than \(q/2+1\). For instance, some examples in these classes include q-ary \([n,n-2k, k+1]\)-quantum MDS codes for cases: (i) \(q\equiv -1\bmod {5}, n=(q^2+4)/5\) and \(1\le k\le (3q-2)/5\); (ii) \(q\equiv -1\bmod {7}, n=(q^2+6)/7\) and \(1\le k\le (4q-3)/7\); (iii) \(2|q, q\equiv -1\bmod {3}, n=2(q^2-1)/3\) and \(1\le k\le (2q-1)/3\); and (iv) \(2|q, q\equiv -1\bmod {5}, n=2(q^2-1)/5\) and \(1\le k\le (3q-2)/5\).     

On the gaps between non-zero Fourier coefficients of cusp forms of higher weight

We show that if a modular cuspidal eigenform f of weight 2k is 2-adically close to an elliptic curve \(E/\mathbb {Q}\), which has a cyclic rational 4-isogeny, then n-th Fourier coefficient of f is non-zero in the short interval \((X, X + cX^{\frac{1}{4}})\) for all \(X \gg 0\) and for some \(c > 0\). We use this fact to produce non-CM cuspidal eigenforms f of level \(N>1\) and weight \(k > 2\) such that \(i_f(n) \ll n^{\frac{1}{4}}\) for all \(n \gg 0\).     

Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-\({\text {PD}}\)-sets of minimum size \(s+1\) for partial permutation decoding for the binary linear Hadamard code \(H_m\) of length \(2^m\), for all \(m\ge 4\) and \(2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1\), are constructed. Moreover, recursive constructions to obtain s-\({\text {PD}}\)-sets of size \(l\ge s+1\) for \(H_{m+1}\) of length \(2^{m+1}\), from an s-\({\text {PD}}\)-set of the same size for \(H_m\), are also described. These results are generalized to find s-\({\text {PD}}\)-sets for the \({\mathbb {Z}}_4\)-linear Hadamard codes \(H_{\gamma , \delta }\) of length \(2^m\), \(m=\gamma +2\delta -1\), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type \(2^\gamma 4^\delta \). Specifically, s-PD-sets of minimum size \(s+1\) for \(H_{\gamma , \delta }\), for all \(\delta \ge 3\) and \(2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1\), are constructed and recursive constructions are described.     

On a Weighted Singular Integral Operator with Shifts and Slowly Oscillating Data

Let \(\alpha ,\beta \) be orientation-preserving diffeomorphism (shifts) of \(\mathbb {R}_+=(0,\infty )\) onto itself with the only fixed points \(0\) and \(\infty \) and \(U_\alpha ,U_\beta \) be the isometric shift operators on \(L^p(\mathbb {R}_+)\) given by \(U_\alpha f=(\alpha ')^{1/p}(f\circ \alpha )\), \(U_\beta f=(\beta ')^{1/p}(f\circ \beta )\), and \(P_2^\pm =(I\pm S_2)/2\) where

$$\begin{aligned} (S_2 f)(t):=\frac{1}{\pi i}\int \limits _0^\infty \left( \frac{t}{\tau }\right) ^{1/2-1/p}\frac{f(\tau )}{\tau -t}\,d\tau , \quad t\in \mathbb {R}_+, \end{aligned}$$

is the weighted Cauchy singular integral operator. We prove that if \(\alpha ',\beta '\) and \(c,d\) are continuous on \(\mathbb {R}_+\) and slowly oscillating at \(0\) and \(\infty \), and

$$\begin{aligned} \limsup _{t\rightarrow s}|c(t)|<1, \quad \limsup _{t\rightarrow s}|d(t)|<1, \quad s\in \{0,\infty \}, \end{aligned}$$

then the operator \((I-cU_\alpha )P_2^++(I-dU_\beta )P_2^-\) is Fredholm on \(L^p(\mathbb {R}_+)\) and its index is equal to zero. Moreover, its regularizers are described.

     

Blow-up solutions for Hardy–Sobolev equations on compact Riemannian manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension \(n\ge 6\), \(\xi _0\in M\), and we are concerned with the following Hardy–Sobolev elliptic equations:

$$\begin{aligned} -\Delta _gu+h(x)u=\frac{u^{2^{*}(s)-1-\epsilon }}{d_{g}(x,\xi _0)^s},\ \ \ \ u>0\ \ \mathrm{in} \ \ M, \end{aligned}$$

(0.1)

where \(\Delta _g\,=\,\mathrm{div}_g(\nabla )\) is the Laplace–Beltrami operator on M, h(x) is a \(C^1\) function on M, \(\epsilon \) is a sufficiently small real parameter, \(2^{*}(s):=\frac{2(n-s)}{n-2}\) is the critical Hardy–Sobolev exponent with \(s\in (0,2)\), and \(d_{g}\) is the Riemannian distance on M. Performing the Lyapunov–Schmidt reduction procedure, we obtain the existence of blow-up families of positive solutions of problem (0.1).

     

New nonbinary code bounds based on divisibility arguments

For \(q,n,d \in \mathbb {N}\), let \(A_q(n,d)\) be the maximum size of a code \(C \subseteq [q]^n\) with minimum distance at least d. We give a divisibility argument resulting in the new upper bounds \(A_5(8,6) \le 65\), \(A_4(11,8)\le 60\) and \(A_3(16,11) \le 29\). These in turn imply the new upper bounds \(A_5(9,6) \le 325\), \(A_5(10,6) \le 1625\), \(A_5(11,6) \le 8125\) and \(A_4(12,8) \le 240\). Furthermore, we prove that for \(\mu ,q \in \mathbb {N}\), there is a 1–1-correspondence between symmetric \((\mu ,q)\)-nets (which are certain designs) and codes \(C \subseteq [q]^{\mu q}\) of size \(\mu q^2\) with minimum distance at least \(\mu q - \mu \). We derive the new upper bounds \(A_4(9,6) \le 120\) and \(A_4(10,6) \le 480\) from these ‘symmetric net’ codes.     

Extremal Problems for Mappings with Generalized Parametric Representation in $${{\mathbb {C}}}^{n}$$

In this paper we are concerned with the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) (\(t\ge 0\)) of normalized biholomorphic mappings on the Euclidean unit ball \(\mathbb {B}^n\) in \({\mathbb {C}}^n\) that can be embedded in normal Loewner chains whose normalizations are given by time-dependent operators \(A\in \widetilde{\mathcal {A}}\), where \(\widetilde{\mathcal {A}}\) is a family of measurable mappings from \([0,\infty )\) into \(L({\mathbb {C}}^n)\) which satisfy certain natural assumptions. In particular, we consider extreme points and support points associated with the compact family \(\widetilde{S}^t_A(\mathbb {B}^n)\), where \(A\in \widetilde{\mathcal {A}}\). We prove that if \(f(z,t)=V(t)^{-1}z+\cdots \) is a normal Loewner chain such that \(V(s)f(\cdot ,s)\in \mathrm{ex}\,\widetilde{S}^s_A(\mathbb {B}^n)\) (resp. \(V(s)f(\cdot ,s)\in \mathrm{supp}\,\widetilde{S}^s_A(\mathbb {B}^n)\)), then \(V(t)f(\cdot ,t)\in \mathrm{ex}\, \widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\) (resp. \(V(t)f(\cdot ,t)\in \mathrm{supp}\,\widetilde{S}^t_A(\mathbb {B}^n)\), for all \(t\ge s\)), where V(t) is the unique solution on \([0,\infty )\) of the initial value problem: \(\frac{d V}{d t}(t)=-A(t)V(t)\), a.e. \(t\ge 0\), \(V(0)=I_n\). Also, we obtain an example of a bounded support point for the family \(\widetilde{S}_A^t(\mathbb {B}^2)\), where \(A\in \widetilde{\mathcal {A}}\) is a certain time-dependent operator. We also consider the notion of a reachable family with respect to time-dependent linear operators \(A\in \widetilde{\mathcal {A}}\), and obtain characterizations of extreme/support points associated with these families of bounded biholomorphic mappings on \(\mathbb {B}^n\). Useful examples and applications yield that the study of the family \(\widetilde{S}^t_A(\mathbb {B}^n)\) for time-dependent operators \(A\in \widetilde{\mathcal {A}}\) is basically different from that in the case of constant time-dependent linear operators.     

A concave–convex problem with a variable operator

We study the following elliptic problem \(-A(u) = \lambda u^q\) with Dirichlet boundary conditions, where \(A(u) (x) = \Delta u (x) \chi _{D_1} (x)+ \Delta _p u(x) \chi _{D_2}(x)\) is the Laplacian in one part of the domain, \(D_1\), and the p-Laplacian (with \(p>2\)) in the rest of the domain, \(D_2 \). We show that this problem exhibits a concave–convex nature for \(1<q<p-1\). In fact, we prove that there exists a positive value \(\lambda ^*\) such that the problem has no positive solution for \(\lambda > \lambda ^*\) and a minimal positive solution for \(0<\lambda < \lambda ^*\). If in addition we assume that p is subcritical, that is, \(p<2N/(N-2)\) then there are at least two positive solutions for almost every \(0<\lambda < \lambda ^*\), the first one (that exists for all \(0<\lambda < \lambda ^*\)) is obtained minimizing a suitable functional and the second one (that is proven to exist for almost every \(0<\lambda < \lambda ^*\)) comes from an appropriate (and delicate) mountain pass argument.     

Monochromatic Solutions for Multi-Term Unknowns

Given integers \(k\ge 2\), \(n \ge 2\), \(m \ge 2\) and \( a_1,a_2,\ldots ,a_m \in {\mathbb {Z}}{\backslash }{\{0\}}\), and let \(f(z)= \sum _{j=0}^{n}c_jz^j\) be a polynomial of integer coefficients with \(c_n>0\) and \((\sum _{i=1}^ma_i)|f(z)\) for some integer z. For a k-coloring of \([N]=\{1,2,\ldots ,N\}\), we say that there is a monochromatic solution of the equation \(a_1x_1+a_2x_2+\cdots +a_mx_m=f(z)\) if there exist pairwise distinct \(x_1,x_2,\ldots ,x_m\in [N]\) all of the same color such that the equation holds for some \(z\in \mathbb {Z}\). Problems of this type are often referred to as Ramsey-type problems. In this paper, it is shown that if \(a_i>0\) for \(1\le i\le m\), then there exists an integer \(N_0=N(k,m,n)\) such that for \(N\ge N_0\), each k-coloring of [N] contains a monochromatic solution \(x_1,x_2,\ldots ,x_m\) of the equation \(a_1x_1+a_2x_2+ \cdots +a_mx_m= f(z)\). Moreover, if n is odd and there are \(a_i\) and \(a_j\) such that \(a_ia_j<0\) for some \(1 \le i\ne j\le m\), then the assertion holds similarly.     

Generalized Frobenius partitions with 6 colors

We present the generating function for \(c\phi _6(n)\), the number of generalized Frobenius partitions of \(n\) with \(6\) colors, in terms of Ramanujan’s theta functions and exhibit \(2\), and \(3\)-dissections of it that yield the congruences \(c\phi _6(2n+1)\equiv 0~(\text {mod}~4)\), \(c\phi _6(3n+1)\equiv 0~(\text {mod}~3^2)\) and \(c\phi _6(3n+2)\equiv 0~(\text {mod}~3^2)\).     

A potential well for an integral functional with singular Lagrangian

This paper concerns a functional of the form

$$\begin{aligned} \Phi (u)=\int _\Omega L(x,u(x),\nabla u(x))\, dx \end{aligned}$$

on the Sobolev space \(H_0^1(\Omega )\) where \(\Omega \) is a bounded open subset of \({\mathbb {R}}^N\) with \(N\ge 3\) and \(0\in \Omega \). The hypotheses on L ensure that \(u\equiv 0\) is a critical point of \(\Phi \), but allow the Lagrangian to be singular at \(x=0\). It is shown that, under these assumptions, the usual conditions associated with Jacobi (positive definiteness of the second variation of \(\Phi \) at \(u\equiv 0\)), Legendre (ellipticity at \(u\equiv 0\)) and Weierstrass [strict convexity of \(L(x,s,\xi )\) with respect to \(\xi \)] from the calculus of variations are not sufficient ensure that \(u\equiv 0\) is a local minimum of \(\Phi \). Using recent criteria for the existence of a potential well of a \(C^1\)-functional on a real Hilbert space, conditions implying that \(u\equiv 0\) lies in a potential well of \(\Phi \) are established. They are shown to be sharp in some cases.