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We are concerned with the following nonlinear Hartree system − Δ u + P 1 (| x |) u = α 1 | x | − 1 ∗ u 2 u + β | x | − 1 ∗ v 2 u in R 3 , − Δ v + P 2 (| x |) v = α 2 | x | − 1 ∗ v 2 v + β | x | − 1 ∗ u 2 v in R 3 , where P1(r) and P2(r) are positive radial potentials, α1 > 0, α2 > 0 and β ∈ R is a coupling constant. We first study nondegeneracy of ground states for the limit system of the above problem. As applications, we show that the nonlinear Hartree system has infinitely many non-radial positive synchronized solutions, whose energy can be arbitrarily large. [ABSTRACT FROM AUTHOR] more...

In this article, we consider the following elliptic system of Hamiltonian-type on a bounded domain:−Δu=K1(∣y∣)∣v∣p−1v,inB1(0),−Δv=K2(∣y∣)∣u∣q−1u,inB1(0),u=v=0on∂B1(0),\left\{\begin{array}{ll}-\Delta u={K}_{1}\left(| y| ){| v| }^{p-1}v,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ -\Delta v={K}_{2}\left(| y| ){| u| }^{q-1}u,\hspace{1.0em}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=v=0\hspace{1.0em}& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where K1(r){K}_{1}\left(r) and K2(r){K}_{2}\left(r) are positive bounded functions defined in [0,1]\left[0,1], B1(0){B}_{1}\left(0) is the unit ball in RN{{\mathbb{R}}}^{N}, and (p,q)\left(p,q) is a pair of positive numbers lying on the critical hyperbola 1p+1+1q+1=N−2N.\frac{1}{p+1}+\frac{1}{q+1}=\frac{N-2}{N}. Under some suitable further assumptions on the functions K1(r){K}_{1}\left(r) and K2(r){K}_{2}\left(r), we prove the existence of infinitely many nonradial positive solutions whose energy can be made arbitrarily large. Our proof is based on the reduction method. The most ingredients of the article are using the Green representation and estimating the Green function and its regular part very carefully. For this purpose, some more extra ideas and techniques are needed. We believe that our method and techniques can be applied to other related problems. more...

In this article, we study the following nonlinear Schrödinger system −Δu1+V1(x)u1=αu1u2+μu1,x∈R4,−Δu2+V2(x)u2=α2u12+βu22+μu2,x∈R4,\left\{\begin{array}{ll}-\Delta {u}_{1}+{V}_{1}\left(x){u}_{1}=\alpha {u}_{1}{u}_{2}+\mu {u}_{1},& x\in {{\mathbb{R}}}^{4},\\ -\Delta {u}_{2}+{V}_{2}\left(x){u}_{2}=\frac{\alpha }{2}{u}_{1}^{2}+\beta {u}_{2}^{2}+\mu {u}_{2},& x\in {{\mathbb{R}}}^{4},\end{array}\right. with the constraint ∫R4(u12+u22)dx=1{\int }_{{{\mathbb{R}}}^{4}}\left({u}_{1}^{2}+{u}_{2}^{2}){\rm{d}}x=1, where α>0\alpha \gt 0 and α>β\alpha \gt \beta , μ∈R\mu \in {\mathbb{R}}, V1(x){V}_{1}\left(x), and V2(x){V}_{2}\left(x) are bounded functions. Under some mild assumptions on V1(x){V}_{1}\left(x) and V2(x){V}_{2}\left(x), we prove the existence of normalized peak solutions by using the finite dimensional reduction method, combined with the local Pohozaev identities. Because of the interspecies interaction between the components, we aim to obtain some new technical estimates. more...

Let k ≥ 2. A generalization of the well-known Pell sequence is the k-Pell sequence whose first k terms are 0,..., 0, 1 and each term afterwards is given by the linear recurrence p n (k) = 2 P n − 1 (k) + P n − 2 (k) + ⋯ + P n − k (k) . The goal of this paper is to show that 11, 33, 55, 88 and 99 are all repdigits expressible as sum or difference of two k-Pell. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a modified version of Baker-Davenport reduction method (due to Dujella and Pethő). This extends a result of Bravo and Herrera [Repdigits in generalized Pell sequences, Arch. Math. (Brno) 56(4) (2020), 249–262]. [ABSTRACT FROM AUTHOR] more...

Let {Pn}n⩾0 be the Padovan sequence with initial conditions P0=0, P1=1, and P2=1 and the recurrence relation Pn+3=Pn+1 + Pn. Its companion sequence is known as the Perrin sequence {En}n⩾0 that satisfies the same above recurrence relation with the initial conditions E0=3, E1=0 and E2=2. In this paper, we determine all Padovan and Perrin numbers that are concatenations of two distinct base b repdigits with 2 ⩽ b ⩽ 9. As corollary, we prove that the largest Padovan and Perrin numbers which can be representable as a concatenations of two distinct base b repdigits are P 26 = 816 = 2244 ‾ 7 and E 24 = 853 = 31111 ‾ 4 . [ABSTRACT FROM AUTHOR] more...

We are concerned with the following Schrödinger system with coupled quadratic nonlinearity

We consider the following Hénon-type problem with critical growth:

Let Pk and Qk be the k-th terms of the Pell and Pell-Lucas sequences respectively. In this paper, we find all the solutions of the Diophantine equations P n + 1 x ± P n − 1 x = Q m in nonnegative integer variables (m, n, x). [ABSTRACT FROM AUTHOR] more...

Let (Pn)n≥0 be the sequence of Padovan numbers defined by P0 = 0, P1 = 1 = P2, and Pn+3 = Pn+1 +Pn for all n ≥ 0. In this paper, we find all Padovan numbers that are concatenations of two distinct repdigits. [ABSTRACT FROM AUTHOR] more...

In this paper, we find all the solutions of the title Diophantine equation in positive integers (m, n, k, x), where Pi is the ith term of the Pell sequence. [ABSTRACT FROM AUTHOR]

In this paper, we find all the solutions of the title Diophantine equation in positive integers (m, n, k, x), where Pi is the ith term of the Pell sequence. [ABSTRACT FROM AUTHOR] more...

In this paper, by using lower bounds for linear forms in logarithms of algebraic numbers and the theory of continued fractions, we find all Fibonacci numbers that appear in generalized Pell sequences. Some interesting estimations involving generalized Pell numbers, that we believe are of independent interest, are also deduced. This paper continues a previous work that searched for Fibonacci numbers in the Pell sequence. [ABSTRACT FROM AUTHOR] more...

Let {Bn}n≥0 be the sequence of Balancing numbers defined by B0 = 0, B1 = 1, and Bn+2 = 6Bn+1 − Bn for all n ≥ 0. In this paper, we find all repdigits in base 10 which can be written as a sum of three Balancing numbers. [ABSTRACT FROM AUTHOR] more...