Multi-Peak Solutions for Coupled Nonlinear Schrodinger Systems in Low Dimensions

Applied Mathematics & Optimization (2023) 88:4 https://doi.org/10.1007/s00245-023-09974-4 Multi-Peak Solutions for Coupled Nonlinear Schrödinger Systems in Low Dimensions Maoding Zhen1·Binlin Zhang2·Vicen¸tiu D. R˘adulescu3,4,5 Accepted: 1 February 2023 / Published online: 10 April 2023 © The Author(s) 2023 Abstract In this paper, we construct the solutions to the following nonlinear Schrödinger system −2u+P(x)u=μ1up+βup−1 2vp+1 2in RN, −2v +Q(x)v =μ2vp+βup+1 2vp−1 2in RN, where 3 <p<+∞,N∈{1,2},>0 is a small parameter, the potentials P,Q satisfy 0 <P0≤P(x)≤P1and Q(x)satisfies 0 <Q0≤Q(x)≤Q1. We construct the solution for attractive and repulsive cases. When x0is a local maximum point of the potentials Pand Qand P(x0)=Q(x0), we construct kspikes concentrating near the local maximum point x0. When x0is a local maximum point of Pand x0 is a local maximum point of Q, we construct kspikes of uconcentrating at the local maximum point x0and mspikes of vconcentrating at the local maximum point x0 when x0= x0.This paper extends the main results established by Peng and Wang BVicen¸tiu D. R˘adulescu [email protected].ro Maoding Zhen [email protected] Binlin Zhang [email protected] 1School of Mathematics, Hefei University of Technology, Hefei 230009, People’s Republic of China 2College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, People’s Republic of China 3Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland 4Faculty of Electrical Engineering and Communication, Brno University of Technology, Technická 3058/10, 61600 Brno, Czech Republic 5Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania 123 4Page 2 of 56 Applied Mathematics & Optimization (2023) 88 :4 (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case N=3, p=3. Keywords Nonlinear Schrödinger system ·Lyapunov–Schmidt reduction · Singularity ·Perturbation Mathematics Subject Classification 35B40 ·35B20 ·35J47 ·35Q40 1 Introduction and Main Results In this paper, we construct the solutions for the following nonlinear Schrödinger system −2u+P(x)u=μ1up+βup−1 2vp+1 2in RN, −2v +Q(x)v =μ2vp+βup+1 2vp−1 2in RN,(1.1) where >0 is a small parameter, 5 ≤p<+∞,N∈{1,2}, and the potentials P,Q satisfy 0 <P0≤P(x)≤P1,Q(x), respectively 0 <Q0≤Q(x)≤Q1. The use of the Lyapunov–Schmidt reduction method to construct solutions for the nonlinear Schrödinger equation attracted much attention in the last decade, starting from the pioneering contribution by Floer and Weinstein [11]. Noussair and Yan [21] considered multi-peak solutions for the following problem −2u+u=Q(x)|u|q−2u,x∈RN, u∈H1(RN), (1.2) when x0is a local maximum point of Q(x)and is sufficiently small, they proved that for each positive integer k, problem (1.2) has a positive solution with k-peaks concentrating near x0. Wei and Yan [32] studied the following nonlinear Schroödinger equation −u+V(|x|)u=up,u>0x∈RN, u∈H1(RN), (1.3) when 1 <p<(N+2)/(N−2)and V(|x|)is positive function with following expansion V(|x|)=V0+a rm+O1 rm+θas r→+∞. They proved that problem (1.3) has infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. For more results about the nonlinear Schrödinger equation, we refer the reader to [1,6,8–10,12] and the references therein. 123 Applied Mathematics & Optimization (2023) 88 :4 Page 3 of 56 4 Inspired by the work of [21,32], when N=3,p=3,=1, Peng and Wang [23] considered system (1.1), where P(x)and Q(x)satisfy the following hypotheses: (P) There are constants a∈R,m>1,and θ>0 such that as r→+∞ P(|x|)=1+a |x|m+O1 |x|m+θ,as |x|→∞.(H1) (Q) There are constants b∈R,n>1,and >0 such that as r→+∞ Q(|x|)=1+b |x|n+O1 |x|n+,as |x|→∞.(H2) By using the number of the bumps of the solutions as a parameter, for the repulsive case, they constructed non-radial positive vector solutions of segregated type, for the attractive case, they constructed non-radial positive vector solutions of synchronized type. When N=3,p=3, Peng and Pi [22] constructed kinteracting spikes for unear the local maximum point x0of P(x)and minteracting spikes for vnear the local maximum point x0of Q(x), respectively, when x0= x0. Tang and Xie in [27] constructed synchronized positive vector solutions for small. Since it seems to be difficult to provide a complete list of references, we just refer the readers to [2–4,7, 13,14,17–20,22–26,28–30,33,34] and the references therein. In this paper, we have been inspired by the analysis developed in [21–23,31,32] for scalar nonlinear elliptic equations (systems), in particular, by the ideas introduced by Noussair and Yan [21] to deal with nonlinear elliptic equations. Compared with the single scalar equation, we encounter some new difficulties in estimates due to the nonlinear coupling. Firstly, we need to establish non-degenerate results for the solutions of the coupled system, which will be used to prove the invertibility of the operator for the repulsive case. The difficulty is that we need to give an exact integral estimate for the coupled term and the system is more complicated than single equations. We point out that the sign of βhas great influence on the structure of the solutions. Roughly speaking, for the repulsive case, the solutions are small perturbations of (U,V), where (U,V)are scaling and translation of the solution of −u+λu=up. For the attractive case, the solutions are small perturbations of (Uμ,Uν), where Uμ are scaling and translation of the solution of −u+λu=μupand Uνare scaling and translation of the solution of −u+λu=νup. In this paper, we examine how potentials and the interspecies scattering length β influence the structure of solutions to problem (1.1), which improves the results of [19,22] for least energy solutions. We study the existence of high energy solutions to problem (1.1) and provide not only the locations of spikes, but also much finer information on the interaction of spikes. Furthermore, we prove the attractive phenomenon for β<0 and the repulsive phenomenon for β>0. Define the function space H=(u,v)∈H1(RN)×H1(RN):RNP(x)u2dx <+∞,RNQ(x)v2dx <+∞ 123 4Page 4 of 56 Applied Mathematics & Optimization (2023) 88 :4 endowed with the following norm: (u,v)2=(u,v),(u,v)=u2 ,P+v2 ,Q,∀(u,v)∈H, where H1=H1(RN)is the usual Sobolev space, u,P=u,u,P=RN2|∇u|2+P(x)u2dx and v,P=v,v,Q=RN2|∇v|2+Q(x)v2dx. Define E=(ϕ, ψ) ∈H1(RN)×H1(RN), (ϕ, ψ), ∂U,x, j ∂yi ,∂V,x, j ∂yi=0 j=1,2···k;i=1,2···N, where (u,v),(g,h)=RN (2∇u∇g+P(x)ug +2∇v∇h+Q(x)vh)dx. Consider the following system −u+λu=μ1up+βup−1 2vp+1 2in RN, −v +λv =μ2vp+βup+1 2vp−1 2in RN.(1.4) Let Wbe the unique solution of −u+λu=up,in RN, u>0inRN,u(x)→0as|x|→+∞.(1.5) Then (U,V)=(k1W,τ 0k1W) in H1(RN)×H1(RN)is a solution of (1.4), where λ:= P(x0)=Q(x0)and τ0 satisfies μ1+βτ p+1 2 0−μ2τp−1 0−βτ p−3 2 0=0,kp−1 1=μ1+βτ p+1 2 0−1 .(1.6) 123 Applied Mathematics & Optimization (2023) 88 :4 Page 5 of 56 4 Let U,xj, (x), V,xj, (x)=Ux−xj, ,Vx−xj, . In the sequel, we will use U,xj, (x), V,xj, (x)to build up the solutions of (1.1). To show our main results, we first recall some known results from [15]. In the following, let 2∗=+∞,N=1,2, 2N N−2,N≥3. Proposition 1.1 Suppose that 1<p<2∗−1,μ 1>μ 2>0,β>0and one of the following conditions holds (A1)3<p<2∗−1,0<β≤(p−1 2)μ1; (A2)3<p<2∗−1,μ 1≥μ2 2(p+1 p−1)p−1 2,β>( p−1 2)μ1; (A3)3<p<2∗−1,μ 1<μ2 2(p+1 p−1)p−1 2,( p−1 2)μ1≤β≤β0or β≥ max{β1,(p−1 2)μ1}. Then problem (1.4)has s positive solution (U,V)=(k1W,τ 0k1W)in H1(RN)× H1(RN)which is non-degenerate, where τ0satisfies μ1+βτ p+1 2 0−μ2τp−1 0−βτ p−3 2 0=0,kp−1 1=μ1+βτ p+1 2 0−1 . Proposition 1.2 Suppose 1<p<2∗−1,μ1>μ 2>0,β<0.Then there exists a decreasing sequence {βk}⊂(−√μ1μ2,0)such that for β∈(−√μ1μ2,0)\{βk}, problem (1.4)has s positive solution (U,V)=(k1W,τ 0k1W)in H1(RN)×H1(RN) which is non-degenerate, where τ0satisfies μ1+βτ p+1 2 0−μ2τp−1 0−βτ p−3 2 0=0,kp−1 1=μ1+βτ p+1 2 0−1 . Remark 1.1 Although the authors in [15] dealt with the existence and the nondegeneracy of positive solutions for the fractional Schrödinger system, the main results are still true for the classical Schrödinger system with s=1. The main results in this paper are stated in what follows. Theorem 1.1 Assume the conditions in Propositions 1.1 or 1.2.Ifx 0is a local maximum point of P(x),Q(x)and P(x0)=Q(x0), then there exists 0>0such that for any ∈(0, 0], problem (1.1)has a solution of the form u= k  j=1 U,xj, +ϕ,v = k  j=1 V,xj, +ψ, 123 4Page 6 of 56 Applied Mathematics & Optimization (2023) 88 :4 for some x j, ∈Bδ(x0)and (ϕ,ψ )=O( N 2+1). Moreover, as →0,xj, → x0,|xi, −xj, | →+∞if i = j. Set λ=P(x0),λ=Q(x0), it is easy to see that Uλ,μ =λ 1 p−1μ−1 p−1W(√λx)is a solution of −u+λu=μup,in RN, u>0inRN,u(x)→0as|x|→+∞, and Uλ,ν =λ 1 p−1ν−1 p−1W(√λx)is a solution of −u+λu=νup,in RN, u>0inRN,u(x)→0as|x|→+∞. Let U,xj,μ(x), U,zj,ν(x)=Uλj,μ x−xj ,Uλj,ν x−zj , where xj∈Bδ(x0), zj∈Bδ(x0),  E=(ϕ, ψ) ∈H1(RN)×H1(RN), ϕ, ∂U,xj,μ ∂xj,l=0, ψ, ∂U,zj,ν ∂zj,l=0,j=1,2...k;l=1,2...N. We will use U,xj,μ(x), U,zj,ν(x)to build up the solutions of problem (1.1). Theorem 1.2 Suppose that x0is a local maximum point of P(x)and x0is a local maximum point of Q(x), with x0= x0. Then there exists β∗>0depending on x0and x0such that for all β<β ∗, there exists 0>0such that for any ∈(0, 0], problem (1.1)has a solution of the form u= k  j=1 U,xj, ,μ(x)+ϕ,v = m  j=1 U,zj, ,ν(x)+ψ for some x j, ∈Bδ(x0),z j, ∈Bδ(x0),(ϕ, ψ)=O( N 2+1). Moreover, as →0,xj, →x0,|xi, −xj, | →+∞if i = j,i=1,2,...k,and z j, → x0,|zi, −zj, | →+∞i= j,i=1,2,...m. Remark 1.2 The conditions in Theorem 1.1 ensure the existence and the nondegeneracy of positive solutions (U,V)to the related limit problem (1.4), hence the reduction procedure can be carried out successfully. The condition β<β ∗in 123 Applied Mathematics & Optimization (2023) 88 :4 Page 7 of 56 4 Theorem 1.2 is necessary to prove that QB(see (2.5)) is invertible and the inverse operator is bounded. Remark 1.3 Compared with the well-studied case N=3,p=3, in order to get an accurate error estimate and use the Contraction Mapping Theorem to prove that problem (2.54) has a unique solution, our situation is much more complicated. In this sense, we complement the main results established by Peng and Wang (Arch Ration Mech Anal 2013) and Peng and Pi (Discrete Contin Dyn Syst 2016), where the authors considered the case N=3,p=3. The paper is organized as follows. In Sect. 2, we introduce some preliminaries that will be used to prove Theorems 1.1–1.2. In Sect. 3, we prove Theorem 1.1. In Sect. 4, we prove Theorem 1.2. Finally, we give some elementary computations in Appendix A. Throughout this paper, C,Ci,i=1,2,... will always denote various generic positive constants, while O(t)and o(t)denote C1≤|O(t)| |t|≤C2and |o(t)| |t|→0as t→0,respectively. 2 Preliminary Results We first give the definition of multi-peak solutions of system (1.1). Definition 2.1 Let k∈N,1 ≤j≤k.We say that (u,v )is k-peak solutions of system (1.1) concentrated at {x1,x2,...,xk,}if (u,v )satisfies the following properties. (i) (u,v )has k local maximum points xj, ∈RN,j=1,2,...,ksatisfying xj, →xjas →0 for each j. (ii) For any given τ>0, there exists R1, such that |u(x)|≤τ, |v(x)|≤τfor x∈RN\∪k jBR(xj,). (iii) There exists C>0 such that RN 2(|∇u|2+|∇v|2)+u2 +v2 ≤CN. Let x0be the local maximum points of P(x),Q(x)and P(x0)=Q(x0). We want to construct a solution (u,v )of the following form u= k  j=1 U,xj, +ϕ,v = k  j=1 V,xj, +ψ 123 4Page 8 of 56 Applied Mathematics & Optimization (2023) 88 :4 where xj, →x0and (ϕ,ψ )2=o(N)as →0. Then, (ϕ,ψ )satisfies the following equation B(ϕ,ψ )+l=R(ϕ,ψ ), x∈RN, (ϕ,ψ )∈H1(RN)×H1(RN), (2.1) where Bis a bounded linear operator in H1(RN)×H1(RN), defined by B(ϕ,ψ ), (g,h) =RN⎛ ⎜ ⎝2∇ϕ∇g+P(x)ϕg−pμ⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−1 ϕg⎞ ⎟ ⎠dx +RN⎛ ⎜ ⎝2∇ψ∇h+Q(x)ψh−pν⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−1 ψh⎞ ⎟ ⎠dx −βRN⎛ ⎜ ⎝p−1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−3 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p+1 2 ϕg +p−1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p+1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−3 2 ψh⎞ ⎟ ⎠dx −βRN⎛ ⎜ ⎝p+1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−1 2 ψg +p+1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−1 2 ϕh⎞ ⎟ ⎠dx (2.2) for all (g,h)∈H1(RN)×H1(RN). l∈H1(RN)×H1(RN)satisfying l,(g,h) = k  j=1RN (P(x)−λ)U,xj, gdx +μRN ( k  j=1 Up ,xj, −⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p gdx + k  j=1RN (Q(x)−λ)V,xj, hdx +νRN ( k  j=1 Vp ,xj, −⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p hdx 123 Applied Mathematics & Optimization (2023) 88 :4 Page 9 of 56 4 +βRN⎛ ⎜ ⎝ k  j=1 U p−1 2 ,xj, V p+1 2 ,xj, −⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p+1 2⎞ ⎟ ⎠gdx +βRN⎛ ⎜ ⎝ k  j=1 U p+1 2 ,xj, V p−1 2 ,xj, −⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p+1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−1 2⎞ ⎟ ⎠hdx, (2.3) for all (g,h)∈H1(RN)×H1(RN). R(ϕ,ψ ), (g,h) =RN⎛ ⎜ ⎝μ⎛ ⎝ k  j=1 U,xj, +ϕ⎞ ⎠ p +β⎛ ⎝ k  j=1 U,xj, +ϕ⎞ ⎠ p−1 2⎛ ⎝ k  j=1 V,xj, +ψ⎞ ⎠ p+1 2⎞ ⎟ ⎠gdx +RN⎛ ⎜ ⎝ν⎛ ⎝ k  j=1 V,xj, +ψ⎞ ⎠ p +β⎛ ⎝ k  j=1 U,xj, +ϕ⎞ ⎠ p+1 2⎛ ⎝ k  j=1 V,xj, +ψ⎞ ⎠ p−1 2⎞ ⎟ ⎠hdx −μRN⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p gdx −βRN⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p+1 2 gdx −νRN⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p h−βRN⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p+1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−1 2 hdx −βRN p−1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−3 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p+1 2 ϕgdx −βRN p−1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p+1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−3 2 ψhdx −βRN p+1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−1 2 ψgdx −βRN p+1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−1 2 ϕhdx,(2.4) for all (g,h)∈H1(RN)×H1(RN). From [8,9], we have following estimates. Lemma 2.1 For any α>0,β >0and l = j, there exists a constant τ>0such that 123 4Page 16 of 56 Applied Mathematics & Optimization (2023) 88 :4 ϕn,m(x)=ϕn(nx+xn,m),  ψn,m(x)=ψn(nx+xn,m), then RN|∇ϕn,m(x)|2+P(nx+xn,m)| ψn,m(x)|2+|∇ϕn,m(x)|2 +Q(nx+xn,m)| ψn,m(x)|2dx =−N nRN (2 n|∇ϕn(x)|2+λ|ψn(x)|2+2 n|∇ϕn(x)|2 +λ|ψn(x)|2)dx =o(1)≤C.(2.21) Therefore, (ϕn,m, ψn,m)(ϕ, ψ) weakly in H1(RN)×H1(RN), (ϕn,m, ψn,m)→(ϕ, ψ) strongly in L2 loc(RN)×L2 loc(RN). Moreover, (ϕ, ψ) satisfies RN∇∂Un,xn,m ∂xl∇ϕ+λ∂Un,xn,m ∂xl ϕ +RN∇∂Vn,xn,m ∂xl∇ψ+λ∂Vn,xn,m ∂xl ψ=0,l=1,2,...N.(2.22) To prove (2.20), we only need to show that (ϕ, ψ) =(0,0). Remark that relation (2.13) holds just for (g,h)∈Ennot for all (g,h)∈H1(RN)×H1(RN).For (g,h)∈H1(RN)×H1(RN),wetake Qn(g,h)=(g,h)− k  j=1 N  i=1 bn,i,j∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xi∈En.(2.23) Then bn,h,m= k  j=1 N  i=1 an,i,j(g,h), ∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xi , 123 Applied Mathematics & Optimization (2023) 88 :4 Page 17 of 56 4 for some constant a,i,j.From(2.23), we have RN (2 n∇ϕn∇g+P(x)ϕng−pμ⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 ϕng)dx +RN⎛ ⎜ ⎝2 n∇ψn∇h+Q(x)ψnh−pν⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 ψnh⎞ ⎟ ⎠dx −βRN p−1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−3 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p+1 2 ϕngdx −βRN p−1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p+1 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−3 2 ψnhdx −βRN p+1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 2 ψngdx −βRN p+1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 2 ϕnhdx =Bn(ϕn,ψ n), (g,h)n=Bn(ϕn,ψ n), Qn(g,h)n + k  j=1 N  i=1 bn,i,jBn(ϕn,ψ n), ∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xin . (2.24) Since Bn(ϕn,ψ n), Qn(g,h)n=QnBn(ϕn,ψ n), Qn(g,h)n =o(1)(ϕn,ψ n)nQn(g,h)n=o N 2 n(g,h)n, (2.25) k  j=1 N  i=1 bn,i,jBn(ϕn,ψ n), ∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xin = k  j=1 N  i=1 a,i,j(g,h), ∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xin ×QnBn(ϕn,ψ n), ∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xin 123 4Page 18 of 56 Applied Mathematics & Optimization (2023) 88 :4 + k  j=1 N  i=1 c,i,j(g,h), ∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xin ×(∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xi ), (∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xi )n = k  j=1 N  i=1 γn,i,j(g,h), ∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xin .(2.26) Substitute (2.25), (2.26)into(2.24), we obtain RN (2 n∇ϕn∇g+P(x)ϕng−pμ⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 ϕng)dx +RN (2 n∇ψn∇h+Q(x)ψnh−pν⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 ψnh)dx −βRN p−1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−3 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p+1 2 ϕngdx −βRN p−1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p+1 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−3 2 ψnhdx −βRN p+1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 2 ψngdx −βRN p+1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 2 ϕnhdx =o( N 2 n)(g,h)n+ k  j=1 N  i=1 γn,i,j(g,h), ∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xin .(2.27) By (2.27) and (ϕn,ψ n)∈En, we can estimate γn,i,jas following k  j=1 N  i=1 γn,i,j∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xi,∂Un,xn,m ∂xl ,∂Vn,xn,m ∂xln+o(N−1 n) =−RN pμ⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 ϕn ∂Un,xn,m ∂xl )dx 123 Applied Mathematics & Optimization (2023) 88 :4 Page 19 of 56 4 −RN pν⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 ψn ∂Vn,xn,m ∂xl )dx −βRN p−1 2( k  j=1 Un,xn,j)p−3 2( k  j=1 Vn,xn,j)p+1 2ϕn ∂Un,xn,m ∂xl dx −βRN p−1 2( k  j=1 Un,xn,j)p+1 2( k  j=1 Vn,xn,j)p−3 2ψn ∂Vn,xn,m ∂xl dx −βRN p+1 2( k  j=1 Un,xn,j)p−1 2( k  j=1 Vn,xn,j)p−1 2ψn ∂Un,xn,m ∂xl dx −βRN p+1 2( k  j=1 Un,xn,j)p−1 2( k  j=1 Vn,xn,j)p−1 2ϕn ∂Vn,xn,m ∂xl dx =: A1+A2+A3+A4+A5+A6.(2.28) On the other hand, from (2.8) and (ϕn,ψ n)∈En,wehave RN pμ(Un,xn,m)p−1ϕn ∂Un,xn,m ∂xl )dx +RN pν(Vn,xn,m)p−1ψn ∂Vn,xn,m ∂xl )dx +βRN p−1 2(Un,xn,m)p−3 2(Vn,xn,m)p+1 2ϕn ∂Un,xn,m ∂xl +βRN p−1 2(Un,xn,m)p+1 2(Vn,xn,m)p−3 2ψn ∂Vn,xn,m ∂xl +βRN p+1 2(Un,xn,m)p−1 2(Vn,xn,m)p−1 2ψn ∂Un,xn,m ∂xl +βRN p+1 2(Un,xn,m)p−1 2(Vn,xn,m)p−1 2ϕn ∂Vn,xn,m ∂xl =RN(λ −P(x))∂Un,xn,m ∂xl ϕn+(λ −Q(x))∂Vn,xn,m ∂xl ψn =O⎛ ⎝RN (λ −P(x))2∂Un,xn,m ∂xl21 2⎞ ⎠(ϕn,ψ n) +O⎛ ⎝RN (λ −Q(x))2∂Vn,xn,m ∂xl21 2⎞ ⎠(ϕn,ψ n) =εN 2O⎛ ⎝RN (λ −P(x+xn,m))2∂Un,xn,m(εx+xn,m) ∂xl21 2⎞ ⎠ (ϕn,ψ n) 123 4Page 20 of 56 Applied Mathematics & Optimization (2023) 88 :4 +εN 2O⎛ ⎝RN (λ −Q(x+xn,m))2∂Vn,xn,m(x+xn,m) ∂xl21 2⎞ ⎠ (ϕn,ψ n) =N nO(|P(xn,m)−λ|+|Q(xn,m)−λ|+n)=o(N−1 n). (2.29) There is a constant σ>0 such that ⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 −Un,xn,mp−1=O⎛ ⎝ k  j=m Uσ n,xn,j⎞ ⎠,(2.30) ⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 −Vn,xn,mp−1 =τp−1 0⎛ ⎜ ⎝⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 −Un,xn,mp−1⎞ ⎟ ⎠=O⎛ ⎝ k  j=m Uσ n,xn,j⎞ ⎠, (2.31) ⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−3 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p+1 2 −Un,xn,jp−3 2(Vn,xn,j)p+1 2 =τ p+1 2 0⎛ ⎜ ⎝⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 −Un,xn,jp−1⎞ ⎟ ⎠=O⎛ ⎝ k  j=m Uσ n,xn,j⎞ ⎠,(2.32) ⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 2 −Un,xn,jp−1 2Vn,xn,jp−1 2 =τ p−1 2 0⎛ ⎜ ⎝⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 −Un,xn,jp−1⎞ ⎟ ⎠=O⎛ ⎝ k  j=m Uσ n,xn,j⎞ ⎠.(2.33) By (2.30)–(2.33) and (2.29), we obtain A1=−RN pμ⎛ ⎜ ⎝⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 −Un,xn,mp−1⎞ ⎟ ⎠ϕn ∂Un,xn,m ∂xl )dx 123 Applied Mathematics & Optimization (2023) 88 :4 Page 21 of 56 4 −RN pμ(Un,xn,m)p−1ϕn ∂Un,xn,m ∂xl dx =O(e−τ n)(ϕn,0)n−RN pμ(Un,xn,m)p−1ϕn ∂Un,xn,m ∂xl dx, A2=−RN pν⎛ ⎜ ⎝⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 −Vn,xn,mp−1⎞ ⎟ ⎠ψn ∂Vn,xn,m ∂xl dx −RN pνVn,xn,mp−1ψn ∂Vn,xn,m ∂xl dx =Oe−τ n(0,ψ n)n−RN pν(Vn,xn,m)p−1ψn ∂Vn,xn,m ∂xl dx, A3=−p−1 2βRNk  j=1 Un,xn,j)p−3 2( k  j=1 Vn,xn,j)p+1 2 −(Un,xn,m)p−3 2(Vn,xn,m)p+1 2ϕn ∂Un,xn,m ∂xl dx −p−1 2βRN (Un,xn,m)p−3 2(Vn,xn,m)p+1 2ϕn ∂Un,xn,m ∂xl dx =O(e−τ n)(ϕn,0)n−p−1 2β ×RN (Un,xn,m)p−3 2(Vn,xn,m)p+1 2ϕn ∂Un,xn,m ∂xl dx, A4=−p−1 2βRN( k  j=1 Un,xn,j)p+1 2( k  j=1 Vn,xn,j)p−3 2 −(Un,xn,m)p+1 2(Vn,xn,m)p−3 2ψn ∂Vn,xn,m ∂xl dx −p−1 2βRN (Un,xn,m)p+1 2(Vn,xn,m)p−3 2ψn ∂Vn,xn,m ∂xl dx =O(e−τ n)(0,ψ n)n−p−1 2β ×RN (Un,xn,m)p+1 2(Vn,xn,m)p−3 2ψn ∂Vn,xn,m ∂xl dx, A5=−p+1 2βRN( k  j=1 Un,xn,j)p−1 2( k  j=1 Vn,xn,j)p−1 2 −(Un,xn,m)p−1 2(Vn,xn,m)p−1 2ψn ∂Un,xn,m ∂xl dx 123 4Page 22 of 56 Applied Mathematics & Optimization (2023) 88 :4 −p+1 2βRN (Un,xn,m)p−1 2(Vn,xn,m)p−1 2ψn ∂Un,xn,m ∂xl dx =O(e−τ n)(0,ψ n)n−p+1 2β RN (Un,xn,m)p−1 2(Vn,xn,m)p−1 2ψn ∂Un,xn,m ∂xl dx, A6=−p+1 2βRN( k  j=1 Un,xn,j)p−1 2( k  j=1 Vn,xn,j)p−1 2 −(Un,xn,m)p−1 2(Vn,xn,m)p−1 2ϕn ∂Vn,xn,m ∂xl dx −p+1 2βRN (Un,xn,m)p−1 2(Vn,xn,m)p−1 2ϕn ∂Vn,xn,m ∂xl dx =O(e−τ n)(ϕn,0)n−p+1 2β RN (Un,xn,m)p−1 2(Vn,xn,m)p−1 2ϕn ∂Vn,xn,m ∂xl dx. Combining (2.28), (2.29) with above A1to A6,wehave k  j=1 N  i=1 γn,i,j∂Un,xn,j ∂xi ,∂Vn,xn,j ∂xi,∂Un,xn,m ∂xl ,∂Vn,xn,m ∂xln=oN−1 n. (2.34) So, by (2.9) and (2.10), we have γn,i,j=o(n). (2.35) Thus, (2.27) becomes RN⎛ ⎜ ⎝2 n∇ϕn∇g+P(x)ϕng−pμ⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 ϕng⎞ ⎟ ⎠dx +RN⎛ ⎜ ⎝2 n∇ψn∇h+Q(x)ψnh−pν⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 ψnh⎞ ⎟ ⎠dx −βRN p−1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−3 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p+1 2 ϕngdx 123 Applied Mathematics & Optimization (2023) 88 :4 Page 23 of 56 4 −βRN p−1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p+1 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−3 2 ψnhdx −βRN p+1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 2 ψngdx −βRN p+1 2⎛ ⎝ k  j=1 Un,xn,j⎞ ⎠ p−1 2⎛ ⎝ k  j=1 Vn,xn,j⎞ ⎠ p−1 2 ϕnhdx =(ϕn,ψ n)n(g,h)n,∀(g,h)∈H1(RN)×H1(RN). (2.36) For any (g,h)∈H1(RN)×H1(RN),weletgn(x)=g(x−xn,m n).Using(2.36), we have RN∇ϕn,m∇g+P(nx+xn,m)ϕn,mg −pμ⎛ ⎝ k  j=1 Un,xn,j(nx+xn,m)⎞ ⎠ p−1 ϕn,mg⎞ ⎟ ⎠dx +RN∇ ψn,m∇h+Q(nx+xn,m) ψn,mh −pν⎛ ⎝ k  j=1 Vn,xn,j(nx+xn,m)⎞ ⎠ p−1  ψn,mh⎞ ⎟ ⎠dx −βRN p−1 2⎛ ⎝( k  j=1 Un,xn,j(nx+xn,m)⎞ ⎠ p−3 2 ⎛ ⎝ k  j=1 Vn,xn,j(nx+xn,m)⎞ ⎠ p+1 2 ϕn,mgdxβRN p−1 2 ⎛ ⎝( k  j=1 Un,xn,j(nx+xn,m)⎞ ⎠ p+1 2 ⎛ ⎝ k  j=1 Vn,xn,j(nx+xn,m)⎞ ⎠ p−3 2  ψn,mhdx 123 4Page 24 of 56 Applied Mathematics & Optimization (2023) 88 :4 −βRN p+1 2⎛ ⎝( k  j=1 Un,xn,j(nx+xn,m)⎞ ⎠ p−1 2 ⎛ ⎝ k  j=1 Vn,xn,j(nx+xn,m)⎞ ⎠ p−1 2  ψn,mgdx −βRN p+1 2 ⎛ ⎝( k  j=1 Un,xn,j(nx+xn,m)⎞ ⎠ p−1 2 ⎛ ⎝ k  j=1 Vn,xn,j(nx+xn,m)⎞ ⎠ p−1 2 ϕn,mhdx =−N n(ϕn,ψ n)n(gn, hn)n=o(1), ∀(g,h)∈H1(RN)×H1(RN). (2.37) Therefore, (ϕ, ψ) satisfies ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ −ϕ +λϕ −pμUλ(x)p−1ϕ−βp−1 2Uλ(x)p−3 2Vλ(x)p+1 2ϕ −βp+1 2Uλ(x)p−1 2Vλ(x)p−1 2ψ=0inRN, −ψ +λψ −pνVλ(x)p−1ψ−βp−1 2Uλ(x)p+1 2Vλ(x)p−3 2ψ −βp+1 2Uλ(x)p−1 2Vλ(x)p−1 2ϕ=0inRN. (2.38) From proposition 2.1, the solution of (Uλ,Vλ)gives ϕ= N  l=1 cl ∂Uλ ∂xl ,ψ= N  l=1 dl ∂Vλ ∂xl .(2.39) On the other hand, from (ϕn,ψ n)∈Enand (2.29), we have RN pμ(Un,xn,m)p−1ϕn ∂Un,xn,m ∂xl dx +RN pν(Vn,xn,m)p−1ψn ∂Vn,xn,m ∂xl dx +βRN p−1 2(Un,xn,m)p−3 2(Vn,xn,m)p+1 2ϕn ∂Un,xn,m ∂xl dx +βRN p−1 2(Un,xn,m)p+1 2(Vn,xn,m)p−3 2ψn ∂Vn,xn,m ∂xl dx +βRN p+1 2(Un,xn,m)p−1 2(Vn,xn,m)p−1 2ψn ∂Un,xn,m ∂xl dx +βRN p+1 2(Un,xn,m)p−1 2(Vn,xn,m)p−1 2ϕn ∂Vn,xn,m ∂xl dx 123 Applied Mathematics & Optimization (2023) 88 :4 Page 25 of 56 4 =o(N−1 n). (2.40) Thus, RN pμUp−1 λϕ∂Uλ ∂xl dx +RN pνVp−1 λψn ∂Vλ ∂xl dx +βRN p−1 2U p−3 2 λV p+1 2 λϕ∂Uλ ∂xl dx +βRN p−1 2U p+1 2 λV p−3 2 λψ∂Vλ ∂xl dx +βRN p+1 2U p−1 2 λV p−1 2 λψ∂Uλ ∂xl dx +βRN p+1 2U p−1 2 λV p−1 2 λϕ∂Vλ ∂xl dx =0.(2.41) By (2.39), (2.41) and Uλ=1 τ0Vλ,we have cl=dl=0,l=1,2,...N. Thus, (ϕ, ψ) =(0,0). So, we have prove there exist 0,θ 0>0,ρ > 0,independent of xj,j=1,2,···k such that for any ∈(0, 0]and xj∈Bθ0(y0), QB(ϕ,ψ )is bijective in E. Moreover, it holds QB(ϕ, ψ)≥ρ(ϕ, ψ),∀(ϕ, ψ) ∈E. Thus the proof is complete.  Next, we give the error estimate for land R(ϕ,ψ ). Lemma 2.3 There is a constant C >0independent of , such that l≤C⎛ ⎝N+2 2+N 2 k  j=1 (|P(xj,)−λ|+|Q(xj,)−λ|)⎞ ⎠ +N 2 i=j e−√α|xi, −xj, | . Proof Observe that l,(g,h) = k  j=1RN (P(x)−λ)U,xj, gdx +μRN ( k  j=1 Up ,xj, −( k  j=1 U,xj, )pgdx 123 4Page 32 of 56 Applied Mathematics & Optimization (2023) 88 :4 +C⎛ ⎝RN ( k  j=1 U,xj, )p+1dx⎞ ⎠ p−2 p+1RN|ϕ|p+1dx2 p+1RN|h|p+1dx1 p+1 ≤CN×p−2 p+1N(1−p+1 2)3 p+1(ϕ, ψ)2 (g,h)≤C−N 2(ϕ, ψ)2 (g,h). Similarly, we have C2≤C−N 2(ϕ, ψ)2 (g,h), C4≤C−N 2(ϕ, ψ)2 (g,h). C3≤C⎛ ⎝RN ( k  j=1 U,xj, )p+1dx⎞ ⎠ p−3 p+1 RN|ψ|p+1dx2 p+1RN|ϕ|p+1dx1 p+1RN|g|p+11 p+1 +C⎛ ⎝RN ( k  j=1 U,xj, )p+1dx⎞ ⎠ p−3 p+1 RN|ϕ|p+1dx2 p+1RN|ψ|p+1dx1 p+1RN|h|p+1dx1 p+1 ≤C−N(ϕ, ψ)3 (g,h). By the similar argument as C3,wehave C5≤C−N(ϕ, ψ)3 (g,h). We also have C6≤C⎛ ⎝RN ( k  j=1 U,xj, )p+1dx⎞ ⎠ p−4 p+1 RN|ψ|p+1dx2 p+1RN|ϕ|p+1dx2 p+1RN|g|p+1dx1 p+1 +C⎛ ⎝RN ( k  j=1 U,xj, )p+1dx⎞ ⎠ p−4 p+1 RN|ϕ|p+1dx2 p+1RN|ψ|p+1dx2 p+1RN|h|p+1dx1 p+1 123 Applied Mathematics & Optimization (2023) 88 :4 Page 33 of 56 4 ≤C−3N 2(ϕ, ψ)4 (g,h). Combining (2.50)–(2.53), the estimates for C1–C6and (2.48), we obtain R(ϕ, ψ)≤C−N 2(ϕ, ψ)2 +−N(ϕ, ψ)3 +−3N 2(ϕ, ψ)4 . This finishes the proof.  Now, we consider the following projection problem QB(ϕ, ψ) +Ql=QR(ϕ, ψ), (2.54) by using the contraction mapping theorem, we give the following lemma. Lemma 2.5 There exists 0>0, such that for any ∈(0, 0],xj∈Bθ(x0), then the problem (2.54)has a unique (ϕ,ψ )∈Eand (ϕ,ψ )≤Cl≤C⎛ ⎝N+2 2+N 2 k  j=1 (|P(xj,)−λ|+|Q(xj,)−λ|)⎞ ⎠ +N 2 i=j e−√α|xi, −xj, | . Proof From Lemma 2.2, we can rewrite (2.54)asfollows: (ϕ, ψ) =B(ϕ, ψ) := (QB)−1Ql+(QB)−1QR(ϕ, ψ). By Lemmas 2.2 and 2.3,wehave (QB)−1Ql≤CQl≤Cl ≤C⎛ ⎝N+2 2+N 2 k  j=1 (|P(xj,)−λ|+|Q(xj,)−λ|)⎞ ⎠+N 2 i=j e−√α|xi, −xj, | . (2.55) Next, we will use the contraction mapping theorem in a ball whose radius is slightly bigger than Cl.Sowetake S:= (ϕ, ψ) :(ϕ, ψ) ∈E, (ϕ, ψ)≤C⎛ ⎝N+2 2−τ+N 2 k  j=1 (|P(xj,)−λ|1−τ+|Q(xj,)−λ|1−τ)⎞ ⎠ +N 2 i=j e−√α|xi, −xj, | , 123 4Page 34 of 56 Applied Mathematics & Optimization (2023) 88 :4 where τ>0 is a fixed small constant. Step 1 B is a map from Sto S. In fact, from Lemmas 2.2,2.3 and 2.4,wehave B(ϕ, ψ)≤Cl+CR(ϕ, ψ) ≤C⎛ ⎝N+2 2−τ+N 2 k  j=1 (|P(xj,)−λ|1−τ+|Q(xj, )−λ|1−τ)⎞ ⎠+N 2 i=j e−√α|xi, −xj, |  +C−N 2(ϕ, ψ)|2 +−N(ϕ, ψ)|3 +−3N 2(ϕ, ψ)|4  ≤C⎛ ⎝N+2 2−τ+N 2 k  j=1 (|P(xj,)−λ|1−τ+|Q(xj, )−λ|1−τ)⎞ ⎠+N 2 i=j e−√α|xi, −xj, |  +C⎛ ⎝N 2+2(1−τ) +N 2 k  j=1 (|P(xj, )−λ|2(1−τ) +|Q(xj, )−λ|2(1−τ))⎞ ⎠ +N 2 i=j e−√α|xi, −xj, |  +C⎛ ⎝N 2+3(1−τ) +N 2 k  j=1 (|P(xj,)−λ|3(1−τ) +|Q(xj, )−λ|3(1−τ))⎞ ⎠ +N 2 i=j e−√α|xi, −xj, |  +C⎛ ⎝N 2+4(1−τ) +N 2 k  j=1 (|P(xj, )−λ|4(1−τ) +|Q(xj, )−λ|4(1−τ))⎞ ⎠ +N 2 i=j e−√α|xi, −xj, |  ≤C⎛ ⎝N+2 2−τ+N 2 k  j=1 (|P(xj,)−λ|1−τ+|Q(xj, )−λ|1−τ)⎞ ⎠ +N 2 i=j e−√α|xi, −xj, | . Thus, Bis a map from Sto S. Step 2 B is a contraction map. For any (ϕ1,ψ 1)∈S,wehave (ϕ1,ψ 1)≤C⎛ ⎝N+2 2−τ+N 2 k  j=1 (|P(xj,)−λ|1−τ+|Q(xj,)−λ|1−τ)⎞ ⎠ +N 2 i=j e−√α|xi, −xj, | , 123 Applied Mathematics & Optimization (2023) 88 :4 Page 35 of 56 4 (ϕ2,ψ 2)≤C⎛ ⎝N+2 2−τ+N 2 k  j=1 (|P(xj,)−λ|1−τ+|Q(xj,)−λ|1−τ)⎞ ⎠ +N 2 i=j e−√α|xi, −xj, | . Since RNR(ϕ1,ψ 1)−R(ϕ2,ψ 2), (g,h)dx =D1+D2+D3−D4+D5 −D6−D7−D8−D9−D10,(2.56) where D1=RN⎛ ⎝μ⎛ ⎝ k  j=1 U,xj, +ϕ1⎞ ⎠ p −μ⎛ ⎝ k  j=1 U,xj, +ϕ2⎞ ⎠ p⎞ ⎠gdx, D2=RN⎛ ⎝ν⎛ ⎝ k  j=1 V,xj, +ψ1⎞ ⎠ p −ν⎛ ⎝ k  j=1 V,xj, +ψ2⎞ ⎠ p⎞ ⎠hdx, D3=RN β⎛ ⎝ k  j=1 U,xj, +ϕ1⎞ ⎠ p−1 2⎛ ⎝ k  j=1 V,xj, +ψ1⎞ ⎠ p+1 2 gdx, D4=RN β⎛ ⎝ k  j=1 U,xj, +ϕ2⎞ ⎠ p−1 2⎛ ⎝ k  j=1 V,xj, +ψ2⎞ ⎠ p+1 2 gdx, D5=RN β⎛ ⎝ k  j=1 U,xj, +ϕ1⎞ ⎠ p+1 2⎛ ⎝ k  j=1 V,xj, +ψ1⎞ ⎠ p−1 2 hdx, D6=RN β⎛ ⎝ k  j=1 U,xj, +ϕ2⎞ ⎠ p+1 2⎛ ⎝ k  j=1 V,xj, +ψ2⎞ ⎠ p−1 2 hdx, 123 4Page 36 of 56 Applied Mathematics & Optimization (2023) 88 :4 D7=βRN p−1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−3 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p+1 2 (ϕ1−ϕ2)gdx, D8=βRN p−1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p+1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−3 2 (ψ1−ψ2)hdx, D9=βRN p+1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−1 2 (ψ1−ψ2)gdx, D10 =βRN p+1 2⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−1 2⎛ ⎝ k  j=1 V,xj, ⎞ ⎠ p−1 2 (ϕ1−ϕ2)hdx. Since p≥3, we have D1=RN pμ⎛ ⎝ k  j=1 U,xj, +ϕ1+t(ϕ1−ϕ2)⎞ ⎠ p−1 (ϕ1−ϕ2)gdx ≤CRN⎛ ⎝ k  j=1 U,xj, ⎞ ⎠ p−2 (|ϕ1|+|ϕ2|)(|ϕ1−ϕ2|)gdx ≤C⎛ ⎝RN ( k  j=1 U,xj, )p+1dx⎞ ⎠ p−2 p+1 (ϕ1Lp+1+ϕ2Lp+1)ϕ1−ϕ2Lp+1gLp+1 ≤CN×p−2 p+1N(1−p+1 2)×3 p+1(ϕ1Lp+1+ϕ2Lp+1)ϕ1−ϕ2Lp+1gLp+1 ≤−N 2(ϕ1+ϕ2)ϕ1−ϕ2g. Similarly, we have D2=RN⎛ ⎝ν( k  j=1 V,xj, +ψ1)p−ν( k  j=1 V,xj, +ψ2)p⎞ ⎠hdx ≤−N 2(ψ1+ψ2)ψ1−ψ2h. 123 Applied Mathematics & Optimization (2023) 88 :4 Page 37 of 56 4 Thus, D1+D2≤C−N 2((ϕ1,ψ 1)+(ϕ2,ψ 2))(ϕ1−ϕ2,ψ 1−ψ2(g,h). (2.57) We also have D3−D4=RN βp−1 2( k  j=1 U,xj, +ϕ2)p−3 2( k  j=1 V,xj, +ψ2)p+1 2(ϕ1−ϕ2)gdx +RN βp+1 2( k  j=1 U,xj, +ϕ2)p−1 2( k  j=1 V,xj, +ψ2)p−1 2(ψ1−ψ2)gdx +o(RN (|ϕ1−ϕ2|2)gdx)+o(RN o(|ψ1−ψ2|2)gdx) := E1+E2+o(RN (|ϕ1−ϕ2|2)gdx)+o(RN o(|ψ1−ψ2|2)gdx), (2.58) D5−D6=RN βp+1 2( k  j=1 U,xj, +ϕ2)p−1 2( k  j=1 V,xj, +ψ2)p−1 2(ϕ1−ϕ2)hdx +RN βp−1 2( k  j=1 U,xj, +ϕ2)p+1 2( k  j=1 V,xj, +ψ2)p−3 2(ψ1−ψ2)hdx +o(RN (|ϕ1−ϕ2|2)hdx)+o(RN o(|ψ1−ψ2|2)hdx) := E3+E4+o(RN (|ϕ1−ϕ2|2)hdx)+o(RN o(|ψ1−ψ2|2)hdx). (2.59) Thus, E1−D7=βp−1 2 p−3 2RN ( k  j=1 U,xj, )p−5 2( k  j=1 V,xj, )p+1 2ϕ2(ϕ1−ϕ2)gdx +βp−1 2 p+1 2RN ( k  j=1 U,xj, )p−3 2( k  j=1 V,xj, )p−1 2ψ2(ϕ1−ϕ2)gdx +o(RN|ϕ2|2(ϕ1−ϕ2)gdx)+o(RN|ψ2|2(ϕ1−ϕ2)gdx) ≤C⎛ ⎝RN ( k  j=1 U,xj, )p+1⎞ ⎠ p−2 p+1 ×ϕ2Lp+1ϕ1−ϕ2Lp+1+ψ2Lp+1)ψ1−ψ2Lp+1gLp+1 123 4Page 38 of 56 Applied Mathematics & Optimization (2023) 88 :4 ≤C−N 2ϕ2,ψ 2)(ϕ1−ϕ2,ψ 1−ψ2(g,h).(2.60) Similarly, we have E2−D9≤C−N 2ϕ2,ψ 2)(ϕ1−ϕ2,ψ 1−ψ2(g,h),(2.61) E3−D10 ≤C−N 2ϕ2,ψ 2)(ϕ1−ϕ2,ψ 1−ψ2(g,h),(2.62) E4−D8≤C−N 2ϕ2,ψ 2)(ϕ1−ϕ2,ψ 1−ψ2(g,h).(2.63) Combining (2.57)–(2.63) with (2.56), we have RN R(ϕ1,ψ 1)−R(ϕ2,ψ 2), (g,h)dx ≤C−N 2ϕ2,ψ 2)(ϕ1−ϕ2,ψ 1−ψ2(g,h). (2.64) Thus, B(ϕ1,ψ 1)−B(ϕ2,ψ 2)≤1 2(ϕ1,ψ 1)−(ϕ2,ψ 2). So, Bis a contraction map. By the contraction mapping theorem, we conclude that for any ∈(0, 0],xj∈ Bθ(x0), there is a (ϕ,ψ )∈Edepending only on xjand such that (ϕ,ψ )=B(ϕ,ψ ). Moreover, from Lemma 2.3 and Lemma 2.4,wehave (ϕ,ψ ) =B(ϕ,ψ )≤Cl+CR(ϕ, ψ) ≤Cl+C−N 2(ϕ, ψ)|2 +−N(ϕ, ψ)|3 +−3N 2(ϕ, ψ)|4  ≤Cl+C−N 2(ϕ, ψ)|+−N(ϕ, ψ)|2 +−3N 2(ϕ, ψ)|3 (ϕ, ψ)| ≤C⎛ ⎝N+2 2+N 2 k  j=1 (|P(xj,)−λ|+|Q(xj,)−λ|)⎞ ⎠+N 2 i=j e−√α|xi, −xj, | . As desired.  123 Applied Mathematics & Optimization (2023) 88 :4 Page 39 of 56 4 Next, we solve equation (2.1). Since QB(ϕ, ψ) +Ql−QR(ϕ, ψ) =B(ϕ, ψ) +l−R(ϕ, ψ) − k  j=1 N  i=1 b,i,j∂U,xj, ∂xi ,∂V,xj, ∂xi. From Lemma 2.5, we know the following equation QB(ϕ, ψ) +Ql=QR(ϕ, ψ) has a unique solution (ϕ,ψ ).So B(ϕ,ψ )+l−R(ϕ,ψ )= k  j=1 N  i=1 b,i,j∂U,xj, ∂xi ,∂V,xj, ∂xi(2.65) for some constant b,i,j.Next, we should to choose suitable xjsuch that b,i,j= 0,i=1,2...N.j=1,2,...k. Firstly, it is easy to see that the right hand of (2.65) belongs to E, if the left hand of of (2.65) belongs to E, then the right hand of (2.65) must be zero. Let u= k  j=1 U,xj, +ϕ,v = k  j=1 V,xj, +ψ, then B(ϕ,ψ )+l−R(ϕ,ψ ), (g,h) =RN (2∇u∇g+P(x)ug+2∇v∇v+Q(x)vh)dx −RNμup g+βu p−1 2 v p+1 2 g+νvp h+βu p+1 2 v p−1 2 hdx for any (g,h)∈H1(RN)×H1(RN). Lemma 2.6 Suppose that x j, satisfies RN (2∇u∇∂U,xj, ∂xi+P(x)u ∂U,xj, ∂xi+2∇v∇∂V,xj, ∂xi+Q(x)v ∂U,xj, ∂xi )dx −RNμup  ∂U,xj, ∂xi+βu p−1 2 v p+1 2  ∂U,xj, ∂xi+νvp  ∂V,xj, ∂xi+βu p+1 2 v p−1 2  ∂V,xj, ∂xi dx =0i=1,2,...N,j=1,2,...k.(2.66) 123 4Page 40 of 56 Applied Mathematics & Optimization (2023) 88 :4 then b,i,j=0,i=1,2,...N,j=1,2,···k. Proof If (2.66) holds, then k  m=1 N  h=1 b,h,m (∂U,xj, ∂xi ,∂V,xj, ∂xi ), (∂U,xm ∂yh ,∂V,xm ∂yh )=0,i=1,2,...N,j=1,2,...k. (2.67) By (2.9) and (2.10), we obtain b,i,j=0,i=1,2,...N,j=1,2,...k.  3 Proof of Theorem 1.1 In this section, we give the proof of Theorem 1.1. Proof In order to solve (2.65), we define a function as following K(x)=I⎛ ⎝ k  j=1 U,xj, +ϕ, k  j=1 V,xj, +ψ⎞ ⎠, where I(u,v)=1 2RN (2|∇u|2+P(x)u2+2|∇v|2+Q(x)v2)dx −1 p+1RN (μup+1+νup+1 +2βup+1 2vp+1 2)dx,∀(u,v)∈H1(RN)×H1(RN). (3.1) Then, from Lemmas 2.3,2.4 and Proposition 5.1, there exists a small constant σ>0 such that K(x)=I⎛ ⎝ k  j=1 U,xj, , k  j=1 V,xj, ⎞ ⎠+l,(ϕ ,ψ )+O(l(ϕ,ψ )+ϕ,ψ 2) =I⎛ ⎝ k  j=1 U,xj, , k  j=1 V,xj, ⎞ ⎠+ON(P(xj,)−λ)2+(Q(xj, )−λ)2+2 123 Applied Mathematics & Optimization (2023) 88 :4 Page 41 of 56 4 =(1 2−1 p+1)kNλ1−N 2A−CN(λ −P(xj,)) +(λ −Q(xj, )) + −CN k  i=j e−√λ|xi, −xj, | +O⎛ ⎝N k  i=j e−2(√λ−σ)|xi, −xj, | ⎞ ⎠ +ON(P(xj, )−λ)2+(Q(xj,)−λ)2+2.(3.2) We use the ideas introduced in [22] to consider the following maximizing problem max x∈DK(x), where D={x:xj∈Bθ(x0), j=1,2,...k,|xm−xj|≥θ|ln |1 2,m= j}. We prove that if maxx∈DK(x)is achieved by some xin D, then xis an interior point of D. Taking xi=x0+θ|ln |ej,(j=1,2,...k), then |xi−xj|=θ|ln |, which means that x=(x1,x2,...,xk)∈D, thus, we have 1 2−1 p+1kNλ1−N 2A−CN+1|ln |≤K(x)≤K(x) ≤1 2−1 p+1kNλ1−N 2A−CN ⎛ ⎝ k  j=1 (λ −P(xj,)) + k  j=1 (λ −Q(xj,)) +⎞ ⎠ −CN k  i=j e−√α|x,i−xj, | . Thus, k  j=1 (λ −P(xj,)) + k  j=1 (λ −Q(xj,)) ++ k  i=j e−√α|x,i−xj, | ≤C|ln |, so k  j=1 (λ −P(xj,)) ≤C|ln |, k  j=1 (λ −Q(xj,)) ≤C|ln |, √α|x,i−xj, | ≥C|ln |≥|ln |1 2. 123 4Page 48 of 56 Applied Mathematics & Optimization (2023) 88 :4 −CN m  i=j e−min{√λi,√λj}−σ)|zi−zj| +O⎛ ⎝N m  i=j e−(2min{√λi,√λj}−σ)|zi−zj| ⎞ ⎠ −CN k  i=1 m  j=1 e−min{√λi,√λj}|xi−zj| +O⎛ ⎝ k  i=1 m  j=1 e−(2min{√λi,√λj}−σ)|xi−zj| ⎞ ⎠ +O(N+1). Consider the following maximizing problem max (x,y)∈D1×D2 K(x,z), where D1={x:xj∈Bθ(x0), j=1,2,...k,|xi−xj|≥θ|ln |1 2,i= j}, D2={z:zj∈Bθ(x0), j=1,2,...m,|zi−zj|≥θ|ln |1 2,i= j}. We prove that if max(x,y)∈D1×D2K(x,z), is achieved by some (x,z)in D1×D2, then (x,z)is an interior point of D1×D2. Taking xi=x0+θ ln |ej,(j= 1,2,...k), zi=x0+θln |ej,(j=1,2,...m), then |xi−xj|=θ|ln |,|zi− zj|=θ|ln |, which means that (x=(x1,x2,...,xk), z=(z1,z2,...,zm)) ∈ D1×D2, thus, we have =1 2−1 p+1A⎛ ⎝ k  j=1 Nλ p+1 p−1−N 2 jμ−2 p−1+ m  j=1 Nλ p+1 p−1−N 2 jν−2 p−1⎞ ⎠−CN+1|ln | ≤K(x,z)=(1 2−1 p+1)A⎛ ⎝ k  j=1 Nλ p+1 p−1−N 2 j, μ−2 p−1+ m  j=1 Nλ p+1 p−1−N 2 j, ν−2 p−1⎞ ⎠ −CN k  i=j e−min{√λi, ,√λj, }|xi, −xj, | +O⎛ ⎝N k  i=j e−(2min{√λi, ,√λj, }−σ)|xi, −xj, | ⎞ ⎠ −CN m  i=j e−min{√λi, ,√λj, }|zi, −zj, | +O⎛ ⎝N m  i=j e−(2min{√λi, ,√λj, }−σ)|zi, −zj, | ⎞ ⎠. Thus, we can get 1 2−1 p+1AN⎛ ⎝ k  j=1λ p+1 p−1−N 2 j−λ p+1 p−1−N 2 j, μ−2 p−1 + m  j=1λ p+1 p−1−N 2 j−λ p+1 p−1−N 2 j, ν−2 p−1⎞ ⎠≤C|ln | 123 Applied Mathematics & Optimization (2023) 88 :4 Page 49 of 56 4 and m  i=j e−min{√λi, ,√λj, }|zi, −zj, | ≤C|ln |, k  i=j e− min{√λi, ,√λj, }|xi, −xj, | ≤C|ln |, which imply that m  i=j |xi, −xj,| ≥C|ln |≥|ln |1 2, k  i=j |zi, −zj,| ≥C|ln |≥|ln |1 2. Therefore, (x,z)is an interior point of D1×D2, which implies that (u,v )=⎛ ⎝ k  j=1 U,xj, ,μ +ϕ, k  j=1 U,zj, ,ν +ψ⎞ ⎠ is a critical point of K(x,z). So, (1.1) has a solution of the form u= k  j=1 U,xj, ,μ +ϕ,v = k  j=1 U,zj, ,ν +ψ for some xj, ∈Bθ(pj), z, j∈Bθ(pj)and (ϕ, ψ)=O( N 2+1).  Author Contributions All authors contribute equally to this paper. All authors read and approved the final manuscript. Funding Maoding Zhen was supported by the NSFC (Grant No. 12201167) and the Fundamental Research Funds for the Central University of China (Grant No. JZ2021HGTA0177, JZ2022HGQA0155). Binlin Zhang was supported by the National Natural Science Foundation of China (Grant No. 11871199), the Heilongjiang Province Postdoctoral Startup Foundation, PR China (Grant No. LBH-Q18109), and the Cultivation Project of Young and Innovative Talents in Universities of Shandong Province. The research of Vicen¸tiu D. R˘adulescu was supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI-UEFISCDI, project number PCE 137/2021, within PNCDI III. Data Availability This paper has no associated data and material. Declarations Conflict of interest The authors have no relevant financial or non-financial interests to disclose. 123 4Page 50 of 56 Applied Mathematics & Optimization (2023) 88 :4 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. 123 Applied Mathematics & Optimization (2023) 88 :4 Page 51 of 56 4 Appendix A: Error Estimate Proposition 5.1 There exist positive constants C1,C2,C3and C4, such that I⎛ ⎝ k  j=1 U,xj, , k  j=1 V,xj, ⎞ ⎠=1 2−1 p+1kNλ1−N 2A +CN⎛ ⎝ k  j=1 (P(xj,)−λ) + k  j=1 (Q(xj, )−λ) +2⎞ ⎠ −CN k  i=j e−√λ|xi, −xj, | +O(N k  i=j e−2(√λ−σ)|xi, −xj, | ), where A =(a2 λ+b2 λ)RNWp+1dx. Proof By the elementary computations, we have I⎛ ⎝ k  j=1 U,xj, , k  j=1 V,xj, ⎞ ⎠ =1 2−1 p+1k  j=1RN(μUp+1 ,xj, +νVp+1 ,xj, +2βU p+1 2 ,xj, V p+1 2 ,xj, )dx +1 2RN k  j=1(P(x)−P(x0))U2 ,xj, +(Q(x)−Q(x0))V2 ,xj, dx +1 2RN k  j=i(P(x)−P(x0))U,xj, U,x,i+(Q(x)−Q(x0))V,xj, V,x,idx −μ p+1RN⎛ ⎝( k  j=1 U,xj, )p+1− k  j=1 Up+1 ,xj, −p+1 2 k  j=i Up ,xj, U,x,i⎞ ⎠dx −ν p+1RN⎛ ⎝( k  j=1 V,xj, )p+1− k  j=1 Vp+1 ,xj, −p+1 2 k  j=i Vp ,xj, V,x,i⎞ ⎠dx −2β p+1RN( k  j=1 U,xj, )p+1 2( k  j=1 V,xj, )p+1 2− k  j=1 U p+1 2 ,xj, V p+1 2 ,xj, dx +2β p+1RN p+1 4k  j=i U p−1 2 ,xj, V p+1 2 ,x,iU,x,i+ k  j=i U p+1 2 ,xj, V p−1 2 ,x,iV,x,idx. 123 4Page 52 of 56 Applied Mathematics & Optimization (2023) 88 :4 By the definition of U,xj, and V,xj, , we get 1 2−1 p+1k  j=1RN (μUp+1 ,xj, +νVp+1 ,xj, +2βU p+1 2 ,xj, V p+1 2 ,xj, )dx =1 2−1 p+1kNλ1−N 2(a2 λ+b2 λ)RN Wp+1dx. From (2.42), we know 1 2RN k  j=1(P(x)−P(x0))U2 ,xj, +(Q(x)−Q(x0))V2 ,xj, dx +1 2RN k  j=i(P(x)−P(x0))U,xj, U,x,i+(Q(x)−Q(x0))V,xj, V,x,idx =CN⎛ ⎝ k  j=1 (P(xj,)−λ) + k  j=1 (Q(xj, )−λ) +2⎞ ⎠. Since U,xj, (x)≤Ce−√α|x−xj, | ,V,xj, (x)≤Ce−√α|x−xj, | , by the similar argument as (2.43)–(2.45) we have there exists a σ>0 such that μ p+1RN⎛ ⎝( k  j=1 U,xj, )p+1− k  j=1 Up+1 ,xj, −p+1 2 k  j=i Up ,xj, U,xi, ⎞ ⎠dx =CN k  i=j e−√α|xi, −xj, | +O(N k  i=j e−2(√α−σ)|xi, −xj, | ), μ p+1RN⎛ ⎝( k  j=1 U,xj, )p+1− k  j=1 Up+1 ,xj, −p+1 2 k  j=i Up ,xj, U,xi, ⎞ ⎠dx =CN k  i=j e−√α|xi, −xj, | +O(N k  i=j e−2(√α−σ)|xi, −xj, | ), −2β p+1RN( k  j=1 U,xj, )p+1 2( k  j=1 V,xj, )p+1 2− k  j=1 U p+1 2 ,xj, V p+1 2 ,xj, dx +2β p+1RN p+1 4k  j=i U p−1 2 ,xj, V p+1 2 ,xi, U,x,i+ k  j=i U p+1 2 ,xj, V p−1 2 ,xi, V,x,idx =CN k  i=j e−√α|xi, −xj, | +O(N k  i=j e−2(√α−σ)|xi, −xj, | ). 123 Applied Mathematics & Optimization (2023) 88 :4 Page 53 of 56 4 Thus, I⎛ ⎝ k  j=1 U,xj, , k  j=1 V,xj, ⎞ ⎠=(1 2−1 p+1)kNλ1−N 2A +CN⎛ ⎝ k  j=1 (P(xj,)−λ) + k  j=1 (Q(xj,)−λ) +⎞ ⎠ −CN k  i=j e−√α|xi, −xj, | +O(N k  i=j e−2(√α−σ)|xi, −xj, | ).  Proposition 5.2 There exist positive constants C1,C2,C3and C4, such that I⎛ ⎝ k  j=1 U,xj, , k  j=1 V,xj, ⎞ ⎠ =1 2−1 p+1A⎛ ⎝ k  j=1 Nλ p+1 p−1−N 2 jμ−2 p−1+ m  j=1 Nλ p+1 p−1−N 2 jν−2 p−1⎞ ⎠ −CN k  i=j e− min{√λi,√λj}|xi, −xj, | +O⎛ ⎝N k  i=j e− (2min{√λi,√λj}−σ)|xi, −xj, | ⎞ ⎠ −CN m  i=j e−min{√λi,√λj}|zi, −zj, | +O⎛ ⎝N m  i=j e− (2min{√λi,√λj}−σ)|zi, −zj, | ⎞ ⎠ −CN k  i=1 m  j=1 e−min{√λi,√λj}|xi, −zj, | +O⎛ ⎝ k  i=1 m  j=1 e− (2min{√λi,√λj}−σ)|xi, −zj, | ⎞ ⎠ +O(N+1), where A =RNWp+1dx and σ>0is a small constant. Proof By direct computations, we have I⎛ ⎝ k  j=1 U,xj,μ, k  j=1 V,zj,ν⎞ ⎠ =1 2−1 p+1k  j=1RNμUp+1 ,xj,μ +1 2−1 p+1m  j=1RNνUp+1 ,zj,ν +1 2RN k  j=1(P(x)−P(xj))U2 ,xj,μ+ m  j=1(Q(x)−Q(zj))U2 ,zj,νdx 123 4Page 54 of 56 Applied Mathematics & Optimization (2023) 88 :4 +1 2RN k  j=i(P(x)−P(xj))U,xj,μU,xi,μ+ m  j=i (Q(x)−Q(zj))U,zj,νU,zi,νdx −μ p+1RN⎛ ⎝( k  j=1 U,xj,μ)p+1− k  j=1 Up+1 ,xj,μ −p+1 2 k  j=i Up ,xj,μU,xi,μ⎞ ⎠dx −ν p+1RN⎛ ⎝( k  j=1 U,zj,ν)p+1− k  j=1 Up+1 ,zj,ν −p+1 2 k  j=i Up ,zj,νU,zi,ν⎞ ⎠dx −2β p+1RN( k  j=1 U,xj,μ)p+1 2( m  j=1 U,zj,ν)p+1 2dx. By the definition of U,xj,μ and U,zj,ν, we get 1 2−1 p+1k  j=1RN μUp+1 ,xj,μ +(1 2−1 p+1) m  j=1RN νUp+1 ,zj,νdx =1 2−1 p+1k  j=1 Nλ p+1 p−1−N 2 jμ−2 p−1RN Wp+1dx +1 2−1 p+1m  j=1 Nλ p+1 p−1−N 2 jν−2 p−1RN Wp+1dx. By the similar arguments as (2.42), we can get 1 2RN k  j=1(P(x)−P(xj))U2 ,xj,μ+ m  j=1(Q(x)−Q(zj))U2 ,zj,νdx +1 2RN k  j=i(P(x)−P(xj))U,xj,μU,xi,μ+ m  j=i (Q(x)−Q(zj))U,zj,νU,zi,νdx =CN⎛ ⎝ k  i=j (P(xi)−P(xj)) + m  i=j (Q(xi)−Q(xj)) +⎞ ⎠=O(N+1). Since U,xj,μ ≤Ce−√λi|x−xj| ,U,zj,ν ≤Ce−√λi|x−zj| , where λi=P(xi), λi= Q(zi), by the similar arguments as (2.43)–(2.45), we have μ p+1RN⎛ ⎝( k  j=1 U,xj,μ)p+1− k  j=1 Up+1 ,xj,μ −p+1 2 k  j=i Up ,xj,μU,xi,μ⎞ ⎠dx =CN k  i=j e− min{√λi,√λj}|xi−xj| +O⎛ ⎝N k  i=j e− (2min{√λi,√λj}−σ)|xi−xj| ⎞ ⎠, 123 Applied Mathematics & Optimization (2023) 88 :4 Page 55 of 56 4 ν p+1RN⎛ ⎝( k  j=1 U,zj,ν)p+1− k  j=1 Up+1 ,zj,ν −p+1 2 k  j=i Up ,zj,νU,zi,ν⎞ ⎠dx =CN m  i=j e−min{√λi,√λj}|zi−zj| +O⎛ ⎝N m  i=j e− (2min{√λi,√λj}−σ)|zi−zj| ⎞ ⎠, 2β p+1RN( k  j=1 U,xj,μ)p+1 2( m  j=1 U,zj,ν)p+1 2dx =CN k  i=1 m  j=1 e−min{√λi,√λj}|xi−zj| +O⎛ ⎝ k  i=1 m  j=1 e− (2min{√λi,√λj}−σ)|xi−zj| ⎞ ⎠. 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