Principal Eigenvalue of p-Laplacian Operator in Exterior Domain

We consider an eigenvalue problem of the form $$\left.\begin{array}{cl}-\Delta_{p} u = \lambda\, K(x)|u|^{p-2}u \quad \mbox{in}\quad \Omega^e\\ u(x) =0 \quad \mbox{for}\quad \partial \Omega\\ u(x) \to 0 \quad \mbox{as}\quad |x| \to \infty,\end{array} \right \}$$ where \({\Omega \subset \mathrm{I\!R\!}^N}\) is a simply connected bounded domain, containing the origin, with C ² boundary \({\partial \Omega}\) and \({\Omega^e:=\mathrm{I\!R\!^N} \setminus \overline{\Omega}}\) is the exterior domain, \({1 < p < N, \Delta_{p}u:={\rm div}(|\nabla u|^{p-2} \nabla u)}\) is the p-Laplacian operator and \({K \in L^{\infty}(\Omega^e) \cap L^{N/p}(\Omega^e)}\) is a positive function. Existence and properties of principal eigenvalue λ ₁ and its corresponding eigenfunction are established which are generally known in bounded domain or in \({\mathrm{I\!R\!}^N}\) . We also establish the decay rate of positive eigenfunction as \({|x| \to \infty}\) as well as near ?Ω.     

Regularity lifting of weak solutions for nonlinear sub-Laplace equations on homogeneous groups

Let G be a homogeneous group, and let X ₁, X ₂, … , X _m be left invariant real vector fields being homogeneous of degree one on G. We consider the following Dirichlet boundary value problem of the sub-Laplace equation involving the critical exponent and singular term: $$\left\{\begin{array}{ll}-\sum_{j=1}^{m}X_j^2u(x)-\frac{a}{\|x\|^\nu}u(x)=u^{\frac{Q+2}{Q-2}}(x), x\in\Omega,\\ u(x)=0, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\, x\in \partial\Omega,\end{array}\right.$$ where ${\Omega\subset G}$ is a bounded domain with smooth boundary and ${\mathbf{0}\in\Omega}$ , Q is the homogeneous dimension of G, ${a\in \mathbb{R},\ \nu <2 }$ . We boost u to ${L^p(\Omega)}$ for any ${1\leq p < \infty}$ if ${u\in S^{1,2}_0(\Omega)}$ is a weak solution of the problem above.     

Sums of cubes of primes in short intervals

In this paper, we are able to sharpen Hua??s result by proving that almost all integers satisfying some necessary congruence conditions can be represented as $$N=p_1^3+ \cdots+p_s^3 \quad \mbox{with } \biggl \vert p_j-\sqrt[3]{\frac {N}{s}}\biggr \vert \leqslant U, j=1,\ldots, s, $$ where p _j are primes and $U=N^{\frac{1}{3}-\delta_{s}+\varepsilon }$ with $\delta_{s}=\frac{s-4}{6s+72}$ , where s=5,6,7,8.     

Error function inequalities

We present various inequalities for the error function. One of our theorems states: Let α?≥?1. For all x,y?>?0 we have $$ \delta_{\alpha} < \frac{ \mbox{erf} \left( x+ \mbox{erf}(y)^{\alpha}\right) +\mbox{erf}\left( y+ \mbox{erf}(x)^{\alpha}\right) } {\mbox{erf}\left( \mbox{erf}(x)+\mbox{erf}(y)\right) } < \Delta_{\alpha} $$ with the best possible bounds $$ \delta_{\alpha}= \left\{ \begin{array}{ll} 1+\sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha=1$,}\\ \sqrt{\pi}/2, & \ \ \textrm{{if} $\alpha>1$,}\\ \end{array}\right. \quad{\mbox{and} \,\,\,\,\, \Delta_{\alpha}=1+\frac{1}{\mbox{erf}(1)}.} $$      

Inequalities for the harmonic numbers

We present various inequalities for the harmonic numbers defined by ${H_n=1+1/2 +\ldots +1/n\,(n\in{\bf N})}$ . One of our results states that we have for all integers n ???2: $$\alpha \, \frac{\log(\log{n}+\gamma)}{n^2} \leq H_n^{1/n} -H_{n+1}^{1/(n+1)} < \beta \, \frac{\log(\log{n}+\gamma)}{n^2}$$ with the best possible constant factors $$\alpha= \frac{6 \sqrt{6}-2 \sqrt[3]{396}}{3 \log(\log{2}+\gamma)}=0.0140\ldots \quad\mbox{and} \quad\beta=1.$$ Here, ?? denotes Euler??s constant.     

Perturbed asymptotically linear problems

The aim of this paper is to investigate the existence of solutions of the semilinear elliptic problem $$\left\{\begin{array}{ll} -\Delta u\ =\ p(x, u) + \varepsilon g(x, u)\quad {\rm in}\,\, \Omega, \\ u=0 \quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,\,\,{\rm on}\,\, \partial\Omega, \end{array} \right. \quad\quad\quad(0.1) $$ where Ω is an open bounded domain of ${\mathbb{R}^N}$ , ${\varepsilon\in\mathbb{R}, p}$ is subcritical and asymptotically linear at infinity, and g is just a continuous function. Even when this problem has not a variational structure on ${H^1_0(\Omega)}$ , suitable procedures and estimates allow us to prove that the number of distinct critical levels of the functional associated to the unperturbed problem is “stable” under small perturbations, in particular obtaining multiplicity results if p is odd, both in the non-resonant and in the resonant case.     

Strong Regularizing Effect of a Gradient Term in the Heat Equation with a Weight

We deal with the following parabolic problem, $$(P)\left\{\begin{array}{lll} u_t - \Delta{u} + |\nabla{u}|^q \quad=\quad \lambda{g}(x)u + f(x, t),\quad u > 0 \; {\rm in} \; \Omega \; \times \; (0, T),\\ \qquad\quad\quad\; u(x, t) \quad=\quad 0 \quad{\rm on}\; {\partial}{\Omega}\; \times ; (0, T),\\ \qquad\quad\quad\; u(x, 0) \quad=\quad u_{0}(x), \quad x \in {\Omega},\end{array}\right.$$ where is a bounded regular domain or ${\Omega = \mathbb{R}^N}$ , ${1 < q \leq 2, \lambda > 0\; {\rm and}\; f \geq 0, u_{0} \geq 0}$ are in a suitable class of functions. We give assumptions on g with respect to q for which for all λ >  0 and all ${f \in L^1(\Omega_T ), f \geq 0}$ , problem (P) has a positive solution. Under some additional conditions on the data, the Cauchy problem and the asymptotic behavior of the solution are also considered.     

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.     

Infinitely many solutions for the prescribed curvature problem of polyharmonic operator

We consider the following prescribed curvature problem for polyharmonic operator: $$\left\{\begin{array}{llll} D_{m} u = \tilde{K}(y)|u|^{m^*-2}u\; {\rm in}\; \mathbb{S}^N\\ u \quad\; >0\qquad\quad\quad\quad\quad{\rm on}\; \mathbb{S}^N\\ u \quad\; \in H^{m}(\mathbb{S}^N), \end{array} \right.$$ where ${m^*=\frac{2N}{N-2m}, N\geq 2m+1,m \in \mathbb{N}_{+}, \tilde{K}}$ is positive and rationally symmetric, ${\mathbb{S}^N}$ is the unit sphere with the induced Riemannian metric ${g=g_{\mathbb{S}^N},}$ and D _m is the elliptic differential operator of 2m order given by $$\begin{array}{lll}D_m={\prod\limits_{k=1}^m}{\left(-\Delta_g+\frac{1}{4}(N-2k)(N+2k-2)\right)}\end{array}$$ where Δ _g is the Laplace-Beltrami operator on ${\mathbb{S}^N}$ . We will show that problem (P) has infinitely many non-radial positive solutions, whose energy can be arbitrary large.     

Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix

Let {X _k,i ; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let {p _n ; n ≥ 1} be a sequence of positive integers such that n/p _n is bounded away from 0 and ∞. In this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry ${L_{n}={\rm max}_{1\leq i < j\leq p_{n}}|\hat{\rho}^{(n)}_{i,j}|}$ of the sample correlation matrix ${{\bf {\Gamma}}_{n}=(\hat{\rho}_{i,j}^{(n)})_{1\leq i,j\leq p_{n}}}$ where ${\hat{\rho}^{(n)}_{i,j}}$ denotes the Pearson correlation coefficient between (X _1,i , ..., X _n,i )′ and (X _1,j ,...,X _n,j )′. Write ${F(x)= \mathbb{P}(|X_{1,1}|\leq x), x\geq0}$ , ${W_{c,n}={\rm max}_{1\leq i < j\leq p_{n}}|\sum_{k=1}^{n}(X_{k,i}-c)(X_{k,j}-c)|}$ , and ${W_{n}=W_{0,n},n\geq1,c\in(-\infty,\infty)}$ . Under the assumption that ${\mathbb{E}|X_{1,1}|^{2+\delta} < \infty}$ for some δ > 0, we show that the following six statements are equivalent: $$ {\bf (i)} \quad \lim_{n \to \infty} n^{2}\int\limits_{(n \log n)^{1/4}}^{\infty}\left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x}\right) \right) dF(x) = 0,$$ $$ {\bf (ii)}\quad n \mathbb{P}\left ( \max_{1 \leq i < j \leq n}|X_{1,i}X_{1,j} | \geq \sqrt{n \log n}\right ) \to 0 \quad{\rm as}\,n \to \infty,$$ $$ {\bf (iii)}\quad \frac{W_{\mu, n}}{\sqrt {n \log n}}\stackrel{\mathbb{P}}{\rightarrow} 2\sigma^{2},$$ $$ {\bf (iv)}\quad \left ( \frac{n}{\log n}\right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2,$$ $$ {\bf (v)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (\frac{W_{\mu, n}^{2}}{n \sigma^{4}} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8\pi}} e^{-t/2}\right \}, - \infty < t < \infty,$$ $$ {\bf (vi)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (n L_{n}^{2} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 \pi}} e^{-t/2}\right \}, - \infty < t < \infty$$ where ${\mu=\mathbb{E}X_{1,1}, \sigma^{2}=\mathbb{E}(X_{1,1} - \mu)^{2}}$ , and a _n  = 4 log p _n ? log log p _n . The equivalences between (i), (ii), (iii), and (v) assume that only ${\mathbb{E}X_{1,1}^{2} < \infty}$ . Weak laws of large numbers for W _n and L _n , n ≥  1, are also established and these are of the form ${W_{n}/n^{\alpha}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(\alpha > 1/2)$ and ${n^{1-\alpha}L_{n}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(1/2 < \alpha \leq 1)$ , respectively. The current work thus provides weak limit analogues of the strong limit theorems of Li and Rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of L _n obtained by Jiang. Some open problems are also posed.     

A perturbation result for semi-linear stochastic differential equations in UMD Banach spaces

We consider the effect of perturbations of A on the solution to the following semi-linear parabolic stochastic partial differential equation: $$\left\{\begin{array}{ll}{\rm d}U(t) & = AU(t)\,{\rm d}t + F(t,U(t))\,{\rm d}t + G(t,U(t))\,{\rm d}W_H(t), \quad t > 0;\\U(0)& = x_0. \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad({\rm SDE})\end{array} \right.$$ Here, A is the generator of an analytic C ₀-semigroup on a UMD Banach space X, H is a Hilbert space, W _H is an H-cylindrical Brownian motion, ${G:[0,T]\times X\rightarrow \mathcal{L}(H, X_{\theta_G}^{A})}$ , and ${F : [0, T]\times X \rightarrow X_{\theta_F}^{A}}$ for some ${\theta_G > -\frac{1}{2}, \theta_F > -\frac{3}{2}+\frac{1}{\tau}}$ , where ${\tau\in [1, 2]}$ denotes the type of the Banach space and ${X_{\theta_F}^{A}}$ denotes the fractional domain space or extrapolation space corresponding to A. We assume F and G to satisfy certain global Lipschitz and linear growth conditions. Let A ₀ denote the perturbed operator and U ₀ the solution to (SDE) with A substituted by A ₀. We provide estimates for ${\|U - U_0\|_{L^p(\Omega;C([0,T];X))}}$ in terms of ${D_{\delta}(A, A_0) := \|R(\lambda : A) - R(\lambda : A_0)\|_{\mathcal{L}(X^{A}_{\delta-1},X)}}$ . Here, ${\delta\in [0, 1]}$ is assumed to satisfy ${0\leq \delta < {\rm min}\{\frac{3}{2} - \frac{1}{\tau} + \theta_F,\, \frac{1}{2} - \frac{1}{p} + \theta_G \}}$ . The work is inspired by the desire to prove convergence of space approximations of (SDE). In this article, we prove convergence rates for the case that A is approximated by its Yosida approximation.     

Constant-Sign Periodic and Almost Periodic Solutions for a System of Integral Equations

We consider the following system of integral equations $${u_{i}(t)=\int\nolimits_{I} g_{i}(t, s)f(s, u_{1}(s), u_{2}(s), \cdots, u_{n}(s))ds, \quad t \in I, \ 1 \leq i\leq n}$$ where I is an interval of $\mathbb{R}$ . Our aim is to establish criteria such that the above system has a constant-sign periodic and almost periodic solution (u ₁, u ₂,…,u _n ) when I is an infinite interval of $\mathbb{R}$ , and a constant-sign periodic solution when I is a finite interval of $\mathbb{R}$ . The above problem is also extended to that on $\mathbb{R}$ $$u_{i} {\left( t \right)} = {\int_\mathbb{R} {g_{i} {\left( {t,s} \right)}f_{i} {\left( {s,u_{1} {\left( s \right)},u_{2} {\left( s \right)}, \cdots ,u_{n} {\left( s \right)}} \right)}ds\quad t \in \mathbb{R},\quad 1 \leqslant i \leqslant n.} }$$      

Global Calderón–Zygmund theory for nonlinear parabolic systems

We establish a global Calderón–Zygmund theory for solutions to a large class of nonlinear parabolic systems whose model is the inhomogeneous parabolic \(p\) -Laplacian system $$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u - {{\mathrm{div}}}(|Du|^{p-2}Du) = {{\mathrm{div}}}(|F|^{p-2}F) &{}\quad \hbox {in }\quad \Omega _T:=\Omega \times (0,T)\\ u=g &{}\quad \hbox {on }\quad \partial \Omega \times (0,T)\cup {\overline{\Omega }}\times \{0\} \end{array} \right. \end{aligned}$$ with given functions \(F\) and \(g\) . Our main result states that the spatial gradient of the solution is as integrable as the data \(F\) and \(g\) up to the lateral boundary of \(\Omega _T\) , i.e. $$\begin{aligned} F,Dg\in L^q(\Omega _T),\ \partial _t g\in L^{\frac{q(n+2)}{p(n+2)-n}}(\Omega _T) \quad \Rightarrow \quad Du\in L^q(\Omega \times (\delta ,T)) \end{aligned}$$ for any \(q>p\) and \(\delta \in (0,T)\) , together with quantitative estimates. This result is proved in a much more general setting, i.e. for asymptotically regular parabolic systems.     

Inequalities for the Volume of the Unit Ball in {\mathbb{R}^{n}}

The volume of the unit ball in ${\mathbb{R}^{n}}$ is defined by $$\Omega_{n} = \frac{\pi^{n/2}}{\Gamma(n/2+1)},\qquad n = 1,2,3,\ldots,$$ where Γ denotes the classical gamma function of Euler. In several recently published papers numerous authors studied properties of Ω _n . In particular, various inequalities involving Ω _n are given in the literature. In this paper, we continue the work on this subject and offer new inequalities. More precisely, we offer sharp upper and lower bounds for $$\frac{\Omega_{n}^{2}}{\Omega_{n-1} \Omega_{n+1}},\quad\frac{\Omega_{n}}{\Omega_{n-1}+\Omega_{n+1}} \quad {\rm and} \quad\Omega_{n}.$$      

On the solutions of a singular elliptic equation concentrating on two orthogonal spheres

Let \({A=\{x\in \mathbb{R}^{2m}: 0 < a < |x| < b\}}\) be an annulus. We consider the following singularly perturbed elliptic problem on A $$\left\{\begin{array}{lll}-\varepsilon ^2{\Delta u} + |x|^{\eta}u =|x|^{\eta}u^p, \quad {\rm in} A,\\ u > 0, \quad \quad \quad \quad \quad \quad \quad {\rm in} A, \\ u=0, \quad \quad \quad \quad \quad \quad \quad {\rm on}\partial A,\end{array}\right. $$ where \({1 < p < \frac{m+3}{m-1}}\) . We shall prove the existence of a positive solution \({u_\epsilon }\) which concentrates on two different orthogonal spheres of dimension (m?1) as \({\varepsilon \to 0}\) . We achieve this by studying a reduced problem on an annular domain in \({\mathbb{R}^{m+1}}\) and analysing the profile of a two point concentrating solution in this domain.     

The Left Hilbert BVP for h-Regular Functions in Clifford Analysis

Consider the real Clifford algebra ${\mathbb{R}_{0,n}}$ generated by e ₁, e ₂, . . . , e _n satisfying ${e_{i}e_{j} + e_{j}e_{i} = -2\delta_{ij} , i, j = 1, 2, . . . , n, e_{0}}$ is the unit element. Let ${\Omega}$ be an open set in ${\mathbb{R}^{n+1}}$ . u(x) is called an h-regular function in ${\Omega}$ if $$D_{x}u(x) + \widehat{u}(x)h = 0, \quad\quad (0.1)$$ where ${D_x = \sum\limits_{i=0}^{n} e_{i}\partial_{xi}}$ is the Dirac operator in ${\mathbb{R}^{n+1}}$ , and ${\widehat{u}(x) = \sum \limits_{A} (-1)^{\#A}u_{A}(x)e_{A}, \#A}$ denotes the cardinality of A and ${h = \sum\limits_{k=0}^{n} h_{k}e_{k}}$ is a constant paravector. In this paper, we mainly consider the Hilbert boundary value problem (BVP) for h-regular functions in ${\mathbb{R}_{+}^{n+1}}$ .     

Existence Results for Critical Semi-linear Equations on Heisenberg Group Domains

Following the work of G. Citti and F. Uguzzoni who studied Yamabe type problems on Heisenberg group domains, we consider here the following critical semi-linear equation on domains of the Heisenberg group ${{\mathbb{H}^1}}$ : $$(P) \left\{\begin{array}{lll}-{\Delta_{H}}u\quad =\quad K{u^{3}}\quad\,{\rm in}\,\,\Omega,\\ \quad\quad\,{u}\quad > \quad0\qquad\,\,\,\,{\rm in}\,\,\Omega,\\ \quad\quad\,{u}\quad = \quad 0 \quad\quad\,\,\,{\rm on}\,\partial \Omega, \end{array}\right. $$ where Δ _H is the sublaplacian on ${{\mathbb{H}^1}}$ and K is a C ³ positive function defined on Ω. Using a version of the Morse Lemma at infinity, we give necessary conditions on K to insure the existence of solutions for (P).     

Parabolic power concavity and parabolic boundary value problems

This paper is concerned with power concavity properties of the solution to the parabolic boundary value problem $$\begin{aligned} (P)\quad \left\{ \begin{array}{l@{\quad }l} \partial _t u=\varDelta u +f(x,t,u,\nabla u) &{} \text{ in }\quad \varOmega \times (0,\infty ),\\ u(x,t)=0 &{} \text{ on }\quad \partial \varOmega \times (0,\infty ),\\ u(x,0)=0 &{} \text{ in }\quad \varOmega , \end{array} \right. \end{aligned}$$ where $\varOmega $ is a bounded convex domain in $\mathbf{R}^n$ and $f$ is a nonnegative continuous function in $\varOmega \times (0,\infty )\times \mathbf{R}\times \mathbf{R}^n$ . We give a sufficient condition for the solution of $(P)$ to be parabolically power concave in $\overline{\varOmega }\times [0,\infty )$ .     

A local mountain pass type result for a system of nonlinear Schr?dinger equations

We consider a singular perturbation problem for a system of nonlinear Schr?dinger equations: $$ \begin{array}{l} -\varepsilon^2\Delta v_1 +V_1(x)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\varepsilon^2\Delta v_2 +V_2(x)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N), \end{array} \quad\quad\quad\quad\quad (*) $$ where N?=?2, 3, ?? ₁, ?? ₂, ?? > 0 and V ₁(x), V ₂(x): R ^N ?? (0, ??) are positive continuous functions. We consider the case where the interaction ?? > 0 is relatively small and we define for ${P\in{\bf R}^N}$ the least energy level m(P) for non-trivial vector solutions of the rescaled ??limit?? problem: $$ \begin{array}{l} -\Delta v_1 +V_1(P)v_1 = \mu_1 v_1^3 + \beta v_1v_2^2 \quad {\rm in}\,\,{\bf R}^N, \\ -\Delta v_2 +V_2(P)v_2 = \mu_2 v_2^3 + \beta v_1^2v_2 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x) >0 \quad {\rm in}\,\,{\bf R}^N, \\ \null\ v_1(x), \ v_2(x)\in H^1({\bf R}^N). \end{array} \quad\quad\quad\quad\quad\quad (**) $$ We assume that there exists an open bounded set ${\Lambda\subset{\bf R}^N}$ satisfying $$ {\mathop {\rm inf} _{P\in\Lambda} m(P)} < {\mathop {\rm inf}_{P\in\partial\Lambda} m(P)}. $$ We show that (*) possesses a family of non-trivial vector positive solutions ${\{(v_{1\varepsilon}(x), v_{2\varepsilon} (x))\}_{\varepsilon\in (0,\varepsilon_0]}}$ which concentrates??after extracting a subsequence ?? _n ?? 0??to a point ${P_0\in\Lambda}$ with ${m(P_0)={\rm inf}_{P\in\Lambda}m(P)}$ . Moreover (v _1?? (x), v _2?? (x)) converges to a least energy non-trivial vector solution of (**) after a suitable rescaling.