equations perturbed by nonhomogeneous potential in the super critical case

perturbed by a non-homogeneous potential V when p ∈ [pc , N+2 N−2 ), where pc is the Joseph-Ludgren exponent. When p ∈ ( N N−2 , pc), the fast decaying solution could be approached by super and sub solutions, which are constructed by the stability of the k-fast decaying solution wk of −∆u = up in RN \ {0} by authors in [9]. While the fast decaying solution wk is unstable for p ∈ (pc , N+2 N−2 ), so these fast decaying solutions seem not able to disturbed like (0.1) by non-homogeneous potential V. A surprising observation that there exists a bounded sub solution of (0.1) from the extremal solution of −∆u = u N+2 N−2 in RN and then a sequence of fast decaying solutions and slow decaying solutions could be derived under appropriated restrictions for V.


Introduction
Our concern in this paper is to consider fast decaying solutions of weighted Lane-Emden equation in punc- where p > , N and the potential V is a locally Hölder continuous function in R N \ { }. During the last years there has been a renewed and increasing interest in the study of the semilinear elliptic equations with potentials, motivated by great applications in mathematical elds and physical elds, e.g. the well known scalar curvature equation in the study of Riemannian geometry, the scalar eld equation for standing wave of nonlinear Schrödinger and Klein-Görden equations, the Matukuma equation, see a survey [17,21] and more references on decaying solutions at in nity see [1, 5-7, 10, 14]. For Lane-Emden equation (1.1) involving nonhomogeneous potential V(x) = |x| α ( + |x|) β−α , the authors in [3,4] showed the nonexistence provided β > − and p N+β N− , also see [1,Theorem 3.1]. In [8], the in nitely many positive solutions of problem (1.1) are constructed for p ∈ ( N+β N− , N+α N− ) ∩ ( , +∞) with α ∈ (−N, +∞) and β ∈ (−∞, α ), by dealing with the distributional solutions of −∆u = Vu p + κδ in R N , (1.2) where k > , δ is Dirac mass at the origin and p = N+α N− is the critical exponent named Serrin exponent, the value for problem (1.2) with recoverable isolated singularities. Compared to the case V ≡ , problem (1.1) would have totally di erent isolated singular solution structure for the super critical case p ≥ N+α N− , due to the behavior of potential at in nity.
When V = , equation (1.1) is well known as Lane-Emden-Fowler equation which has been extensively studied in the last decades. The authors in [4] showed the nonexistence of positive solutions of problem (1.3) for p N N− ; and when p > N N− , problem (1.3) always has a singular solution , a branch of fast decaying solutions of (1.3) can be derived by phase analysis as following: Here and in the sequel, a function u ∈ C (R N \ { }) is called a ν-fast decaying if u has the asymptotic behavior at in nity lim |x|→+∞ u(x)|x| N− = ν for ν > . It is worth noting that there is a critical exponent, named by Joseph-Ludgren exponent: (1.5) such that the fast decaying solutions w p,k is stable for p ∈ ( N N− , pc), semistable for p = pc and unstable for p ∈ (pc , N+ N− ). Thanks to the stability of {w k } k for p ∈ ( N N− , pc), solutions with multiple singular points are derived for Lane-Emden equation in a bounded smooth domain, see the references [7,18,19,23]. Moreover, for the supercritical case that p ≥ N+ N− , problem (1.3) has been studied in [11,12,18] and the references therein. In particular, the authors in [12,13] constructed in nitely many solutions of (1.3) with p > N+ N− in an exterior domain by analyzing the related linearization problem at wp.
Involving a nonhomogeneous potential V, we can't transform (1.1) into ODE to obtain the symmetric solutions by using the phase analysis, nor the variational method fails to apply due to the singularity at the origin. Thanks to the stability of k-fast fast decaying solution w k of (1.3) for p ∈ ( N N− , pc), the Schauder xed point theorem could be applied to obtain fast decaying solution of (1.1) by constructing a solution v k of the problem for k > su ciently small. And then aν k -fast decaying solution uν k := v k + w k of (1.1) is derived, see the reference [9]. Precisely, (a) let p ∈ N N− , pc , potential V is a Hölder function verifying that near the origin, Then there exists ν > such that for any ν ∈ ( , ν ), problem (1.1) has a ν-fast decaying solution uν, which has singularity at the origin as lim |x|→ uν(x)|x| p− = cp . (b) When V is radially symmetric and decreasing with respect to |x|, γ |x| α ≤ V(|x|) ≤ γ|x| α for |x| > .
Unlike the case of p ∈ ( N N− , pc), the Schauder xed point theorem fails to build the lower bound for problem (1.1), due to lack the stability of fast decaying solution w k of (1.3) for p ∈ [pc , N+ N− ). Note that construct pairs sub-super solution (U µ * , w k ) with µ * = (N(N − )) − and k ≥ k * for some k * > and iterating procedure with initial data w k could be applied to approach a sequence of fast decaying.
Our nal interest is to study the limit of {uν}ν as ν → +∞ and the result states as follows.
The rest of this paper is organized as follows. In Section 2, we show qualitative properties of the solutions to elliptic problem with homogeneous potential and some basic estimates. Section 3 is devoted to build fast decaying solutions of (1.1) by iteration method. Section 4 is devoted to the slow decaying solution as the limit of fast decaying solutions.

Preliminary . Singularity at the origin
Since the lower bound U µ * does not blow up at the origin, so we have to provide the classi cation of singularity at the origin of positive solution of (1.1). (1.6). Let u be a positive solution of (1.1), then u is removable at the origin or Proof. In order to apply [16,Theorem 3.3], we need to check that V veri es the conditions: Let V( ) = and from (1.6), there exists c > such that and so we have that |∇V( )| ≤ c. Now we apply [16,Theorem 3.3] to obtain that u is removable at the origin or there exists c > such that Finally, we improve the singularity when u is not removable. Let It is shown in Appendix B in [16] that (2.3) has only solutions v∞ ≡ or v∞ ≡ cp. Note that the limit set of a C function is connected, then we have that v(t, ω) → or v(t, w) → cp as t → −∞ Therefore, we obtain (2.1).

. Basic estimate
In this subsection, some estimates are introduced, which play important roles in our construction of fastdecaying solutions for problem (1.1). Denote which is the fundamental solution of −∆Γ = δ in R N and c N > is a normalized constant.
Proof. By the decay condition of f , we have that for any ϵ > , there exists R > r such that for R large, which yields that for |x| large, Passing to the limit as ϵ → and letting R = |x|/ → +∞, we see that |x| → +∞ and then (2.4) holds.

2
We remark that k-fast decaying solution w k of (1.3) veri es the integral equation w k = Γ * (w p k ) and For µ > , denote , we denote which is radially symmetric and decreasing with respect to r = |x|. then , w k be the k-fast decaying solution of (1.3) with k > and U µ * is de ned in (2.5). Then there exists k * > such that for any k ≥ k * Step 1: we show Uµ < wp in R N \ { } for µ = µ * . This is equivalent to that which achieve the minimum at point r = a −a µ − and we want to show In fact, f µ * (r ) > is equivalent to This could be written as which is true for p ∈ N N− , N+ N− by direct computation.

Fast decaying solutions
Proof of Theorem 1.1. Our proof divides into ve steps.
Step 1. Existence by iteration method. We initiate from v := w k , denote by vn iteratively the unique solu- As Inductively, we can deduce that vn ≤ v n− in R N \ { }. Thus, the sequence {vn}n is decreasing. Now we show that U µ * is a lower bound for {vn}n for k ∈ [k * , +∞). From (2.7) and the assumption that we obtain that U µ * is a sub solution of (1.1), i.e.
Inductively, we see that for any n ∈ N, we have that so {vn}n has a lower barrier U µ * . Therefore, the sequence {vn}n converges. Denote uν k = lim n→∞ vn, then for any compact set K in R N \ { }, uν k veri es the equation −∆u = Vu p in K, and then uν k is a classical solution of (1.1) verifying Here we let we also replace ν k by ν k (V) if it is not confusing. Then Step 2: the mapping k → ν k is increasing and V → ν k (V) is increasing in the sense that ν k (V ) ≥ ν k (V ) if V ≥ V . For k * ≤ k < k , by the increasing monotonicity of k → w k , we have that w k < w k . Let {v n,k i } be sequence of (3.1) with the initial data v i, = w k i , here i = , . Let We see that Inductively, we have that for any n ∈ N, ν ,n ≥ ν ,n , which implies that the limit uν k of {v n,k } and the limit uν k of {v n,k } as n → +∞ veri es that As a conclusion, for any k ∈ [k * , +∞), there exists a ν * := ν k * > such that problem (1.1) has a solution uν k such that lim That means that the mapping k → ν k is increasing.
Similarly, we can obtain that the mapping: V → ν k (V) is increasing.
Step 3: we prove that the mapping k ∈ (k * , +∞) → ν k is continuous. For < k < k , we have that w k < w k . Let {v n,k i } be sequence of (3.1) with the initial data v ,i = w k i , here i = , . Let We see that Inductively, we have that for any n ∈ N, ≤ ν n, − ν n, ≤ k − k , which implies that the limit uν k of {v n,k } and the limit uν k of {v n,k } as n → +∞ veri es that As a conclusion, k → ν k is increasing and continuous. Let ν∞ = lim k→+∞ ν k , then we have that for any k ∈ [k * , +∞), there exists ν ∈ [ν * , ν∞) (ν = ν * if ν∞ = ν * ) such that problem (1.1) has a solution uν k such that lim |x|→+∞ uν k (x)|x| N− = ν k .
By contradiction, we may assume that and denote Letk be the number such that νk =ν. By direct computation, we have that  7). From Theorem 2.1, we only have to rule out the case that uν k is removable at the origin. If uν k is removable at the origin. then uν k is a bounded classical sub solution of −∆u = u p in R N .
Since uν k ≤ w k and uν k is bounded at the origin, there exists |x | > small enough such that Now consider the sequence vn = Γ * (v p n− ) with the initial data v = uν k , and this sequence is increasing and controlled by function w k and wp(x + ·), the limit of {vn} will be a bounded positive classical solution of −∆u = u p in R N .
Step 1. Radial symmetry. We recall that the solution uν is approaching by sequence vn = Γ * (Vv p n− ) with initial data v = w k . Note that w k and V are radially symmetric and decreasing in r = |x|, so is {vn} for any n ∈ N. Therefore, uν is radially symmetric and decreasing in r = |x|.
Step 2. Uniform estimates. It is standard to show that uν is a very weak solution of (1.1) in the distributional sense that uν ∈ L loc (R N ) ∩ L p loc (R N , Vdx) satis es the identity We claim that there exists c > independent of ν such that Indeed, we recall that For ϵ ∈ ( , ), we denote where η : [ , +∞) → [ , ] is a smooth increasing function such that Take U η ϵ as a test function of (4.1), then by Hölder inequality, we have that where c > depends on ϵ and V, but it is independent of ν. Then we have that Step 3. the limit of {uν}ν. From Theorem 1.1, the mapping ν ∈ [ν * , ∞) → uν is increasing and uniformly bounded in uν → u∞ as ν → +∞ a.e. in Ω and in L loc (R N ) ∩ L p loc (R N , Vdx).
It is known in [20] that uν is also a weak solution of (1.1), i.e.
Passing to the limit of (4.3), we obtain that u∞ is a weak solution of (1.1) in the sense of (4.1). Note that uν is radially symmetric and decreasing with respect to |x|, so is u∞. Then we have that u∞ ∈ L ∞ loc (R N \ { }) and then Vu p ∞ is in L ∞ loc (R N \ { }). By standard regularity results, we have that u is a classical solution of (1.1).
Since uν veri es (1.7) at the origin for any ν > and u∞ is the limit of an increasing sequence {uν}ν, then we have that lim inf |x|→ u∞(x)|x| p− ≥ cp .