Asymptotic behaviour of solutions to the anisotropic doubly critical equation

The aim of this paper is to deal with the anisotropic doubly critical equation $$-\Delta_p^H u - \frac{\gamma}{[H^\circ(x)]^p} u^{p-1} = u^{p^*-1} \qquad \text{in } \R^N,$$ where $H$ is in some cases called Finsler norm, $H^\circ$ is the dual norm, $1<p<N$, $0 \leq \gamma<\left((N-p)/p\right)^p$ and $p^*=Np/(N-p)$. In particular, we provide a complete asymptotic analysis of $u \in \mathcal{D}^{1,p}(\R^N)$ near the origin and at infinity, showing that this solution has the same features of its euclidean counterpart. Some of the techniques used in the proofs are new even in the Euclidean framework.


Introduction and main results
This work is devoted to the study of the following anisotropic doubly critical problem where 1 < p < N, p * := Np/(N − p) is the Sobolev critical exponent, 0 ≤ γ < C H := ((N − p)/p) p is the Hardy constant and (1.1) ∆ H p u := div(H p−1 (∇u)∇H(∇u)), where ∆ H p is the so-called anisotropic p-Laplacian or Finsler p-Laplacian.We point out that H is a Finsler type norm and H • is its the dual norm (H satisfies assumptions (h H ), see Section 2 for further details).In particular, when H(ξ) = |ξ| = H • (ξ) the Finsler type p-Laplacian coincides with the classical p-Laplacian, and, hence it is singular when 1 < p < 2 and degenerate when p > 2.Then, according with standard regularity theory [16,35] and the regularity results in the anisotropic framework [3,8], we say that any solution of (P H ) has to be understood in the weak distributional meaning, i.e. u ∈ D 1,p (R N ) satisfies the following integral equality (1.2) The literature about critical problems is really huge.Going back to the Euclidean framework, i.e. when we consider H(ξ) = |ξ| = H • (ξ) in (P H ), we deal with In the seminal paper [7], Caffarelli, Gidas and Spruck classified any positive solution to (1.3) with p = 2, N ≥ 3, and γ = 0. We point out that a first result, under stronger assumption on the decay of solutions, was obtained by Gidas, Ni and Nirenberg in [17].Moreover, in this setting a complete answer in the subcritical case was done in the celebrasted work of Gidas and Spruck [18], where the authors proved Liouville-type theorems.
In the quasilinear framework, the situation is much more involved due to the nonlinear nature of the operator.Recently, a classification result of positive solutions to (1.3) with p > 2, γ = 0 and u ∈ D 1,p (R N ) := {u ∈ L p * (R N ) | ∇u ∈ L p (R N )} was obtained in [31].The proof of this result is based on a refined version of the well-known moving plane method of Alexandrov-Serrin [2,30] and on some a priori estimates of the solutions and their gradients, proved in [38].To be more precise, we note that the classification result of positive solution to the Sobolev critical quasilinear equation with finite energy started in [14] in the case, and then was extended in [38] for every 1 < p < 2. Subsequently the full case was obtained in [31].Recently, we refer to the papers [9,27,39] for new partial results on the classification of positive solutions without a priori assumption on the energy of solutions.In the anisotropic setting, Ciraolo, Figalli and Roncoroni [11], obtained a complete classification result for positive solution to (P H ) with γ = 0 using different techniques that do not require the use of the moving plane method, which could not be used in the anisotropic context due to the lack of invariance.
When γ = 0 the situation is really different.In the seminal paper of Terracini [33], it was proved for the first time the classification result for positive solutions to (1.3) in the case p = 2.The author firstly showed the existence of solutions to this problem with a minimization argument based on the concentration and compactness principle.Subsequently, she proved that any solution to this problem is radial and radially decreasing about the origin combining the moving-plane technique and the use of the Kelvin transformation, in the same spirit of [7].The case p = 2 and γ = 0 is much more involved and it is available in [26], where the techniques used are mainly based on a fine asymptotic analysis at infinity and refined versions of the moving plane procedure, and also on some asymptotic estimates proved in [40,41].
Our aim is to prove some decay estimates for positive weak solutions to (P H ) in the anisotropic framework 1 < p < N and γ = 0.More precisely, our first main result is the following: Theorem 1.1.Let u ∈ D 1,p (R N ) be a weak solution of (P H ) with 1 < p < N, 0 ≤ γ < C H . Then there exist positive constants 0 < R 1 < 1 < R 2 depending on N, p, γ and u, such that and where µ 1 , µ 2 are the solutions of (1.6) µ p−2 [(p − 1)µ 2 − (N − p)µ] + γ = 0, C 1 , C 2 are positive constants depending on N, p, γ, H and u, c 1 is a positive constant depending on N, p, γ, H, R 1 , µ 1 and u, c 2 is a positive constant depending on N, p, γ, H, R 2 , µ 2 and u.
Remark 1.2.In the following we shall assume that µ 1 < µ 2 and it is easy to see that furthermore B H • R is the dual anisotropic ball also known as Frank diagram (see Section 2 for further details).
In the proof we will also exploit some clever ideas from [40] facing the difficulties of the anisotropic issue.A different approach is in fact needed for the study of the asymptotic behaviour of the gradient.In particular, the fact that the moving plane plane technique cannot be applied, a crucial point is given by the following classification result: ) be a positive weak solution of the equation where 0 ≤ γ < C H . Assume that there exist two positive constants C and c such that where µ i (i = 1, 2) are the roots of (1.6) and suppose that there exists a positive constant Ĉ such that for some c > 0.
Theorem 1.3 is new and interesting in itself.The proof is very much different than the ones available in the euclidean case H(ξ) = |ξ|.Here we shall exploit it to deduce the precise asymptotic estimates for the gradient.more precisely we have the following: ) be a weak solution of (P H ) with 1 < p < N, and 0 ≤ γ < C H . Then there exist positive constants c, C depending on N, p, γ, H and u such that and where µ 1 , µ 2 are roots of (1.6) as in Theorem 1.1, and 0 < R 1 < 1 < R 2 are constants depending on N, p, γ and u.
The paper is structured as follows: • In Section 2 we recall some notions about Finsler type anisotropic geometry, and we prove some technical lemmas that will be crucial in the proof of the main results.• In Section 3 we prove some preliminary estimates, elliptic estimates and weak comparison principles in bounded and exterior domains that will be essential in the proof of Theorem 1.1.• In Section 4 we give the proof of decay estimates of solutions to (P H ) near the origin and at infinity, i.e. we prove Theorem 1.1.The, using this result we also prove decay estimates for the gradient of positive weak solutions to (P H ) near the origin and at infinity, i.e. we prove Theorem 1.4.• Although the existence of solutions can be easy deduced in the radial-anisotropic setting, in the Appendix A we show that problem (P H ) admits at least a positive solution u ∈ D 1,p (R N ) that minimizes the Hardy-Sobolev anisotropic inequality.This result follows using classical arguments (see also [33]) that we decide to add for the readers' convenience.

Preliminaries
Notation.Generic fixed and numerical constants will be denoted by C (with subscript in some case) and they will be allowed to vary within a single line or formula.By |A| we will denote the Lebesgue measure of a measurable set A.
The aim of this section is to recall some properties and geometrical tools about the anisotropic elliptic operator defined above.For a, b ∈ R N we denote by a ⊗ b the matrix whose entries are (a ⊗ b) ij = a i b j .We remark that for any v, w ∈ R N it holds that: Now, we recall the definition of anisotropic norm.
In all the paper we assume that H is a anisotropic norm if it satisfies the following set of assumptions: A set is said uniformly convex if the principal curvatures of its boundary are all strictly positive.Moreover, assumption (iii) is equivalent to assume that D 2 (H 2 ) is definite positive.
The dual norm H • : R N → [0, +∞) is defined as: It is possible to show that H • is also a Finsler norm and it has the same regularity properties of H.Moreover, it holds (H • ) • = H.For R > 0 and x ∈ R N we define: For simplicity of exposition, when x = 0, we set: R and B H • R are also called "Wulff shape" and "Frank diagram" respectively.We remark that there holds the following identities: We refer the reader to [5,10] for further details.Observe also that H is a norm equivalent to the euclidean one, i.e. there exist α 1 , α 2 > 0 such that: (2.4) Moreover, recalling that H is 1-homogeneous, by the Euler's Theorem it follows Since H is 1-homogeneous, we have that ∇H is 0-homogeneous and it satisfies Hence, by the previous equality, we infer that there exists M > 0 such that For the same reasons there exists a constant M > 0 such that: (2.7) where | • | denotes the usual Euclidean norm of a matrix, and We start with some elliptic estimates that can be proved in the same spirit of the Euclidean framework.
We state now the Hardy inequality for the anisotropic operator ∆ H p u, defined in (1.1).We refer to [36,Proposition 7.5].

Theorem 2.2 (Hardy inequality).
For any H satisfying the assumption (h H ) and any u ∈ D 1,p (R N ) and 1 < p < N, where C H = ((N − p)/p) p is optimal.
Now we prove a technical lemma that will be very important in the proof of the asymptotic estimates.
Lemma 2.3.Let p > 1 and a, b ≥ 0.Then, for all δ > 0 there exist C δ > 0 such that (2.25) Proof.Let us consider p > 1 as follows: where ⌊•⌋ is the floor function and {•} is the mantissa function.Without loss of generality we assume that {p} = 0 and, moreover, we set m := ⌊p⌋.Hence, we have Using this inequality in (2.26), we deduce where we used the fact that p = m + {p}.Now, we can apply the weighted Young's inequality to each member of the first sum with conjugate exponents (p/(p − k), p/k) and to each member of the second sum with conjugate exponents (p/(m − k), p/(k + {p})) as follows (2.28) Hence, using this estimate we deduce where we renamed with C δ := C δ /(1 + 2 p+1 δ), and hence the thesis (2.25).
Finally, we recall a lemma (see Lemma 4.19 in [19]) that will be very useful in the proofs of our results.

Preliminary asymptotic estimates and comparison principles
The aim of this section is to prove some preliminary estimates that will be crucial in the proofs of the main results.
Lemma 3.1.There exists a positive constant τ depending only on N, p and γ such that for any R > 0 and for any solution u to problem (P H ) satisfying R ) ≤ τ, there exists a positive constant C depending only on N, p, γ and R such that where σ 1 , σ 2 are two positive constants depending on N, p and γ.
Proof.We start proving (3.2).To this aim let us consider R > 0 and a cut-off function By density argument it is possible to put ϕ = η p u as test function in (1.2), so that we obtain First of all, using Euler's Theorem (2.5), the 0-homogeneity of ∇H (2.6) and Schwarz's inequality, equation (3.5) becomes Recalling that H is 1-homogeneous function, using the weighted Young's inequality ab ≤ εa p p−1 + C ε b p on the first term of the right hand side of (3.6), for any 0 < ε < 1 we have where C(p, M, ε) := (pM) p C ε .Now, noticing that ∇(ηu) = u∇η + η∇u, by the triangular inequality, we deduce that for every p > 1 it holds Thanks to (3.8) and applying Lemma 2.3 with a = H(η∇u) and b = H(u∇η), we deduce that (3.9) Now, applying the anisotropic Hardy inequality (see Theorem 2.2 or [36]) and (2.4) we have Let us fix ε, δ > 0 sufficiently small such that where Now, using the Sobolev inequality in the left hand side of (3.13), the Hölder inequality and (3.4) in the right hand side, we obtain hence we deduce where C(p, N) is a positive constant depending on p and N. Setting and choosing R > 0 sufficiently small such that (3.1) holds, then u p * −p p * p and it depends only on N, p and γ.Denoting with where ϑ = C/( C + 1) ∈ (0, 1), depends only on N, p and γ.Now, by Lemma 2.4 it follows that In a similar way, we can deduce (3.3).

Now, we denote by
. There exists a positive constant σ = σ(N, p, γ, t) such that for any solution u to problem (P H ) and for any R > 0 satisfying the following inequality and C is a positive constant depending only on N, p, γ, R and t.
Let us now prove the following: Let u be a weak solution of (P H ). Then there exists a positive constant where σ 1 , σ 2 are givem in Lemma 3.1 and R 1 , R 2 > 0 are constants depending on N, p, γ and u.
Proof.Let us fix t := (p * + N/µ 1 )/2 ∈ (p * , N/µ 1 ) as in Lemma 3.2 and κ := min{τ, σ}, where τ and σ are respectively as in Lemma 3.1 and Lemma 3.2.Let R > 0 such that (3.1) holds for κ and let us consider û(x) = u(Rx), for R > 0 fixed.We note that û satisfies the equation ) with q = t/(p * − p) > N/p due to Lemma 3.2.Hence, as in the proof of [29, Theorem 1] a classical Moser iteration argument yields (3.36) sup ).We claim that V R L q (D 1 ) is uniformly bounded with respect to R. Indeed from Lemma 3.2, since where C is a positive constant depending on N, p, γ, q and R.
Using a covering argument we deduce that Noticing that û(x) = u(Rx), by (3.38) we obtain that (3.39) sup for each 0 < R ≤ R/8 or R ≥ 8/ R. By applying the Hölder's inequality in (3.39), we get for each 0 < R ≤ R/8 or R ≥ 8/ R and C depends only on N, p, γ, q, R and u L p * (R N ) .

Now we note that, since
c for any R ≥ 8/ R, there exist, by Lemma 3.1, σ 1 , σ 2 > 0 depending only on N, p, γ such that and that (3.40) we get the thesis.
The next result is devoted to show the existence of some special supersolutions of our problem, in order to perform a comparison between them and the solutions of the doubly critical equation (P H ). Proposition 3.4.Given two constants A > 0 and α < p, there exist constants 0 < ε, δ < 1, depending on N, p, γ, A, α, such that is a positive supersolution to equation for some positive constant 0 < R 1 < 1 depending only on N, p, γ, A and α, where g(x) is a positive function that belongs to In a similar way, given A > 0 and α > p, there exist 0 < ε, δ < 1 such that is a positive supersolution to equation for some positive constant R 2 > 1 depending only on N, p, γ and α, where g(x) is a positive function that belongs to Proof.Let us consider µ, δ, ε > 0 and let us define the function where s(t 2), we now compute where in the last line we used the fact that ∇H due to (2.2) and (2.3).
) and it satisfies (3.43).The other case is similar.Now, we consider the following equation where Ω is an open subset of R N , w > 0 and w ∈ D1,p (Ω).Let us start with a comparison principle in bounded domains.
The first result is given by the following pointwise estimate, in the same spirit of [26,40].
Proposition 3.5.Let u, v two weakly differentiable strictly positive functions on a domain Ω 1 .We have that: for some positive constant C p depending only on p; for some positive constant C p depending only on p.
Proof.Let u, v two weakly differentiable positive functions and consider the following Then, thanks to the Euler's theorem for 1-homogeneous functions, and since ∇H is 0homogeneus, we deduce that (3.56) (i) p ≥ 2. We recall that when p ≥ 2, it holds (2.10), i.e. (3.57) Hence we can apply this inequality, in order to give an estimate from below for (3.55) and (3.56): (3.59) Adding both these two inequalities, we obtain (3.60) (ii) 1 < p < 2. We recall that when 1 < p < 2, it holds (2.11), i.e. (3.61) where C p is a positive constant depending only on p. Now we proceed exactly as in the previous case to get an estimate from below for (3.55) and (3.56): (3.63) Adding both these two inequalities, we obtain Now, we are ready to prove the comparison principles in bounded and exteriors domains.
Proposition 3.6.Let Ω be an open bounded smooth domain of R N and f ∈ L N p (Ω).Let u ∈ D 1,p (Ω) be a weak positive subsolution to (3.52) and v ∈ D 1,p (Ω) be a weak positive supersolution of Proof.We will give the proof of this result in the case p ≥ 2. The case 1 < p < 2 is similar.
Let us define It is quite standard to show that η 1 and η 2 are good test function that we can use in the weak formulations of (3.52) and (3.65).Taking both these test function and subtracting the two equations, we obtain (3.67) Applying (3.53) in (3.67) and making some computations we obtain but this implies that For the right hand side of (3.69), we have where Hence, passing to the limit for m → +∞ in (3.69) we obtain that for some positive constant K.By our assumptions inf x∈Ω v > 0 and u ≤ v on ∂Ω, hence it follows that K = 1.But this implies that u ≤ v in Ω and this complete the proof of this result in the case p ≥ 2. The case 1 < p < 2 follows repeating verbatim the proof of the case p ≥ 2, but applying inequality (3.54) instead of (3.53).
Now we want to prove the corresponding result of Proposition 3.6 in exterior domains.
Proposition 3.7.Let Ω be an exterior domain such that R N \Ω is bounded and f ∈ L N p (Ω).Let u ∈ D 1,p (Ω) be a weak positive subsolutions to (3.52) and v ∈ D 1,p (Ω) be a positive supersolution of Proof.In the same spirit of Proposition 3.6 we prove our result in the case p ≥ 2. The other case is similar and it can be shown using the same arguments.To this aim, let

Let us define
where m > 1.As pointed out in the proof of previous proposition, it is possible to show, by standard arguments, that η 1 and η 2 are good test functions for the weak formulations (3.52) and (3.72).Hence, we obtain where By Proposition 3.5 and using the definition of ϕ R , it follows that there exits a positive constant depending only on p such that (3.76) Now we are going to give estimates for I 2 , I 3 and I 4 .We start with I 2 .Using (2.6) and the Cauchy-Schwarz inequality, setting Ω1 := {x ∈ Ω : v p ≤ u p }, we have where in the last line we applied the Hölder inequality with conjugate exponent (N, p/(p − 1), p * ) and C := 2M.Passing to the limit for R that goes to +∞ in the right hand side of (3.77), using also assumption (3.73) and (2.4), we deduce that I 2 goes to zero.Now we proceed with the estimate of the term I 3 .By (2.5), we have where Ω2 := {x ∈ Ω | u p ≥ m}.Using this definition and also the properties of ϕ R we deduce that where C := 2M.Passing to the limit for m, R that go to +∞, we deduce that For the last term I 4 , by (2.5), recalling Ω2 := {x ∈ Ω : u p ≥ m}, we have Hence, passing to the limit the right hand side of (3.80), by (3.73) we have that I 4 goes to zero when R tends to +∞.

Proof of the main results
This section is dedicated to the proof of our main results: Theorem 1.1, Theorem 1.3 and Theorem 1.4.
Proof of Theorem 1.1.We start by proving (1.4).To this aim, let us consider u a solution of (4.1) In particular, we have that u is a subsolution of (4.1) in any bounded domain R 1 due to Theorem 3.3.By Proposition 3.4 we have that the function and where 0 < δ, ε, R 1 < 1 are positive constants depending only on N, p, γ, A and α.
It is easy to chek that w is a positive supersolution of (3.65) Hence, by Proposition 3.6 we deduce that u ≤ w in B H • R 1 .Passing to the limit for Γ → 0 we obtain that where C = M • N .Now we have to show the estimate from below.Let u be a weak solution of (P H ), then u is a supersolution of We set R 1 , we conclude by using Proposition 3.6 to obtain u ≥ w in B H • R 1 , and hence combining the estimates from above and below we deduce that (1.4) is proved.Now, our aim is to prove (1.5).Let us consider u a subsolution of and where 0 < δ, ε < 1 and R 2 > 1 are positive constants depending only on N, p, γ, A and α.
We note that w is a positive supersolution of (3.72) . We verify the condition (3.73).Since |∇ log v(x)| ≤ C|x| −1 , by Hölder inequality we have lim sup where C is a constant independent of R.
Hence, by Proposition 3.7 we deduce that u ≤ w in (B H • R 2 ) c .Passing to the limit for Γ → 0 we obtain that where We conclude with the estimate from below.Let u be a weak solution of (P H ). Then u is a supersolution of We claim that (4.6) for R sufficiently large and constant C independent of R. Indeed, by (4.5), we have nonnegative function, and taking in (4.7) we have By Hölder inequality and by 0-homogeneity of ∇H we get (4.9) .
Taking a standard cutoff function ζ in (4.9) we get the claim (4.6).

Now we set
a weak solution of (4.5).Moreover the condition (3.73) is verified.Indeed by Hölder inequality and (4.6) we have since µ 2 > (N − p)/p.Applying the Proposition 3.7 we conclude that and therefore the thesis.Now we prove Theorem 1.3 that will be essential to prove the asymptotic behavior of the gradient of solutions to (P H ). For the reader convenience we state a more detailed statement contained in the following: where 0 ≤ γ < C H . Assume that there exist two positive constants C and c such that and suppose that there exists a positive constant Ĉ such that On the other hand, suppose that there exist two positive constants C and c such that and suppose that there exists a positive constant C such that Proof.We prove this result only for p ≥ 2. The other case is similar.Suppose that (4.12) holds.From definition of c 1 , there exist r (sufficiently small) such that given a n → 0.
On the other hand, there exist sequences of radii R n and points x n with R n tending to zero and ) and, since w n satisfies the equation (4.11), by [3,20,21], it is also uniformly bounded in C 1,α (K), for 0 < α < 1 and for any compact set . By (4.18), we have and by (4.19), there exist a point x ∈ ∂B H • 1 such that w ∞ (x) = c 1 .By the strong comparison principle [13], that holds under our assumption on H (see Section 2), we have and we note that it solves (4.11).Fix R > 0 sufficiently large and let ϕ R ∈ C ∞ c (B 2R ) be a standard cut-off function such that (4.23) Let us define (4.24) where m > 1.
We remark that ϕ 1 and ϕ 2 are good test function in any domain 0 ∈ Ω. Testing (4.24) in (4.11) on the domain R N \ B H • Rn , and, since the stress field H(∇u Rn ) (see [24]), exploiting the divergence theorem, we have where η n is the inner unite normal vector at ∂B H • Rn .Now we set A := {0 ≤ v p − vp ≤ m} and B := {v p − vp ≥ m}.Then (4.25) becomes: (4.28) By (4.22) we have where we used the Hölder inequality.Since v has the right summability at the infinity, for R that goes to infinity in the right hand side of (4.28), we obtain that I 3 + I 4 goes to zero.Now we proceed with the estimate of the term I 5 .Recalling that v p ≥ m in Ω 2 , we get Using the properties of ϕ R we obtain that Passing to the limit for m, R that go to +∞, we deduce that I 5 goes to zero since the set Ω 2 vanishes as m → +∞.
For the last term I 6 we obtain Hence, passing to the limit in the right hand side of (4.31) and using (4.29) we have that the right hand of (4.31) tends to zero when R tends to the infinity.Now we estimate the left hand of (4.26).By (4.24) and Proposition 2.1 (in particular see (2.11)), we get where, in the last line, we used (2.4) and (2.6).
We set x = R n y, with y ∈ ∂B H • 1 .Using this change of variables and recalling (4.12), (4.20) and (4.21), J 1 can be estimated as where we choose m = R −ǫ n , for ǫ > 0 fixed sufficiently small, and C1 is a positive constant.In a similar way J 2 is estimated as where C2 is a positive constant.
To prove that v ≤ v, let us consider (4.35) where m > 1 and ϕ R is the standard cutoff function defined in (4.23).Using (4.35) in the weak formulation of (4.11), proceeding in a similar way as above we obtain (4.14).The only thing to check is that Assumption (4.13) ensures (4.36), and therefore we are done.Now, assume that (4.15) holds and suppose that p ≥ 2, the other case is similar.The proof is similar to the previous one and, for this reason, we omit some details.From definition of c 2 , there exist r (sufficiently large) such that given a n → 0.
On the other hand, there exist sequences of radii R n and points x n with R n tending to infinity and As in the previous case we obtain (4.40) and we remark that it solves (4.11).We show that v = v in R N .To this aim, fix ε > 0 sufficiently small and let ϕ ε ∈ C ∞ (R N ) be a function such that (4.41) Let us define (4.42) where m > 1.
We note that ϕ where η n is the outward unite normal vector at ∂B H • Rn .By (4.42) and Proposition 2.1, the right hand of (4.43) becomes where, in the last line, we used (2.4) and (2.6).
We consider the following change of variables x = y/R n , with y ∈ ∂B H • 1 .Using this fact, by (4.39) and (4.40), we have where C 1 is a positive constant.
In a similar way J2 is estimated as where C 2 is a positive constant.We note that −N + µ 1 p + p < 0 since 0 ≤ µ 1 < (N − p)/p.Proceeding as in the case (4.12), we prove that (4.47) which implies v ≤ v in Ω as we concluded in the proof of Proposition 3.6.
To prove that v ≤ v, let us consider (4.48) where m > 1 and ϕ ε is defined as (4.41).Testing (4.48) in (4.11), using the assumption (4.16) and proceeding in a similar way as above we get (4.17).
At this point we are ready to prove Proof of Theorem 1.4.We prove (1.12), the other case is similar and it can be proved in a similar way.We start by showing the estimate from above.For R n tending to infinity, let us consider (4.49) ) and it weakly solves [3,20,21], w n is also uniformly bounded in C 1,α (K), for 0 < α < 1 and for any compact set For R n sufficiently large we get the estimate from above in (1.12).Now we prove the estimate from below.Suppose by contradiction that there exist sequences of points x n such that For 0 < a < A fixed, let us consider For n sufficiently large, from Theorem 1.1, relabeling the constants, we have Furthermore, recalling the estimate from above of the gradients of the weak solution u, proved previously, we get For a, A fixed, by [3,20,21], w n is also uniformly bounded in C 1,α (K), for 0 < α < 1 and for any compact set Recalling (A.1), using the weak lower semicontinuity of the norm and (A.3), we deduce Using Sobolev inequality we have that u n → u 0 a.e. in R N .Therefore by the Breis-Lieb result [6], it follows that Then from (A.5) we obtain and Using these last inequalities in (A.6) we get where in the last line we used (A.3).Taking the limit superior of (A.7) we get a contradicition, i.e.S(γ) < S(γ) unless the sequence , we finally get that u n → u 0 in D 1,p (R N ).Therefore passing to the limit in (A.1), we obtain , namely u 0 (eventually redefining it as Cu 0 , for some positive constant) is a weak solution to (P H ). Now we prove that actually (A.4) holds, concluding indeed the proof.Let {u n } the minimizing sequence such that (A.3) holds.For every n let us take a sequence of radii R n such that (A.8) and let us define the rescaled sequence (A.9) Using assumptions (h H ), (A.8) and (A.9) we deduce that (A.10) Now we prove the following Lemma A.2. Let {u n } be a minimizing sequence, weakly converging to zero.Then, for every ball B r and for every ε ∈ (−r, r) there exists ρ ∈ (0, ε) ∪ (ε, 0) such that for a subsequence Proof.By the homogeneity of the quotient in (A.1) we can assume that the minimizing sequence {u n } is such that u n L p * (R N ) = 1, so that L(u n ) → S(γ).By Ekeland's ε−principle, we can suppose that the minimizing sequence has the Palais-Smale property, that is (A. with both embedding compact, we can assume that a subsequence converges strongly to some limit, say u in the trace space W 1− 1 p ,p ((r + ρ)S N −1 ).Using the fact that the trace operator has a continuous embedding from W 1,p (B r+ρ ) into W 1− 1 p ,p ((r + ρ)S N −1 ), by the weak convergence to zero of {u n }, we deduce that indeed u ≡ 0.

Now we show the following
Let us define the two auxiliary sequences of D 1,p (R N ) as follows: In the same way, using u The other case of (A.11) can be proved arguing in the same as we have done above, assuming that {u 2,n } does not converge to zero.This concludes the proof of the lemma.
Using the invariance of the problem under the scaling R (p−N )/p n u(x/R n ), the sequence {w n } (see (A.9) and (A.10)) is still a minimizing sequence bounded in D 1,p (R N ): hence it admits (up to a subsequence) a weakly convergence sequence.We want to show that the weak limit cannot be zero.We argue by contradiction and we apply Lemma A.2 twice choosing r = 1 and ε = ±1/4 respectively.We find the existence of ρ + ∈ (0, 1/4) and ρ − ∈ (−1/4, 0) such that (A.11) holds.Using the alternative (A.11) together with (A.10), we obtain that  .
We claim that (A.25) is not possible.Indeed let u 0 be a minimizer of (A.1) for γ = 0 ( [11]).Therefore we clearly deduce the following