FOCUSING NLKG EQUATION WITH SINGULAR POTENTIAL

. We study the dynamics for the focusing nonlinear Klein Gordon equation with a positive, singular, radial potential and initial data in energy space. More precisely, we deal with with 0 < a < 2. In dimension d ≥ 3, we establish the existence and uniqueness of the ground state solution that enables us to deﬁne a threshold size for the initial data that separates global existence and blow-up. We ﬁnd a critical exponent depending on a . We establish a global existence result for subcritical exponents and subcritical energy data. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary sets.

In [5] the authors consider the case m = 0, V (x) = 1 and 1 + 4/d < p ≤ 2 * − 1. They give a similar dichotomy result by using the ground state of the wave equation. The same authors cover the critical exponent at energy threshold level in [6]. For the case d = 2, they take a different nonlinear term.
Here we deal with a singular potential, so that our equation is the relativistic version of the Schrödinger equation considered in [1]: The presence of the potential |x| −a suggests that the critical value p 0 = 1+4/(d−2) has to be shifted. It is reasonable to concentrate on the case p < p 0 (a) = 1 + (4 − 2a)/(d − 2) as in [1]. This equation has many application in nonlinear optics and in that paper a dichotomy result is established. More precisely the size of the initial data is compared with the size of the ground state in the energy space. We expect that for such potentials and Klein Gordon operator the dichotomy is preserved by using the ground state energy. In turn this energy threshold will depend on the potential. Here we consider the subcritical case, that is p < 1 + (4 − 2a)/(d − 2).
Since p is energy-subcritical and since we are interesting in the existence result only, avoiding the treatment of the scattering problem, we can avoid concentrationcompactness principle, used in [7]. More precisely, the key point in our approach is the use of the uniqueness result of ground state solutions from [12]. Let us compare our technique with the one used in [5] for V = 1. The proof of such uniqueness result in the case of KG equation u tt − ∆u + m 2 u = f (u) with combined nonlinearities f (u) = c 1 |u| p1−1 u + c 2 |u| p1−1 u meets essential difficulties. Therefore, we have chosen a situation different from [5] and we have a simplified argument. We underline that we shall prove a global existence for subcritical exponent below the energy threshold. The restriction 0 < a we impose says that we treat a case essentially different from the case V = 1 and our results are not applicable in that situation.
Finally we prove a dichotomy result in the subcritical exponent and threshold energy according to the relation between the H 1 norm of u 0 with respect to the V (x)|u 0 | p+1 d x. This is a novelty with respect to [7] where this case is not considered.
In a forthcoming paper we will deal with a non-symmetric potential and the uniqueness property will be more delicate, since the scale invariance property of the equation is lost.
Similar arguments with scale-invariant nonlinear term are treated in [13] and [3]. In particular in [3] the authors consider m = |x| −2 and V ≡ 1. The paper [2] overcomes the scale-invariance requirement but deals with the specific cases d = 2, 3 and p = 1 + 4/d and V ∈ L q with d 2 /4 ≤ q < ∞. On the other hand, in that paper the nonlinear exponent is independent of decay power of the potential. In this work we observe a possible dependence of the energy critical threshold for the nonlinear exponent on the decay of the potential at infinity.
1.1. Main theorems. By the aid of Strichartz estimates and nonlinear pseudodifferential inequalities in [4], we prove the the following local existence result.
Theorem 1.1. Let d ≥ 3 and Then for R > 0 there exists T = T (R) > 0 and a Banach space X T such that Our nonlinear problem (1) has energy quantity This quantity is conserved, so that Our first aim is to neglect the time dependence and deal with the ground states associated with the minimization of the energy functional In particular . Namely, let µ > 0, we consider the minimization problem inf g∈N (µ) where N (µ) is the constraint determined by the nonlinear term Due to our assumption 0 < a < 2 and to the next Lemma 1.2, for any g ∈ H 1 (R d ), one has In particular N (µ) is well defined. Let us state and prove this Gagliardo-Nirenberg type inequality. Lemma 1.2. Let d ≥ 3, 0 < a < 2 and p < p GN given by There exists C > 0 and θ 1 , Proof. Lemma 3 in [1] with u = ξ = ϕ and v = 0 gives Being p < p GN the interval for γ is not-empty. Recalling the usual Gagliardo-Nirenberg inequality 2γ and θ 2 = d(p−1) 2(p+1) , we have the thesis. Our first result is the following. Theorem 1.3. Suppose that d ≥ 3 and Then we have the following properties: a) for any µ > 0 one can find a unique radial, positive, decaying exponentially to 0 as r → ∞ function Q µ ∈ N (µ), which solves the problem (2), that is Moreover the function Q µ satisfies the equation b) there exists a unique µ 0 > 0 and a unique radial, positive, decaying exponentially to 0 as r → ∞ function Q = Q µ0 ∈ N (µ 0 ) which is solution of the equation The key tool in the proof is the following uniqueness result.
has at most one positive, radial solution Q(x), decaying exponentially to 0 as r → ∞.
Another characterization of the "mountain"is the following link to the classical Payne-Sattinger energy critical level where K(g) determines the first Pohozaev relation We have the following result.
Our next global existence theorem shows that in the energy subcritical case, for suitable controlled energy data, the solution given in Theorem 1.1 is a global solution.
Theorem 1.6. Suppose that d ≥ 3 and Let µ 0 > 0 and Q µ0 be determined in Theorem 1.3 b). For any initial data (u 0 , , and |x| −a |u 0 | p+1 d x < µ 0 then we have the existence of the unique global solution to the Cauchy problem (1), moreover Let (−T − , T + ) be the maximal existence interval for the solution to the Cauchy problem (1) u ∈ C((−T − , T + ); H 1 ×L 2 ) and let us concentrate on the energy critical case In this case the threshold separating global existence (T − = T + = ∞) and blow-up (at least one of T − , T + is finite) is determined by the functional K defined in (7).
First, we consider the case Theorem 1.7. Suppose that d ≥ 3, and (8) and (9) we have unique global solution to the Cauchy problem (1).
If K(u 0 ) < 0 (10) then we have the following blow-up result.

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VLADIMIR GEORGIEV AND SANDRA LUCENTE Theorem 1.8. Suppose that d ≥ 3, and Let µ 0 > 0 and Q µ0 be determined in Theorem 1.3 b). If the initial data (u 0 , u 1 ) satisfy (8) and (10), then we have at least one of the following possibilities 2. Existence and uniqueness of ground states.

2.1.
Uniqueness of radial positive solutions. One can use the uniqueness result of Yanagida [12], Theorem 2.2, to get uniqueness of radial positive decreasing solutions to (5). We rewrite this result having in mind our particular application.
We can use further the estimate (see Lemma 7.17 in [8]) This estimate and (iv) Section 3.3 in [8], imply Moreover the constraint domain is invariant under the action of the Schwartz symmetrization, i.e.
Due to the positivity, we have I(µ) ≥ 0. We can find a minimizing sequence Being such sequence bounded in H 1 rad , there exists a weak limit g ∈ H 1 rad , that is g n g ∈ H 1 rad Using the weakly lower-semicontinuous property of the norm we have Hence g is the good candidate to the minimum. It remains to show that so that g ∈ N rad (µ), and I(µ) ≤ g 2 . It sufficies to show that, a part of passing to subsequence, one has Here we need Theorem 1 in [11].
One can derive the Euler-Lagrange equation for the minimizers of I(µ) recalling the sharpest constraint condition |x| −a |g(x)| p+1 dx = µ. Considering the functional where ω = ω(µ) is the Lagrange multiplier. If Q µ is a minimizer of (2), then it is a critical point of VLADIMIR GEORGIEV AND SANDRA LUCENTE at ε = 0 and for any test function h(x). We get the equation Due to maximum principle we can conclude that Q µ is strictly positive. It remains to show that Q µ has an exponential decay. The exponential decay of the solution Q(r) to (22) can be obtained, by setting and using the relation On the other hand, the radial Strauss lemma assures that, for any H 1 radial function, it holds . From this and (23), we can deduce that Moreover T (r) > 0 for any r > 0. Hence the relation (24) implies that there exists R 0 > 0 such that Now we observe that T ∈ H 1 (R 0 , +∞).
This information, combined with T (r) > 0 for r ≥ R 0 assures that T (r) < 0 for r > R 0 . Indeed if there exists R > 0 such that T (R) > 0 then it remains positive for any r > R, in particular T (r) is strictly increasing in (R, +∞), and T is not in L 2 (0, ∞). We can summarize these properties as T (r) > 0 is a decreasing function for r ≥ R 0 ; T (r) < 0 is a increasing function for r ≥ R 0 ; Therefore we define the simplified "energy" functional E(r) = (T (r)) 2 2 − m 2 (T (r)) 2 8 so using (25) we see that E (r) < 0, ∀r ≥ R 0 .
Using the fact that T ∈ H 1 (R 0 , ∞) we see that E(r) ∈ L 1 (R 0 , ∞). Being E(r) strictly decreasing, the only possibility to remain summable is that E(r) > 0 and E(r) → 0 for r → ∞. By using the positivity of E(r) we conclude that −T (r) = |T (r)| ≥ mT (r) 2 for any r ≥ R 0 .
Integrating this inequality, we get and as a consequence the exponential decay of Q µ . Let us discuss the behaviour of ω(µ) with respect to µ. By homogeneity argument, we can rewrite However, the minimizers of I(1) (and hence the minimizers of I(µ)) is not unique since the homogeneous of order zero functional is invariant under multiplication by any non zero complex number. In order to have uniqueness, we recall the constraint condition Let Q µ be a positive radial solution to Multiplying by Q µ and Q 1 respectively, applying the constrained conditions, we get ω(µ)µ = 2I(µ) = 2µ Proof of Theorem 1.5. Let K − = g ∈ H 1 \ {0} | K(g) ≤ 0 with K(g) defined in (7). Since g ∈ K − =⇒ g * ∈ K − , then we can proceed as in the proof of Theorem 1.3 and find Q ∈ H 1 rad (R d ) radial, positive and exponentially decaying function such that To be more precise, if {Q k } is a minimizing radially symmetric non-negative sequence, such that then Q k tends weakly to Q and satisfies the estimate To the left side, we apply the Gagliardo-Nirenberg inequality (3) and deduce By using Q k ≡ 0, we find .
Recalling that p is subcritical, passing to the limit we deduce the nonvanishing property Q = 0. If K(Q) < 0, then Q satisfies the equation −∆Q + m 2 Q = 0 and then we get Q = 0 and this contradicts the nonvanishing property Q = 0. Hence we have K(Q) = 0 and then Indeed rewriting (4) in Q µ0 and multiplying by Q µ0 we find K(Q µ0 ) = 0 so that I K ≤ I(µ 0 ). Conversely, the minimizer Q of and since K(Q) = 0. Therefore, we have Multiplying the equation (28) by Q, we get Λ = 1/p and substituting Λ in (28) gives Due to the uniqueness proved in Lemma 1.4, we arrive at Q = Q µ0 ∈ N (µ 0 ), so that I K ≥ I(µ 0 ) and (27) is established. This completes the proof.
3. Global existence in the energy subcritical case.
Proof of Theorem 1.6. Let T > 0 be the maximal time existence for the solution of (1).

Recalling the energy conservation law E[u](t) = E[u](0) <Ẽ[Q µ0
], and the fact that Q µ0 is the unique radial, positive, exponentially decreasing minimizer of we obtain the chain of inequalities Hence we arrive at a trivial contradiction, showing that (29) is fulfilled. Now we see how this implies the global existence result. Let t ∈ [0, T ), we have This assures that the H 1 norm of the solution is uniformly bounded, hence T = +∞.
4. Threshold manifold in the critical case. In all this section we assume The threshold that separates the blow-up and global existence behaviour on the energy critical level (30) is determined by the functional and K(u 0 ) < 0 (resp. K(u 0 ) ≥ 0) then for any solution u(t) ∈ C((−T − , T + ); H 1 × L 2 ) to the Cauchy problem (1) we have K(u(t)) < 0 (resp. K(u(t)) ≥ 0).
By using Gagliardo-Nirenberg inequalities, we see that there exists C 0 > 0 such that u(τ ) H1 > C 0 > 0, passing to the limit for t → t 1 , we can conclude K(u(t 1 )) = 0 , u(t 1 ) ≡ 0. (32) Being Q the unique radial positive minimizer of we can write the inequalities Using further (32) and K(Q) = 0, we get The energy conservation and the assumption (8) implỹ We conclude that This identity, the assumption K(u(t 1 )) = 0 and the relation K(Q) = 0 imply that Using (16) and (17), we obtain K(u * (t 1 )) ≤ K(u(t 1 )) = 0. SinceẼ[u](t 1 ) =Ẽ[Q] and u(t 1 ) ≡ 0, we can conclude that u * (t 1 ) is a minimizer of I K . To this end, from Theorem 1.5, we use the uniqueness result for the minimizers of I K and deduce Using Theorem 3.4 in [8], we know that the estimate We can deduce In particular |u( ∞)), hence the identity |u(t 1 , |x|)| = Q(|x|) > 0 between continuous one variable functions implies On the other hand u(t 1 , x) = Q(x) is the global solution on R of the Cauchy Problem hence we can conclude that In particular K(u 0 ) = K(±Q) = 0 and this contradicts (31).
Summarizing, we proved that the set is invariant under the flow of (1). Next we consider the case If by absurd K(u(τ )) < 0 for some τ ≥ 0, then u(τ ) ∈ G, hence u(t) ∈ G for any t ∈ (−T − , T + ) in particular u 0 ∈ G against our assumption (33).
Proof of Theorem 1.7. Let K(u 0 ) ≥ 0, u 0 ≡ 0. From the previous lemma we have Recalling thatẼ Due to (34), we conclude This estimate shows that This uniform H 1 bound implies that one can extend the solution up to T = +∞.
Recalling that E[u](t) =Ẽ[Q] = 1 2 − 1 p+1 µ 0 , we can deduce Now we shall use the relation (kind of virial identity) d dt which is given by integration of the identity . For the right side of this expression, we have the following different cases: a): there exists T 1 ∈ (0, T + ) so that and Remark 1. We can immediately discard the case b), since in this case (39) and and this obviously contradicts the inequality (35).
In the same way we can get the blow-up result for time interval (−T, 0], concluding the proof of Theorem 1.8. The same argument of the proof of Theorem 4.2 guarantees that c 0 = 0. Integrating the equation (37), we get u(t) 2 L 2 ≥ −2c 0 (t − T 2 + 1) > 0, ∀t < T 2 − 1.
At this point the proof follows exactly the one of Theorem 4.2.