Ground state for the Schr\"odinger operater with the weighted Hardy potential

We establish the existence of ground states on Euclidean space for the Laplace operator involving the Hardy type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.


Introduction
In this paper we investigate the existence of ground states of the Schrödinger operator associated with the quadratic form where V belongs to the Lorentz space L N 2 ,∞ (R N ) and Λ V is the largest constant (whenever exists) for which the form Q V is nonnegative. This assumption implies that the potential term R N V (x)u 2 dx is continuous in D 1,2 (R N ), where D 1,2 (R N ) is the Sobolev space obtained as the completion of C ∞ • (R N ) with respect to the norm We are mainly interested in the case of the Hardy type potential V (x) = m(x) |x| 2 with m ∈ L ∞ (R N ). Assuming that V is positive on a set of positive measure, the constant Λ V is given by the variational problem and the continuity of R N V (x)u 2 dx implies that Λ V > 0. If problem (1.2) has a minimizer u, then it satisfies the equation A solution of (1.3) is understood in the weak sense for every φ ∈ D 1,2 (R N ).
Since |u| is also a minimizer for Λ V , we may assume that u ≥ 0 a.e. on R N . In particular, when V (x) = m(x) |x| 2 with m ∈ L ∞ (R N ), then u > 0 on R N by the Harnack inequality [14]. If the potential term is weakly continuous in D 1,2 (R N ), for example, when V (x) = m(x) |x| 2 with m ∈ L ∞ (R N ) and lim |x|→∞ m(x) = lim x→0 m(x) = 0, then there exists a minimizer for Λ V . We will call the minimizer of (1.2) a ground state of finite energy. In general, (1.2) may not have a minimizer. This is the case for the Hardy potential V (x) = 1 |x| 2 with the corresponding optimal constant Λ V = Λ N = N −2 2 2 . In fact, the ground state of finite energy is a particular case of the generalized ground state, defined as follows (see [24], [26] and [27]).

Definition 1.1
Let Ω ⊂ R N be an open set, and let Q V be as in (1.1). A sequence of nonnegative functions v k ∈ C ∞ • (Ω) is said to be a null-sequence for the functional Q V if Q V (v k ) → 0, as k → ∞, and there exists a nonnegative function ψ ∈ C ∞ • (Ω) such that Ω ψv k dx = 1 for each k.
Let us recall that the capacity of a compact set E relative to an open set Ω ⊂ R N , with E ⊂ Ω, is given by cap (E, Ω) = inf{ Ω |∇u| 2 dx; u ∈ C ∞ • (Ω), with u(x) ≥ 1 on E}.
In the case Ω = R N we use notation cap (E) (see [23]).
Theorem 1.2 Let V be a measurable function bounded on every compact subset of Ω = R N − Z, where Z is a closed set of capacity zero, and assume that Q V (u) ≥ 0 for all u ∈ C ∞ • (Ω). Then, if Q V admits a null sequence v k , then the sequence v k converges weakly in H 1 loc (R N ) to a unique (up to a multiplicative constant ) positive solution of (1.3).
This theorem gives rise to the definition of the generalized ground state. If V (x) = 1 |x| 2 , the functional Q V has a ground state v(x) = |x| It is important to note that the functional Q V with the optimal constant Λ V does not necessarily have a ground state. We quote the following statement from [27]. Theorem 1.4 Let V be a measurable function bounded on every compact subset of Ω = R N − Z, where Z is a closed set of capacity zero, and assume that Q V (u) ≥ 0 for all u ∈ C ∞ • (Ω). Then either Q V admits a null sequence, or there exists a function W , positive and continuous on Ω, such that For example, let m be a continuous function on Obviously, ground states of finite D 1,2 norm are principal eigenfunctions of (1.3). There is a quite extensive literature on principal eigenfunctions with indefinite weight functions for elliptic operators on R N , or on unbounded domains of R N , with the Dirichlet boundary conditions. We mention papers [2], [6], [7], [15], [19], [24], [29], [30], [31], where the existence of principal eigenfunctions has been established under various assumptions on weight functions. These conditions require that a potential belongs to some Lebesgue space, for example L p (R N ) with p > N 2 . These results have been recently greatly improved in papers [3] and [33], where potentials from the Lorentz spaces have been considered. To describe the results from [3] and [33] we recall the definition of the Lorentz space [5], [18], [21]. Let f : R N → R be a measurable function. We define the distribution function α f and a nonincreasing rearrangement f * of f in the following way We now set The functional f * (p,q) is only a quasi-norm. To obtain a norm we replace f by f * * (t) = , that is, the norm is given by In paper [33] the existence of principal eigenfunctions has been established for weights belonging to 1≤q<∞ L N 2 ,q (R N ). This was extended in [3] to a larger class of weights F N 2 obtained as the completion of C ∞ • (R N ) in norm · N 2 ,∞ . However, these conditions do not cover the singular weight functions considered in this paper. By contrast, in our approach we give an exact upper bound for the principal eigenvalue which allows us to prove the existence of the principal eigenfunction. We point out that if V ∈ L N 2 ,∞ (R N ), then the functional R N V (x)u 2 dx is continuous on D 1,2 (R N ), but not necessarily weakly continuous.
The paper is organized as follows. In Section 2 we prove the existence of minimizers with finite norm D 1,2 (R N ) and also with infinite norm D 1,2 (R N ). In Section 3 we discuss perturbation of a given quadratic form . We show that if Q V• has ground state, then this property is stable under small perturbations of V • . This is not true if Q V• does not have a ground state; rather it is stable under larger perturbation of V • . The final Section is devoted to a higher integrability property of minimizers of Q V• in the case where V • (x) = m(x) |x| 2 with m ∈ L ∞ (R N ). We also examine the behaviour of the principal eigenfunction around 0.
Throughout this paper, in a given Banach space we denote strong convergence by " → " and weak convergence by " ⇀ ". The norms in the Lebesgue space L p (Ω), 1 ≤ p ≤ ∞, are denoted by u p .

Existence of minimizers
We consider the Hardy type potential V (x) = m(x) |x| 2 with m ∈ L ∞ (R N ). In Theorem 2.2 we formulate conditions on m guaranteeing the existence of a principal eigenfunction. Let γ + > 1 and γ − > 1. In our approach to problem (1.2) the following two limits play an important role: it is assumed that the following limits exist a.e.
Both functions m ± satisfy m ± (γ ± x) = m ± (x), that is, m ± are homogeneous of degree 0. We now define the following infima: , (we use the notation Λ m instead of Λ V ) and The following holds true Letting j → ∞ and using the Lebesgue dominated convergence theorem, we obtain The inequality Λ m ≤ Λ + follows. The proof of the inequality Λ m ≤ Λ − is similar. In the case when the inequality (2.4) is strict problem (2.2) has a minimizer.
Proof Let {u k } ⊂ D 1,2 (R N ) be a minimizing sequence for Λ m , that is, We can assume, up to a subsequence, that u k ⇀ w in D 1,2 (R N ), L 2 (R N , dx |x| 2 ) and u k → w in L 2 loc (R N ) for some w ∈ D 1,2 (R N ). Let v k = u k − w. We then have and We define a radial function χ j It is clear that J 2 is a quantity of type o (j) k→∞ (1). Therefore, we have In a similar way we obtain for j sufficiently large. We now fix j ∈ N so that (2.9) and (2.10) hold. Consequently, we have We now estimate R N |∇v k | 2 dx in the following way Since v k → 0 in L 2 loc (R N ) we obtain the following estimate This, combined with (2.6), gives the following estimate Let Λ * = min Λ − , Λ + . We deduce from (2.11) and (2.12) that It then follows from (2.11) that In what follows, we use denote by m(∞) = lim |x|→∞ m(x), assuming that this limit exists. As a direct consequence of Theorem 2.2 we obtain the following result.
, then there exists a minimizer for Λ m .
Remark 2.4 Λ m has a minimizer also in the following cases, corresponding formally to Λ + or Λ − taking the value +∞. We point out that Theorem 2.3 and the results described in Remark 2.4 can be deduced from Theorem 1.2 in [32]. Unlike in paper [32], to obtain Theorem 2.3 we avoided the use of the concentration -compactness principle.
We now give examples of weight functions m satisfying conditions of Theorems 2.2 and 2.3. In general, functions satisfying this condition have large local maxima.
where A > 0 is a constant to be chosen later and m 1 : Further we assume that Both limits are uniform. Since m − and m + are bounded, Λ − and Λ + are positive and finite. We have for A large. By Theorem 2.2, Λ m with m = m A has a minimizer.
Then m k (0) = B and m k (∞) = A for k = 1, 2, . . .. We show that for k sufficiently large m k satisfies the conditions of Theorem 2.3. Let u(x) = exp(−|x|) (one can take any other as k → ∞. So we can find k • ≥ 1 so that In Proposition 2.7, below, we described a class of weight functions m satisfying conditions of Theorem 2.3.
|x| 2 u 2 dx > 0} we deduce from the above inequality that where λ D 1 (B(x M , r)) denotes the first eigenvalue for " − ∆" in B(x M , r) with the Dirichlet boundary conditions. We now estimate λ D Combining this with (2.14) we derive Therefore Λ m < Λ N min 1 m(0) , 1 m(∞) if (2.13) holds.

2.
The estimate (2.13) has terms that are easy to compute, but are of course not optimal. In particular, the factor (N +1)(N +2) 2 can be replaced by the first eigenvalue of the Laplacian on a unit ball with Dirichlet boundary conditions.
If m(x) is a continuous bounded and nonnegative function such that m(x) ≤ m(0) on R N and m(0) > 0 (or m(x) ≤ m(∞) on R N , m(∞) > 0), then Λ m does not have a minimizer. Indeed, suppose that m(x) ≤ m(0) on R N and that Λ m has a minimizer u. Then by the Hardy inequality we obtain So u is a minimizer for Λ N , which is impossible.
We now construct a ground state with infinite D 1,2 norm.
Theorem 2.8 Let γ > 1 and assume that the function m ∈ L ∞ (R N ) satisfies Then the form The function v is uniquely defined by its values on A γ = {x ∈ R N ; 1 < |x| < γ} and moreover the function v |Aγ is a minimizer for the problem Proof The problem (2.17) is a compact variational problem that has a minimizer v which satisfies the equation with the Neumann boundary conditions. Since the test functions satisfy u(γx) = γ 2−N 2 u(x) for |x| = 1, one has Note that |v| is also a minimizer, so we may assume that v is nonnegative. We now extend the function v from A γ to R N − {0} by using (2.16) and denote the extended function again by v. Since v satisfies (2.17), the extended function v is of class C 1 (R N − {0}) and satisfies the equation in a weak sense. From this and the Harnack inequality on bounded subsets of R N − {0} it follows that v is positive on R N − {0} and subsequently there exists a constant C > 0 such that We can now explain the choice of the exponent 2−N 2 in the constraint u(γx) = γ 2−N 2 u(x) from (2.17): with any other choice the resulting Neumann condition would not yield the continuity of the derivatives of the extended function v on the spheres |x| = γ j , j ∈ N. Finally, we show that v is a ground state for the corresponding quadratic form Using the ground state formula (2.7) from [28] and (2.19), we have with w k (x) = |x| 1 k for |x| ≤ 1 and w k (x) = |x| − 1 k for |x| ≥ 1, as k → ∞. Since vw k → v uniformly on compact sets, this implies that v is a ground state for Q. By (2.19) and the Sobolev inequality, v ∈ D 1,2 (R N ). 2

Perturbations from virtual ground states
In this section we show that if a potential term admits a (generalized or large or virtual) ground state, then its weakly continuous perturbations in the suitable direction will admit a ground state with the finite D 1,2 norm. Then we investigate potentials that do not give rise to a ground state with finite D 1,2 norm.
We need the following existence result.
is positive on a set of positive measure and that the functional If Λ 1 < Λ • , then there exists a minimizer for Λ 1 .
Proof Let {u k } ⊂ D 1,2 (R N ) be a minimizing sequence for (3.2), that is, R N V 1 (x)u 2 k dx = 1 and R N |∇u k | 2 dx → Λ 1 . We may assume that, up to a subsequence, u k ⇀ w in D 1,2 (R N ) and Assuming that t < 1 we get From this we deduce that Λ 1 ≥ Λ • which is impossible. Hence R N V 1 (x)w 2 dx = 1. From this and the lower semi-continuity of the norm with respect to weak convergence, we derive that w is a minimizer and u k → w in D 1,2 (R N ).
Proposition 3.1 is related to Theorem 1.7 in [32] which asserts that a potential of the form V (x) = 1 |x| 2 + g(x), with a subcritical potential g (for the definition of a subcritical potential see [32]) has a principal eigenfunction. This follows from the fact that g is weakly continuous in D 1,2 (R N ) (see [30]) and the potential g admits a principal eigenfunction.
We now give a sufficient condition for the inequality Λ 1 < Λ • .
Proof It suffices to show that the inequality for sufficiently large k, which completes the proof of the theorem.   (1.5). Then for every t ∈ 0, 1 Λ• the functional Q V•+tW has no ground state and Λ V•+tW = Λ V• . Furthermore, if the functional R N W (x)u 2 dx is weakly continuous in D 1,2 (R N ), the the same conclusion holds for −∞ < t < 0.
Proof First we observe that the constants Λ • and Λ 1 corresponding to V • and V 1 = V • +tW , respectively, are equal. Indeed, since V 1 > V • , one has Λ 1 ≤ Λ • by monotonicity. On the other hand, it follows from (1.5) that which implies that,up to subsequence, v k → 0 a.e. If v k were a null sequence, it would converge in H 1 loc (R N ) and it would have a limit zero. Therefore Q V 1 admits no null sequence and consequently no ground state. Assume now that the functional R N W (x)u 2 dx is weakly continuous in D 1,2 (R N ). Let {w k } ⊂ D 1,2 (R N ) be a minimizing sequence for Λ • . If {w k } has a subsequence weakly convergent in D 1,2 (R N ) to some w = 0, then it is easy to see that |w| would be a minimizer for Λ • and thus a ground state for Q Λ• . Therefore w k ⇀ 0. By the weak continuity of R N W (x)u 2 dx we get and thus This yields Λ 1 ≤ Λ • . Then Since t < 0, this implies that Q V 1 has no ground state. 2 Theorem 3.5 concerns with small perturbations of a potential that does not change the constant Λ or the absence of a ground state. The next theorem shows that a compact perturbation of the potential term yields a ground state of finite D 1,2 (R N ) norm.
Theorem 3.6 Assume that V • satisfies conditions of Proposition 3.1 and that W ∈ L 2,∞ (R N ) is such that the functional R N W (x)u 2 dx is weakly continuous in D 1,2 (R N ). Then for every λ ∈ 0, Λ • there exists σ ∈ R such that Q V•+σW has a ground state of finite D 1,2 (R N ) norm corresponding to the energy constant (3.2).
Proof Assume without loss of generality that W is positive on a set of positive measure. Let 0 < λ < Λ • and consider σ = inf defines an equivalent norm on D 1,2 (R N ) it is easy to show that there exists a minimizer for σ. It is clear that this minimizer is also a ground state of Q V•+σW corresponding to the optimal constant λ. 2 If we assume additionally that W is positive on a set of positive measure, then it is easy to show that σ is a continuous decreasing function of λ with lim λ→0 σ(λ) = +∞ and σ • = lim λ→Λ• σ(λ) ≥ 0. In particular, if (1.5) holds with a weight W • satisfying W • ≥ αW , then σ • ≥ α. In other words, given V • and W as in Theorem 3.6, the potential V • + σW admits a ground state whenever σ ≥ σ • .
For further results of that nature we refer to paper [32].

Behaviour of a ground state around 0
In what follows we consider the potential of the Hardy type V (x) = m(x) |x| 2 , where m(x) is continuous and m(0) > 0 and m(∞) > 0. The corresponding ground state, if it exists, is denoted by φ 1 , which is chosen to be positive on R N . Obviously the ground state φ satisfies the equation We need the following extension of the Hardy inequality: let Ω ⊂ R N be a bounded domain and 0 ∈Ω, then for every δ > 0 there exists a constant A(δ, Ω) > 0 such that for every u ∈ H 1 (Ω) (see [11]).
Proof Let Φ ∈ C 1 (R N ) be such that Φ(x) = 1 on B(0, r), Φ(x) = 0 on R N − B(0, 2r), 0 ≤ Φ(x) ≤ 1 on R N and |∇Φ(x)| ≤ 2 r . For simplicity we set λ = Λ m , u = φ 1 . We define v = Φ 2 u min(u, L) p−2 = Φ 2 uu p−2 L , where L > 0 and p > 2. Testing (4.1) with v, we get Applying the Young inequality to the third term on the left side, we get where η > 0 is a small number to be suitably chosen. Since the second integral on the left side is nonnegative, this inequality can be rewritten in the following form Multiplying this inequality by p+2 4 and noting that p+2 4 > 1, we get We now observe that Hence (4.3) takes the form Since λm(0) Λ N < 1, we can choose ǫ 1 > 0 so that λ Λ N (m(0) + ǫ 1 ) < 1. By the continuity of m there exists 0 < r 1 < r such that m(x) ≤ m(0) + ǫ 1 for x ∈ B(0, r 1 ). This is now used to estimate the first integral on the right side of (4.4): Applying the Hardy inequality (4.2), we get for every ǫ > 0. Inserting this estimate into (4.4) we obtain (4.5) . We put p = 2 + δ, δ > 0. We now observe that we can choose δ and ǫ so small that We point out that we have used here the inequality λ Λ N (m(0) + ǫ 1 ) < 1. With this choice of ǫ and δ we now choose η > 0 so small that Finally, we apply the Sobolev inequality in H 1 (B(0, r)) and deduce where S denotes the best Sobolev constant of the embedding of H 1 (B(0, r)) into L 2 * (B(0, r)).
Letting L → ∞ we deduce that u ∈ L 2 * (1+ δ 2 ) (B(0, r)). So the assertion holds with δ • = δ 2 . 2 We now establish the higher integrability property of the principal eigenfunction on R N \ B(0, R). Although this will not be used in the sequel, we add it for the sake of completness. We denote by D 1,2 (R N \ B(0, R)) the Sobolev space defined by for every u ∈ D 1,2 (R N \ B(0, R)).
We now consider the above equation in a small ball B(0, r). Since   Then there exists r > 0 such that (4.10) for x ∈ B(0, r) and some constants M 1 > 0, M 2 > 0.
The lower bound follows from Proposition 2.2 in [13]. To apply it we need inequality (4.9).