GROUNDSTATES FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH POTENTIAL VANISHING AT INFINITY

Transcription

1 GROUNDSTATES FOR THE NONLINEAR SCHRÖDINGER EQUATION WITH POTENTIAL VANISHING AT INFINITY DENIS BONHEURE AND JEAN VAN SCHAFTINGEN Abstract. Groundstates of the stationary nonlinear Schrödinger equation u + V u = Ku, are studied when the nonnegative function V and K are neither bounded away from zero, nor bounded from above. A secial care is aid to the case of a otential V that goes to 0 at infinity. Conditions on comact embeddings that allow to rove in articular the existence of groundstates are established. The fact that the solution is in L (R N ) is studied and decay estimates are derived using Moser iteration scheme. The results deend on whether V decays slower than x at infinity. Contents. Introduction. Embedding theorems 7.. Concentration function method 7.. Marcinkiewicz saces method 7.3. Trace-tye inequalities 8 3. Decay estimates Linear estimates Nonlinear estimates Moser iteration scheme 3.4. Proof of Theorem Further comments Fast decay for exloding otential Divergence-form oerators Nonuniformly ellitic oerators 5 References 6 Date: May 7, Mathematics Subject Classification. 35J0 (35B65, 35J60, 35Q55). Key words and hrases. Stationnary nonlinear Schrödinger equation; decay of solutions; weighted Sobolev saces; regularity theory; Moser iteration scheme; trace inequalities; degenerate otentials. Both authors were suorted by the Fonds de la Recherche Scientifique FNRS (Communauté française de Belgique). JVS was suorted by the Fonds séciaux de recherche (Université catholique de Louvain).

2 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN. Introduction In this aer, we consider the following roblem for the time-indeendent nonlinear Schrödinger equation: u + V u = Ku in R N, lim u(x) = 0. (P V,K ) x Here u : R N R is an unknown function, while V : R N R + and K : R N R + are given otentials. Solutions to (P V,K ) can be used to reresent a standing wave to the time-deendent nonlinear Schödinger equation; they also aear as stationary solutions in models of cross-diffusion []. The study of such roblems was initiated by Floer and Weinstein [9] by erturbation methods. Afterwards, Rabinowitz showed how the variational methods could be alied to this roblem. Indeed, the solutions of (P V,K ) are at least formally critical oints of the action functional RN u I(u) = + V u K u. The quadratic art of the functional naturally defines the Hilbert sace HV (R N ) = { u W, loc (RN ) u + V u < } ; R N the functional I : HV (RN ) R { } is then well-defined. The groundstate is the nontrivial weak solution to (P V,K ) in HV (RN ) which has the least energy I(u) among all solutions in HV. The classical scheme to rove the existence of groundstates consists in minimizing I on the Nehari manifold { } N = u HV (R N ) u + V u = K u R N R N The articularization of one result of Rabinowitz to our setting is Theorem (Rabinowitz [6]). Let V C(R N ; R + 0 ) and K C(RN ; R). If < < N/(N ), (i) su R N K <, (ii) inf R N V > 0, (iii) lim x V (x) = +, then roblem (P V,K ) has a groundstate u H V (RN ). Rabinowitz could also handle cases in which V is bounded from above on R N. Further alications of variational methods have yield existence of solutions that are not groundstates, for roblems that might also not have a groundstate, see e.g. [7, 8]. All the works mentioned are built on the assumtion that V has a ostive lower bound and that K is bounded. In a recent work, Ambrosetti, Felli and Malchiodi have investigated groundstates when V tends to zero at infinity. One of the roblems arising is that the natural sace H V (RN ) is not anymore embedded in L (R N ). They obtained

3 GROUNDSTATES WITH VANISHING POTENTIAL 3 Theorem (Ambrosetti, Felli and Malchiodi []). Assume N 3, V C(R N ; R + 0 ) and K C(RN ; R). If < < N/(N ), 0 < α <, ( ( N )) β > ( α) N, () (i) su x R N ( + x ) β K < +, (ii) inf x R N ( + x ) α V (x) > 0, then roblem (P V,K ) has a groundstate u H V (RN ). Moreover, u L (R ) and u(x) Ce λ x α for some C > 0 and λ > 0. One should note that the solution is constructed as an element of HV (RN ), and need therefore not be a riori in L (R N ). However, some regularity theory allows to show afterwards that u is indeed square integrable. The fact that u L (R N ) has an interretation in the model of the nonlinear Schrödinger equation: since u corresonds to the robability density of a article, this means that the article is localized, and that the solution corresonds to a boundstate. The study of boundstates which are not necessary groundstates with otentials vanishing at infinity has also been recently studied [3, 5]. The aim of the resent work consists in giving more insights on Theorem. A first question is the existence question: What conditions should V and K satisfy so that roblem (P V,K ) has a groundstate? A second question is whether the groundstate solution is in L (R N ). We rovide here an unified aroach which allows to handle otentials V that vanish at infinity or otentials K that exlode at infinity. Unbounded otentials have been considered by several authors, see e.g. [8]. A classical tool to rove the existence of groundstates of (P V,µ ) is Theorem 3. If one has the continuous embedding H V (R N ) L (R N, KL N ), then the functional I : HV (RN ) R defined by u I(u) = + V u R N u dµ R N is well-defined and continuously differentiable on H V (RN ). If moreover this embedding is comact, then there exists a groundstate solution to roblem (P V,µ ). The alicability of Theorem 3 deends just on the answer to a question about continuous and comact embeddings. The assumtions of Theorem are one way to ensure these embeddings, but there are other ways. A first tool is the function K(x) W(x) = V (x). N ( N ) Using Hölder s inequality and Sobolev inequality, one can rove the following result.

4 4 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN Theorem 4. Let K : R N R + and V : R N R + be measurable functions. i) If W L (R N ) and N N, then one has the continuous embedding H V (R N ) L (R N, KL N ). ii) If moreover K L loc (RN ), < N N then this embedding is comact. and for every ε > 0, L N ({x R N W(x) > ε} <, Theorem 4 is related to Theorem 8.6 in [4] by which H V (RN ) L K (RN ) when there exists R > 0 and r : R N \ B(0, R) R + such that Since r(x) x V (x) 3 0 < c r(y) r(x) c su x R N \B(0,R) y B(x,r(x)) for every x R N \ B(0, R), for every x RN \ B(0, R) and y B(x, r(x), su W(x) K(x)r(x) N ( N ) K(y)r(x) N ( N ) <. su K(y)r(x) N ( N ), y B(x,r(x) these assumtions are stronger than those of Theorem 4, and that they may fail for highly oscillating otentials while those of Theorem 4 hold. In the case where V (x) = ( + x ) α, Theorem 4 allows for otentials K such that lim x x β K(x) = 0, with ( β = ( α) N ( N )), () which is a small imrovement in view of Theorem. In the case of unbounded otentials, we recover the embeddings of [8]. While the condition of Theorem 4 allows V and K to oscillate strongly, their oscillation should be coordinated. A second tool rovides embedding theorems with a condition without interlay between K and V, in terms of Marcinkiewicz saces. Setting f L r, = su f, E R N L N (E) r E for >, recall that the sace L r, (R N ) is the sace of measurable functions f : R N R such that f L r, < +. Its subsace L,r 0 (R N ) is the closure of (L L )(R N ) in L r, (R N ). In the sequel, we denote by Ḣ (R N ) the homogeneous Sobolev sace, i.e. HV (RN ) with V 0. Theorem 5. Assume N 3. i) If N N ( ) + N r =

5 GROUNDSTATES WITH VANISHING POTENTIAL 5 and K L r, (R N, R + ), then the embedding Ḣ (R N ) L (R N, KL N ) is continuous. ii) If moreover < N N and K Lr, 0 (R N ), then this embedding is comact. The first art of the result has been obtained by Visciglia [0]. Whereas the combination of Theorems 4 and 5 allows K not to be controlled ointwise by V, it still requires when V is bounded that K should not be locally worst than a function in L r,. On the other hand, when is small enough, trace theorems show that u is locally integrable on subsurfaces. This brings us to embeddings theorem for a general measure. Here we state the result with an exlicit shae of a model otential V. Define { µ(b(x, ρ)) [µ] α = su ρ N x R N and 0 < ρ < } ( + x ) α. (3) Theorem 6. Let N 3, α 0, V (x) = ( + x ) α and µ be a Radon measure. Then, (i) [µ] α is finite if and only if there exists c > 0 such that for every u H V (RN ), u L (R N,µ) c u H V, the quantity [µ] α being equivalent to the otimal constant in the inequality ; (ii) the embedding H V (RN ) L (R N, µ) is comact if and only if { µ(b(x, ρ)) lim su δ 0 ρ N { µ(b(x, ρ)) lim su x ρ N } x R N and 0 < ρ < δ( + x ) α = 0, (4) 0 < ρ < } ( + x ) α = 0. (5) When α = 0, then HV (RN ) = D, (R N ); then the continuity art of Theorem 6 was roven by Maz ja [, Theorem.4.4/] and the comactness art by Schneider [7, Theorem.]. When α =, it is due to Maz ja [, Theorems.4.4/ and.4.6/]. Whereas we do not have counterarts of Theorems 4 and 5 when N =, Theorem 6 remains true when N = rovided ρ N is relaced by (log ρ(+ x ) α ) everywhere in the statement (see Theorem ). When < N N, Theorem 6 allows the measure to be singular with resect to the Lebesgue measure. Another situation in which Theorem 6 works while the revious theorems fail is the following: α = and K L r loc (RN )\L (R N ) is eriodic. We now draw our interest to the question whether the solutions to u + V u = u µ in R N, lim u(x) = 0. (P V,µ ) x are in L (R N ), as it is the case in Theorem. Observe that we have relaced the otential K by a ostive Radon measure µ. The solution is then understood in the distributional sense.

6 6 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN Let us first oint out a necessary condition. Indeed, if u 0, and lim su V (x) x < λ(λ + N), (6) x then, by the maximum rincile, we have, for some c > 0, c u(x) ( + x ) λ. In articular, if (6) holds with λ = N, then u L (R N ). This decay of V is in fact critical for u to be square-integrable. Theorem 7. Assume that H V (RN ) L (R N ), and that then u L (R N ). lim inf x x V (x) > ( N ) > 0, (7) The roof roceeds by multilication of the equation by a test function of the form u( + x ). We will go further in this analysis, and try to catch as much information as ossible about the decay of a solution. Theorem 8. Assume that H (R N, V ) L (R N, µ) and u H V (RN ) solves (i) If there exists λ > 0 such that then there exists C < such that u + V u = u µ. lim inf V x (x) x > λ(λ + N), u(x) C ( + x ) λ. (ii) If moreover there exists α > 0 and λ > 0 such that then there exists C < such that lim inf V x (x) x α > λ, u(x) Ce λ(+ x )α. Theorem gives the same decay rate than the last art of the theorem. However, our result allows equality in () rovided a solution exists. The limit case where equality holds in () brings us some comlications in the roof. In the revious situation, the condition () imlies that HV (RN ) L q (R N, µ) for some q >. This allows to start immediately a bootstra argument. In the resent setting, a first ste is required to rove that HV (RN ) L q (R N, µ) for some q >. The sequel of the aer is organized as follows. In Section, we work out the continuous and comact embeddings ; in articular, we rove Theorems 4, 5 and 6. Section 3 is devoted to decay estimates and contains the roofs of Theorems 7 and 8. Finaly, Section 4 deals with some extensions of our decay estimates to other frameworks that we do not cover with details.

7 GROUNDSTATES WITH VANISHING POTENTIAL 7. Embedding theorems In this section, we consider conditions that ensure continuity or comactness of the imbedding of H V (RN ) into L (R N, KL N ). We shall use three different methods: one based on the concentration function, the second based on Marcinkiewicz weak L saces and the last on the measure of balls, which will lead resectively to Theorems 4, 5 and 6 which are indeendent... Concentration function method. A first technique to obtain embeddings of HV (RN ) consists in interolating between L (R N, V L N ) and a sace in which HV (RN ) is contained : L N N (R N ). Proof of Theorem 4. For every measurable set A R N, since, using Hölder s inequality, we infer that for any u H V (RN ), A ( ) N K u W L (A) V u A ( N ) ( A ) ( u N )( N ) N. (8) Taking A = R N, we deduce the first statement of the Theorem from the Sobolev inequality. To rove the second statement, it is sufficient to show that for any ε > 0, there exists a set A R N of finite-measure such that for every u H V (RN ) with u H V, A c K(x) u < ε. Choosing A δ = {x R N W(x) δ} in (8), we get ( ) N K(x) u δ V u ( N ) (RN ) ( u N )( N ) N, R N \A δ R N so that our claim follows from the Sobolev inequality. As mentioned in the introduction, Theorem 4 imlies that H V (RN ) L (R N, KL N ) when V (x) = x α and K(x) = x β, with β given by (). It should be ointed out that not only the roof of Theorem 4 fails in dimension : one can find counter-examles. A weaker statement will be roved in Section Marcinkiewicz saces method. Another oint of view to obtain embedding, consists in using only the information about the Sobolev embedding of H V (RN ). Proof of Theorem 5. By [5], see also [, Chater ], the Sobolev sace Ḣ (R N ) is continuousloy embedded in the Lorentz sace L N N, (R N ), i.e. the estimate u L N N, C u L holds. One has then, by Hölder s inequality for Lorentz saces and by the embedding L N N, (R N ) L N N, (R N ), and for every measurable set A

8 8 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN R N A K u K L r, (A) u L N N, K L r, (A) u L N N, C K L r, (A) u L (R N ). Under assumtion ii), the comactness of the embedding can be roved easily. Let us comare Theorems 4 and 5 in the case where V (x) ( + x ) α and K(x) ( + x ) β. The first gives a continuous embedding when β ( α) (N ( N ) ) while the latter requires β N ( N ). If α 0, the condition of Theorem 4 is weaker than the condition of Theorem 5; when α 0, one has the converse situation. The criticality of the rate α = 0 can be exlained by the Hardy inequality: H V (RN ) is a strict subsace of Ḣ (R N ) if, and only if, α > 0. As a byroduct of Theorems 4 and 5, one has Corollary.. Assume that ( ) N with N N + s + t N =, and t > 0. i) If KV t L s, (R N ), then the embedding HV (RN ) L (R N, KL N ) holds. ii) If < N N and KV t L s, 0 (R N ), this embedding is comact. Proof. Taking θ = ( N t ( N )) and using Hölder s inequality, we infer ( K u V N ( N ) u ) ( θ ( KV t ) θ θ u ) θ. R N R N R N One checks that the first factor is bounded by Theorem 4 while the second is bounded by Theorem 5. We then conclude that K u C KV t L s, u. R N HV Under the assumtions in ii), one obtains the comactness in a straightforward way..3. Trace-tye inequalities. We now examine the secial case where V (x) = ( + x ) α. In this case, one can find necessary and sufficient conditions on a Radon measure µ so that one has the continuous embedding H V (RN ) L (R N, µ), or so that it is comact. This aroach is based on the corresonding work of Maz ja on Ḣ (R N ). We first exlain how the case N > is treated before sketching out how to adat the arguments to the dimension N =.

9 GROUNDSTATES WITH VANISHING POTENTIAL The subcritical case. A first tool in the roof of Theorem 6 is a characterizations of the measures for which HV (RN ) L (R N, µ) when N >. Define { µ(b(x, ρ)) } [µ] = su x R N and ρ > 0. ρ N Theorem 9 (Adams [], Maz ja [, Theorems.4.4/ and.4.6/]). Let N >, µ be a Radon measure and >. Then, (i) [µ] is finite if and only if there exists C > 0 such that for every u Ḣ (R N ), u L (R N,µ) C u L, the quantity [µ] being equivalent to the otimal constant in the inequality ; (ii) the embedding Ḣ (R n ) L (R N, µ) is comact if and only if { µ(b(x, ρ)) lim su δ 0 ρ N } x R N and 0 < ρ < δ = 0, { µ(b(x, ρ)) lim su } ρ > 0 = 0. x ρ N Remark. Since for every Radon measure µ 0, lim inf ρ 0 µ(b(x, ρ)) su x R N ρ N > 0, Theorem 9 essentially alies only if < N N. In order to rove Theorem 6, we first rove that Theorem 9 alies to the restriction of the measure µ to the ball B(x, ( + x )α ). Recall that [µ] α has been defined in (3). Lemma.. Under the assumtions of Theorem 6, one has (i) for every x, y R N and ρ > 0, where r = ( + x ) α ; µ(b(y, ρ) B(x, r)) ρ N C[µ] α, (ii) for every R > 0 and δ > 0, { µ(b(x, ρ) B(0, R)) su ρ N } x R N and ρ < δ { µ(b(x, ρ) B(0, R)) su x R N ρ N ( + x ) α } and ρ < δ min(, ( + δ + R) α ) (9)

10 0 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN and { µ(b(x, ρ) \ B(0, R)) su ρ N { µ(b(x, ρ)) su x R N and ρ < } ( + x ) α x > R and 0 < ρ < ρ N 3 ( + x ) α}. (0) Proof. When ρ < ( + y )+α, one has trivially µ(b(y, ρ) B(x, r)) ρ N µ(b(y, ρ)) [µ] ρ N α. Assume now that ρ ( + y ) α. If 3 ( + x ) ( + y ) 3( + x ), one has ρ 3 α r, and thus µ(b(y, ρ) B(x, r)) ρ N µ(b(x, r)) ρ N 3 α ( N ) µ(b(x, r) 3 α ( N ) [µ] α. r N () If 3( + y ) < + x, assume without loss of generality that B(x, r) B(y, ρ). One has then, since r ( + x ), so that x x r < y + ρ x 3 ρ x + > r 6 3. Reasoning as in (), one obtains + ρ µ(b(y, ρ) B(x, r)) 3 ( N ρ N ) [µ] α. Finally, when 3( + x ) < + y and B(x, r) B(y, ρ), one has so that and, as before, 3 x + ρ y ρ < x + r 3 x + ρ 3 ( x + ) > 3r, µ(b(y, ρ) B(x, r)) ρ N 3 [µ] α. N For the second statement, assume that ρ δ and B(x, ρ) B(0, R). One has then x ρ + R δ + R, so that ( + x ) α ρ δ min(, ( + δ + R) α ).

11 GROUNDSTATES WITH VANISHING POTENTIAL For the last statement, if B(x, ρ) B(0, R), then R x + ρ (3 x + )/ and x (3R )/; the conclusion follows. The third tool to rove Theorem 6 is Theorem 0 (Besicovitch s covering theorem, see e.g. [0, Theorem.7]). If A R N is bounded and B is a family of closed balls such that each oint of A is the center of some ball of B, then there exists a finite or countable collection of balls B i B that covers A and such that every oint of R N belong to at most P (N) balls. We can now rove the main result of this section Proof of Theorem 6. By Lemma. and Theorem 9, for every x R N and v Ḣ (R N ), v L (B(x,r/),µ) v L (B(x,r),µ) C[µ] α v, R N where r = ( + x ) α. Recall that every u H (B(0, /)) has an extension v H (R N ) such that v C u + u. R N B(0,/) By translation and scaling, every u H (B(x, r/)) has an extension v H (R N ) such that v C u + r u. R N B(x,r/) By the choice of r, for every y B(x, r), so that 3 ( + x ) α ( + x ) + x ( + x ) α + x + v C R N B(x,r/) + y 3 ( + x ), u + V u. One has thus, for every u H V (RN ), ( B(x,r/) u ) C[µ] α B(x,r/) u + V u. For every R > 0, alying now Theorem 0 to A = B(0, R) and B = B(x, (+ x ) α ), one obtains a collection of balls ( B(x i, r i /)) i I such that A i I B(x i, r i /), with r i = ( + x i ) α and i I χ B(x i,r i /) P (N),

12 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN so that ( B(0,R) ) u dµ ( i I ( B(x i,r i ) i I C[µ] α B(x i,r i ) i I ) u dµ u dµ B(x i,r i ) ) u + V u CP (N)[µ] α R N u + V u. One obtains the continuous embedding by letting R. For the converse statement, let ϕ be a comactly suorted smooth function such that ϕ = on B(0, ) and su ϕ B(0, 3/4) and set ϕ x,ρ(y) = ϕ((x y)/ρ). If ρ < ( + x ) α, then ( + y ) ( + x ) ( + y ) for y B(x, ρ), so that V ϕ x,ρ Cρ N R N ( + x ) α C ρ N. () One has thus µ(b(x, ρ)) ( ) ϕ x,ρ L (R N,µ) c ϕ x,ρ + V ϕ x,ρ R N ccρ N. For the comactness art, first note that we deduce from (9) of Lemma. and Theorem 9 that Ḣ (R N ) is comactly embedded in L (B(0, R), µ) for every R > 0. Therefore the ma u χ B(0,R) u is a comact oerator from H V (RN ) to L (R N, µ). By the first art of this theorem and (0) of Lemma., u χ B(0,R) u L (R N,µ) u H V { µ(b(x, ρ) \ B(0, R)) su ρ N x R N and ρ < } ( + x ) α 0 as R. Therefore the embedding HV (RN ) L (R N, µ) is comact as a limit in the oerator norm of comact oerators. For the necessity art, let δ k 0 and (x k ) k R N. Set ρ k = δ k (+ x ) α. The sequence u k = ρ (N )/ k ϕ xk,ρ k is bounded in HV (RN ) (see ()) and converges weakly to 0. Since HV (RN ) is embedded comactly in L (R N, µ), one obtains µ(b(x k, ρ k )) ρ N k C u k L (R N,µ) 0. as k. This roves (4). Assuming that x k and taking δ k = instead of δ k 0, one obtains similarly (5).

13 GROUNDSTATES WITH VANISHING POTENTIAL 3 Remark. In view of [], it is clear that similar results aly to the Sobolev saces W,q (R N ), with q < N. For examle, one has where ( ) ( k u D i u ) q dµ [µ] q,α R N R N ( + x ) ( α)(k i), { µ(b(x, ρ)) [µ] α,q = su ρ N k i=0 x R N and 0 < ρ < } ( + x ) α. Remark 3. One can also consider saces with a weight on the gradient. For examle, set H = {u W, loc R N ( + x ) τ u + ( + x ) α+τ u }. One has then H L (R N, µ) if and only if { µ(b(x, ρ)) su ρ N ( + x ) x τ RN and 0 < ρ < } ( + x ) α <..3.. The critical case. In two dimensions, one has a similar result. Define [µ] α, = su{ log ρ µ(b(x, ρ( + x ) α )) x R N and 0 < ρ < }. Theorem. Assume α 0, V (x) = ( + x ) α and let µ be a Radon measure. Then, (i) [µ] α, is finite if and only if there exixts C > 0 such that for every u Ḣ (R ), u L (R,µ) C u H V, the quantity [µ] α, being equivalent to the otimal constant in the inequality ; (ii) the embedding H V (R ) L (R, µ) is comact if and only if lim su{ log ρ µ(b(x, ρ( + δ 0 x ) α )) x R N and 0 < ρ < δ} = 0, lim x su{ log ρ µ(b(x, ρ( + x ) α )) 0 < ρ < } = 0. Instead of Theorem 9, the main tool to rove Theorem is Theorem (see [, Corollary 8.6/]). Let µ be a Radon measure, > and [µ] = su{ log ρ µ(b(x, ρ)) x R N and 0 < ρ < }. Then, (i) [µ] is finite if and only if there exixts C > 0 such that for every u H (R ), u L (R,µ) C( u L + u L ),

14 4 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN the quantity [µ] being equivalent to the otimal constant in the inequality ; (ii) the embedding H (R ) L (R, µ) is comact if and only if lim δ 0 su{ log ρ µ(b(x, ρ)) x R N and 0 < ρ < δ} = 0, lim su{ log ρ µ(b(x, ρ)) 0 < ρ < } = 0. x Proof of Theorem 6. By a variant of Lemma. and Theorem together with a scaling argument, one obtains that for every v H (R N ) and x R N, v L (B(x,R/),µ) C[µ] v + v R N R, where R = (+ x ) α. The roof continues then as the roof of Theorem 6. Remark 4. Remark still alies for W k,q (R N ), with kq = N and { [µ] q,α = su log ρ q µ(b(x, ρ)) x R N and 0 < ρ < ( + x ) α} Power-like otentials. When N and V (x) = ( + x ) α, Theorems 6 and, show that when K(x) = ( + x ) β, where β is given by (), H V (R N ) L (R N, KL N ). While Theorem 4 fails when N =, the receding conclusion holds in this articular case. We rove it as a lemma that we kee for future reference in section 3. As this remains true when N =, we rovide a direct roof that works for all dimensions: N N Lemma.3. Let N, α > 0, if N 3 and < otherwise, and β be given by (). If <, V (x) = ( + x ) α and K(x) = ( + x ) β, then HV (RN ) L (R N, KL N ). Proof. First note that by Gagliardo Nirenberg s inequality [3] and by scale invariance, for every R > 0 B(0,R)\B(0,R) u(x) x β ( u(x) ) N dx C B(0,R)\B(0,R) x α dx ( B(0,R)\B(0,R) u(x) + u(x) x ( N ) ) ( dx ) N. Summing this for R = k, k 0, we obtain since α 0, u(x) ( u(x) ) N R N \B(0,) x β dx C R N \B(0,) x α dx ( N ) ( u(x) + u(x) ) ( R N \B(0,) x dx ) N ( C u(x) + u(x) ) R N \B(0,) x α dx. The conclusion follows then from Sobolev s embedding Theorem.

15 GROUNDSTATES WITH VANISHING POTENTIAL 5 One could similarly obtain some conditions for the comactness of the embedding. As a corollary, one has in R, R u(x) ( R x dx C u(x) + u(x) ) x dx. In contrast with the higher-dimensional case, the revious lemma cannot be imroved when N = and α > 0 by relacing ( + x ) by x. If one sets V (x) = ( + x α ) and K(x) = + x β, then the conclusion of the Lemma holds rovided α Decay estimates We now turn out to the decay roerty of solutions to (P V,µ ). The first imrovement is to obtain that u multilied by some function is still in the energy sace H V (RN ). The latter method also allows that the same holds for a small ower of u. By Moser s iteration technique, we show then that a solution u satisfies some decay estimates at infinity. 3.. Linear estimates. We begin by considering the L theory of decay of finite-energy. These are secial cases of the sequel, but give an useful insight on the roof of the exact decay estimates. Assumtion. Let µ be a Radon measure, f L /( ) (R N, µ) and u H V (RN ) be such that (i) the embedding H V L (R N, µ) is continuous, (ii) u satisfies u + V u = fuµ. (3) Proosition 3.. Under Assumtion, if then ( + x ) λ u H V (RN ). ν := lim inf x x V (x) > λ ( N ) > 0, (4) Let us first show how Theorem 7 follows: Proof of Theorem 7. Under the assumtions of Theorem 7, the assumtions of Proosition 3. hold with f = u L (R N, µ) and λ =. We have thus ( + x )u HV and it easily follows that u L (R N ). The roof roughly goes as follow. Take x λ u as a test function in (3), integrate on R N \ B(0, R) and aly Hölder s inequality to obtain ( x λ u) + V (x) x λ u R N \B(0,R) ( R N \B(0,R) ) / ( f /( ) dµ + λ R N \B(0,R) R N u x λ x + ) / x λ u B(0,R) u ν (u x λ ). When R is large enough, by the assumtion on f, µ and λ, the two first terms in the right-hand side can be absorbed, so that the conclusion follows.

16 6 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN As usual, we need to be careful in the estimates of quantities that might not be finite. Proof of Proosition 3.. For every Ω R N and for every ϕ W, 0 (Ω) such that ϕ has comact suort in Ω, recall that ϕ u and ϕu H V (RN ), (ϕu) = u (ϕ u) + ϕ u. (5) so that, by Hölder s inequality and the embedding HV (RN ) L (R N, µ), we get (ϕu) + V ϕu = fϕ u dµ + ϕ u R N R ( ) N f ( ) dµ ϕu dµ + ϕ u Ω R N R ( ) N C f ( dµ (ϕu) + V ϕu ) + ϕ u. Ω R N R N ( ) Let δ = C Ω f dµ. Since f L (R N, µ), we can choose Ω = R N \ B(0, R) in such a way that 0 < δ <. The receding estimates then yield a control on the norm of ϕu ( δ) (ϕu) + V ϕu R N ϕ u. R N (6) Taking (4) into account and increasing R if necessary, we can assume that for every x Ω, V (x) ν δ x (7) and (ν δ)( δ) λ δ ( δ) ( N ), (8) where we recall that ν = lim inf x x V (x). Choose now ψ Cc (Ω) such that ψ on R N \ B(0, R) and, for k > 0, set ϕ k (x) = ψ(x) min(k, x λ ). We infer from (6) and (7) that ( ( δ) (ϕ k u) + δv + ( δ) ν δ ) R N x ϕ k u ϕ k u R N RN λ x ϕ ku + C u, B(0,R)\B(0,R) where the constant C deends only on ψ, R and λ. Therefore, ( (ϕ k u) + δv + (( δ)(ν δ) λ ) ) R N δ x ϕ k u C δ B(0,R)\B(0,R) u.

17 GROUNDSTATES WITH VANISHING POTENTIAL 7 Now, using (8), we infer that ( ) δ (ϕ k u) + V ϕ k u R N ( ( N ) + ( δ) (ϕ k u) R N ϕk u ) x C B(0,R)\B(0,R) and Hardy s inequality then yields ( ) δ (ϕ k u) + V ϕ k u C u. R N B(0,R)\B(0,R) By letting k, we deduce from Fatou s lemma that (ϕu) + V ϕu C u, R N B(0,R)\B(0,R) with ϕ(x) = ψ(x) x λ. Since local estimates are straightforward, we easily conclude that x λ u H V (RN \ B(0, )). To comlete the roof, we need to show that (( + x ) λ u) L (R N ). For this urose, it is enough to observe that ( + x ) λ u = and to use the fact that ( x λ u) L (R N ). ( + x )λ x λ x λ u A similar method works in the case where V decays slowly at the infinity: Proosition 3.. Under Assumtion, if then e λ(+ x )α u H V (RN ). u ν α := lim inf x x α V (x) > λ, (9) Proof. Arguing as in the roof of Proosition 3., we choose the radius R in such a way that δ <, λ ν α > + δ. (0) ( δ) and V (x) > ν α δ, () x α for every x U. Let ψ Cc (U) be such that ψ on R N \ B(0, R) and, for k > 0, set ϕ k (x) = ψ(x) min(k, e λ x α ). By (6), (0) and (), we deduce that (ϕ k u) + V ϕ k u C u. R N B(0,R)\B(0,R) Letting k and alying Fatou s lemma, we conclude that (ϕu) + V ϕu C u, R N B(0,R)\B(0,R) with ϕ(x) = ψ(x)e λ x α. One concludes therefrom and from local estimates that e λ (+ x ) α u HV (RN ) for every λ < λ.

18 8 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN Remark 5. The statement uses the weight e λ(+ x )α instead of the simler one e λ x α because the latter is not Lischitz when 0 < α <. 3.. Nonlinear estimates. The method of roof of Proositions 3. and 3. allows in fact to obtain information about ((+ x ) λ u) γ or (e λ(+ x )α u) γ for γ >. Lemma 3.3. Under Assumtion, assuming moreover that γ >, u L γ loc (RN ) and one of the following hyothesis holds (i) λ < ( N γ ) γ γ, and ν = lim inf x x V (x) > ( λ + γ γ ( N )) ( N ) > 0, (ii) ν > ( + (γ ) γ )λ, we have (( + x ) λ u) γ HV (RN ). The statement of Theorem 3.3 is a erturbation of Proosition 3. in the sense that for every λ that satisfies (4), there exists γ(ν, λ) > such that Theorem 3.3 is alicable for γ < γ(ν, λ). On the other hand, Theorem 3.3 will only be useful when γ is small. Indeed, starting with u Hloc, Sobolev s embedding Theorem only says u Lγ loc (RN ) for γ N/(N ). Iterating the Lemma, one obtains successively that u L γ k loc (RN ) for γ k = N k /(N ) k for every k. For every fixed λ > 0, the iteration rocess will cease giving global integrability information about (( + x ) λ u) γ when γ is too large. The roof of Lemma 3.3 follows the strategy used to rove that solutions u H (B(0, )) of the critical roblem u = u N+ N are in L q (B(0, )) for q < N /(N ) [4, 6, 9]. The roof roceeds as follows. We first establish by integration by arts the inequality (5). A suitable choice of test functions yields that (( + x ) λ u) γ HV (RN \ B(0, R)) for some large R > 0. Finally we rove that one also has that for every y R N, (( + x ) λ u) γ HV (B(y, ρ)) for some ρ > 0. Since by Besicovitch s covering theorem, R N can be written as the union of a finite collection of such balls together with R N \ B(0, R), the claim will follow. Proof of Lemma 3.3. First note that if v H loc (RN ) is locally bounded and if ϕ is locally Lischitz, one has ((ϕv) γ ) = and thus, for every η > 0, ( η γ γ γ ) ((ϕv)γ ) γ γ v (ϕγ v γ ) + γ γ γ vγ ϕ γ ϕ (ϕv) γ + γ γ ϕ v γ ϕ γ () γ γ v (ϕγ v γ ) + ( γ γ + γ γ η γ ) ϕ v γ ϕ γ. (3)

19 GROUNDSTATES WITH VANISHING POTENTIAL 9 On the other hand, by (5), and since γ >, ( η γ γ γ ) (ϕv) γ γ (ϕv) = γ γ v (ϕ v) + γ γ ϕ v (4) γ γ v (ϕ v) + ( γ γ + γ γ η γ ) ϕ v. We will use this last estimates successively to obtain a first estimate at infinity and a second one on small balls. First ste - a basic inequality. Define the truncation sequences (v k ) k and (w k ) k by v k = min((uϕ k ) γ, kuϕ k ) and w k = min((uϕ k ) γ, k uϕ k ), where the choice of ϕ k will be secified later. By alying successively (3) and (4) to v k, we get the estimate ( η γ γ γ ) v k γ γ u (ϕ kw k ) + ( γ γ + γ γ η γ ) ϕ k vk. If the suort of ϕ k lies in some oen set Ω R N, choosing ϕ k w k as test function, alying Hölder s inequality and the embedding HV (RN ) L (R N, µ), we infer that ( γ R N γ ( Ω η γ γ ) v k + V v k f dµ ( C f dµ Ω R N f v k dµ + ( + η ) ) ( R N v k dµ ( Let us set again δ = C Ω f dµ to ( γ η γ γ γ R δ) v k +( δ) N ) γ γ ) ϕ k R N ϕ k v k + ( + η ( R N v k +V v k ) +(+ η ) ϕ k γ γ ) ϕ k R N ϕ k v k γ γ ) ϕ k R N ϕ k v k. (5). The receding estimate then leads R N V v k (+ η γ γ ) ϕ R N ϕ v k. (6) Second ste - An estimate at infinity. Assume first that (i) holds. We then choose η = λ/( N ). Since f L (R N, µ), we can take Ω = R N \B(0, R) in such a way that δ( δ) γ λ γ γ N γ. On the other hand, increasing R if necessary, we can assume that and (ν δ) (λ + γ γ ( N )) ( δ) ( N ) (7) V (x) ν δ x,

20 0 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN for every x Ω. Let ψ Cc (Ω) be such that ψ on R N \ B(0, R). For k > 0, set ϕ k (x) = ψ(x) min(k, x λ ). By (6), for k and R large enough, we have ( γ R N γ ( η γ γ δ) v k + ( δ)δv + ( δ) ν δ ) x v k ( + η ( + η γ γ ) ϕ k R N ϕ k v k γ γ ) ( R N λ x v k + C B(0,R)\B(0,R) u γ), where the constant C does not deend on k. Taking (7) into account, we deduce that ( γ γ η γ γ ( RN δ( δ)) v k ( N v k ) ) x ( ) + ( δ)δ v k + V v k R N C Alying Hardy s inequality yields v k + V v k C R N and letting k, we conclude that (ϕu) γ + V (ϕu) γ C R N B(0,R)\B(0,R) B(0,R)\B(0,R) u γ, B(0,R)\B(0,R) u, u γ with ϕ(x) = ψ(x) x λ. Arguing as in the roof of Proosition 3., we deduce that (( + x ) λ u) γ H V (RN \ B(0, R)). If (ii) holds, we roceed similarly, choosing the radius R sufficiently large and η > 0 such that η γ γ instead of (7). + δ δ γ γ, λ ( + η γ γ ) (ν δ)( δ) Third ste - the local estimates. Keeing the same notations, we now fix x 0 R N, choose η = /(γ ), Ω = B(x 0, ρ), ϕ C c (Ω) such that ϕ = on B(x 0, ρ/) and we set ψ k = ϕ for every k. Taking ρ in such a way that δ γ γ, we deduce from (6) that v k + V v k C γ γ B(x 0,ρ) B(x 0,ρ) v k C B(x 0,ρ) u γ. Letting k, we conclude that (u γ ) L (B(x 0, ρ/)), and therefore (( + x ) λ u) γ H V (B(x 0, ρ/).

21 GROUNDSTATES WITH VANISHING POTENTIAL Conclusion. Taking all the revious estimates into account, the conclusion now follows from a standard alication of Besicovitch s covering theorem. In view of Theorem 8, one would have exected to have conditions (i) or (ii) relaced by the weaker assumtion ν > (λ γ γ ( N )) ( N ). Observe that the sign in front of γ γ has changed. This can be exlained artially roughly as follows. If λ is otimal, one exects u to behave as x λ ( N )/γ and u γ x λ(γ ) x λ ( x λ u) γ λ(n ) x N. When assing from () to (3), the latter quantity can be bounded by η (u x λ ) γ + η uγ x λ(γ ) x λ so that choosing η = λ/( N ) as in the roof, yields λ(n )/ x N, i.e. the oosite quantity. (One would like thus to take η = λ/( N ).) The method of roof also works for < γ <. In this case, the second term on the right-hand side of () has a negative coefficient, so that one (3) holds for η < 0. The conditions on γ, λ and ν are the same exceted that the second inequality in (i) becomes ν > ( λ γ γ ( N )) ( N ). In view of the revious remark, the case γ < is slightly better. Finally, in the same fashion, one obtains the counterart of Proosition 3.: Lemma 3.4. Under Assumtion, if u L γ loc (RN ) with γ >, and if ν α = lim inf x x α V (x) > ( ) + (γ ) γ λ, then ( e λ(+ x )α u ) γ H V (R N ). As for Lemma 3.3, the condition on ν α and λ are stonger than the condition ν α > λ that is stated in Theorem 8. Whereas Lemma 3.3 lays a crucial role in the sequel, Lemma 3.4 is not really needed, since Lemma 3.6 only requires information on the integrability of u with a ower-tye weight Moser iteration scheme. We now show that whenever u and f are in slighlty better saces than H V (RN ) and L /( ) (R N, µ), this information can be ugraded to a uniform decay of u at infinity. Lemma 3.5. Assume that (4) holds, H (R N, V ) L (R N, µ) and where f( + x ) (N )(η ) L q (R N, µ), η = ( ) >. q

22 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN Then, if ( + x ) λ u HV (RN ) and u solves (3), there exists C < such that C u(x) ( + x ) λ+(n )/. Proof. Assume that (( + x ) σ u) γ H V (RN ) for some γ and σ > 0. Setting γ = ηγ, and one has, see (4), σ = σ + ( N )η γ, w(x) = u γ ( + x ) γ σ v(x) = (( + x ) σ u) γ, v = γ γ u w + σ γ (γ ) v γ + x so that x v x v γ γ u w + γ σ ( + (γ ) ) By a suitable limiting argument, one has therefore v γ R N γ fv dµ γ R N γ V v R N + γ γ σ v ( + x ), v ( + x ). RN + γ σ ( + v ) (γ ) ( + x ). One has by Hölder s inequality and the embedding HV L (R N, µ) fv dµ = f( + x ) (N )(η ) u(x)( + x ) σ γ dµ R N R ( N ) ( ) C f( + x ) (N )(η ) q q dµ ( + x ) σ u γ q dµ R N R ( ) N C f( + x ) (N )(η ) q q dµ (( + x ) σ u) γ η. R N HV Observing that η < N/(N ) and combining this with (4), we infer that Lemma.3 is alicable and yields RN v RN ( + x ) = ( ( + x ) σ u γ ) η ( + x ) C (( + ( N )(η ) x )σ u) γ η. HV One concludes thus that (( + x ) σ u) γ H V Setting now γ k = η k and C( + γ + σ γ ) (( + x ) σ u) γ η. HV σ k = λ + ( η k )N,

23 GROUNDSTATES WITH VANISHING POTENTIAL 3 we get (( + x ) σ k+ u) γ k+ /γ k+ HV Therefore, the quantity [C( + η (k+) )] /ηk+ (( + x ) σ k u) γ /γ k. HV (( + x ) σ k u) γ k /γ k H V is bounded uniformly in k. In articular, by Lemma.3 again, we infer that ( RN (( + x ) λ+(n )/ u) ηk ( + x ) N ) /(ηk) is bounded uniformly in k, so that ( + x ) λ+(n )/ u L (R N ). The same can be done when the otential decays slowly at infinity. Lemma 3.6. Assume (9) holds, H (R N, V ) L (R N, µ) and f( + x ) ( α)(n )(η ) L q (R N, µ), where η = ( ) >. q If e λ(+ x )α u HV (RN ) and u solves (3), then there exists C < such that Ce λ(+ x )α u(x) ( + x ) ( α)(n )/. Proof. We argue as in the roof of the revious lemma, taking γ = ηγ, σ = σ + ( α)( N )η γ and One obtains similarly R N v γ γ w(x) = ( + x ) γ σ e γ λ(+ x ) α u γ (x), v(x) = (( + x ) σ e λ(+ x )α u(x)) γ. R N fv dµ γ γ R N V v + γ ( σ + λα) ( + (γ ) ) RN v ( + x ) α. From the embedding H V L (R N, µ) and Lemma.3, we deduce (( + x ) σ u) γ H V Setting now γ k = η k and C( + γ + ( σ + λα)γ ) (( + x ) σ u) γ η. HV σ k = λ + ( α)( η k )N, and iterating as before, one has that (e λ(+ x )α u) γ k /γ k H V

24 4 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN is bounded uniformly in k. In articular, by Lemma.3 ( RN (( + x ) ( α)(n )/ e λ(+ x )α u) ηk ( + x ) N( α) ) /(ηk) is bounded uniformly in k, so that ( + x ) ( α)(n )/ e λ(+ x )α u L (R N ) Proof of Theorem 8. We can now bring together the results of the revious sections in order to deduce the decay at infinity. Proof of Theorem 8. Consider first the statement (i). Since we know that u L /( ) (R N, µ) and, by assumtion, we have lim inf x V (x) x > (λ ( N )) ( N ), we deduce from Proosition 3. that u( + x ) λ ( N ) H V (RN ). Next, when γ > is sufficiently small, Lemma 3.3 shows that Setting q = γ and γ (u( + x ) γ ( N ) ) γ HV (R N ) L (R N, µ). η = ( q ) = + γ γ ( ), one reaches the conclusion by using Lemma 3.5. The roof of (ii) is similar. We start from Proosition 3. which states e λ(+ x )α u HV (RN ). On the other hand, in view of Lemma 3.3, there exists γ > such that γ (u( + x ) γ (α )( N ) ) γ HV (R N ) Taking q and η as above, by Lemma 3.6, ( + x ) ( α)(n )/ e λ(+ x )α u L (R N ). This gives the conclusion if α. Otherwise, one just need to notice that the range of admissible λ is oen. 4. Further comments The method that we have followed is known to be very flexible. Let us highlight some similar situations that can be treated as above. 4.. Fast decay for exloding otential. By the Kelvin transform the estimates around infinity are equivalent to local estimates with a singular otential. Indeed, if u HV (RN ) satisfies (P V,µ ), then ( x ) ū(x) = x N u x. satisfies where ū + V u = u µ, V (x) = x 4 V ( x x )

25 GROUNDSTATES WITH VANISHING POTENTIAL 5 and the measure µ is defined by ϕ d µ = R N R N ϕ ( x ) x x As a consequence of Theorem 8, one has that if (N ) dµ. lim inf x 0 x V (x) > λ(λ + N ) for λ > 0, then in a a neighbourhood of 0, u(x) C x λ. Similarly, if lim inf x 0 x +α V (x) > λ, then u(x) e λ/ x α in a a neighbourhood of Divergence-form oerators. The Lalacian can be relaced by an ellitic oerator in divergence form. Assume that u solves, div A u + V u = u uµ, where A : R N R N N is measurable and A(x) is symmetric for every x R N and there exist 0 < a a < such that If a ξ ξ Aξ a ξ. (8) lim inf x x V (x) > aλ a( N ) > 0 then ( + x ) λ H V (RN ). Similarily, if lim inf x x α V (x) > aλ, then e λ(+ x )α u H V (RN ). The roof of Lemmas 3.5 and 3.6 aly directy, so that u(x) C(+ x ) λ+ N and u(x) Ce λ(+ x )α (+ x ) (α )( N ) Nonuniformly ellitic oerators. If the matrix A is not anymore uniformly ellitic, but satisfies a ( + x ) τ ξ ξ Aξ a ( + x ) τ ξ, instead of (8). One has then the following extension: if lim inf x x V (x) > aλ a( N τ ) > 0, then ( + x ) λ u H, where H is defined in Remark 3, and if lim inf x x α V (x) > aλ, then e λ(+ x )α u H. Suitable adatations of Lemmas 3.5 allow also to show that u(x) C( + x ) λ ( N τ) and resectively. u(x) Ce λ(+ x )α ( + x ) (α )( N τ)

26 6 DENIS BONHEURE AND JEAN VAN SCHAFTINGEN References [] D. Adams, A trace inequality for generalized otentials, Studia Math. 48 (973), [] A. Ambrosetti, V. Felli, and A. Malchiodi, Ground states of nonlinear Schrödinger equations with otentials vanishing at infinity, J. Eur. Math. Soc. (JEMS) 7 (005), no., [3] A. Ambrosetti, A. Malchiodi, D. Ruiz, Bound states of Nonlinear Schrödinger Equations with Potentials Vanishing at Infinity, J. Anal. Math. 98 (006), [4] T. Aubin and Y. Y. Li, On the best Sobolev inequality, J. Math. Pures Al. (9) 78 (999), no. 4, [5] D. Bonheure and J. Van Schaftinegn, Bound state solutions for a class of nonlinear Schrödinger equations, Rev. Mat. Iberoamericana, 4 (008), [6] H. Brezis and T. Kato, Remarks on the Schrödinger oerator with singular comlex otentials, J. Math. Pures Al. (9) 58 (979), no., [7] M. del Pino, P. Felmer, Local mountain asses for semilinear ellitic roblems in unbounded domains. Calc. Var. Partial Differential Equations 4 (996), -37. [8] N. del Pino, P. Felmer, Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 34 (00), no., -3. [9] A. Floer, and A. Weinstein, Nonsreading wave ackets for the cubic Schrödinger equation with a bounded otential, J. Funct. Anal. 69 (986), no. 3, [0] P. Mattila, Geometry of sets and measures in Euclidean saces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 995, Fractals and rectifiability. [] V. G. Maz ja, Sobolev saces, Sringer Series in Soviet Mathematics, Sringer-Verlag, Berlin, 985, Translated from the Russian by T. O. Shaoshnikova. [] W-M Ni, Diffusion, cross-diffusion, and their sike-layer steady states, Notices Amer. Math. Soc. 45 (998), no., 9 8. [3] L. Nirenberg, On ellitic artial differential equations, Ann. Scuola Norm. Su. Pisa (3) 3 959, 5 6. [4] B. Oic and A. Kufner, Hardy-tye inequalities, Pitman Research Notes in Mathematics Series, 9, Longman Scientific & Technical, Harlow, 990. [5] J. Peetre, Esaces d interolation et théorème de Soboleff, Annales de l institut Fourier, 6 (966) no., [6] P. H. Rabinowitz, On a class of nonlinear Schrödinger equations, Z. Angew. Math. Phys. 43 (99), no., [7] M. Schneider, Comact embeddings and indefinite semilinear ellitic roblems, Nonlinear Anal. 5 (00), no., Ser. A: Theory Methods, [8] B. Sirakov, Existence and multilicity of solutions of semi-linear ellitic equations in R N, Calc. Var and PDE (000), 9 4. [9] N. S. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on comact manifolds, Ann. Scuola Norm. Su. Pisa (3) (968), [0] N. Visciglia, A note about the generalized Hardy-Sobolev inequality with otential in L,d (R n ), Calc. Var. Partial Differential Equations 4 (005), no., [] W. Ziemer, Weakly differentiable functions. Sobolev saces and functions of bounded variation, Graduate Texts in Mathematics, 0, Sringer-Verlag, New York, 989. Déartement de Mathématique, Université libre de Bruxelles, CP 4, Boulevard du Triomhe, B-050 Bruxelles, Belgium address: denis.bonheure@ulb.ac.be Université Catholique de Louvain, Déartement de Mathématique, Chemin du Cyclotron,, 348 Louvain-la-Neuve, Belgium address: Jean.VanSchaftingen@uclouvain.be