VANISHING-CONCENTRATION-COMPACTNESS ALTERNATIVE FOR THE TRUDINGER-MOSER INEQUALITY IN R N
|
|
- Randell Dennis
- 5 years ago
- Views:
Transcription
1 VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE FOR THE TRUDIGER-MOSER IEQUALITY I R Abstract. Let 2, a > 0 0 < b. Our aim is to clarify the influence of the constraint S a,b = { u W 1, (R ) u a + u b = 1 } on concentration phenomena of (spherically symmetric non-increasing) maximizing sequences for the Trudinger- Moser supremum d (a, b) = sup φ ( α u ) dx u S a,b R where α is the sharp exponent of Moser, i.e. α = ω 1/() ω is the surface measure of the ( 1)-dimensional unit sphere in R. We obtain a vanishingconcentration-compactness alternative showing that maximizing sequences for d (a, b) cannot concentrate either when b or when b = a > 0 is sufficiently small. From this alternative, we deduce the attainability of d (a, b) for special values of the parameters a b. Keywords: Trudinger-Moser inequality, vanishing, concentration Contents 1. Introduction 1 2. A Vanishing-Concentration-Compactness alternative 5 3. Alternative (I) Vanishing 9 4. Alternative (II) Concentration Alternative (III) Compactness The case of maximizing sequences for the Trudinger-Moser inequality Proof of Theorem 1.2 Attainability of the supremum 23 References Introduction Let a, b > 0 let us denote by d,α (a, b) the supremum corresponding to the Trudinger- Moser inequality in the whole space R, 2, with exponent α > 0 constraint u a + u b = 1 u W 1, (R ) (1.1) 1
2 2 More precisely, d,α (a, b) = sup φ u W 1, (R ), u a + u b =1 R ( α u ) dx (1.2) where α > 0 φ (t) = e t 2 t k t 0 k=0 k! When a = b =, we set d,α = d,α (, ) the above supremum d,α corresponds to the classical Trudinger-Moser inequality in R. It is well known that d,α < + (1.3) where α = ω 1/() ω is the surface measure of the ( 1)-dimensional unit sphere in R. Moreover, the exponent α is sharp in the sense that d,α = + α > α In the 2-dimensional case, the study of the attainability of the supremum d 2,α is due to B. Ruf [21] M. Ishiwata [13]. Roughly speaking, from the delicate analysis carried out in [21] [13], we can deduce that, given a (spherically symmetric non-increasing) maximizing sequence {u } W 1,2 (R 2 ) for d 2,α with 0 < α α, the following alternative occurs: either the weak it u in W 1,2 (R 2 ) of the maximizing sequence {u } is nontrivial (compactness) it is a maximizer for d 2,α or u = 0. In the latter case, the loss of compactness can be caused by either vanishing phenomena, i.e. u 2 dx = 0 R 2 u 2 dx = 1 R 2 or concentration phenomena, i.e. R 2 u 2 dx = 1, R 2 u 2 dx = 0 (1.4) u 2 dx = 0 for any fixed R > 0 (1.5) R 2 B R The proper understing of the above alternative was a priori not obvious. However, the most valuable results obtained in [21] [13] cannot be summarized in this way are clearly more involved. In the critical case α = α 2 = 4π, as showed in [13], it is possible to rule out vanishing behaviors of maximizing sequences for d 2,4π the most hard inspiring part of the result in [21] is to exclude concentration phenomena. In particular Theorem A ([21]). In the 2-dimensional case, the level of normalized concentrating sequences for the Trudinger-Moser functional is exactly eπ. More precisely, sup { R 2(e4πu2 1) dx {u } is a normalized concentrating sequence } = eπ
3 VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE 3 where a normalized concentrating sequence is a (spherically symmetric non-increasing) sequence {u } W 1,2 (R 2 ) such that u u 2 2 = 1 satisfying (1.4) (1.5). In the subcritical case 0 < α < α 2 = 4π, concentration cannot occur, due to the fact that one can always gain some L p -uniform integrability with p > 1, loss of compactness can be caused only by the failure of the compact embedding of W 1,2 (R 2 ) in L 2 (R 2 ), i.e. by the fact that the embedding W 1,2 (R 2 ) L 2 (R 2 ) is continuous but not compact. Therefore vanishing phenomena prevail, as enlightened in [13], provoke the non-attainability of d 2,α when α > 0 is sufficiently small. Theorem B ([13], Theorem 1.2). In the 2-dimensional case, if α > 0 is sufficiently small then the Trudinger-Moser supremum d 2,α is not attained. Concerning the higher dimensional case 3, the study of the attainability of d,α is due to Y. Li B. Ruf [18]. Even if, from [18], one can deduce that no loss of compactness of maximizing sequences occurs, a very careful blow-up analysis, as developed in [18], is needed to prove that d,α is attained. Differently from the 2-dimensional case due to the method of proof adopted in [18] which is based on blow-up analysis, one cannot deduce from [18] a precise estimate of the level of (spherically symmetric non-increasing) concentrating sequences for the Trudinger-Moser functional in higher dimensions 3. This problem is heavily non-trivial still open. In contrast (at least apparently) with the 2-dimensional case, in the subcritical case 0 < α < α when 3, the supremum d,α is always attained also vanishing phenomena do not play any role. Actually, even in the higher dimensional case 3, the attainability of d,α (a, b) with (a, b) (, ) heavily depends on the value of the exponent 0 < α < α, as showed by M. Ishiwata H. Wadade in [14] (see also [15] Remark 1.2). From [14], we can deduce that the constraint (1.1) has an effect on vanishing phenomena. Then one may wonder How does the constraint (1.1) influence concentration phenomena? In a very recent paper,. Lam, G. Lu L. Zhang [17] proved that, when d,α (a, b) < +, the exponent α is sharp for the corresponding Trudinger-Moser inequality it is not affected by the values of a b Theorem C ([17], Theorem 1.2). Let 2 a, b > 0. Then d,α (a, b) < + if only if b. Moreover, if 0 < b then the exponent α is sharp in the sense that d,α (a, b) = + α > α Remark 1.1. If a > 0 b > then it is not difficult to see that for any 0 < α < α there exists a constant C α > 0 such that sup φ u W 1, (R ), u a + u b =1 R ( α u ) dx Cα (1.6)
4 4, summarizing, we have C α if 0 < α < α a > 0 b > d,α (a, b) = + if α α Therefore, when a > 0 b >, we have a family of subcritical Trudinger-Moser type inequalities for which the exponent α is not admissible, as in the case of Adachi-Tanaka type inequalities, see [1]. Our aim is to clarify the influence of the constraint (1.1) on concentration phenomena of (spherically symmetric non-increasing) maximizing sequences for the corresponding Trudinger-Moser inequality. Since when a > 0 b >, the range of the exponent α for the validity of (1.6) is an open interval, i.e. α (0, α ), it is not difficult to exclude concentration behaviors of maximizing sequences. For this reason, from now on we will ust consider the supremum d,α (a, b), with 2, when a > 0 0 < b, to simplify notations, we will denote by d (a, b) the Trudinger-Moser supremum d,α (a, b) defined by (1.2) with exponent α = α, i.e. d (a, b) = d,α (a, b) (1.7) Definition 1.1. Let {u } W 1, (R ) be a spherically symmetric non-increasing sequence assume that each u satisfies the constraint (1.1), i.e. u a + u b = 1 1 (I) We say that {u } is a normalized vanishing sequence if u = 0 u = 1 (1.8) (II) We say that {u } is a normalized concentrating sequence if u = 1, u = 0 R B R u dx = 0 for any fixed R > 0 Our main result is a vanishing-concentration-compactness alternative for normalized (spherically symmetric non-increasing) sequences in W 1, (R ), that we will state in Section 2 (see Lemma 2.2). In particular, this alternative entails the following precise estimates of the energy level of normalized vanishing concentrating sequences. Theorem 1.1. Let 2, a > 0 0 < b. Then any normalized vanishing sequence {u } W 1, (R ) satisfies φ α ( α u ) dx = R ( 1)! (1.9)
5 VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE 5 If we assume in addition that b, i.e. a > 0 0 < b <, then any normalized concentrating sequence {u } W 1, (R ) satisfies φ ( α u ) dx = 0 R We call the attention to the fact that, when a > 0 0 < b <, the energy level of normalized concentrating sequences is zero, see also Remark 2.4. Therefore, the corresponding problem of the attainability of the Trudinger-Moser supremum d (a, b), with this range of the parameters a b defining the constraint (1.1), becomes much easier: if a > 0 0 < b < then maximizing sequences for d (a, b) cannot concentrate. In other words, concentration phenomena may occur only when b =, while when a > 0 0 < b <, if we restrict our attention to maximizing sequences for d (a, b) then the vanishingconcentration-compactness alternative expressed by Lemma 2.2 reduces to a vanishingcompactness alternative, see Section 6 more precisely Lemma 6.2. Exploiting Lemma 6.2, in particular the energy level of vanishing sequences (1.9), we will prove the following attainability result Theorem 1.2. Let 2, a > 0 < b < be fixed. Then the Trudinger-Moser supremum d (a, b) defined by (1.7) is attained. ote that, the additional condition a > 1 is meant to exclude possible vanishing behaviors of maximizing sequences for d (a, b), see Section 7. Remark 1.2. The case 0 < a is beyond our aims, since it is significant for vanishing phenomena, as one can deduce from the interesting analysis carried out in [14]. We also refer the reader to the new result by M. Ishiwata H. Wadade [15], where the authors address explicitly the problem of the attainability of d,α (γ, γ), with subcritical exponent 0 < α < α γ > 0, showing that vanishing phenomena may prevent the subcritical supremum d,α (γ, γ) to be attained. 2. A Vanishing-Concentration-Compactness alternative In our analysis, the following improved version of the Adachi-Tanaka inequality [1] will be crucial Theorem 2.1 ([6] [17]). Let 2. Then there exists a constant C > 0 such that for any γ (0, 1) we have φ C R ( α γ u ) dx 1 γ u u W 1, (R ) with u 1 (2.1)
6 6 We recall that (2.1), in the 2-dimensional case, was deduced in [6, Theorem 1.2] as a direct consequence of the critical Trudinger-Moser inequality in R 2, i.e. (1.3) with = 2.. Lam, G. Lu L. Zhang in [17, Theorem 1.1] obtained the generalization to the higher dimensional case 3, without assuming a priori the validity of (1.3). Remark 2.1. In this framework, it is important to mention also a Lions-type result [12, Theorem 1.1] in the whole space R. This result tells us that, if a sequence {u } W 1, (R ), satisfying the constraint (1.1) with a = b =, i.e. u + u = 1 1, weakly converges to a non-trivial function u 0 then an inequality of Trudinger-Moser type holds along the sequence with an exponent larger than α. More precisely, sup φ R ( α p u ) dx < + for any 0 < p < [ 1 ( u + u ) ] 1 (2.2) The scale invariant inequality (2.1) implies only a weaker version of (2.2) but it is more flexible to treat the case of normalized sequences with respect to the constraint (1.1). Moreover, inequality (2.1) will enable us to describe the effect of the constraint (1.1) on concentration phenomena. Let 2, a > 0 0 < b be fixed. We begin considering a (spherically symmetric non-increasing) maximizing sequence {u } W 1, (R ) for the Trudinger-Moser supremum d (a, b) defined by (1.7), i.e. u 0 a.e. in R for any 1, u a + u b = 1 1 φ ( α u ) dx = d (a, b) R Remark 2.2. By Schwarz symmetrization, it is well know that given a maximizing sequence {u } W 1, (R ) for d (a, b), one may always assume that each u is non-negative, spherically symmetric non-increasing with respect to the radial variable. We will set θ = u b (0, 1), so that u a = 1 θ. Since {θ } (0, 1), without loss of generality, we may assume θ = θ [0, 1], it becomes natural to distinguish three cases according to θ = 1, θ = 0 θ (0, 1). Intuitively, in terms of the maximizing sequence {u } for d (a, b), this suggests the following alternative: (I) (vanishing) if θ = 1 then {u } is a vanishing maximizing sequence for d (a, b); (II) (concentration) if θ = 0 then {u } is a concentrating maximizing sequence for d (a, b);
7 VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE 7 (III) (compactness) If θ (0, 1) then u = (1 θ) 1 a (0, 1) u = θ 1 b (0, 1) {u } weakly converges in W 1, (R ) to a maximizer of d (a, b). In other words, d (a, b) is attained. In fact, in Section 6, we will show that this intuition can be derived from the following vanishing-concentration-compactness alternative Lemma 2.2. Let 2, a > 0 0 < b be fixed. We consider a (spherically symmetric non-increasing) sequence {u } W 1, (R ) satisfying the constraint (1.1), i.e. u a + u b = 1 1 we assume that u u in W 1, (R ). Then (I) either {u } is a normalized vanishing sequence More precisely, (II) or {u } is such that φ ( α u ) dx = 0 for any fixed R > 0 B R φ α ( α u ) dx = R ( 1)! u = 1 u = 0 In this case there is a striking difference between the range 0 < b < b =. If 0 < b <, independently of a > 0, we have φ ( α u ) dx = 0 R While, if b = either {u } is a normalized concentrating sequence φ ( α u ) dx = 0 for any fixed R > 0 R B R or φ ( α u ) dx = 0 R (III) Finally, if both (I) (II) do not occur then φ ( α u ) dx = φ α R R ( α u ) dx + ( 1)! ( θ b u ) where, up to subsequences, θ b = u
8 8 In particular, if u u in L (R ) then φ ( α u ) dx = φ R R ( α u ) dx It is important to recall that, in the pioneering work [19], P.-L. Lions developed a version of his Concentration-Compactness Principle for the iting case of the Sobolev embedding theorem, i.e. for the Trudinger-Moser case. The result of P.-L. Lions concerns bounded domains has been sharpened by R. Cerný, A. Cianchi S. Hencl [8]. The approach introduced in [8] is different from Lions technique yields to deal, not only with functions vanishing on the boundary but, with functions with unrestricted boundary values on a fixed bounded domain. The case of unbounded domains has already been considered in [3], [12] (see Remark 2.1) [7]. L. Battaglia G. Mancini in [3] focused their attention to the 2-dimensional case in particular to the planar strip R ( 1, 1) R 2. R. Cerný in [7] obtained a version of the Concetration-Compactness Principle for the Trudinger-Moser functional on the whole space R, i.e. with respect to the constraint u W 1, (R ) φ R ( α u ) dx S M = { u W 1, (R ) u 1 u M }, M > 0, which is different from (1.1). We mention that Lions Concentration-Compactness Principle [19] inspired Adimurthi O. Druet [2] to study a valuable improvement of the Trudinger-Moser inequality on bounded domains of R 2. The improved inequality by Adimurthi O. Druet [2] has been extended to the higher dimensional case by Y. Yang [23, 24]. Related partial results in the whole space R have been approached in [9] [10]. A new interpretation a further generalization of [2] has been introduced by C. Tintarev [22], see also Y. Yang [25] for a study of the corresponding problem of attainability. Remark 2.3. Definition 1.1 of vanishing sequences is apparently different from the notion of vanishing introduced by M. Ishiwata [13, Definition 2.1]. Recall that, a (spherically symmetric non-increasing) sequence {u } W 1, (R ) is vanishing according to Ishiwata [13] if each u satisfies the constraint (1.1) R + φ ( α u ) dx = 0 (2.3) B R evertheless, as a consequence of (I) of Lemma 2.2, condition (1.8) implies (2.3). In other words, any normalized vanishing sequence is also a vanishing sequence according to Ishiwata. Let {u } W 1, (R ) be a (spherically symmetric non-increasing) sequence satisfying the constraint (1.1) suppose u u in W 1, (R )
9 VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE 9 As mentioned above, if we set θ = u b (0, 1), so that u a = 1 θ, then we may assume, without loss of generality, θ = θ [0, 1] (2.4) In the next Sections, we will prove Lemma 2.2 through the following steps: if θ = 1 then (I) occurs (see Section 3); if θ = 0 then (II) holds (see Section 4). More precisely in this case, either u dx = 0 for any fixed R > 0 R B R or φ ( α u ) dx = 0 (2.5) R In particular, if 0 < b < then (2.5) holds also for normalized concentrating sequences. Finally, in Section 4, we will show that if θ (0, 1) then we have the convergence result expressed by (III), but we do not know a priori whether or not u u in L (R ). Remark 2.4. Looking at case (I) case (II) of Lemma 2.2, we can say that the constraint (1.1) has not effects on the energy level of normalized vanishing sequences while, in contrast, the energy level corresponding to normalized concentrating sequences is heavily influenced by the constraint (1.1). On one h, in view of (I) of Lemma 2.2, the level of normalized vanishing sequences is always α ( 1)! independently of the parameters a > 0 0 < b defining the constraint (1.1). On the other h, taking into consideration (II) of Lemma 2.2, if 0 < b < then the level of normalized concentrating sequences is zero, independently of a > 0. The same is not true when b =, at least in general. This cannot be deduced from (II) of Lemma 2.2 but, as mentioned in the Introduction (see Theorem A), it is well known that in the 2-dimensional case when a = b = 2 then sup { R 2(e4πu2 1) dx {u } is a normalized concentrating sequence } = eπ It is important to point out that, even if the value of the parameters a > 0 0 < b does not influence the energy level of normalized vanishing sequences, we cannot deduce that the constraint (1.1) has not effects on vanishing phenomena. 3. Alternative (I) Vanishing In this Section, we consider the case of normalized vanishing sequences, i.e. (spherically symmetric non-increasing) sequences {u } W 1, (R ) such that each u satisfies the constraint (1.1), namely u a + u b = 1 1,
10 10 u = 0 u = 1 First, we will show that the energy of any normalized vanishing sequence {u } W 1, (R ) can be localized in the exterior of any fixed ball of radius R > 0; more precisely, φ ( α u ) dx = 0 for any fixed R > 0 B R To this aim, we will use the classical Trudinger-Moser inequality on balls B R R of radius R > 0 centered at the origin, i.e. Theorem 3.1 ([20]). There exists a constant C > 0 such that for any R > 0 φ ( α u ) dx C R u u W 1, 0 (B R ) {0} with u 1 (3.1) BR We point out that the local estimate expressed by (3.1) is not the original version of Moser s inequality [20], but it can be deduced directly from the famous inequality in [20] with the aid of the rescaled function ũ = u/ u, see for instance [26, Lemma 2.1]. Lemma 3.2. Let 2, a > 0 0 < b be fixed. If {u } W 1, (R ) is a (spherically symmetric non-increasing) sequence satisfying, for some θ > 0, then u = 0 sup u θ φ ( α u ) dx = 0 for any fixed R > 0 B R Proof. Let R > 0 be arbitrarily fixed. In order to apply Theorem 3.1, the idea is to reconstruct zero-dirichlet boundary conditions on the boundary of B R letting By construction w W 1, 0 (B R ) ote that, for any fixed α > 0, if we set then there exists 1 such that w = u u (R) on B R w L (B R ) = u L (B R ) 0 as + w = ( α ) w α on B R w = ( α α ) w 1 Therefore in view of Lemma 3.1, for any fixed α > 0, there exists 1 such that BR φ ( αw ) dx = BR φ ( α w ) dx C ( α α ) R w (3.2)
11 VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE 11 ext, applying the one-dimensional inequality we get (1 + x) p (1 + ε)x p + ( 1 1 p 1 (1 + ε) 1/(p 1) ) α u = α ( w + u (R) ) C 1 u x 0, p > 1, ε > 0 (3.3) (R) + C 2 w on B R for some constants C 1, C 2 > 0 depending on. We do not need to explicitly write the value of the constant C 2 > 0 the reason for that is essentially (3.2). Summarizing,, if we show that φ ( α u ) dx φ ( C 1 u BR BR φ ( C 1 u B R exp( C 1u (R) + C 2 w ) dx (R) + C 2 w ) dx (R) ) φ ( C 2 w ) dx BR (3.4) then the proof is complete. On one h, by means of the following Radial Lemma, which holds for any spherically symmetric non-increasing function ϕ W 1, (R ) we may estimate ϕ ( x ) ϕ ω x ϕ whenever x 0 (3.5) exp( C 1 u (R) ) exp( C 1 θ u R On the other h, by means of (3.2) with α = C 2 > 0, we get Hence, 1 BR φ ( C 2 w ) dx C R w φ ( α u ) dx exp( C 1u B R ) (R) ) φ ( C 2 w ) dx BR C R w exp( C 1 θ u 1 ) = 0 R To complete the proof, it remains to show that (3.4) holds. To this aim, we begin with an elementary one-dimensional estimate. For any s, t 0, we have φ (s + t) = e s+t ± e s 2 t k 2 k=0 k! k=0 (s + t) k k! e s φ (t) + e s 2 t k k=1 k! + (es 1)
12 12 In particular, when = 2, If we set then A (2) = 0 e s+t 1 = e s (e t 1) + (e s 1) A () = exp( C 1 u 2 C 2 k (R) ) k=1 k! BR φ ( C 1 u B () = [ exp( C 1 u exp( C 1 u (R) + C 2 w ) dx Therefore, it suffices to prove that From the Radial Lemma (3.5), we deduce (3.7) (R) ) 1 ] B R BR w k dx when 3 (R) ) φ ( C 2 w ) dx + A () + B () BR A () = 0 (3.6) B () = 0 (3.7) A () exp( C 1 θ u 1 2 C2 k ) R k=1 k! Moreover, w 0 in W 1, 0 (B R ) the embedding W 1, 0 (B R ) L k (B R ) BR w k is compact for any 1 k 2 with 3. Hence, also (3.6) holds the proof is complete. We complete this Section with a precise estimate of the level of vanishing sequences. Lemma 3.3. Let 2, a > 0 0 < b be fixed. If {u } W 1, (R ) is a (spherically symmetric non-increasing) sequence satisfying, for some θ > 0, dx then u = 0 u = θ φ α ( α u ) dx = R ( 1)! θ
13 Proof. Since it is easy to see that VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE 13 φ ( α u R φ (t) t ( 1)! ) dx t 0 α ( 1)! u = Therefore, the proof of Lemma 3.3 is complete if we show that φ α ( α u ) dx R ( 1)! θ α ( 1)! θ To obtain the preceding estimate from above, let us fix R > 0 let us rewrite φ R ( α u ) dx = + φ ( α u ) dx R B R BR To this aim, it is clear from Lemma 3.2 that, it suffices to show that Using the elementary inequality α φ ( α u ) dx R B R ( 1)! θ (3.8) φ (t) the Radial Lemma (3.5), we get which gives (3.8). t ( 1)! et t 0 (3.9) α φ ( α u ) dx R B R ( 1)! u e α u R B R dx α ( 1)! θ exp( 1 R u 1 ) ote that, the case of normalized vanishing sequence is included in Lemma 3.2 Lemma 3.3 with θ = 1 hence Corollary 3.4. Let 2, a > 0 0 < b be fixed. If {u } W 1, (R ) is a (spherically symmetric non-increasing) normalized vanishing sequence then Moreover, φ ( α u ) dx = 0 for any fixed R > 0 B R φ α ( α u ) dx = R ( 1)!
14 14 4. Alternative (II) Concentration Let {u } W 1, (R ) be a (spherically symmetric non-increasing) sequence satisfying the constraint (1.1), i.e. u a + u b = 1 1, u = 1 u = 0 which is the case of normalized concentrating sequences. In fact, we have Lemma 4.1. Let 2, a > 0 0 < b be fixed. If {u } W 1, (R ) is a (spherically symmetric non-increasing) sequence satisfying the constraint (1.1) then u = 1 u = 0 either {u } is a normalized concentrating sequence or φ ( α u ) dx = 0 for any fixed R > 0 (4.1) R B R φ ( α u ) dx = 0 (4.2) R Proof. First, recalling the elementary inequality (3.9) the Radial Lemma (3.5), we may estimate for any fixed R > 0 α φ ( α u ) dx R B R ( 1)! u e α u R B R dx α ( 1)! ec()/r u 0 as + (4.3) this gives (4.1). ext, we consider the case when the sequence {u } is not a normalized concentrating sequence. More precisely, we assume the existence of R > 0, δ (0, 1) 1 such that R B R u dx δ Under this assumption, if we show that (4.2) holds then the proof is complete. Even if the arguments are stard, we give a sketch for the convenience of the reader. ote that, from (4.3), we deduce for any fixed R > 0. φ ( α u ) dx = R φ ( α u ) dx B R
15 VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE 15 To obtain a uniform estimate of the integral on balls of fixed radius 0 < R R, we argue as in the proof of Lemma 3.2 we set w = u u (R) W 1, 0 (B R ) 1 Applying the one-dimensional inequality (3.3), with p = Radial Lemma (3.5), we may estimate φ ( α u ) dx exp( α u BR BR If 0 < R R then exp( C(δ)u ) dx > ε = δ 2 (R) ) exp( α (1 + δ BR 2 ) w ) dx > 0, the exp( C(δ, ) 1 R u ) exp( α (1 + δ BR 2 ) w ) dx w L (B R ) = w L (B R ) u L (B R ) = ( 1 u b ) a u L (R B R ) ( 1 u b ) a δ where we also used the constraint (1.1), Therefore, there exists 1 such that w L (B R ) 1 δ w L (B R ) 1 δ 2, from the classical Trudinger-Moser inequality on bounded domains, we deduce sup BR exp( α (1 + δ 2 ) w ) dx sup v W 1, 0 (B R ), v =1 In conclusion, for any fixed 0 < R R, exp( α v ) dx C R BR φ ( α u ) dx C R exp( C(δ, ) 1 B R R u ) = C R φ ( α u ) dx C R R from which we deduce (4.2), letting R 0. for any fixed 0 < R R We will see that the alternative expressed by Lemma 4.1 is meaningful only when b =. In fact, we are going to show that if 0 < b < then the level of normalized concentrating sequences is zero; more precisely, if 0 < b < then (4.2) holds also for normalized concentrating sequences.
16 16 Let {u } W 1, (R ) be a (spherically symmetric non-increasing) sequence satisfying the assumptions of Lemma 4.1. If we set θ = u b (0, 1) then, by assumptions, θ = 0 u a = 1 θ. In order to apply the improved version of the Adachi-Tanaka inequality (2.1), it turns out to be convenient to introduce the normalized sequence with respect to the Dirichlet norm, so that u v = u C φ R ( α (1 θ ) 1 a v ) dx 1 (1 θ ) a ote that Hence, C 1 (1 θ ) a θ b (1 θ ) a v = ac = φ ( α u ) dx = R φ ( α (1 θ ) R ac In the case 0 < b <, the above estimate yields Therefore, if b we have θ b 1 φ ( α u ) dx = 0 R C 1 (1 θ ) a θ b 1 1 a v ) dx θ b (1 θ ) a (4.4) Lemma 4.2. Let 2, a > 0 0 < b < be fixed. If {u } W 1, (R ) is a (spherically symmetric non-increasing) sequence satisfying the constraint (1.1) then u = 1 u = 0 φ ( α u ) dx = 0 R Remark 4.1. If b = then (4.4) is not useful to obtain a precise estimate of the level of normalized concentrating sequences. However, from (4.4), we can deduce that for any fixed δ > 0 there exists a = a(δ) > 0 such that if b = 0 < a a then φ ( α u ) dx < δ R
17 VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE Alternative (III) Compactness We will use the following convergence result, which holds for sequences which are neither vanishing nor concentrating, i.e. in particular θ 1 θ 0, see (2.4). Lemma 5.1. Let 2, a > 0 0 < b be fixed. We consider a (spherically symmetric non-increasing) sequence {u } W 1, (R ) satisfying the constraint (1.1) with θ = u b, so that u a = 1 θ. Assume, up to subsequences, u u in W 1, (R ) Then θ = θ (0, 1) φ ( α u ) dx = φ α R R ( α u ) dx + ( 1)! ( θ b u ) Proof. It is enough to show that [ φ α ( α u ) R ( 1)! u ] dx = [ φ α R ( α u ) ( 1)! u ] dx (5.1) In view of the improved version of the Adachi-Tanaka inequality (2.1), we will obtain (5.1) simply by means of Strauss Lemma (see [4, Theorem A.I]) Since θ (0, 1), we have also that (1 θ) 1 a (0, 1) there exists ε > 0 1 such that α (1 θ) 1 a α (1 ε) Let then t P (t) = φ ( t ) ( 1)! Q (t) = φ ( (1 + ε) t P (t) P (t) = 0 t + Q (t) t 0 Q (t) = 0 ) ext, we introduce the normalized sequence with respect to the Dirichlet norm By construction v = u u = u (1 θ ) 1 a 1 Q R ( α u ) = Q R ( α (1 θ ) 1 a v ) dx
18 18 Applying the improved version of the Adachi-Tanaka inequality (2.1) to the Dirichletnormalized sequence {v }, we get for any Hence, Q R ( α sup (1 θ ) 1 a v ) dx φ R ( α (1 ε 2 )v ) dx C 1 (1 ε 2 ) v C θ = 1 (1 ε 2 ) (1 θ ) a Q R ( α u ) dx C 1 (1 ε 2 sup ) b θ b (1 θ ) a < + Recalling that the sequence {u } is spherically symmetric, we can apply Strauss Lemma (see [4, Theorem A.I]) obtaining that is (5.1). P ( α R u ) dx = P R ( α u ) dx To complete the proof of Lemma 2.2, in view of the analysis carried out in Section 3 Section 4, we ust need to consider the case for which the assumptions of the above convergence result are fulfilled. More precisely, we consider a (spherically symmetric non-increasing) sequence {u } W 1, (R ) satisfying the constraint (1.1) we assume, up to subsequences, that u u in W 1, (R ) u = θ b (0, 1) In this case, we can apply Lemma 5.1, to conclude that alternative (III) holds. Moreover, if u u in L (R ) then u = u, from Lemma 5.1, we deduce φ ( α u ) dx = φ R R ( α u ) dx 6. The case of maximizing sequences for the Trudinger-Moser inequality Let 2, a > 0 0 < b be fixed. We begin this Section deducing a useful lower bound for the Trudinger-Moser supremum d (a, b) defined by (1.7). Recalling that t φ (t) ( 1)! + t t 0,! for any u W 1, (R ) satisfying the constraint (1.1), i.e. u a + u b = 1
19 we may estimate VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE 19 d (a, b) φ α R ( α u ) dx ( 1)! u + α 2! u /() 2 /() If we consider the supremum then it is clear that D (a, b) = Remark 6.1. When a = b, we set = α ( 1)! ( u + α 2 u /() 2 /() ) sup ( u + α 2 u W 1, (R ), u a + u b =1 u /() 2 /() ) d (a, b) D (γ) = D (γ, γ) α ( 1)! D (a, b) (6.1) with 0 < γ This is a particular case of the more general maximizing problem considered by M. Ishiwata H. Wadade in [14]. As pointed out in [14], the attainability of the supremum d (γ, γ) associated with the Trudiger-Moser inequality is closely related to the behavior of D (γ). In fact, we can observe that the constant appearing on the right h side of (6.1) α ( 1)! corresponds to the level of normalized vanishing sequences (see (I) of Lemma 2.2). Intuitively, when we look at maximizing sequences for d (γ, γ) then the behavior of D (γ) could be crucial to exclude possible vanishing phenomena. More precisely, if 0 < γ is such that D (γ) > 1 then maximing sequences for d (γ, γ) cannot vanish, i.e. cannot be normalized vanishing sequences. The careful study developed in [14] shows that both the behavior of D (γ) its attainability are intimately related to the value of γ in the range (0, ]. We mention that the attainability of D (γ) is not only interesting in the iting case of the Sobolev embedding theorem but also in the classical Sobolev case. We refer the reader to [16], where the authors approach the study of the existence of maximizers for with 2, 1 < p <, p < q < p p sup ( u p p + α u q q ) u W 1,p (R ), u γ p+ u γ p=1 α, γ > 0. Following the arguments introduced by M. Ishiwata H. Wadade [14], it is not difficult to show that
20 20 Lemma 6.1. Let 2, a > 0 0 < b be fixed. Then D (a, b) 1 (6.2) hence, the Trudinger-Moser supremum d (a, b) defined by (1.7) satisfies d (a, b) α ( 1)! (6.3) Proof. The proof of (6.2) can be deduced arguing exactly as in [14]; for the convenience of the reader, we briefly sketch it. As showed in [14, Section 2.2], given any u W 1, (R ) satisfying the constraint (1.1), i.e. u a + u b = 1 the family of comparison functions w t W 1, (R ) depending on the parameter t (0, 1) defined by 1 (1 t) a w t (x) = (1 t) 1 a u( λ t x ) λ t = u still satisfies the constraint (1.1). In fact, w t = (1 t) 1 a w t = (1 t) 1 a 1 u = t 1 b u λ t Therefore, we may estimate t 1 b u u (6.4) D (a, b) w t + α w t 2 /() 2 /() w t = t b t (0, 1) (6.5) (6.2) follows. Remark 6.2. We mention that (6.3) can also be directly deduced from [17, Theorem 1.2]. In fact, denoting by 1 AT (γ) = sup u W 1, (R ) {0} u φ R ( α γ u ) dx γ (0, 1),. Lam, G. Lu L. Zhang in [17] obtained the following more precise version of (2.1) d (a, b) = sup γ (0,1) ( 1 γ ow, a simple scaling argument shows that AT (γ) = a ) b γ AT (γ) a > 0, 0 < b (6.6) sup u W 1, (R ), u = u =1 φ R ( α γ u ) dx (6.7) ote that the supremum on the right h side of the above identity corresponds to the inequality first studied in [11]. Combining (6.6) with (6.7), it is easy to see that (6.3) follows.
21 VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE 21 ext, we consider a (spherically symmetric non-increasing) maximizing sequence {u } W 1, (R ) for the Trudinger-Moser supremum d (a, b) defined by (1.7), i.e. u 0 a.e. in R for any 1, u a + u b = 1 1 φ ( α u ) dx = d (a, b) R Dealing with a maximizing sequence for d (a, b), the alternative expressed by Lemma 2.2 becomes simpler. Case (I) If {u } is a vanishing maximizing sequence then d (a, b) = φ α ( α u ) dx = R ( 1)! Case (II) If 0 < b < then the following conditions cannot hold, since otherwise u = 1 u = 0 (6.8) d (a, b) = φ ( α u ) dx = 0 R which would contradict Lemma 6.1. While when b =, if (6.8) holds then {u } must be a concentrating maximizing sequence φ ( α u ) dx = 0 for any fixed R > 0 R B R Moreover, in the latter case, combining Remark 4.1 with the estimate from below of d (a, ) (i.e. Lemma 6.1), we deduce the existence of a > 0 such that if 0 < a < a (6.8) holds then α ( 1)! d (a, ) = φ ( α u R ) dx < α ( 1)! which is a contradiction. Consequently, if b = a > 0 is sufficiently small then concentration cannot occur. Case (III) Finally, let θ = u b (0, 1) let us consider a subsequence still denoted by {θ } such that θ = θ Since we already discussed the cases θ = 1 θ = 0, without loss of generality, we may assume θ (0, 1). We may also assume, up to subsequences, From Lemma 2.2, we deduce that u u in W 1, (R ) φ ( α u ) dx = φ α R R ( α u ) dx + ( 1)! ( θ b u ) (6.9)
22 22 In particular, this implies that u 0. In fact, if not then α d (a, b) = φ ( α u ) dx = R ( 1)! θ b < α ( 1)! contradicting Lemma 6.1. Therefore, we can define τ = θ 1 b u = 1 u u 1 ote that, in view of Brezis-Lieb Lemma [5], if we show that τ = 1 then we can conclude that u u in L (R ) d (a, b) = φ ( α u ) dx = φ R R ( α u ) dx from which we deduce that u is a maximizer for d (a, b). Let u τ (x) = u( x τ ) x R so that u τ = u, u τ = τ u = θ 1 b Therefore, we may estimate u τ a + u τ b = u a + θ ( u a + u b ) = 1 d (a, b) φ R ( α u τ ) dx = τ R φ ( α u ) dx, using (6.9), we get = φ R ( α u ) dx + (τ 1) φ R ( α u ) dx d (a, b) d (a, b) α ( 1)! ( θ b u ) + (τ 1) φ R ( α u ) dx = d (a, b) α ( 1)! u (τ 1) + (τ 1) φ R ( α u ) dx = d (a, b) + (τ 1) φ R +1( α u ) dx (6.10) Since u 0, we have φ R +1( α u ) dx = ( eα u R α k k=0 k! u k ) dx > 0 Consequently, τ = 1. In fact, if not then τ > 1 (6.10) gives a contradiction. Summarizing,
23 VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE 23 Lemma 6.2. Let 2, a > 0 0 < b be fixed. We consider a (spherically symmetric non-increasing) maximizing sequence {u } W 1, (R ) for the Trudinger- Moser supremum defined by (1.7) we assume that u u in W 1, (R ). If b = then one of the following alternatives occurs: (I) either {u } is a vanishing maximizing sequence d (a, b) = φ α ( α u ) dx = R ( 1)! (II) or {u } is a concentrating maximizing sequence φ ( α u ) dx = 0 for any fixed R > 0 R B R (III) or the weak it u is non-trivial it is a maximizer for d (a, b). If either 0 < b < a > 0 or b = 0 < a << 1 then maximizing sequences for d (a, b) cannot concentrate we have ust two alternatives: either vanishing (I) or attainability (III) occurs. or When either 0 < b < a > 0 b = 0 < a << 1 since concentration cannot occur, the lack of compactness of maximizing sequences for d (a, b) can be only caused by vanishing phenomena. This possible lack of compactness may prevent the supremum d (a, b) to be attained. In this respect, the analysis carried out in [14] plays a crucial role, see also [15] Remark Proof of Theorem 1.2 Attainability of the supremum Let 2, a > 0 < b <. To prove the attainability of the Trudinger-Moser supremum d (a, b) defined by (1.7), we will follow the arguments introduced by M. Ishiwata H. Wadade in [14] (see also [15]). In fact, since 0 < b < Lemma 6.2 expresses a vanishing-compactness alternative for (spherically symmetric non-increasing) maximizing sequences of d (a, b). More precisely, if 0 < b < {u } W 1, (R ) is a (spherically symmetric non-increasing) maximizing sequence for d (a, b) then {u } cannot concentrate if it would be possible to exclude vanishing phenomena then d (a, b) would be attained. In this respect, the restriction to the case a > 1
24 24 plays a crucial role. If a > then it is possible to improve the lower bound of d (a, b) expressed by Lemma 6.1 showing that (I) of Lemma 6.2 cannot occur. Proposition 7.1. Let 2, a > d (a, b) defined by (1.7) satisfies 0 < b. Then the Trudinger-Moser supremum d (a, b) > If in addition b then d (a, b) is attained. α ( 1)! Proof. As already mentioned, in view of Lemma 6.2, if 0 < b < then the validity of (7.1) would enable to conclude the attainability of d (a, b). Therefore, we ust need to prove (7.1) for this, we have to restrict the range of the parameter a > 0 defining the constraint (1.1). Following [14, Section 2.2], we consider a suitable family of comparison functions {w t } t (0,1) W 1, (R ), generated by a fixed function u W 1, (R ) satisfying the constraint (1.1). We used the same argument in the proof of Lemma 6.1 we refer to (6.4) for the definition of w t with t (0, 1). ote that Let w t p p = t b p (1 t) a B(u) = u p p u p 2/() u 2 /() u u f(t) = f,a,b (t) = w t + α w t 2 /() 2 /() = t b Combining (6.1) with (6.5), we get u > 0 ( 1 + α p d (a, b) α ( 1)! D (a, b) α f(t) t (0, 1) ( 1)! 1 (1 t) a B(u) ) (7.1) If f(t) > 1 for some t (0, 1) (7.2) then it would be possible to conclude that (7.1) holds. ote that f(1) = 1, therefore (7.2) would follow if f (t) < 0 for some t (0, 1) sufficiently close to 1 We can compute f (t) = b t b 1 ( 1 + α, if 1 (1 t) a B(u) ) 1 α 1 1 a t b 1 a 1 < 0, i.e. a > 1, 1 (1 t) a 1 B(u)
25 VAISHIG-COCETRATIO-COMPACTESS ALTERATIVE 25 we have f (t) = t 1 References [1] S. Adachi, K. Tanaka, Trudinger type inequalities in R their best exponents. Proc. Amer. Math. Soc. 128 (2000), no. 7, [2] Adimurthi, O. Druet, Blow-up analysis in dimension 2 a sharp form of Trudinger-Moser inequality. Comm. Partial Differential Equations 29 (2004), no. 1-2, [3] L. Battaglia, G. Mancini, Remarks on the Moser-Trudinger inequality. Adv. onlinear Anal. 2 (2013), no. 4, [4] H. Berestycki, P.-L. Lions, onlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983), no. 4, [5] H. Brezis, E. Lieb, A relation between pointwise convergence of functions convergence of functionals. Proc. Amer. Math. Soc. 88 (1983), no. 3, [6] D. Cassani, F. Sani, C. Tarsi, Equivalent Moser type inequalities in R 2 the zero mass case. J. Funct. Anal. 267 (2014), no. 11, [7] R. Cerný, Concentration-compactness principle for embedding into multiple exponential spaces on unbounded domains. Czechoslovak Math. J. 65 (140) (2015), no. 2, [8] R. Cerný, A. Cianchi, S. Hencl, Concentration-compactness principles for Moser-Trudinger inequalities: new results proofs. Ann. Mat. Pura Appl. (4) 192 (2013), no. 2, [9] M. de Souza, J. M. do Ó, A sharp Trudinger-Moser type inequality in R2. Trans. Amer. Math. Soc. 366 (2014), no. 9, [10] M. de Souza, J. M. do Ó, A sharp inequality of Trudinger-Moser type extremal functions in H 1,n (R n ). J. Differential Equations 258 (2015), no. 11, [11] J. M. do Ó, -Laplacian equations in R with critical growth. Abstr. Appl. Anal. 2 (1997), no. 3-4, [12] J. M. do Ó, M. de Souza, E. de Medeiros, U. Severo, An improvement for the Trudinger-Moser inequality applications. J. Differential Equations 256 (2014), no. 4, [13] M. Ishiwata, Existence nonexistence of maximizers for variational problems associated with Trudinger-Moser type inequalities in R. Math. Ann. 351 (2011), no. 4, [14] M. Ishiwata, H. Wadade, On the effect of equivalent constraints on a maximizing problem associated with the Sobolev type embeddings in R. Math. Ann. to appear [15] M. Ishiwata, H. Wadade, Existence non-existence of maximizers for the Moser-Trudinger inequalities of the inhomogeneous type. Preprint 2015 [16] M. Ishiwata, H. Wadade, On a maximizing problem associated with the Sobolev type embeddings in R under equivalent constraints. Preprint 2015 [17]. Lam, G. Lu, L. Zhang, Equivalence of critical subcritical sharp Trudinger-Moser-Adams inequalities. Preprint 2015 [18] Y. Li, B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in R n. Indiana Univ. Math. J. 57 (2008), no. 1, [19] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The it case. I. Rev. Mat. Iberoamericana 1 (1985), no. 1, [20] J. Moser, A sharp form of an inequality by. Trudinger. Indiana Univ. Math. J. 20 (1970/1971), [21] B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in R 2. J. Funct. Anal. 219 (2005), no. 2,
26 26 [22] C. Tintarev, Trudinger-Moser inequality with remainder terms. J. Funct. Anal. 266 (2014), no. 1, [23] Y. Yang, A sharp form of Moser-Trudinger inequality in high dimension. J. Funct. Anal. 239 (2006), no. 1, [24] Y. Yang, Corrigendum to: A sharp form of Moser-Trudinger inequality in high dimension. J. Funct. Anal. 242 (2007), no. 2, [25] Y. Yang, Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two. J. Differential Equations 258 (2015), no. 9, [26] Y. Yang, X. Zhu, A new proof of subcritical Trudinger-Moser inequalities on the whole Euclidean space. J. Partial Differ. Equ. 26 (2013), no. 4, Universidade Federal da Paraíba, , João Pessoa, PB, Brazil address: mbo@pq.cnpq.br Università degli Studi di Milano, via Cesare Saldini 50, Milano, Italy address: federica.sani@unimi.it Università degli Studi di Milano, via Cesare Saldini 50, Milano, Italy address: cristina.tarsi@unimi.it
NONLINEAR SCHRÖDINGER ELLIPTIC SYSTEMS INVOLVING EXPONENTIAL CRITICAL GROWTH IN R Introduction
Electronic Journal of Differential Equations, Vol. 014 (014), No. 59, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONLINEAR SCHRÖDINGER
More informationEXISTENCE OF SOLUTIONS FOR CROSS CRITICAL EXPONENTIAL N-LAPLACIAN SYSTEMS
Electronic Journal of Differential Equations, Vol. 2014 (2014), o. 28, pp. 1 10. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF SOLUTIOS
More informationMinimization problems on the Hardy-Sobolev inequality
manuscript No. (will be inserted by the editor) Minimization problems on the Hardy-Sobolev inequality Masato Hashizume Received: date / Accepted: date Abstract We study minimization problems on Hardy-Sobolev
More informationSYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS. Massimo Grosi Filomena Pacella S. L. Yadava. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 21, 2003, 211 226 SYMMETRY RESULTS FOR PERTURBED PROBLEMS AND RELATED QUESTIONS Massimo Grosi Filomena Pacella S.
More informationON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT
PORTUGALIAE MATHEMATICA Vol. 56 Fasc. 3 1999 ON NONHOMOGENEOUS BIHARMONIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT M. Guedda Abstract: In this paper we consider the problem u = λ u u + f in, u = u
More informationEXISTENCE OF MULTIPLE SOLUTIONS TO ELLIPTIC PROBLEMS OF KIRCHHOFF TYPE WITH CRITICAL EXPONENTIAL GROWTH
Electronic Journal of Differential Equations, Vol. 2014 (2014, o. 107, pp. 1 12. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTECE OF MULTIPLE
More informationGlobal unbounded solutions of the Fujita equation in the intermediate range
Global unbounded solutions of the Fujita equation in the intermediate range Peter Poláčik School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Eiji Yanagida Department of Mathematics,
More informationSTABILITY RESULTS FOR THE BRUNN-MINKOWSKI INEQUALITY
STABILITY RESULTS FOR THE BRUNN-MINKOWSKI INEQUALITY ALESSIO FIGALLI 1. Introduction The Brunn-Miknowski inequality gives a lower bound on the Lebesgue measure of a sumset in terms of the measures of the
More informationThe effects of a discontinues weight for a problem with a critical nonlinearity
arxiv:1405.7734v1 [math.ap] 9 May 014 The effects of a discontinues weight for a problem with a critical nonlinearity Rejeb Hadiji and Habib Yazidi Abstract { We study the minimizing problem px) u dx,
More informationNonlinear elliptic systems with exponential nonlinearities
22-Fez conference on Partial Differential Equations, Electronic Journal of Differential Equations, Conference 9, 22, pp 139 147. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu
More informationSCATTERING FOR THE TWO-DIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY
SCATTERING FOR THE TWO-DIMENSIONAL NLS WITH EXPONENTIAL NONLINEARITY S. IBRAHIM, M. MAJDOUB, N. MASMOUDI, AND K. NAKANISHI Abstract. We investigate existence and asymptotic completeness of the wave operators
More informationEXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS WITH UNBOUNDED POTENTIAL. 1. Introduction In this article, we consider the Kirchhoff type problem
Electronic Journal of Differential Equations, Vol. 207 (207), No. 84, pp. 2. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
More informationON A CONJECTURE OF P. PUCCI AND J. SERRIN
ON A CONJECTURE OF P. PUCCI AND J. SERRIN Hans-Christoph Grunau Received: AMS-Classification 1991): 35J65, 35J40 We are interested in the critical behaviour of certain dimensions in the semilinear polyharmonic
More informationSYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS. M. Grossi S. Kesavan F. Pacella M. Ramaswamy. 1. Introduction
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 12, 1998, 47 59 SYMMETRY OF POSITIVE SOLUTIONS OF SOME NONLINEAR EQUATIONS M. Grossi S. Kesavan F. Pacella M. Ramaswamy
More informationA Caffarelli-Kohn-Nirenberg type inequality with variable exponent and applications to PDE s
A Caffarelli-Kohn-Nirenberg type ineuality with variable exponent and applications to PDE s Mihai Mihăilescu a,b Vicenţiu Rădulescu a,c Denisa Stancu-Dumitru a a Department of Mathematics, University of
More informationADAMS INEQUALITY WITH THE EXACT GROWTH CONDITION IN R 4
ADAMS INEQUALITY WITH THE EXACT GROWTH CONDITION IN R 4 NADER MASMOUDI AND FEDERICA SANI Contents. Introduction.. Trudinger-Moser inequality.. Adams inequality 3. Main Results 4 3. Preliminaries 6 3..
More informationPolyharmonic Elliptic Problem on Eistein Manifold Involving GJMS Operator
Journal of Applied Mathematics and Computation (JAMC), 2018, 2(11), 513-524 http://www.hillpublisher.org/journal/jamc ISSN Online:2576-0645 ISSN Print:2576-0653 Existence and Multiplicity of Solutions
More informationHardy Rellich inequalities with boundary remainder terms and applications
manuscripta mathematica manuscript No. (will be inserted by the editor) Elvise Berchio Daniele Cassani Filippo Gazzola Hardy Rellich inequalities with boundary remainder terms and applications Received:
More informationOn Nonuniformly Subelliptic Equations of Q sub-laplacian Type with Critical Growth in the Heisenberg Group
Advanced Nonlinear Studies 12 212), 659 681 On Nonuniformly Subelliptic Equations of sub-laplacian Type with Critical Growth in the Heisenberg Group Nguyen Lam, Guozhen Lu Department of Mathematics Wayne
More informationBlow-up solutions for critical Trudinger-Moser equations in R 2
Blow-up solutions for critical Trudinger-Moser equations in R 2 Bernhard Ruf Università degli Studi di Milano The classical Sobolev embeddings We have the following well-known Sobolev inequalities: let
More informationSHARP BOUNDARY TRACE INEQUALITIES. 1. Introduction
SHARP BOUNDARY TRACE INEQUALITIES GILES AUCHMUTY Abstract. This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region R N. The inequalities bound (semi-)norms
More informationHARDY INEQUALITIES WITH BOUNDARY TERMS. x 2 dx u 2 dx. (1.2) u 2 = u 2 dx.
Electronic Journal of Differential Equations, Vol. 003(003), No. 3, pp. 1 8. ISSN: 107-6691. UL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) HADY INEQUALITIES
More informationREGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM. Contents
REGULARITY AND COMPARISON PRINCIPLES FOR p-laplace EQUATIONS WITH VANISHING SOURCE TERM BERARDINO SCIUNZI Abstract. We prove some sharp estimates on the summability properties of the second derivatives
More informationA nodal solution of the scalar field equation at the second minimax level
Bull. London Math. Soc. 46 (2014) 1218 1225 C 2014 London Mathematical Society doi:10.1112/blms/bdu075 A nodal solution of the scalar field equation at the second minimax level Kanishka Perera and Cyril
More informationNon-homogeneous semilinear elliptic equations involving critical Sobolev exponent
Non-homogeneous semilinear elliptic equations involving critical Sobolev exponent Yūki Naito a and Tokushi Sato b a Department of Mathematics, Ehime University, Matsuyama 790-8577, Japan b Mathematical
More informationPositive and Nodal Solutions For a Nonlinear Schrödinger Equation with Indefinite Potential
Advanced Nonlinear Studies 8 (008), 353 373 Positive and Nodal Solutions For a Nonlinear Schrödinger Equation with Indefinite Potential Marcelo F. Furtado, Liliane A. Maia Universidade de Brasília - Departamento
More informationarxiv:math/ v1 [math.dg] 23 Dec 2001
arxiv:math/0112260v1 [math.dg] 23 Dec 2001 VOLUME PRESERVING EMBEDDINGS OF OPEN SUBSETS OF Ê n INTO MANIFOLDS FELIX SCHLENK Abstract. We consider a connected smooth n-dimensional manifold M endowed with
More informationNONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian UNDER NONHOMOGENEOUS NEUMANN BOUNDARY CONDITION
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 210, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONLINEAR FREDHOLM ALTERNATIVE FOR THE p-laplacian
More informationNon-radial solutions to a bi-harmonic equation with negative exponent
Non-radial solutions to a bi-harmonic equation with negative exponent Ali Hyder Department of Mathematics, University of British Columbia, Vancouver BC V6TZ2, Canada ali.hyder@math.ubc.ca Juncheng Wei
More informationEXISTENCE AND REGULARITY RESULTS FOR SOME NONLINEAR PARABOLIC EQUATIONS
EXISTECE AD REGULARITY RESULTS FOR SOME OLIEAR PARABOLIC EUATIOS Lucio BOCCARDO 1 Andrea DALL AGLIO 2 Thierry GALLOUËT3 Luigi ORSIA 1 Abstract We prove summability results for the solutions of nonlinear
More informationON THE SCHRÖDINGER EQUATION INVOLVING A CRITICAL SOBOLEV EXPONENT AND MAGNETIC FIELD. Jan Chabrowski Andrzej Szulkin. 1.
Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center Volume 25, 2005, 3 21 ON THE SCHRÖDINGER EQUATION INVOLVING A CRITICAL SOBOLEV EXPONENT AND MAGNETIC FIELD Jan Chabrowski
More informationGRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS
LE MATEMATICHE Vol. LI (1996) Fasc. II, pp. 335347 GRAND SOBOLEV SPACES AND THEIR APPLICATIONS TO VARIATIONAL PROBLEMS CARLO SBORDONE Dedicated to Professor Francesco Guglielmino on his 7th birthday W
More informationA GLOBAL COMPACTNESS RESULT FOR SINGULAR ELLIPTIC PROBLEMS INVOLVING CRITICAL SOBOLEV EXPONENT
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 6, Pages 1857 1866 S 000-9939(0)0679-1 Article electronically published on October 1, 00 A GLOBAL COMPACTNESS RESULT FOR SINGULAR ELLIPTIC
More informationA one-dimensional nonlinear degenerate elliptic equation
USA-Chile Workshop on Nonlinear Analysis, Electron. J. Diff. Eqns., Conf. 06, 001, pp. 89 99. http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu login: ftp)
More informationHOMOCLINIC SOLUTIONS FOR SECOND-ORDER NON-AUTONOMOUS HAMILTONIAN SYSTEMS WITHOUT GLOBAL AMBROSETTI-RABINOWITZ CONDITIONS
Electronic Journal of Differential Equations, Vol. 010010, No. 9, pp. 1 10. ISSN: 107-6691. UL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu HOMOCLINIC SOLUTIONS FO
More informationPeriodic solutions of weakly coupled superlinear systems
Periodic solutions of weakly coupled superlinear systems Alessandro Fonda and Andrea Sfecci Abstract By the use of a higher dimensional version of the Poincaré Birkhoff theorem, we are able to generalize
More informationOn the structure of Hardy Sobolev Maz ya inequalities
J. Eur. Math. Soc., 65 85 c European Mathematical Society 2009 Stathis Filippas Achilles Tertikas Jesper Tidblom On the structure of Hardy Sobolev Maz ya inequalities Received October, 2007 and in revised
More informationON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 167 176 ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES Piotr Haj lasz and Jani Onninen Warsaw University, Institute of Mathematics
More informationOn the Schrödinger Equation in R N under the Effect of a General Nonlinear Term
On the Schrödinger Equation in under the Effect of a General Nonlinear Term A. AZZOLLINI & A. POMPONIO ABSTRACT. In this paper we prove the existence of a positive solution to the equation u + V(x)u =
More informationarxiv: v1 [math.ap] 28 Mar 2014
GROUNDSTATES OF NONLINEAR CHOQUARD EQUATIONS: HARDY-LITTLEWOOD-SOBOLEV CRITICAL EXPONENT VITALY MOROZ AND JEAN VAN SCHAFTINGEN arxiv:1403.7414v1 [math.ap] 28 Mar 2014 Abstract. We consider nonlinear Choquard
More informationA semilinear Schrödinger equation with magnetic field
A semilinear Schrödinger equation with magnetic field Andrzej Szulkin Department of Mathematics, Stockholm University 106 91 Stockholm, Sweden 1 Introduction In this note we describe some recent results
More informationThe Dirichlet s P rinciple. In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation:
Oct. 1 The Dirichlet s P rinciple In this lecture we discuss an alternative formulation of the Dirichlet problem for the Laplace equation: 1. Dirichlet s Principle. u = in, u = g on. ( 1 ) If we multiply
More informationRegularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains
Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains Ilaria FRAGALÀ Filippo GAZZOLA Dipartimento di Matematica del Politecnico - Piazza L. da Vinci - 20133
More informationMass transportation methods in functional inequalities and a new family of sharp constrained Sobolev inequalities
Mass transportation methods in functional inequalities and a new family of sharp constrained Sobolev inequalities Robin Neumayer Abstract In recent decades, developments in the theory of mass transportation
More informationRegularity and compactness for the DiPerna Lions flow
Regularity and compactness for the DiPerna Lions flow Gianluca Crippa 1 and Camillo De Lellis 2 1 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy. g.crippa@sns.it 2 Institut für Mathematik,
More informationEXISTENCE THEORY FOR NONLINEAR STURM-LIOUVILLE PROBLEMS WITH NON-LOCAL BOUNDARY CONDITIONS. 1. Introduction
EXISTENCE THEORY FOR NONLINEAR STURM-LIOUVILLE PROBLEMS WITH NON-LOCAL BOUNDARY CONDITIONS DANIEL MARONCELLI AND JESÚS RODRÍGUEZ Abstract. In this work we provide conditions for the existence of solutions
More informationExistence and multiplicity of solutions to equations of N-Laplacian type with critical exponential growth in R N
Available online at www.sciencedirect.com Journal of Functional Analysis 262 2012 1132 1165 www.elsevier.com/locate/jfa Existence and multiplicity of solutions to equations of -Laplacian type with critical
More informationBorderline Variational Problems Involving Fractional Laplacians and Critical Singularities
Advanced Nonlinear Studies 15 (015), xxx xxx Borderline Variational Problems Involving Fractional Laplacians and Critical Singularities Nassif Ghoussoub, Shaya Shakerian Department of Mathematics University
More informationUniqueness of ground states for quasilinear elliptic equations in the exponential case
Uniqueness of ground states for quasilinear elliptic equations in the exponential case Patrizia Pucci & James Serrin We consider ground states of the quasilinear equation (.) div(a( Du )Du) + f(u) = 0
More informationHARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 37, 2012, 571 577 HARNACK S INEQUALITY FOR GENERAL SOLUTIONS WITH NONSTANDARD GROWTH Olli Toivanen University of Eastern Finland, Department of
More informationSymmetry and monotonicity of least energy solutions
Symmetry and monotonicity of least energy solutions Jaeyoung BYEO, Louis JEAJEA and Mihai MARIŞ Abstract We give a simple proof of the fact that for a large class of quasilinear elliptic equations and
More informationSOBOLEV S INEQUALITY FOR RIESZ POTENTIALS OF FUNCTIONS IN NON-DOUBLING MORREY SPACES
Mizuta, Y., Shimomura, T. and Sobukawa, T. Osaka J. Math. 46 (2009), 255 27 SOOLEV S INEQUALITY FOR RIESZ POTENTIALS OF FUNCTIONS IN NON-DOULING MORREY SPACES YOSHIHIRO MIZUTA, TETSU SHIMOMURA and TAKUYA
More informationRadial balanced metrics on the unit disk
Radial balanced metrics on the unit disk Antonio Greco and Andrea Loi Dipartimento di Matematica e Informatica Università di Cagliari Via Ospedale 7, 0914 Cagliari Italy e-mail : greco@unica.it, loi@unica.it
More informationOn Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma. Ben Schweizer 1
On Friedrichs inequality, Helmholtz decomposition, vector potentials, and the div-curl lemma Ben Schweizer 1 January 16, 2017 Abstract: We study connections between four different types of results that
More informationPolishness of Weak Topologies Generated by Gap and Excess Functionals
Journal of Convex Analysis Volume 3 (996), No. 2, 283 294 Polishness of Weak Topologies Generated by Gap and Excess Functionals Ľubica Holá Mathematical Institute, Slovak Academy of Sciences, Štefánikovà
More informationNECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES
NECESSARY CONDITIONS FOR WEIGHTED POINTWISE HARDY INEQUALITIES JUHA LEHRBÄCK Abstract. We establish necessary conditions for domains Ω R n which admit the pointwise (p, β)-hardy inequality u(x) Cd Ω(x)
More informationAn introduction to Mathematical Theory of Control
An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018
More informationStationary Kirchhoff equations with powers by Emmanuel Hebey (Université de Cergy-Pontoise)
Stationary Kirchhoff equations with powers by Emmanuel Hebey (Université de Cergy-Pontoise) Lectures at the Riemann center at Varese, at the SNS Pise, at Paris 13 and at the university of Nice. June 2017
More informationDETERMINATION OF THE BLOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION
DETERMINATION OF THE LOW-UP RATE FOR THE SEMILINEAR WAVE EQUATION y FRANK MERLE and HATEM ZAAG Abstract. In this paper, we find the optimal blow-up rate for the semilinear wave equation with a power nonlinearity.
More informationElliptic equations with one-sided critical growth
Electronic Journal of Differential Equations, Vol. 00(00), No. 89, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Elliptic equations
More informationCOMPARISON PRINCIPLES FOR CONSTRAINED SUBHARMONICS PH.D. COURSE - SPRING 2019 UNIVERSITÀ DI MILANO
COMPARISON PRINCIPLES FOR CONSTRAINED SUBHARMONICS PH.D. COURSE - SPRING 2019 UNIVERSITÀ DI MILANO KEVIN R. PAYNE 1. Introduction Constant coefficient differential inequalities and inclusions, constraint
More informationGROUND STATE SOLUTIONS FOR CHOQUARD TYPE EQUATIONS WITH A SINGULAR POTENTIAL. 1. Introduction In this article, we study the Choquard type equation
Electronic Journal of Differential Equations, Vol. 2017 (2017), o. 52, pp. 1 14. ISS: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu GROUD STATE SOLUTIOS FOR CHOQUARD TYPE EQUATIOS
More informationComplex geometrical optics solutions for Lipschitz conductivities
Rev. Mat. Iberoamericana 19 (2003), 57 72 Complex geometrical optics solutions for Lipschitz conductivities Lassi Päivärinta, Alexander Panchenko and Gunther Uhlmann Abstract We prove the existence of
More informationSCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE
SCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE BETSY STOVALL Abstract. This result sharpens the bilinear to linear deduction of Lee and Vargas for extension estimates on the hyperbolic paraboloid
More informationOn semilinear elliptic equations with nonlocal nonlinearity
On semilinear elliptic equations with nonlocal nonlinearity Shinji Kawano Department of Mathematics Hokkaido University Sapporo 060-0810, Japan Abstract We consider the problem 8 < A A + A p ka A 2 dx
More informationHOMEOMORPHISMS OF BOUNDED VARIATION
HOMEOMORPHISMS OF BOUNDED VARIATION STANISLAV HENCL, PEKKA KOSKELA AND JANI ONNINEN Abstract. We show that the inverse of a planar homeomorphism of bounded variation is also of bounded variation. In higher
More informationInequalities of Babuška-Aziz and Friedrichs-Velte for differential forms
Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms Martin Costabel Abstract. For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babuška Aziz
More informationb i (µ, x, s) ei ϕ(x) µ s (dx) ds (2) i=1
NONLINEAR EVOLTION EQATIONS FOR MEASRES ON INFINITE DIMENSIONAL SPACES V.I. Bogachev 1, G. Da Prato 2, M. Röckner 3, S.V. Shaposhnikov 1 The goal of this work is to prove the existence of a solution to
More informationOn the relation between scaling properties of functionals and existence of constrained minimizers
On the relation between scaling properties of functionals and existence of constrained minimizers Jacopo Bellazzini Dipartimento di Matematica Applicata U. Dini Università di Pisa January 11, 2011 J. Bellazzini
More informationHomogenization and error estimates of free boundary velocities in periodic media
Homogenization and error estimates of free boundary velocities in periodic media Inwon C. Kim October 7, 2011 Abstract In this note I describe a recent result ([14]-[15]) on homogenization and error estimates
More informationNonhomogeneous linear differential polynomials generated by solutions of complex differential equations in the unit disc
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 20, Number 1, June 2016 Available online at http://acutm.math.ut.ee Nonhomogeneous linear differential polynomials generated by solutions
More informationChanging sign solutions for the CR-Yamabe equation
Changing sign solutions for the CR-Yamabe equation Ali Maalaoui (1) & Vittorio Martino (2) Abstract In this paper we prove that the CR-Yamabe equation on the Heisenberg group has infinitely many changing
More informationM. Ledoux Université de Toulouse, France
ON MANIFOLDS WITH NON-NEGATIVE RICCI CURVATURE AND SOBOLEV INEQUALITIES M. Ledoux Université de Toulouse, France Abstract. Let M be a complete n-dimensional Riemanian manifold with non-negative Ricci curvature
More informationOPTIMAL CONTROL AND STRANGE TERM FOR A STOKES PROBLEM IN PERFORATED DOMAINS
PORTUGALIAE MATHEMATICA Vol. 59 Fasc. 2 2002 Nova Série OPTIMAL CONTROL AND STRANGE TERM FOR A STOKES PROBLEM IN PERFORATED DOMAINS J. Saint Jean Paulin and H. Zoubairi Abstract: We study a problem of
More informationLiouville-type theorems and decay estimates for solutions to higher order elliptic equations
Liouville-type theorems decay estimates for solutions to higher order elliptic equations Guozhen Lu, Peiyong Wang Jiuyi Zhu Abstract. Liouville-type theorems are powerful tools in partial differential
More informationExistence of a ground state and blow-up problem for a nonlinear Schrödinger equation with critical growth
Existence of a ground state and blow-up problem for a nonlinear Schrödinger equation with critical growth Takafumi Akahori, Slim Ibrahim, Hiroaki Kikuchi and Hayato Nawa 1 Introduction In this paper, we
More informationElliptic stability for stationary Schrödinger equations by Emmanuel Hebey. Part III/VI A priori blow-up theories March 2015
Elliptic stability for stationary Schrödinger equations by Emmanuel Hebey Part III/VI A priori blow-up theories March 2015 Nonlinear analysis arising from geometry and physics Conference in honor of Professor
More informationLiouville Theorems for Integral Systems Related to Fractional Lane-Emden Systems in R N +
Liouville Theorems for Integral Systems Related to Fractional Lane-Emden Systems in R Senping Luo & Wenming Zou Department of Mathematical Sciences, Tsinghua University, Beijing 00084, China Abstract In
More informationGlobal Solutions for a Nonlinear Wave Equation with the p-laplacian Operator
Global Solutions for a Nonlinear Wave Equation with the p-laplacian Operator Hongjun Gao Institute of Applied Physics and Computational Mathematics 188 Beijing, China To Fu Ma Departamento de Matemática
More informationRELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS GENERATED BY MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL EQUATIONS
Kragujevac Journal of Mathematics Volume 38(1) (2014), Pages 147 161. RELATION BETWEEN SMALL FUNCTIONS WITH DIFFERENTIAL POLYNOMIALS GENERATED BY MEROMORPHIC SOLUTIONS OF HIGHER ORDER LINEAR DIFFERENTIAL
More informationEXISTENCE OF NONTRIVIAL SOLUTIONS FOR A QUASILINEAR SCHRÖDINGER EQUATIONS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 05, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF NONTRIVIAL
More informationA REMARK ON LEAST ENERGY SOLUTIONS IN R N. 0. Introduction In this note we study the following nonlinear scalar field equations in R N :
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 8, Pages 399 408 S 000-9939(0)0681-1 Article electronically published on November 13, 00 A REMARK ON LEAST ENERGY SOLUTIONS IN R N LOUIS
More informationON QUALITATIVE PROPERTIES OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH STRONG DEPENDENCE ON THE GRADIENT
GLASNIK MATEMATIČKI Vol. 49(69)(2014), 369 375 ON QUALITATIVE PROPERTIES OF SOLUTIONS OF QUASILINEAR ELLIPTIC EQUATIONS WITH STRONG DEPENDENCE ON THE GRADIENT Jadranka Kraljević University of Zagreb, Croatia
More informationPOSITIVE GROUND STATE SOLUTIONS FOR SOME NON-AUTONOMOUS KIRCHHOFF TYPE PROBLEMS
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 47, Number 1, 2017 POSITIVE GROUND STATE SOLUTIONS FOR SOME NON-AUTONOMOUS KIRCHHOFF TYPE PROBLEMS QILIN XIE AND SHIWANG MA ABSTRACT. In this paper, we study
More informationOn uniqueness of weak solutions to transport equation with non-smooth velocity field
On uniqueness of weak solutions to transport equation with non-smooth velocity field Paolo Bonicatto Abstract Given a bounded, autonomous vector field b: R d R d, we study the uniqueness of bounded solutions
More informationCompactness results and applications to some zero mass elliptic problems
Compactness results and applications to some zero mass elliptic problems A. Azzollini & A. Pomponio 1 Introduction and statement of the main results In this paper we study the elliptic problem, v = f (v)
More informationNONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT
Electronic Journal of Differential Equations, Vol. 016 (016), No. 08, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONHOMOGENEOUS ELLIPTIC
More informationNONEXISTENCE OF MINIMIZER FOR THOMAS-FERMI-DIRAC-VON WEIZSÄCKER MODEL. x y
NONEXISTENCE OF MINIMIZER FOR THOMAS-FERMI-DIRAC-VON WEIZSÄCKER MODEL JIANFENG LU AND FELIX OTTO In this paper, we study the following energy functional () E(ϕ) := ϕ 2 + F(ϕ 2 ) dx + D(ϕ 2,ϕ 2 ), R 3 where
More informationNon-degeneracy of perturbed solutions of semilinear partial differential equations
Non-degeneracy of perturbed solutions of semilinear partial differential equations Robert Magnus, Olivier Moschetta Abstract The equation u + F(V (εx, u = 0 is considered in R n. For small ε > 0 it is
More informationThe De Giorgi-Nash-Moser Estimates
The De Giorgi-Nash-Moser Estimates We are going to discuss the the equation Lu D i (a ij (x)d j u) = 0 in B 4 R n. (1) The a ij, with i, j {1,..., n}, are functions on the ball B 4. Here and in the following
More informationBLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED
BLOWUP THEORY FOR THE CRITICAL NONLINEAR SCHRÖDINGER EQUATIONS REVISITED TAOUFIK HMIDI AND SAHBI KERAANI Abstract. In this note we prove a refined version of compactness lemma adapted to the blowup analysis
More informationNonexistence of solutions for quasilinear elliptic equations with p-growth in the gradient
Electronic Journal of Differential Equations, Vol. 2002(2002), No. 54, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Nonexistence
More informationMetric Spaces and Topology
Chapter 2 Metric Spaces and Topology From an engineering perspective, the most important way to construct a topology on a set is to define the topology in terms of a metric on the set. This approach underlies
More informationarxiv: v1 [math.ap] 21 Dec 2016
arxiv:1612.07051v1 [math.ap] 21 Dec 2016 On the extension to slip boundary conditions of a Bae and Choe regularity criterion for the Navier-Stokes equations. The half-space case. H. Beirão da Veiga, Department
More informationContinuity of Solutions of Linear, Degenerate Elliptic Equations
Continuity of Solutions of Linear, Degenerate Elliptic Equations Jani Onninen Xiao Zhong Abstract We consider the simplest form of a second order, linear, degenerate, divergence structure equation in the
More informationMildly degenerate Kirchhoff equations with weak dissipation: global existence and time decay
arxiv:93.273v [math.ap] 6 Mar 29 Mildly degenerate Kirchhoff equations with weak dissipation: global existence and time decay Marina Ghisi Università degli Studi di Pisa Dipartimento di Matematica Leonida
More informationINSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES. Note on the fast decay property of steady Navier-Stokes flows in the whole space
INSTITUTE OF MATHEMATICS THE CZECH ACADEMY OF SCIENCES Note on the fast decay property of stea Navier-Stokes flows in the whole space Tomoyuki Nakatsuka Preprint No. 15-017 PRAHA 017 Note on the fast
More informationEXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN EQUATION WITH DEGENERATE MOBILITY
Electronic Journal of Differential Equations, Vol. 216 216), No. 329, pp. 1 22. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu EXISTENCE OF SOLUTIONS TO THE CAHN-HILLIARD/ALLEN-CAHN
More informationTWO DIMENSIONAL MINIMAL GRAPHS OVER UNBOUNDED DOMAINS
TWO DMENSONAL MNMAL GRAPHS OVER UNBOUNDED DOMANS JOEL SPRUCK Abstract. n this paper we will study solution pairs (u, D) of the minimal surface equation defined over an unbounded domain D in R, with u =
More informationExistence of Positive Solutions to Semilinear Elliptic Systems Involving Concave and Convex Nonlinearities
Journal of Physical Science Application 5 (2015) 71-81 doi: 10.17265/2159-5348/2015.01.011 D DAVID PUBLISHING Existence of Positive Solutions to Semilinear Elliptic Systems Involving Concave Convex Nonlinearities
More information