Weighted Trudinger-Moser inequalities and associated Liouville type equations

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1 Weighted Trudinger-Moser inequalities and associated Liouville type equations Marta Calanchi, Eugenio Massa and ernhard Ruf Abstract We discuss some Trudinger Moser inequalities with weighted Sobolev norms. Suitable logarithmic weights in these norms allow an improvement in the maximal growth for integrability, when one restricts to radial functions. The main results concern the application of these inequalities to the existence of solutions for certain mean-field equations of Liouville-type. Sharp critical thresholds are found such that for parameters below these thresholds the corresponding functionals are coercive and hence solutions are obtained as global minima of these functionals. In the critical cases the functionals are no longer coercive and solutions may not exist. We also discuss a limiting case, in which the allowed growth is of double exponential type. Surprisingly, we are able to show that in this case a local minimum persists to exist for critical and also for slightly supercritical parameters. This allows to obtain the existence of a second mountain-pass solution, for almost all slightly supercritical parameters, using the Struwe monotonicity trick. This result is in contrast to the non-weighted case, where positive solutions do not exist in star-shaped domains in the critical and supercritical case. Keywords: Trudinger-Moser inequality, Liouville type equations. MSC-class: 35J25, 3533, 46E35 Introduction The well known Trudinger-Moser TM inequality provides continuous embeddings into exponential Orlicz spaces in the borderline cases of the standard Sobolev embeddings, when the embeddings into Lebesgue L p spaces hold for every p < but not for p =. Let us recall Moser s result for the case N = 2: Theorem A Moser Mos7]. Let N = 2; then sup Ω u 2 dx Ω { C Ω if α 4π, e αu2 = + if α > 4π.. A useful variant of the TM inequality is the following logarithmic TM inequality: This is an Accepted Manuscript of an article accepted in Proceedings of the AMS.

2 2 Weighted Trudinger-Moser inequalities and associated Liouville type equations Theorem Moser Mos7]. Let Ω R 2 be a bounded domain. Then there exists a constant C > 0 such that log e u dx u 2 dx + C, Ω 6π Ω u H0 Ω..2 The value is optimal see e.g. CLMP92]. 6π. Weighted Trudinger-Moser inequalities Recent results concern the influence of weights on such type of inequalities. In CT05], AS07], dfdods6], for instance, the authors consider the effect of power weights in the integral term on the maximal growth. On the other hand in Cal4], CR5c], CR5a], CR5b], CRS7] the interest is devoted to the impact of weights in the Sobolev norm. We concentrate our attention on this second type of results. More precisely, let w L Ω be a non-negative function, and consider the weighted Sobolev space { } H0 Ω, w = cl u C0 Ω ; u 2 wxdx <..3 It turns out that for weighted Sobolev spaces of the form.3 logarithmic weights have a particular significance. However, as was observed in CR5a, Proposition 8], one needs to restrict the attention to radial functions in order to obtain an actual improvement of the embedding inequalities. One is therefore lead to consider problems of the following type: let R 2 denote the unit ball in R 2, and consider the weighted Sobolev space of radial functions } H β = H0,rad, w β := cl {u C0,rad ; u 2 β := u 2 w β xdx <, where w β x = Ω log e β, β 0..4 x The following results were obtained in CR5a] and will be fundamental in this paper. Theorem C CR5a]. Let β 0,. Then a e u γ dx < +, for all u H β γ γ β := 2 β and b sup e α u γ β dx < + u H β, u β if and only if α α β := 2 β] β critical growth..5 Remark.. This result extends the Trudinger-Moser inequality.; indeed, for β = 0 we recover the classical TM inequality where γ 0 = 2 and α 0 = 4π. Going to the limiting case β = in Theorem C, one sees that the exponent γ of u in the integral can take any value, that is, we are again in a borderline case. ut again, the embedding does not go into L, in fact, we find a critical growth of double exponential type, as described in the following

3 M. Calanchi, E. Massa and. Ruf 3 Theorem D CR5a]. Let β = i.e. w x = log e x. Then, a e eu2 dx < +, u H = H0,rad, w and b sup e a eu2 dx < + a 2. u H, u Finally, in the case β >, one has the following result Theorem E CR5a]. Let β >. Then we have the following embedding: H β = H 0,rad, w β L. Logarithmic inequalities similar to.2 can be obtained also in this setting. We have the following results, which were partially obtained in a previous paper CR5c], but we recall the proofs for completeness in the Appendix. Proposition.2. a For β 0,, there exists a constant Cβ such that log e u θ β dx 2λ u 2 β + Cβ β u H β,.6 where λ β := π ββ 2 β 2 β 2 β and θ β = 2 2 β..7 b For β =, there exists a constant C M such that log log e e u dx u 2 + log The values 2λ β and 8 + log C M e u 2 in.6 and.8, respectively, are optimal. u H..8 Remark.3. Notice that in the case β = 0 inequality.6 gives the classical logarithmic TM inequality.2, actually, λ 0 = 8π and θ 0 =. Remark.4. The optimality of can be found in CLMP92], while the optimality of and 2λ 0 in.6 and.8, respectively, is new, and it will be a consequence of Theorem.5 in this paper..2 Mean field equations of Liouville type The logarithmic version of the TM inequality is crucial in the study of mean field equations of Liouville type see Lio53] of the form u = λ eu Ω eu in Ω R 2, u = 0 on Ω. 2λ β.9

4 4 Weighted Trudinger-Moser inequalities and associated Liouville type equations Equation.9 was derived by Caglioti, Lions, Marchioro and Pulvirenti in their pioneering works CLMP92, CLMP95] from the mean field limit of point vortices of the Euler flow, see also Chanillo-Kiessling CK94] and Kiessling Kie93]. Equation.9 occurs also in the study of multiple condensate solutions for the Chern-Simons-Higgs theory, see Tarantello Tar96], Tar04]. In particular, it has been shown see also Li99], CL0] that equation.9 has a solution if λ < 8π,.0 while a Pohozhaev identity shows that no solution exists for λ 8π in starshaped domains see e.g. CLMP92]. In view of this, we call the case λ < 8π subcritical, the case λ = 8π critical, and the case λ > 8π supercritical. The existence of a solution in the subcritical case can be proved by using variational methods; in fact, solutions of.9 are critical points of the functional J : H0 Ω R, Ju = 2 u 2 λ log e u dx.. Indeed, for λ < 8π, as a consequence of the logarithmic TM inequality, the functional J is coercive, hence bounded from below, and then admits an absolute minimum. For λ = 8π the functional J is still bounded below, but no longer coercive, and the infimum is not attained..3 Main results In this article we concentrate our attention on some functionals similar to. and related nonlocal equations that generalize.9, under the impact of the above mentioned weighted logarithmic inequalities. We define the following functionals: i for β 0,, let J λ : H β R, J λ u := 2 u 2 β λ log Ω e uθ dx where θ = θ β from Proposition.2, and writing u θ := u θ u and ffl := ii for β =, let I λ : H R, I λ u := 2 u 2 λ log log e eu,.2 ; dx..3 Our purpose in this paper is to study the geometry of these functionals in dependence of the positive parameter λ and, as a consequence, to obtain existence results for some related nonlocal equations. In particular we will prove the following results. Theorem.5. i For β 0,, the functional J λ is coercive for λ 0, λ β and it is bounded from below if and only if λ λ β see expression.7. ii For β =, the functional I λ is coercive for λ 0, π and it is bounded from below if and only if λ π. oth results have a natural application to some weighted mean field equations of Liouville type. As for equation.9 we distinguish the subcritical and critical case.

5 M. Calanchi, E. Massa and. Ruf 5 Theorem.6 Subcritical case. i Let β 0,, and θ = θ β = 2 2 β. Then the equation div w β x u = λ θ u θ e uθ in, euθ u = 0 on,.4 has a positive weak radial solution, which is a global minimizer for J λ, for every value λ 0, λ β. ii The equation div w x u e u = λ log ffl e eu in, eeu eeu u = 0 on, has a positive weak radial solution, which is a global minimizer for I λ, for every λ 0, π..5 In contrast to the situation for equation.9, and somewhat surprisingly, for problem.5 with the double exponential nonlinearity we can also prove an existence result for the critical and slightly supercritical case: Theorem.7 Critical and supercritical case. There exists ε 0 > 0 such that equation.5 has a positive weak radial solution, which is a local minimizer for I λ, also for λ π, π + ε 0. When λ = π the minimum is global. Remark.8. The nonlinearities in the problems above are always nonnegative, and so only trivial or positive solutions may exist. In fact, the trivial solution exists for problem.4 if β 0,, while for β = 0 problem.9 and for β = problem.5 u = 0 is not a solution. We observe that in the supercritical situation, that is for λ π, π + ε 0, the functional I λ has a mountainpass-structure, since we have a local minimum, and directions along which the functional tends to. A direct application of the Mountain-pass Theorem by Ambrosetti- Rabinowitz AR73] seems difficult due to loss of compactness. However, we can apply the so-called monotonicity trick by Struwe Str88] see also ST98, Jea99] to obtain Theorem.9. Let ε 0 as in Theorem.7. Then for a.e. λ π, π + ε 0 equation.5 has a second positive radial solution which is of mountain-pass type. 2 Proofs We first prove the result concerning the geometry of the functionals J λ and I λ. Proof of Theorem.5. Coercivity for λ < λ β resp. λ < π is an immediate consequence of.6 and.8, actually with λ 0 J λ u 2 λ 2λ β u 2 β λcβ

6 6 Weighted Trudinger-Moser inequalities and associated Liouville type equations and I λ u 2 λ u 2 λ log 8 + log C M e u 2 2 λ u 2 λ log 8 + log C M. The above estimates also show that the functionals J λ and I λ are bounded from below when λ λ β resp. λ π. Sharpness is much more delicate. When λ exceeds those critical values, the functionals are not bounded from below: we will produce a sequence along which they tend to. Case β 0,. We evaluate the functional along a generalized Moser sequence see CR5a]: let u k x = ψ kt with x = e t, where β + t β for t k, + k ψ k t = β + k β for t > k. 2. With this definition one has u k β =. We set α k = C β + k β / β, where C will be fixed later, and evaluate the functional.2 along the sequence { } α k u k : using the new variable t the functional reads as J λ α k u k = α2 k 2 λ log α ] k 2 exp ψ k t θ β 2t dt. β 0 0 We estimate α k exp ψ k t β θ β 2t dt = k exp 0 exp C + k β β ψk t θ β 2t dt C + k β β + θ β 2t dt = C 2 β/ β ] 2 exp + k β 2/2 β 2k = C ] ] / β 2 exp 2/2 β + k β 2k. Therefore ] ] ] J λ α k ψ k C 2 + k β β / β π β λ C 2/2 β + k β 2k. We now set C 2/2 β = 2 2 β + 2δ, for some δ > 0. β

7 M. Calanchi, E. Massa and. Ruf 7 Since 2 + δ + k β ] / β 2k when k, we estimate, for k large, Then Jα k ψ k ] C 2/2 β + k β β 2k ] + k β β = 2 2 β β 2 + δ ] + k β ] 2 β + δ ] + k β ] β. 2 2 β ] 2 βπ 2 ]] + 2δ β λ β β + δ. 2.2 Let λ = + ελ β = + ε 2 2 β β ] 2 β π β 2 /2, for some ε > 0. Then 2.2 can be rewritten as where J λ α k ψ k ] + k β β 2 2 β 2 βπ { } β..., β { { } δ β 2 β... = + + ε + δ β }. 2 β 2 Since this term tends to ε as δ 0, the expression in braces is negative for δ > 0 small enough, and then J along this sequence. Case β =. Again we prove that the value λ = π is sharp by considering a generalized Moser sequence: let u k x = ψ kt with x = e t, where now we use the sequence log + t for t k, log + k ψ k t = 2.3 log + k for t > k. β Then u k = and evaluating I λ along the sequence { } α k u k, with αk = C log + k, we obtain I λ α k u k = α2 k λ log log 2 exp e α kψ k / ] 2t dt. 2 We estimate and then 0 exp e α kψ k / ] 2t dt = k ] exp e Clog+k 2t k 0 exp e C log+k log+k/ ] 2t dt = 2 exp + k C 2k ] dt, I λ α k u k C 2 π log + k λ log + k C 2k ]. For λ = π + ε we choose C = + 2δε and for k large we can estimate log + k +2δ 2k log + k +δ,

8 8 Weighted Trudinger-Moser inequalities and associated Liouville type equations and then I λ αu k + 2δ 2 π log + k π + ε + δ log + k ; since for δ > 0 small + 2δ 2 π < π + ε + δ, we have proved that, if λ > π, there exists a sequence along which I λ. In order to prove Theorem.6 we need a compactness result. The following Lemma due to de Figueiredo-Miyagaki-Ruf Lemma 2. in dfmr95] will be needed: Lemma F dfmr95]. Let u n be a sequence of functions in L Ω converging to u in L Ω. Assume that F : R R is measurable, and that F u n x and F ux are also L functions. If F u n x u n x dx C then F u n x converges to F ux in L. The compactness result is in the following Lemma: Ω Lemma 2.. Let β 0, and θ 0, γ β or β and θ > 0. Let u n be a bounded sequence in H β. Then there exists u H β such that up to a subsequence and, if β, log log log e uθ ndx log e eun dx log log e uθ dx as n + e eu dx as n +. Proof. Let u n β C. Then there exists u H β, such that up to a subsequence u n u in H β, u n u in L, u n u a.e., as n +. Observe that the nonlinearity e uθ is subcritical with respect to the maximal growth γ β given by Theorem C, and that there exists a constant C depending on C, θ and β such that te tθ C e α β t γβ C, t R. 2.4 From Theorem C and this estimate we have that e un θ, e u θ L and un e un θ dx C2. We now apply Lemma F using F t = e tθ. For the case β one proceeds in the same way using the inequalities te tθ, te et C e 2e t 2 C, t R, Theorem D or E, and applying Lemma F using F t = e tθ and F t = e et.

9 M. Calanchi, E. Massa and. Ruf 9 We are now able to prove our first existence result. Proof of Theorem.6. Since λ < λ β resp λ < π, the functional J λ resp. I λ is bounded from below by Theorem.5, and one can take a minimizing sequence u n, i.e. lim J λu n = m = inf J λ u, n +, i H β which is trivially bounded in H β by coercivity. Therefore there exists u H β, such that up to a subsequence u n u in H β, u n u in L, u n u a.e., as n + y Lemma 2. and the weak lower semicontinuity of the norm m J λ u lim inf n + J λu n = m. Then u is a global minimizer and therefore a solution of Problem.4. The case β = is analogous. When β = 0 or β = the solution obtained is positive by Remark.8. For β 0, we still have to show that the obtained solution is not trivial. This is the case since the origin is not a minimizer. Indeed, let v H β, v 0, 0 v, t 0, : then e tvθ + tv θ and e tvθ dx + tv θ dx. Since ffl tvθ dx we can use the estimate log + τ 2τ for τ 0, to conclude log e tvθ dx 2 tv θ dx. With this we get J λ tv = t2 2 v 2 β λ log e tvθ dx t2 2 v 2 β λ 2 tθ v θ dx ; since θ, 2, the above expression is negative for t small and then m < 0 = J λ 0. In the next proof we consider problem.5 when λ π. Proof of Theorem.7. eyond the threshold λ = π. The functional I π is still bounded from below by Theorem.5. We need to prove that minimizing sequences are still bounded, despite that in this case coercivity does not hold. However, the particular form of the logarithmic TM inequality will help in this direction. Let u n be a minimizing sequence, that is I π u n m = inf u H I π u.

10 0 Weighted Trudinger-Moser inequalities and associated Liouville type equations We observe first that the infimum cannot be positive, since m I π 0 = 0. On the other hand, from inequality.8 we have I π u n = 2 u n 2 π log log e eun dx π log 8 + log C M. If u n we would have 0 m = lim n + I πu n lim inf n + π log e un log C ] M = π log 8 > 0, e un 2 a contradiction. Then u n is bounded and we are done as in the proof of Theorem.6. Now we prove that a minimum now only local exists also for λ = π + ε > π, ε small. Let R > 0 be such that log C M e u 2 < 8 for u R, 2.5 then for u = R and every ε > 0, one has π + ε log 8 + log C M π + ε log /4 = π + ε log 4 π log 4; 2.6 e u 2 as a consequence, for u = R and ε > 0 small enough, I π+ε u 2 π + ε Let then R = {u H : u < R}. Since R 2 + π log 4 ε R2 + π log 4 > π log 2 > 0. inf I π+ε u I π+ε 0 = 0 < inf I π+ε u, 2.7 u R u R we conclude up to a compactness argument as above that the infimum is attained at a local minimum in R, which then yields a nontrivial positive solution. We observe that the limiting value for ε 0 can be estimated in term of C M : in the argument above but a finer estimate could be obtained R > log8 log C M and then ε 0 < log 2 2 log 2 8 log C M. In order to prove Theorem.9, we will use the following generalization whose proof is given in the appendix of a result by Jeanjean Jea99] which is based on the so-called monotonicity trick by Struwe, see Str88, ST98]. Theorem 2.2. Let X be a anach space equipped with the norm, and let µ : X X be a continuous map. We consider a family I λ λ J J R + is an open interval of C functionals on X of the form and suppose that I λ u = Au λu, λ J,

11 M. Calanchi, E. Massa and. Ruf i u 0 for all u µx ; ii I λ µu I λ u, for all u X and λ J; iii either Au + or u + as u +. Assume that there are two fixed points v 0 and v of µ such that for the family of maps it holds Γ = {γ C0, ], X, γ0 = v 0, γ = v }, 2.8 c λ := inf γ Γ max t 0,] I λγt > max{i λ v, I λ v 0 }, for all λ J. 2.9 Then for almost every λ J there exists a bounded PS-sequence for I λ at level c λ, i.e. there is {u n } n X with a {u n } n is bounded, b I λ u n c λ, c I λ u n 0 in the dual space X of X. Remark 2.3. When µ is the identity, this is exactly Jea99, Theorem.]. Proof of Theorem.9. We apply Theorem 2.2 to our functional I π+ε, ε 0, ε 0, with X = H, and ηu = u. In view of 2.7 and Theorem.5-ii, for all ε 0, ε 0 there exists v 0 such that I π+ε with ε ε, ε 0 satisfies condition 2.9 with v 0 = 0. Hence, we find for a.e. ε ε, ε 0 a sequence {v n } satisfying a, b and c. Due to the arbitrariness of ε this is true for a.e. ε 0, ε 0. Since {v n } is bounded, we find a subsequence converging weakly and a.e. to v H. Then the proof is easily concluded: by Lemma 2. we have ffl eevn dx ffl eev dx >, and similarly one also obtains evn e evn ϕdx ev e ev ϕ dx, for all ϕ H. Thus, from 0 I π+εv n v n v = v n v n vw dx π + ε evn e evn v n vdx log ffl eevn dx eevn dx, we conclude that v n v n vw dx 0. As a consequence v n v strongly and v is a weak solution of equation.5. It is different from the first solution since I π+ε v > 0, and positive by Remark.8. 3 Appendix In this appendix we give, for sake of completeness, the proof of the logarithmic TM inequalities, that were already proved in CR5c], and the proof of Theorem 2.2. Proof of Proposition.2. Let β 0,. y Young inequality, if δ, δ are two conjugate exponents, then for every s, t 0 one has st τsδ δ + tδ, τ > δτ δ

12 2 Weighted Trudinger-Moser inequalities and associated Liouville type equations We need δ, δ to be conjugate exponents and satisfy { θδ = γβ = 2 β θδ = 2. This implies θ = 2 2 β, δ = 2 β β, δ = 2 β. θ Then one has, by taking s = u θ β and t = u u β in 3., and selecting τ so that δτ δ = α β see equation.5 u θ u 2 β 2λ β u γβ + α β, 3.2 u β where λ β is given in.7. Let now Cβ = log sup u H β \{0} e α β γβ u u β dx, which is finite by Theorem C. Then we conclude log e u θ dx log e u 2 β 2λ β e α β u γβ u β dx u 2 β 2λ β + Cβ. Consider now the case β =. Taking a = u, b = u u and ε 2 = π in ab a2 4ε 2 + ε2 b we get u u 2 4π u 2 + π, 3.4 u so that e e u dx exp e 4π u 2 +π u u 2 dx 2 exp e 4π u 2 π u e u dx. Let C M = sup exp 2e u 2 u dx u H \{0} 0 which is finite in virtue of Theorem D.

13 M. Calanchi, E. Massa and. Ruf 3 2 Now taking a = e 4π u 2 π u, b = e u log log e eu dx log log = log and ε 2 = 2 in 3.3, one gets 2 exp 8 e u 2 u + 2e u dx 8 e u 2 + log e 2e log 8 e u 2 + log CM u 2 + log 2 u u dx 8 + log C M. e u 2 Proof of Theorem 2.2. It suffices to show that for every λ J c λ = inf γ Γ max t 0,] I λγt = inf γ Γ max t 0,] I λµ γt =: c µ λ 3.5 Indeed, observe first that given a path γ Γ, also the path µ γ Γ, since µ is continuous and v 0, v are fixed points of µ. Hence, c µ λ c λ because every path µ γ on the right also appears on the left. Condition ii gives the reversed inequality. This, together with i, implies that the map λ c λ is non-increasing and then c λ exists for almost every λ J. Then the proof can be completed as in Jea99], Theorem., proceeding by the following steps: Given λ J at which c λ exists and a sequence {λ n} J with λ n λ, there exist a constant K = Kλ > 0 and a sequence of paths γ n Γ such that a max t 0,] I λγ n t c λ + c λ + 2λ λ n. Moreover, if γ n t satisfies I λ γ n t c λ λ λ n, then γ n t K. In the proof of this step, it is important to observe that, in view of 3.5, the paths γ n can be chosen so that they have image in µx, which allows to use condition i. 2 For α > 0 let F α = {u X : u K + and I λ u c λ α}, where K is the constant of the previous step. Then inf{ I λ u : u F α} = 0, for every α > Then, choosing α = ε n 0, we obtain by 2 a u n F εn satisfies u n K + and I λ u n c λ ε n. such that I λ u n ε n, which

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15 M. Calanchi, E. Massa and. Ruf 5 Jea99] Kie93] Li99] L. Jeanjean, On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on R N, Proc. Roy. Soc. Edinburgh Sect. A , no. 4, M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions, Comm. Pure Appl. Math , no., Y. Y. Li, Harnack type inequality: the method of moving planes, Comm. Math. Phys , no. 2, Lio53] J. Liouville, Sur l equation aux derivées partielles, J. Math. Pure Appl , Mos7] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J /7, ST98] M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory, oll. Unione Mat. Ital. Sez. Artic. Ric. Mat , no., Str88] M. Struwe, Critical points of embeddings of H,n 0 into Orlicz spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 5 988, no. 5, Tar96] G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory, J. Math. Phys , no. 8, Tar04] G. Tarantello, Analytical aspects of Liouville-type equations with singular sources, Stationary partial differential equations. Vol. I, Handb. Differ. Equ., North- Holland, Amsterdam, 2004, pp Marta Calanchi and ernhard Ruf, Dipartimento di Matematica, Università di Milano, Via Saldini 50, 2033 Milano, Italia, marta.calanchi@unimi.it; bernhard.ruf@unimi.it Eugenio Massa, Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo - Campus de São Carlos, Caixa Postal 668, , São Carlos SP, razil, eug.massa@gmail.com Marta Calanchi and ernhard Ruf were partially supported by INdAM-GNAMPA Project 206. Eugenio Massa was supported by: grant #204/ , São Paulo Research Foundation FAPESP and grants #308354/204-, #303447/207-6, CNPq/razil.