On a semilinear elliptic equation with inverse square potentia

Brezis, Haïm

Dupaigne, Louis

Tesei, Alberto

Hal-Diderot

On a semilinear elliptic equation with inverse squarepotentialHa¨ım Brezis, Louis Dupaigne, Alberto TeseiTo cite this version:Ha¨ım Brezis, Louis Dupaigne, Alberto Tesei. On a semilinear elliptic equation with inversesquare potential. 2008. <hal-00288567>HAL Id: hal-00288567https://hal.archives-ouvertes.fr/hal-00288567Submitted on 17 Jun 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.On a semilinear elliptic equation with inverse-squarepotentialHaı¨m Brezis ∗†, Louis Dupaigne ∗ and Alberto Tesei ‡In this paper we study existence and nonexistence of solutions u ≥ 0 of the equation :−∆u = cr2u+ up (1)in a ball B(0, R) of RN , N ≥ 3. Here r = |x|, p > 1 and the coefficient c satisfies theinequality 0 < c ≤ c0, where c0 = (N −2)2/4 is the best constant in the Hardy inequality.In this study an important role is played by the rootsα = α± := (N − 2)/2±√c0 − cof the equationα2 − (N − 2)α+ c = 0. (2)Observe that α+ > α− > 0.Our main result asserts that nontrivial solutions of equation (1) exist if and only ifp < p+ wherep+ = 1 + 2/α−.Theorem 1. Let 0 ≤ c ≤ c0. For any p ∈ (1, p+), there exists a nontrivial solution to equation(1) with up and u/r2 belonging to L1(BR) and (1) holds in D′(BR).The proof of Theorem 1 is straightforward and elementary, except for the limitingvalue c = c0. The conclusion of Theorem 1 was known in many–but presumably notall– cases (see e.g. [12]). Concerning nonexistence we haveTheorem 2. Let 0 < c ≤ c0, p ≥ p+. Assume u ∈ Lploc(BR \ {0}), u ≥ 0 satisfies−∆u− cr2u ≥ upin D′(BR \ {0}). Then u ≡ 0.∗ Laboratoire Jacques-Louis Lions, Universite´ Pierre et Marie Curie, Boıˆte Courrier 187, 4 place Jussieu,75252 Paris Cedex 05, France.brezis@ann.jussieu.fr, dupaigne@ann.jussieu.fr† Department of Mathematics – Hill Center, Rutgers Univ., 110 Frelinghuysen Rd., Piscataway, NJ08854-8019 USA.brezis@math.rutgers.edu‡ Dipartimento di Matematica ”G. Castelnuovo”, Universita` di Roma ”La Sapienza”, P.le A. Moro 5,I-00185 Roma, Italia.tesei@mat.uniroma1.it1Theorem 2 is reminiscent of the nonexistence results of Brezis-Cabre´ [2] concerningthe so-called very weak solutions to the inequality−∆u ≥ upr2, u ≥ 0, u ∈ Lploc(BR \ {0}),for any p > 1. The nonexistence aspect in (1) when p ≥ p+ was first investigated byPohozaev-Tesei [11]. However the concept of solution used there was stronger; ourconcept is the weakest possible.We also observe that Theorem 2 seems (formally) to contradict the Implicit FunctionTheorem since there is no solution of−∆u = (c/|x|2)u+up+ t, even when t > 0 is small.As observed in [1], this is due to the lack of an appropriate functional space in which toapply the IFT.Proof of Theorem 1.Set p− = 1 + 2/α+ and observe that1 < p− <N + 2N − 2 < p+ for any 0 < c < c0 andlimc→0p− =NN − 2 , limc→c0 p− =N + 2N − 2 ,limc→0p+ = +∞, limc→c0p+ =N + 2N − 2 .We distinguish three cases :Case 1 : 0 ≤ c < c0 and p < N+2N−2 .Here the existence of a positive solution u ∈ H10 (BR) of (1) is a standard and straight-forward consequence of the Mountain Pass Theorem. In fact, one can find a radial solu-tion by working in the class of radial functions.Case 2 : 0 ≤ c < c0 and p− < p < p+.Here we have an explicit solution of (1) of the form u = A/rβ with β = 2/(p − 1),A > 0 given byAp−1 = −β2 + (N − 2)β − c > 0,because α−, α+ are the roots of (2) and the restriction α− < β < α+ is equivalent to thecondition p− < p < p+. Since β < N − 2, u satisfies (1) in the sense of D′(BR).2Case 3 : c = c0 and 1 < p < p+ = N+2N−2 .This case is a little more delicate : here we need the improved Hardy inequalitywhich asserts that ∫BR|∇u|2 ≥ c0∫BRu2r2+ cq‖u‖2Lq(BR),for any 1 ≤ q < 2NN−2 and u ∈ C∞0 (BR). See [5]. Let H be the Hilbert space obtained bycompleting C∞0 (BR)with respect to the scalar producta(u, v) =∫BR∇u · ∇v − c0∫BRuv/r2.Clearly H is contained in every Lq(BR) with 1 ≤ q < 2NN−2 with continuous injection.Moreover the injection is compact. This fact is due to H. Brezis and the proof is pre-sented in Lemmas 3.2, 3.3 of [7]. We may then use the Mountain Pass Theorem inH andthe (PS) condition is satisfied.Proof of Theorem 2.We will use the following lemma :Lemma 1. Let Σ ⊂⊂ Ω be a closed set of zero (newtonian) capacity and assume that u, f ∈L1loc(Ω \ Σ) are two nonnegative functions such that−∆u ≥ f in D′(Ω \ Σ).Then u, f ∈ L1loc(Ω) and−∆u ≥ f in D′(Ω).Furthermore given any smooth subdomain Ω′ ⊂⊂ Ω, if v ∈ L1(Ω′) is the solution of{−∆v = f in Ω′v = 0 on ∂Ω′,in the sense that∫v(−∆φ) =∫fφ ∀φ ∈ C2(Ω¯′) such that φ|∂Ω′ ≡ 0,thenu ≥ v a.e. in Ω′. (3)3Proof of Lemma 1This lemma can be seen as a fairly easy consequence of Theorem 7.7 in [9]. It is alsoclosely related to a result in [4]. We provide a proof for completeness. Let uk = min(u, k),k > 0, which by Kato’s Lemma (see [10]) satisfies−∆uk ≥ fk in D′(Ω \ Σ), (4)where fk := fχ{u<k}. Since −∆uk is a nonnegative distribution on Ω \ Σ, it extendsto a nonnegative measure on Ω \ Σ. Since uk is bounded, it follows from a Gagliardo-Nirenberg-type inequality that uk ∈ H1loc(Ω \ Σ). We show next that in fact uk ∈ H1loc(Ω).We first take a nonnegative function φ ∈ C∞0 (Ω) and a sequence φn ∈ C∞0 (Ω \ Σ) con-verging to φ inH1(Ω). This is always possible since capΩ(Σ) = 0 (take e.g. φn = φ(1−χn)where χn = 1 near Σ and ‖χn‖H1 → 0 = capΩ(Σ)). We then have, with Ck = ek,∫|∇uk|2φ2n ≤ Ck∫e−uk |∇uk|2φ2n = −Ck∫φ2n∇(e−uk) · ∇uk= Ck(2∫e−ukφn∇φn · ∇uk +∫e−uk∆ukφ2n)≤ 2C2k∫e−uk |∇φn|2 + 12∫e−uk |∇uk|2φ2n,so that ∫|∇(ukφn)|2 ≤ C ′k∫|∇φn|2.Passing to the limit as n→∞ in the above inequality implies that uk ∈ H1loc(Ω).We next show that−∆uk ≥ fk in D′(Ω). (5)Take φ and φn as above so that by (4),∫uk(−∆φn) ≥∫fkφn. (6)Now, as n→∞,∫uk(−∆φn) =∫∇uk∇φn →∫∇uk∇φ = −∫uk∆φ.Passing to the limit in (6) as n→∞, we thus obtain (5).In particular uk is superharmonic in Ω and given almost any x ∈ Ω and any ballB ⊂ Ω centered at x, we haveuk(x) ≥ 1|B|∫Buk(y) dy. (7)Now, since u ∈ L1loc(Ω\Σ) (and |Σ| = 0), uk → u a.e. in Ω as k →∞ and u is finite almosteverywhere. By Fatou’s Lemma we then conclude from (7) that for almost every ball B,4∫Bu <∞,which means that u ∈ L1loc(Ω). Using this information, we can now easily pass to thelimit in (5) and conclude that f ∈ L1loc(Ω) and that−∆u ≥ f in D′(Ω).It only remains to prove (3). We let ρn be a standard smooth mollifier and let un =u ∗ ρn, fn = f ∗ ρn so that for n large enough −∆un ≥ fn and un ≥ 0 in Ω′. By theMaximum Principleun ≥ vn in Ω′,where vn solves {−∆vn = fn in Ω′vn = 0 on ∂Ω′.As n → ∞, un → u in L1(Ω′), fn → f in L1(Ω′) and (by Lemma 4 in [3]) vn → v inL1(Ω′), which yields the desired conclusion.Proof of Theorem 2We argue by contradiction and assume that u 6≡ 0. By Lemma 1, u ∈ Lploc(BR),u/r2 ∈ L1loc(BR) and by the mean-value formula for superharmonic functions, givenR′ ∈ (0, R), there exists  > 0 such that u ≥  a.e. in BR′ . Let λ := q−1/2 > 0 and v0 bethe solution of {−∆v0 = λ in BR′v0 = 0 on ∂BR′ .Once more by Lemma 1, we have0 ≤ v0 ≤ u. (8)Next, for n ≥ 1, define inductively vn by{ −∆vn = c|x|2vn−1 + 12vpn−1 + λ in BR′vn = 0 on ∂BR′ .In order to have a well-defined solution vn (in the sense of Lemma 4 in [3]) it sufficesto prove that f := c|x|2vn−1 +12vpn−1 ∈ L1(BR′). When n = 0, this follows from (8) andLemma 1 which implies that c|x|2u +12up ∈ L1(BR′). Assume now that vn−1 ∈ L1(BR′) iswell-defined. Using the Maximum Principle, it is easy to see that0 ≤ v0 ≤ v1 ≤ · · · ≤ vn−1 ≤ u,whence f ∈ L1(BR′) and by the Maximum Principle again 0 ≤ vn−1 ≤ vn ≤ u.5By monotone convergence, letting v := limn→∞ vn, we have that−∆v =c|x|2v +12vp + λ in BR′v = 0 on ∂BR′ ,in the sense that given any φ ∈ C2(B¯R′) such that φ|∂BR′ ≡ 0,∫v(−∆φ) =∫c|x|2vφ+12∫vpφ+ λ∫φ.This contradicts Theorem 1 of [6].Remark 1. Theorems 1 and 2 extend to more general situations– for example, when up is re-placed by |x|−βuq. Assume 0 < c ≤ c0 and set q+ = 1 + 2−βα− , where α− is as above. Theconclusions of Theorems 1 and 2 remain valid with p+ replaced by q+.Remark 2. The argument presented in the proof of Theorem 2 may be used to provide a slightlysimpler proof of Theorem 1 in [2].Remark 3. Theorem 2 can be extended to problems of the type−∆u = cdist(x,Σ)2u+ up,where c > 0 is a small constant, Σ is a smooth compact manifold of codimension k ≥ 3 and pis larger than some critical exponent, which can be computed explicitly in terms of k and c. Theargument is the same as in the proof of Theorem 2 except that the result of [6] is replaced by aresult from [8].Acknowledgments. The three authors are partially supported by an ECGrant throughthe RTN Program ”Fronts-Singularities”, HPRN-CT-2002-00274. The first author (H.B.)is also a member of the Institut Universitaire de France.References[1] H. Brezis, Is there failure of the Inverse Function Theorem ?, Proceedings of the Work-shop held at the Morningside Center of Mathematics, Chinese Academy of Science,Beijing, June 1999.[2] H. Brezis and X. Cabre´, Some simple nonlinear PDE’s without solutions, Bull UMI, 1(1998), 223–262.[3] H. Brezis, X. Cabre´, Y. Martel and A. Ramiandrisoa, Blow-up for ut − ∆u = g(u)revisited, Adv. Diff. Eq., 1 (1996), 73–90.6[4] H. Brezis and P.L. Lions, A note on isolated singularities for linear elliptic equations,Mathematical analysis and applications, Adv. in Math. Suppl. Stud., 7a, AcademicPress, New York–London, 1981, 263–266.[5] H. Brezis and J.L. Vazquez, Blow-up solutions of some nonlinear elliptic problems, Rev.Mat. Univ. Complut. Madrid, 10 (1997), 443–469.[6] L. Dupaigne, A nonlinear elliptic PDE with the inverse-square potential, J. d’AnalyseMathe´matique, 86 (2002), 359–398.[7] J. Da´vila and L. Dupaigne, Comparison principles for PDE’s with a singular potential,Proc. Roy. Soc. Edinburgh, 133A (2003), 61–83.[8] L. Dupaigne and G. Nedev, Semilinear elliptic PDE’s with a singular potential, Adv.Differential Equations, 7 (2002), 973–1002.[9] L.L. Helms, Introduction to potential theory, New York, Wiley and Sons, 1969.[10] T. Kato, Schro¨dinger operators with singular potentials, Israel J. Math., 13 (1972), 135–148.[11] S.I. Pohozaev and A. Tesei, Nonexistence of local solutions to semilinear partial differ-ential inequalities, Nota Scientifica 01/28, Dip. Mat. Universita´ ”La Sapienza”, Roma(2001).[12] S. Terracini, On positive entire solutions to a class of equations with a singular coefficientand critical exponent, Adv. Differential Equations, 1 (1996), 241–264.7

On a semilinear elliptic equation with inverse square potential

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On a semilinear elliptic equation with inverse square potential

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