Dipartimento di Matematica

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1 Diartimento di Matematica P. ESPOSITO, M. MUSSO, A. PISTOIA ON THE EXISTENCE AND PROFILE OF NODAL SOLUTIONS FOR A TWO-DIMENSIONAL ELLIPTIC PROBLEM WITH LARGE EXPONENT IN NONLINEARITY Raorto interno N. 7, maggio 005 Politecnico di Torino Corso Duca degli Abruzzi, 4-09 Torino-Italia

2 On the existence and rofile of nodal solutions for a two-dimensional ellitic roblem with large exonent in nonlinearity Pieraolo ESPOSITO, Monica MUSSO, Angela PISTOIA, May 9, 005 Abstract We study the existence of nodal solutions to the boundary value roblem u = u u in a bounded smooth domain in IR, with homogeneous Dirichlet boundary condition, when is a large exonent. We rove that, for large enough, there exist at least two airs of solutions which change sign exactly once and whose nodal lines intersect the boundary of. Keywords: nodal solutions, nodal lines, large exonent. AMS subject classification:35j60, 35B33. 0 Introduction We consider the roblem { u = u u in, u = 0 on, 0. where is a bounded regular domain in IR. In order to state old and new results, we need to recall some well known definitions. The Green s function of the Dirichlet Lalacian can be decomosed into a singular art and a regular art, i.e. Gx, y = Hx, y + π log x y. The regular art H is a harmonic First author is suorted by M.U.R.S.T., roject Variational methods and nonlinear differential equations. Second author is suorted by Fondecyt Chile. Third authors is suorted by M.U.R.S.T., roject Metodi variazionali e toologici nello studio di fenomeni non lineari. Diartimento di Matematica, Università degli Studi Roma Tre, Largo S. Leonardo Murialdo, 0046 Roma, Italy. esosito@mat.uniroma3.it Diartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi, 4 09 Torino, Italy and Deartamento de Matematicas, Pontificia Universidad Catolica de Chile, Avenida Vicuna Mackenna 4860, Macul, Santiago, Chile. monica.musso@olito.it Diartimento di Metodi e Modelli Matematici, Università di Roma La Saienza, 0000 Roma, Italy, istoia@dmmm.uniroma.it.

3 function with the same boundary values as the singular art. The leading term of the regular art of the Green s function Hx, x is called the Robin function of at x. Problem 0. has always a ositive solution u, obtained by minimizing the function I u := u, u H 0, on the shere {u H 0 u + = }. Here u r denotes the L r norm of u. In [4] and [5], it is roved that, as goes to infinity, solution u develoes one eak, namely u aroaches zero excet one oint where it stays away from zero and bounded from above. More recisely, the authors roved that, u to a subsequence, the renormalized function u u concentrates as a Dirac mass around a critical oint of the Robin function. Successively, in [0] and [] the authors give a further descrition of the asymtotic behaviour of u as goes to infinity, by identifying a limit rofile roblem of Liouville-tye u + e u = 0 in IR, e u < + and showing that IR u e as +. In [] it is roved that roblem 0. can have many other ositive solutions which concentrate, as goes to infiniy, at some different oints ξ,..., ξ k of, whose location deends on the geometry of. More recisely, if k is a fixed integer, the authors introduce the function Ψ k : M IR defined by Ψ k ξ,..., ξ k := Hξ i, ξ i + Gξ i, ξ j, 0. i=,...,k i,j=,...,k i j where M = k \ and denotes the diagonal in k, i.e. = {ξ,..., ξ k k : ξ i = ξ j for some i j}. They rove that if Ψ k has a stable critical value c according to Definition.6, then if is large enough there exists a ositive solution u to roblem 0. with k eaks, namely as goes to infinity, u + 8πe k i= δ ξ i weakly in the sense of measure in, where ξ,..., ξ k M, with Ψ k ξ,..., ξk = c. After a detailed study of the existence and roerties of ositive solutions to roblem 0., one is interested in studying the existence and roerties of solutions which change sign. Problem 0. is a articular case of roblems treated in [4] and [7], where the authors study the existence of sign changing solutions and their roerties for a larger class of nonlinearities. In articular, it is roved that roblem 0. for any > has a sequence of distinct solutions ±u n with u n + as n +, u n changes sign if n and it has at most n nodal domains. Moreover, there exists a least energy nodal solution ū to 0. which has recisely two nodal domains. More recisely, if J : H 0 IR is defined by J u = u dx + u + dx and N = {u H 0 u + 0, u 0, J uu + = J uu = 0}, then it holds J ū = min N J. 0.3 In this aer, we are interested in the existence and the rofile of sign changing solutions which concentrate ositively at some different oints ξ,..., ξ h of

4 and concentrate negatively at some other different oints ξ h+,..., ξ k of. First of all let us state our general result. Let k be a fixed ositive integer, let a i {, }, i =,..., k, with a i a j = for some i j, and let Φ : M IR be defined by Φξ,..., ξ k = Hξ i, ξ i + i= i,j=,...,k i j a i a j Gξ i, ξ j. 0.4 Theorem 0. Assume that Φ has a stable critical level c in M, according to Definition.6. Then there exists 0 > 0 such that for any 0 roblem 0. has one sign-changing solution u such that, as +, u u 8πe a i δ ξ i i= for some ξ,..., ξ k M, with Φξ,..., ξ k = c. weakly in the sense of measure in, As far as it concerns, the existence and rofile of sign-changing solutions which concentrate ositively and negatively at two different oints ξ and ξ of, resectively, the following results hold. First of all, in this case the function Φ : M IR introduced in 0.4 reduces to Φξ, ξ = Hξ, ξ + Hξ, ξ Gξ, ξ. 0.5 The first result concerns the existence of a least energy nodal solution. Theorem 0. There exists 0 > 0 such that for any 0 i roblem 0. has a airs of sign-changing solutions u and u such that, as +, u u 8πe δ ξ δ ξ weakly in the sense of measure in, for some ξ, ξ, ξ ξ, such that Φξ, ξ = max Φ; ii the set \ {x : u x = 0} has exactly two connected comonents; iii the set {x : u x = 0} intersects the boundary of. We conjecture that the least energy solution found in the revious theorem coincides with the least energy solution ū found in 0.3. The second result is a multilicity result. In order to state it, let C = {A : #A = } = {x, y : x y} / x, y y, x be the configuration sace of unordered airs of elements of. Theorem 0.3 There exists 0 > 0 such that for any 0 3

5 i roblem 0. has at least catc airs of sign-changing solutions u i and u i, i =,..., catc such that, as +, u i u i 8πe δ ξ i δ ξ i for some ξ i, ξi, with ξ i ξ i ; weakly in the sense of measure in, ii the set \ {x : u i x = 0} has exactly two connected comonents; iii the set {x : u i x = 0} intersects the boundary of. It is easy to see that catc for any oen domain IR see, for examle, [6], so Theorem 0.3 yields at least two airs of solutions. However, as it has been ointed out in [6], if has toology then catc may be larger. We refer the reader to [[5], Section 6] for some recent comutations of catc. Section 3 deals with existence of solutions to 0. with more than two nodal zones when is a ball. Let us make some comments and remarks. Results similar to those in Theorem 0. and Theorem 0.3, have been obtained in [6] for the slightly subcritical roblem { u = u q ɛ u in, 0.6 u = 0 on, where is a bounded regular domain in IR N, q = N+ N, N 3 and ɛ is a small ositive arameter. In [6] the authors study the existence and the rofile of sign changing solutions to roblem 0.6, which blow u ositively at a oint ξ of and blow u negatively at a oint ξ of, with ξ ξ, as the arameter ɛ goes to zero. They introduce the function ϕ : M IR defined by ϕξ, ξ = H / ξ, ξ H / ξ, ξ Gξ, ξ, where G is the Green s function of the Dirichlet Lalacian and H is its regular art, i.e. Gx, y = α N x y N Hx, y, x, y, where α N = /N ω N and ω N denotes the surface area of the unit shere in IR N. Note that in this case H is ositive! They rove that, if ɛ is small enough roblem 0.6 has at least catc airs of sign-changing solutions u i ɛ as ɛ goes to zero, u i ɛ at a oint ξ i and u i ɛ blows u ositively at a oint ξ i, with ξi, ξi, ξ i ξ i and ξ i of ϕ. Moreover the set \ {x : u i ɛ comonents. Finally, if Hξ i, ξi = Hξi, i =,..., catc such that, and blows u negatively, ξi is a critical oint x = 0} has exactly two connected, ξi, 0.7 then the nodal set {x : u ɛ i x = 0} intersects the boundary of. We remark that condition 0.7 is satisfied when is a ball. 4

6 We quote the fact that, as far as it concerns roblem 0., we do not need any assumtion as 0.7 to ensure that nodal lines of solutions found in Theorem 0. and Theorem 0.3 intersect the boundary of. We comare this result with the one found in [], where the authors rove that the nodal line of a least energy solution to 0. intersects the boundary of, rovided is a ball or an annulus. Finally, we would like to oint out that the analogy between the almost critical roblem 0. in IR and the almost critical roblem 0.6 in IR N, with N 3, is not comlete. In fact only the traslation invariance lays a role in the study of 0., while both the traslation invariance and the dilation invariance are concerned in the study of 0.6. The same henomena occurs in the study of the mean field equation, as it has been already observed in [3], [9] and []. The aer is organized as follows. In Section we reduce the roblem to a finite dimensional one and we rove Theorem 0.. In this section we use some technical comutation develoed in [], which are contained in Aendix A and Aendix B. In Section we study the rofile of sign changing solutions and we rove Theorem 0. and Theorem 0.3. In Section 3 we consider the case when is a ball. Existence of nodal solutions Let us consider the roblem { u = g u in, u = 0 on.. where g u := u u. Let us introduce some functions which will be the basic elements to build changing sign solutions to.. Firstly, let us consider the limit roblem u = e u in IR, e u < +.. IR The only solutions to this roblem are given by U δ,ξ y = log 8δ δ + y ξ y IR,.3 with δ > 0 and ξ IR see [8]. Let P U δ,ξ denote the rojection of U δ,ξ onto H 0, namely P U δ,ξ = U δ,ξ in, P U δ,ξ = 0 on. Secondly, let Uy := U,0 y. Let w 0 be a solution to w 0 + e U w 0 = f 0 in IR, f 0 y := euy U y.4 5

7 and let w be a solution to w + e U w = f in IR,.5 f y := e Uy w 0 U w0 3 U 3 8 U 4 + w0 U y..6 For any δ > 0 and ξ IR, we define w 0 δ,ξx := w 0 x ξ δ, w δ,ξx := w x ξ δ, x. Let P wδ,ξ 0 resectively. and P w δ,ξ denote the rojection onto H 0 of wδ,ξ 0 and w δ,ξ, Define now U ξ x := i= where a i {, +} and a i γµ i P U δi,ξ i x + P w0 δ i,ξ i x + P w δ i,ξ i x.7 γ := e..8 Furthermore we assume that ξ = ξ,..., ξ k belongs to O ε, for some ε > 0, where O ε := { ξ k dist ξ i, ɛ, ξ i ξ j ɛ, i, j =,..., k, i j }. and that the arameters δ i satisfy the following relation with µ i := µ i, ξ, i =,..., k, given by log8µ 4 i = 8πHξ i, ξ i +8π j i δ i := δ i, ξ = µ i e 4.9 C 0 4 C 4 a i a j µ i µ j Gξ i, ξ j + log δ i C 0 + C C 0 4 C 4 A direct comutation shows that, for large, µ i thus satisfies..0 µ i = e 3 πhξ 4 e i,ξ i +π a ia j Gξ j,ξ i j i + O.. With this choice of the arameters µ i, we have that, if y = x ξ i δ i U ξ x = γµ i then + Uy + w 0y + w y + Oe 4 y + e 4.. We will look for a solution to. of the form u = U ξ + φ, where φ is a higher order term in the exansion of u. 6

8 It is useful to rewrite roblem. in terms of φ, namely Lφ = [R + Nφ] in, where u = 0 on..3 Lφ := L, ξ, φ = φ + g U ξ,.4 R := R, ξ = U ξ + g U ξ,.5 Nφ := N, ξ, φ = [ g U ξ + φ g U ξ g U ξ φ ]..6 A first ste to solve.3, or equivalently., consists in studying the invertibility roerties of the linear oerator L. In order to do so we introduce the weighted L norm defined by h := su δ i hx x δi + x ξ i 3.7 i= for any h L. With resect to this norm, the error term R, ξ given in.5 can be estimated in the following way. Lemma. Let ɛ > 0 be fixed. There exists c > 0 and 0 > 0 such that for any ξ O ɛ and 0 U ξ + g U ξ c 4. Proof. We give a sketch of the roof. We refer the reader to [] for all the details. A direct comutation of U ξ and the estimates given by Lemmas 4. and 4. readily imly that, far away from the oints ξ i, namely for x ξ i > ε for all i =,..., k, the following estimate holds true δ j δj + x ξ U j 3 ξ + g U ξ x Ce 4 j= for a certain ositive constant C. Let us now fix the index i in {,..., k}. Taking into account. and that =, we get, for x ξ i ε δ i, γµ γδ i i µ i δ j δj + x ξ U j 3 ξ + g U ξ x C 4, j= while, for ε δ i x ξ i ε, we get δ j δj + x ξ j 3 j= U ξ + g U ξ x Ce 4. 7

9 This concludes the roof. Next, we will solve the following rojected linear roblem: given h C 0,α, find a function φ and constants c i,j, for i =,..., k, j =,, such that Lφ = h + i,j c i,j e U δ i,ξ i Zi,j, in,.8 φ = 0, on,.9 e U δ j,ξ j Zi,j φ = 0, for all i =,, j =,..., mk..0 Here we set, for i =,..., k and j =,, Z i,j x := z j x ξi δ i, with z j y := U y j y = 4y j + y.. This linear roblem is uniquely solvable, for sufficiently large, in the set of functions with bounded -norm. This is the content of next Lemma, whose roof is given in Aendix B. Lemma. Let ɛ > 0 be fixed. There exist c > 0 and 0 > 0 such that for any > 0 and ξ O ɛ there is a unique solution φ to roblem which satisfies φ c h.. Let us now introduce the following nonlinear auxiliary roblem U ξ + φ + g U ξ + φ = c i,j e U δ i,ξ i Z i,j in, i,j= φ = 0 on, e U δ i,ξ i Z i,j φ = 0 if i =,..., k, j =,,.3 for some coefficients c i,j. The following result holds. Proosition.3 Let ɛ > 0 be fixed. There exist c > 0 and 0 > 0 such that for any > 0 and ξ O ε roblem.3 has a unique solution φ ξ which satisfies φ ξ c 3. Furthermore, the function ξ φ ξ is a C function in L and in H, 0. Proof. Using.4,.5 and.6 we can rewrite roblem.3 in the following way Lφ = R + Nφ + c i,j e U δ i,ξ i Zi,j. i,j Let us denote by L the function sace L endowed with the norm. Lemma. ensures that the unique solution φ = T h of.8-.0 defines a 8

10 continuous linear ma from the Banach sace L into L, with a norm bounded by a multile of. Then, roblem.3 becomes φ = Aφ := T [ R + Nφ ]..4 } Let B r := {φ C : φ r, for some r > 0. Arguing as in [], we 3 can rove that the following estimates hold for any φ, φ, φ B r Nφ c φ, Nφ Nφ c max i=, φ i φ φ..5 By.5, Lemma. and Lemma., we easily deduce that A is a contraction maing of B r for a suitable r > 0. Finally, a unique fixed oint of A exists in B r. Now, let us rove the regularity of the ma ξ φ ξ. Let ξ, ξ O ε. Since φ ξ φ ξ + g U ξ φ ξ φ ξ = g U ξ + φ ξ g U ξ + φ ξ + g U ξ + φ ξ g U ξ + φ ξ g U ξ φ ξ φ ξ + i,j + i,j c i,j ξ c i,j ξ e U δ j,ξ j ξ Z i,j ξ c i,j ξ e U δ j,ξ j ξ Z i,j ξ e U δ j,ξ j ξ Z i,j ξ.6 and since g U ξ + φ ξ g U ξ + φ ξ g U ξ φ ξ φ ξ C φ ξ φ ξ g U ξ + O 3 = o φ ξ φ ξ uniformly in O ε, by Proosition. we get φ ξ φ ξ C g U ξ + φ ξ g U ξ + φ ξ + C e U δ j,ξ j ξ Z i,j ξ e U δ j,ξ j ξ Z i,j ξ, for any 0 and ξ, ξ O ε here, is with resect to ξ. Hence, for fixed 0, the ma ξ φ ξ is continuous in L and, in view of Remark 5., in H0. Further, this ma is a C -function in H0 as it follows by the Imlicit Function Theorem alied to the equation Gξ, φ := Π ξ [ Uξ + Π ξ φ + g Uξ + Π ξ φ ] + Π ξ φ = 0, where Π ξ, Π ξ are the orthogonal rojections in H 0 resectively onto K ξ = san {P Z i,j : i =,, j =,..., m} 9

11 and its orthogonal Kξ according to the notations in Ste 6 in the roof of Proosition.. Indeed, Gξ, φ ξ = 0 and the linearized oerator: G φ ξ, φ ξ = Π ξ [ Id + g U ξ + φ ξπ ] ξ + Πξ.7 is invertible for large. In fact, easily we reduce the invertibility roerty to uniquely solve the equation G φ ξ, φ ξ[φ] = h in Kξ for any h Kξ. By Fredholm s alternative, we need to show that in Kξ only the trivial solution occur for the equation G φ ξ, φ ξ[φ] = 0, or equivalently for Lφ = g U ξ g U ξ + φ ξ φ + i,j c i,j e U δ j,ξ j Zij, for any coefficients c ij. By Proosition., we derive that φ C g U ξ g U ξ + φ ξ φ C φ ξ φ g U ξ C φ g U ξ C φ < φ and hence, φ = 0. Finally, since ξ φ ξ is a C ma in H 0, it is ossible to write down the equation satisfied by the difference between the incremental ratio in ξ of φ ξ and the associated derivative and by Remark 5. obtain that ξ φ ξ is a C ma also in L. The roof is now comlete. After roblem.3 has been solved, we find a solution to roblem.3 hence to the original roblem. if we find a oint ξ such that coefficients c ij ξ in.3 satisfy c ij ξ = 0 for i =,..., k, j =,..8 Let us introduce the energy functional J : H 0 IR given by J u := u dx u + dx,.9 + whose critical oints are solutions to.. We also introduce the finite dimensional restriction J : k IR given by The following result holds. J ξ := J Uξ + φ ξ..30 Lemma.4 If ξ is a critical oint of J then U ξ + φ ξ is a critical oint of J, namely a solution to roblem.. 0

12 Proof. The function J is of class C since the ma ξ φ ξ is C. Then, D ξ F ξ = 0 is equivalent to c i,j ξ e U δj,ξj Zi,j D ξ U ξ + c i,j ξ D ξ e U δ j,ξ j Zi,j φ ξ = 0 i,j i,j taking into account.3. Direct comutations then show that, for large and uniformly in ξ O ε, D 0 = c γδ j µ i,j ξ + O c l,h ξ γδ j j l,h where D is a given ositive constant. This fact imlies that c i,j ξ = 0 for all i, j. Next we need to write the exansion of J as goes to +, Lemma.5 It holds J ξ = k a + k a a 3 Φξ + R ξ, where R = O 3 uniformly with resect to ξ in comact sets of M := k \, where = {ξ k : ξ i = ξ j for some i, j}. Here a := 4πe, a := 4πe + e U U w 0 ydy, a 3 := 3π e.3 IR and the function Φ : M IR is defined by see 0.4 Φξ,..., ξ k = Hξ i, ξ i + a i a j Gξ i, ξ j. i=,...,k i,j=,...,k i j Proof. Multilying equation in.3 by U ξ + φ ξ and integrating by arts, we get that J ξ = + U ξ + φ ξ + c i,j ξ e U δ i,ξ i Zi,j U ξ. i,j Now, from equation 5. contained in the roof of Lemma., we have that Hence we have J ξ = + c i,j ξ = O Nφ + R + φ = O 4. U ξ + U ξ φ ξ + φ ξ + O 4.

13 We exand the term U ξ : in view of. we have that U ξ = δ j = = j= w x ξ j δ j γµ j Bξ j,ε e U δ j,ξ j δj w 0 x ξ j δ j + Oe U ξ + O e 8 j= γ µ 4 B0, ε δ + y w0 w + Oe j j + U + w0 + w + Oe 4 y + e 4 + O e j= = 8πk γ γ µ 4 j 3π γ 8π + IR 8 + y U w0 + O log µ j + k 8 γ IR + y U w0 + O 3 j= since µ 4 j = 4 log µ j + O. Recalling the exression of µ j in.0 and., we get that U ξ = 8πk γ + 64π γ Φξ,..., ξ k + 4πk γ + k 8 γ IR + y U w0 + O 3.3 uniformly for oints in O ε. By Proosition. and Remark 5. we get that φ ξ H 0 C φ ξ + Nφ ξ + R = O 3 since Nφ ξ = O φ ξ and R = O 4. Hence we get that U ξ φ ξ + φ ξ = O Finally, inserting in the exression of J, we get that J ξ = 4πk γ + k γ + 3π γ Φξ,..., ξ k + 4πk γ 8 + y U w0 + Θ ξ IR where Θ ξ = o uniformly on oints in O ε. Let us introduce the following Definition.

14 Definition.6 c is a stable critical level of Φ in M if there exist an oen set D comactly contained in M and B, B 0 closed subsets of D, with B connected and B 0 B, such that the following conditions hold: and c := inf su γ Γ ξ B Φγξ > su ξ B 0 Φξ.34 {ξ D : Φξ = c} =..35 Here, Γ denotes the class of all mas γ CB, D such that there exists an homotoy Ψ C[0, ] B, D satisfying Ψ0, = Id B, Ψ, = γ, Ψt, B0 = Id B0 for all t [0, ]. Under these assumtions, a critical oint ξ D of Φ exists at level c, as a standard deformation argument shows. As an examle, taking B = B 0 = D, it is easy to check that hold if su ξ D Φξ < su Φξ or inf Φξ > inf Φξ, ξ D ξ D ξ D namely the case of ossibly degenerate local maximum/minimum values of ϕ m. We are now in osition to carry out the roof of our main result: Proof of Theorem 0.. In view of Lemma.4, the function u = U ξ + φ ξ is a solution to roblem. if we show that ξ is a critical oint of the function J. This is equivalent to show that F ξ = a 3 J ξ k a k a has a critical oint ξ. Lemma.5 imlies that F is uniformly close to Φ, as, on comact sets of M. Moreover, assumtions.34 and.35 are stable with resect to C 0 erturbation and still hold for the function F rovided is large enough. Therefore, F has a critical oint ξ D, whose critical value c aroaches c, as. This roves our claim. The case k =. This section is devoted to the roblem of finding nodal solutions with exactly two nodal regions to roblem 0. in a general domain IR. Assume k = and a = + and a =. Then function Φ defined in 0.4 reduces to see 0.5 Φξ = Hξ, ξ + Hξ, ξ + Gξ, ξ, ξ, ξ M, where M = \ and is the diagonal in. First of all, let us rove the existence art i in Theorem 0. and Theorem 0.3. Proof of i of Theorem 0.. 3

15 We oint out that there exists c := max Φ, which is a stable critical value of M the function Φ according to Definition.6, since Φξ as ξ aroaches M. Therefore, the claim follows by Theorem 0.. Proof of i of Theorem 0.3. By Lemma.4, we need to rove that, if is large enough, the function J has at least catc airs of critical oints. In order to rove that J has at least catc airs of critical oints, it is enough to show that the Φ ξ = a 3 J ξ ka ka has at least catc airs of critical oints. From Lemma.5 we see that Φ is uniformly close to Φ on comact sets of M as. Moreover we oint out that Φξ as ξ M.. Now, let M denote the quotient manifold with resect to the equivalence ξ, ξ ξ, ξ. We consider the induced functions Φ, Φ : M IR. Setting m := cat M, there exists a comact set C M such that catc = m. By. we deduce that there exists an oen bounded set U M such that C U and su Φ < min Φ. Therefore if is large enough it follows that su Φ < min Φ. U C U C Now, for j =,..., m, c j := = su { su c : cat Φc M U { min Φ A } j : A U comact, cat M A j Standard arguments based on the Deformation Lemma shows that c j, for j =,..., m are critical levels for Φ. Finally, since M is homotoy equivalent to the configuration sace C, it holds m = cat M = catc. That roves our claim. Let u be a solution to roblem. found in art i of Theorem 0. and Theorem 0.3. We know that P U δ,ξ x + P w0δ,ξ where u x = γ µ µ }. P U δ,ξ x + P + ˆφ w0δ,ξ x,. γ := γ = e γ, as +, e.3 ξ i ξi as +, ξ, ξ, ξ ξ,.4 δ i := δ i, ξ = µ i, ξ e 4,.5 4

16 Here ˆφ := µ i := µ i, ξ e 3 4 e πhξ i,ξ i πgξ,ξ as +,.6 ˆφ C..7 µ P wδ,ξ µ P wδ,ξ + γ φ and the last estimate.7 follows by Proosition.3 and 4.7 of Lemma 4.. Let us rove that u changes sign exactly once. Theorem. Let u be a solution to. as in..7. Then, if is large enough, the set \ {x : u x = 0} has exactly two connected comonents. Proof. First of all, by Lemma 4. and Lemma 4., using also.3.7, we deduce that there exists r > 0 and 0 > 0 such that for any > 0 and u x > 0 for any x B ξ, r..8 u x < 0 for any x B ξ, r..9 Now, let := \ [ B ξ, r B ξ, r ]. Let us comute u c u L..0 Moreover, by.4.7 and Lemma 4. we deduce that u L γ P U δ,ξ + µ L + P w 0 L P δ γ,ξ + w 0 L δ,ξ L µ P U δ,ξ L + L ˆφ γ c + c log δ + log δ + c c γ γ γ,. because by Lemma 4. we easily deduce that Therefore by.0 and. we deduce that lim + u + L L P U δi,ξ i c, i =,. = 0.. Finally, it is clear that \ {x : u x = 0} has at least two connected comonents + {x : u x > 0} and {x : u x < 0}. By.8 and.9 it follows that B ξ, rδ + and B ξ, rδ. By contradiction, we assume that there exists a third connected comonent ω. Therefore, u solves u = u u in ω, u = 0 on ω and the weight a := u satisfies.3, because of.. By Lemma. below it follows that u 0 in ω and a contradiction arises. 5

17 Lemma. Let ω be a bounded domain in IR and assume that a : ω IR satisfies lim + a < 8πe..3 + L ω Then the roblem u = au in ω, u = 0 on ω, has only the trivial solution. Proof. First of all, we oint out that u L ω for any > and u H 0 ω = au a L ω u L ω S a L ω ω u H 0 ω, where S denotes the best Sobolev constant of the Sobolev embedding H 0ω L ω. Therefore, if u is a nontrivial solution the following condition has to be satisfied: S a. L ω + On the other hand, in [4] it was roved that lim S = 8πe. Then, by.3 a contradiction arises. In order to rove that the nodal line of u touches the boundary of, it is useful to describe the asymtotic behaviour of u in a neighbourhood of the boundary, as goes to infinity. Proosition.3 Let u be a solution to. as in..7. It holds as goes to + u x 8π e [Gx, ξ Gx, ξ ] in C loc \ {ξ, ξ }..4 Proof. By Lemma 4. and Lemma 4., using..7, we deduce that estimate.4 holds ointwise in \ {ξ, ξ}. We are going to rove that for some ositive constant c g u L = u L c.5 and g u L ω = u L ω c,.6 where ω is a neighbourhood of. By Lemma.4, we deduce that u C 0,α ω c and the claim follows by Ascoli-Arzelá Theorem. Let us rove.6. Let ω be a neighbourhood of which does not contain ξ and ξ. By.4.7 and Lemma 4. we deduce that u L ω γ µ γ µ P Uδ,ξ + L ω P Uδ,ξ + L ω P w 0 δ,ξ L ω P wδ 0 L,ξ ω + L ˆφ γ ω c c + c log δ + c c + c log δ + c c γ γ γ,.7 6

18 L because by Lemma 4. we easily deduce that P U δi,ξ i c, i =,. ω Finally,.5 follows by because u dx + lim u + = 8πek. + u + dx + We recall the following lemma see Lemma,[3]. c + c, Lemma.4 Let u be a solution to u = f in, u = 0 on. If ω is a neighbourhood of, then u C 0,α ω c f L + f L ω, where α 0, and ω ω is a neighbourhood of. Theorem.5 Let u be a solution to. as in..7. Then if is large enough {x : u x = 0}. Proof. First of all, we remark that if {x : u x = 0} =, then u ν does not change sign on. On the other hand, the normal derivative ν [G, ξ G, ξ] changes sign on, since ν [Gx, ξ Gx, ξ] dx = 0. By Proosition.3 we deduce that u ν x ν [Gx, ξ Gx, ξ] uniformly on as. Therefore, if is large enough u ν also changes sign on and a contradiction arises. Proof of ii and iii of Theorem 0. and Theorem 0.3. ii follows by Theorem. and iii follows by Theorem.5. 3 The symmetric case In this section we describe two ossible configurations for oints with ositive and negative concentration for nodal solutions to roblem 0. when the domain is a ball in IR. In both cases we strongly use the symmetry of the roblem. Let be the ball {x IR : x < R}. In the first examle, we build a solution to 0. with h oints of ositive concentration and h oints of negative concentration located on the vertices of a regular olygon, distributed with alternating sign. Let h be a fixed integer and let k = h. Set ξi = cos πk i, sin πk i for any i =,..., k. 3. 7

19 Theorem 3. For any even integer k there exists k > 0 such that for any k roblem 0. has a sign-changing solution u such that, for some ρ 0, R, as +, u u 8πe i+ δ ρ ξi i= weakly in the sense of measure in. Proof. where We will look for a solution to roblem 0. as u x = U ρ x + φx, U ρ := γ i+ i= µ i P U δi,ξi + P w0δi,ξi + P wδi,ξi, 3. where δ i are given in.9 and the concentration oints are for i =,..., k ξ i := ξρ = ρξi = ρ cos πk i, ρ sin πk i with ρ [0, R], 3.3 and the rest term φ is symmetric with resect to the variable x and is symmetric with resect to each line {tξi t IR} for i =,..., k. Using results obtained in revious sections and taking into account the symmetry of the domain, we reduce the roblem of finding solutions to. to that of finding critical oints of the function J : [0, R] IR defined as in.30 by J ρ := J U ρ + φ ρ. It is not difficult to check that J ρ = k a + k a k a 3 Φρ + O 3, where a, a and a 3 are given in.3 and Since, in this case, Gx, y = π Φρ := Hρξ, ρξ log i Gρξ, ρξi, ρ 0, R. 3.4 i= R y x log R, Hx, x = xy π log R x, function Φ reduces to since k is even { Φρ = logr ρ + i [ log Rρ ξ ξi log R ρ ξ π ξ ] } i i= [ = logr ρ + log ρ i log R ρ cos πk ] π i + π i log R ξ ξi. i= i= 8

20 It is easy to check that lim Φρ = lim Φρ =. Then there exists ρ 0 + ρ R ρ 0, R such that Φρ = max Φρ, which is a critical oint which ersists ρ 0,r under small C 0 erturbation. That roves our claim. Our second examle is a nodal solution to roblem 0. with a negative or ositive oint of concentration at the origin of the ball and again h oints of ositive concentration and h oints of negative concentration located at the vertices of a regular olygon with alternating signs. Theorem 3. For any even integer k there exists k > 0 such that for any k roblem. has a sign-changing solution u such that, for some ρ 0, R, as +, u u 8πe δ 0 + i+ δ ρ ξi weakly in the sense of measure in. i= Proof. Here we will look for a solution to roblem. as u x = U ρ x + φx, where U ρ := γ γ i+ i= µ k+ µ i P U δi,ξi + P w0δi,ξi + P wδi,ξi P U δk+,0 + P w0 δ k+,0 + P w δ k+,0, 3.5 where δ i are given in.9, the concentration oints ξ i are given in 3.3 and the rest term φ is symmetric with resect to the variable x and is symmetric with resect to each line {tξi t IR} for i =,..., k. Using results obtained in revious sections and taking into account the symmetry of the domain, we reduce the roblem of finding solutions to. to that of finding critical oints of the function J : [0, R] IR defined as in.30 by J ρ := J U ρ + φ ρ. It is not difficult to check that J ρ = k a + k a + H0, 0 k a 3 Φρ + O 3, where a, a and a 3 are given in.3 and Φ is defined in 3.4. By the roof of Theorem 3., it follows that Φ has a maximum oint, which ersists under small C 0 erturbation. That roves our claim. 4 Aendix A Let U δ,ξ be the function defined in.3. The following result holds. Lemma 4. We have P U δ,ξ x = U δ,ξ x log 8δ + 8πHx, ξ + O δ as δ

21 uniformly on comact sets of and P U δ,ξ x = 8πGx, ξ + O δ as δ 0 4. uniformly on comact sets of \ {ξ}. Moreover the following global estimate holds. Let ɛ > 0 be fixed. Then there exist δ 0 > 0 and a ositive constant c > 0 such that for any δ 0, δ 0 and ξ with dist ξ, ɛ we have P U δ,ξ U δ,ξ x + log 8δ c. 4.3 Proof. Since, uniformly for x, P U δ,ξ x U δ,ξ x+log 8δ = 4 log x ξ + Oδ as δ 0, by harmonicity 4. readily follows. On the other hand, away from ξ, we have U δ,ξ x log 8δ = 4 log x ξ + Oδ. This fact, together with 4. gives 4.. The uniform estimate 4.3 is consequence of., 4. and the definition of Hx, ξ. Let w 0 and w be the functions defined in.4 and.5-.6, resectively. They are radial functions satisfying resectively w 0 y = C 0 log y + O as y +, 4.4 y and w y = C log y + O y as y +, 4.5 where C 0 := 4 log 8 and C is a suitable ositive constant. See []. By 4.4 and 4.5 we deduce the following exansions. Lemma 4. We have and P w 0 δ,ξx = w 0 δ,ξx C 0 πhx, ξ + C 0 log δ + O δ as δ 0 P w δ,ξx = w δ,ξx C πhx, ξ + C log δ + O δ as δ 0, uniformly on comact sets of. Moreover P w 0 δ,ξx = C 0 log x ξ C 0 πhx, ξ + O δ as δ 0 uniformly on comact sets of \ {ξ}. Moreover the following global estimates holds. For any ɛ > 0 and δ 0 > 0 there exists c > 0 such that for any δ 0, δ 0 and ξ with dist ξ, ɛ we have P w 0 c log δ 4.6 and P w δ,ξ δ,ξ c log δ. 4.7 Proof. The roof follows from the same arguments used to rove Lemma 4. and from estimates 4.4 and

22 5 Aendix B This aendix is devoted to rove Lemma.. First of all, in order to treat the invertibility roerties of the lineal oerator L, we need to estimate g U ξ x. If x ξ i ε for some i =,..., k g U ξ x = a i γµ i + Uy + w0 y + 3 w y +Oe 4 y + e 4 = a i δ i + Uy + w0 y + 3 w y + Oe 4 y + e 4, 5. where we use the notation y = x ξi δ i. In this region, we have then that g U ξ x Ce e Uy = O e U δ i,ξ i x, since Uy. On the other hand, if x ξ i ε for any i =,..., k g U ξ x = O C. Summing u, we have that there exist D > 0 and 0 > 0 such that g U ξ x D j= e U δ j,ξ j x 5. for any ξ O ε and 0. Proof of Lemma.. The roof consists of 6 stes. Ste. L satisfies the maximum rincile in := \ m j= Bξ j, Rδ j for R large and indeendent of, namely if Lψ 0 in and ψ 0 on, then ψ 0 in. Indeed, let where Zx = m j= ax ξj z 0, δ j z 0 y = y + y. 5.3

23 If a is chosen ositive and small and R is chosen large, deending on a but indeendent of, it follows that Z is a ositive function in and, taking into account 5., it satisfies LZx 0 for all x, for sufficiently large. The existence of such a function Z guarantees that L satisfies the maximum rincile in. Ste. Let R be as before. Define φ i = su x m j= Bξj,Rδj φx. Then there is a constant C > 0 such that, if Lφ = h in, then Indeed, consider the solution ψ j x of the roblem δ j x ξ j 3 φ C[ φ i + h ]. 5.4 ψ j = in Rδ j < x ξ j < M ψ j x = 0 on x ξ j = Rδ j and x ξ j = M. Here M = diam. The function ψ j x is a ositive function, which is uniformly bounded from above for sufficiently large. Define now the function φx = φ i Zx + h ψ j x, where Z was defined in the revious Ste. From the definition of Z, choosing R larger if necessary, we see that j= φx φx for x ξ j = Rδ j, j =,..., k, and by the ositivity of Zx and ψ j x Furthermore, direct comutation yields φx 0 = φx for x. L φx Lφx rovided that is large enough. Hence, by the maximum rincile established in Ste we obtain that φx φx for x, and therefore φ C[ φ i + h ]. Ste 3. Given h, assume φ is a solution of roblem Lφ = h in, φ = 0 on, when φ satisfies.0 and in addition the orthogonality conditions e U δ j,ξ j Z0,j φ = 0, for j =,..., k 5.5

24 where Z 0,j x = z 0 x ξ j δ j see 5.3. We rove that there exists a ositive constant C such that for any ξ O ε φ C h, 5.6 for sufficiently large. By contradiction, assume the existence of sequences n, oints ξ n O ε, functions h n and associated solutions φ n such that h n 0 and φ n =. Since φ n =, Ste shows that lim inf n + φ n i > 0. Let us set ˆφ n j y = φ n δj ny + ξn j. Ellitic estimates readily imly that ˆφ n j converges uniformly over comact sets to a non trivial bounded solution ˆφ j of the equation in IR φ y φ = 0. This imlies that ˆφ j is a linear combination of the functions z i, i = 0,, see. and 5.3. Since ˆφ n j, by Lebesgue theorem the orthogonality conditions.0 and 5.5 on φ n ass to the limit and yield 8 IR + y z i y ˆφ j = 0 for any i = 0,,. Hence, ˆφ j 0. Contradiction. Ste 4. We rove that there exists a ositive constant C > 0 such that any solution φ of equation Lφ = h in, φ = 0 on, satisfies. when we assume on φ only the orthogonality conditions.0. Proceeding by contradiction as in Ste 3, we can suose further that n h n 0 as n but we loss in the limit the condition 8 IR + y z 0 y ˆφ j that = 0. Hence, we have ˆφ n j C j y y + in C0 loc IR 5.8 for some constants C j. Testing.8 against roerly chosen test functions, one can show that C j = 0 for any j =,..., k. Ste 5. We establish the validity of the a riori estimate.. The revious ste yields φ C h + c i,j 5.9 i,j since e U δ j,ξ j Zi,j e U δ j,ξ j C. 3

25 Hence, roceeding by contradiction as in Ste 3, we can suose further that n h n 0, n c n ij δ > 0 as n i,j We omit the deendence on n. It suffices to estimate the values of the constants c ij. Let P Z i,j be the rojection on H, 0 of the functions Z i,j. Testing equation.8 against P Z i,j and integrating by arts one gets Dc ij + O e c lh + O h = ε B0, δj where D = 64 IR Hence, we obtain that 3y i + y 3 l,h w 0 U U ˆφ j + O φ 5. y + y 4, ˆφ j y = φδ j y + ξ j and w 0 is given by.4. c lh = O h + φ. 5. l,h Since l,h c lh = o, as in Ste 4 we have that for some constants C j. Lebesgue theorem the term ε B0, δj ˆφ j C j y y + in C0 loc IR 5.3 Hence, in 5. we have a better estimate since by 3y i + y 3 w 0 U U y ˆφ j y converges to C j IR 3y i y + y 4 w 0 U U y = 0. Therefore, we get that l,h c lh = O h + o. This contradicts i,j c i,j δ > 0, and the claim is established. Ste 6. We rove the solvability for.8,.9,.0. To this urose we consider the sace K ξ = san {P Z i,j : i =,, j =,..., k} 4

26 and its orthogonal sace { Kξ = φ H0 : e U δ j,ξ j Zi,j φ = 0 } for i =,, j =,..., k, endowed with the usual inner roduct. Let Π ξ and Π ξ be the associated orthogonal rojections in H0. Problem.8-.0 exressed in weak form is equivalent to that of finding φ Kξ, such that φ, ψ H 0 = W φ h ψ dx, for all ψ Kξ. With the aid of Riesz s reresentation theorem, this equation gets rewritten in in the oeratorial form K ξ Id Kφ = h, 5.4 where h = Π ξ h and Kφ = Π ξ W φ is a linear comact oerator in Kξ. The homogeneous equation φ = Kφ in K ξ, which is equivalent to.8-.0 with h 0, has only the trivial solution in view of the a riori estimate.. Now, Fredholm s alternative guarantees unique solvability of 5.4 for any h Kξ. Remark 5. Given h C0 α, let φ be the solution of.8-.0 given by Proosition.. Multilying.8 by φ and integrating by arts, we get φ H 0 = W φ hφ. Since 5. holds true, we get φ H 0 C φ + h. On the other hand, we get W φ δ φ i for some δ > 0. Ste in the roof of the revious Proosition then imlies φ C φ i + h + i,j c ij and in view of 5. we get Hence, we have that φ i φ C φ i + h. δ φ i φ H 0 + C δ h + δ φ i and in turn φ C φ H 0 + h. 5

27 References [] Adimurthi, M. Grossi, Asymtotic estimates for a two-dimensional roblem with olynomial nonlinearity, Proc. Amer. Math. Soc , no. 4, [] A. Aftalion, F. Pacella, Qualitative roerties of nodal solutions of semilinear ellitic equations in radially symmetric domains C. R. Acad. Sci. Paris 004 to aear. [3] S. Baraket, F. Pacard, Construction of singular limits for a semilinear ellitic equation in dimension. Calc. Var. Partial Differential Equations 6 998, no., 38. [4] T. Bartsch Critical oint theory on artially ordered Hilbert saces. J. Funct. Anal , 7 5. [5] T. Bartsch, M. Cla, T. Weth, Configuration saces, transfer, and - nodal solutions of a semiclassical nonlinear Schrödinger equation. Prerint. [6] T. Bartsch, A.M. Micheletti, A. Pistoia, On the existence and the rofile of nodal solutions of ellitic equations involving critical growth. Prerint [7] T. Bartsch, T. Weth, A note on additional roerties of sign changing solutions to suerlinear ellitic equations. To. Meth. Nonlin. Anal. 003, 4. [8] W. Chen, C. Li, Classification of solutions of some nonlinear ellitic equations. Duke Math. J , [9] M. del Pino, M. Kowalczyk, M. Musso, Singular limits in Liouvilletye equations. Calc. Var. Partial Differential Equations to aear [0] K. El Mehdi, M. Grossi, Asymtotic estimates and qualitative roerties of an ellitic roblem in dimension two. Adv. Nonlinear Stud , no., [] P. Esosito, M. Grossi, A. Pistoia, On the existence of blowing-u solutions for a mean field equation. Ann. IHP Analyse Non Linéaire, no., [] P. Esosito, M. Musso, A. Pistoia, Concentrating solutions for a lanar ellitic roblem involving nonlinearities with large exonent, rerint 004 [3] Z.C. Han, Asymtotic aroach to singular solutions for nonlinear ellitic equations involving critical Sobolev exonent. Ann. IHP Analyse Non Linéaire 8 99, [4] X. Ren, J. Wei, On a two-dimensional ellitic roblem with large exonent in nonlinearity. Trans. Amer. Math. Soc , no.,

28 [5] X. Ren, J. Wei, Single oint condensation and least energy solutions. Proc. Amer. Math. Soc.4 996, 0. 7