A 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 Tintarev Abstract We prove the existence of a sign-changing eigenfunction at the second minimax level of the eigenvalue problem for the scalar field equation under a slow decay condition on the potential near infinity. The proof involves constructing a set consisting of sign-changing functions that is dual to the second minimax class. We also obtain a nonradial sign-changing eigenfunction at this level when the potential is radial. 1. Introduction Consider the eigenvalue problem for the scalar field equation where N 2, V L ( ) satisfies Δu + V (x)u = λ u p 2 u, u H 1 ( ), (1.1) V (x) 0 x, (1.2) lim V (x) =V > 0, (1.3) x and p (2, 2 ). Here 2 =2N/(N 2) if N 3 and 2 = if N =2.Let I(u) = u p, J(u) = u 2 + V (x)u 2,u H 1 ( ). Then the eigenfunctions of (1.1) on the manifold M = {u H 1 ( ):I(u) =1}, and the corresponding eigenvalues coincide with the critical points and the corresponding critical values of the constrained functional J M, respectively. Equation (1.1) has been studied extensively for more than three decades (see Bahri and Lions [1] for a detailed account). The main difficulty here is the lack of compactness inherent in this problem. This lack of compactness originates from the invariance of under the action of the noncompact group of translations, and manifests itself in the noncompactness of the Sobolev imbedding H 1 ( ) L p ( ). This in turn implies that the manifold M is not weakly closed in H 1 ( ) and that J M does not satisfy the usual Palais Smale compactness condition at all energy levels. Least energy solutions, also called ground states, are well-understood. In general, the infimum λ 1 := inf J(u) u M is not attained. For the autonomous problem at infinity, Δu + V u = λ u p 2 u, u H 1 ( ), Received 11 February 2014; published online 10 September 2014. 2010 Mathematics Subject Classification 35J61 (primary), 35P30, 35J20 (secondary).

A NODAL SOLUTION OF THE SCALAR FIELD EQUATION 1219 the corresponding functional J (u) = u 2 + V u 2, u H 1 ( ) attains its infimum λ 1 := inf J (u) > 0, u M at a radial function w1 > 0 (see Berestycki and Lions [5]), and this minimizer is unique up to translations (see Kwong [17]). For the nonautonomous problem, we have λ 1 λ 1 by (1.3) and the translation invariance of J,andλ 1 is attained if the inequality is strict (see Lions [19, 20]). As for higher energy solutions, also called bound states, radial solutions have been extensively studied when the potential V is radially symmetric (see, for example, Berestycki and Lions [6], Jones and Küpper [16], Grillakis [14], Bartsch and Willem [4], and Conti et al. [10]). The subspace Hr 1 ( )ofh 1 ( ) consisting of radially symmetric functions is compactly imbedded into L p ( ) for p (2, 2 ) by a compactness result of Strauss [23]. So in this case, the restrictions of J and J to M Hr 1 ( ) have increasing and unbounded sequences of critical values given by a standard minimax scheme. Furthermore, Sobolev imbeddings remain compact for subspaces with any sufficiently robust symmetry (see, for example, Bartsch and Willem [3], Bartsch and Wang [2], and Devillanova and Solimini [12]). As for multiplicity in the nonsymmetric case, Zhu [25], Hirano [15], Clapp and Weth [8], and Perera [21] have given sufficient conditions for the existence of 2, 3, N/2 + 1, and N 1 pairs of solutions, respectively (see also Li [18]). There is also an extensive literature on multiple solutions of scalar field equations in topologically nontrivial unbounded domains (see the survey paper of Cerami [7]). The second minimax level is defined as λ 2 := inf max J(u), γ Γ 2 u γ(s 1 ) where Γ 2 is the class of all odd continuous maps γ : S 1 Mand S 1 is the unit circle. In the present paper, we address the question of whether solutions of (1.1) at this level, previously obtained in Perera and Tintarev [22], are nodal (sign-changing), as it would be expected in the linearized problem where the nonlinearity u p 2 u is replaced by U(x)v with the decaying potential U(x) = u p 2. Ultimately, nodality of a solution u follows from the orthogonality relation (2.11), which can be understood as u,v 1 (u) = 0, where v 1 (u) is the ground state of the linearized problem with the potential U(x) = u p 2 and u = u p 2 u is the duality conjugate of u. A similar argument has been used in Tarantello [24] to obtain a nodal solution of a critical exponent problem in a bounded domain (see also Coffman [9]). Nodality (and furthermore, the number of nodes, that is, domains of constant sign) is an important qualitative characteristic of solutions, both in terms of significance for applications (for example, some physical quantities such as temperature or concentration cannot change sign and nodality of a solution indicates change of sign of another physical quantity, such as electric charge) and in terms of classification of solutions, as the number of nodes is connected to the Morse index of the solution and the type of minimax involved. Recall that w1 e V x (x) C 0 as x, x (N 1)/2 for some constant C 0 > 0, and that there are constants 0 <a 0 V and C>0such that if λ 1 is attained at w 1 0, then w 1 (x) Ce a0 x x

1220 KANISHKA PERERA AND CYRIL TINTAREV (see Gidas et al. [13]). Write V (x) =V W (x), so that W (x) 0as x by (1.3), and write q for the norm in L q ( ). Our main result is the following. Theorem 1.1. Assume that V L ( ) satisfies (1.2) and (1.3), p (2, 2 ), and W (x) c 0 e a x x, for some constants 0 <a<a 0 and c 0 > 0. IfW L p/(p 2) ( ) and W p/(p 2) < (2 (p 2)/p 1)λ 1, then equation (1.1) has a nodal solution on M for λ = λ 2. Nodal solutions to a closely related problem have been obtained in Zhu [25] and Hirano [15] under assumptions different from those in Theorem 1.1. We will prove this theorem by constructing a set F Mconsisting of sign-changing functions that is dual to the class Γ 2. As a corollary, we obtain a nonradial nodal solution of (1.1) at the level λ 2 when V is radial. Let Γ 2,r denote the class of all odd continuous maps from S 1 to M r = M H 1 r ( ) and set λ 2,r := inf γ Γ 2,r max J(u), u γ(s 1 ) λ 2,r := inf γ Γ 2,r max J (u). u γ(s 1 ) Since the imbedding Hr 1 ( ) L p ( ) is compact by the result of Strauss [23], these levels are critical for J Mr and J Mr, respectively. Since Γ 2,r Γ 2 and λ 2 := inf max J (u) γ Γ 2 u γ(s 1 ) is not critical for J M (see, for example, Cerami [7]), we have λ 2 λ 2,r and λ 2 <λ 2,r. It was shown in Perera and Tintarev [22, Theorem 1.3] that if W L p/(p 2) ( )and W p/(p 2) <λ 2,r λ 2, then λ 2 <λ 2,r and every nodal solution of (1.1) onm with λ = λ 2 is nonradial. Combining Theorem 1.1 with this result now gives us the following corollary. Corollary 1.2. Assume that V L ( ) is radial and satisfies (1.2) and (1.3), p (2, 2 ), and W (x) c 0 e a x x, for some constants 0 <a<a 0 and c 0 > 0. IfW L p/(p 2) ( ) and W p/(p 2) < min{(2 (p 2)/p 1)λ 1,λ 2,r λ 2 }, then λ 2 <λ 2,r and equation (1.1) has a nonradial nodal solution on M for λ = λ 2. We will give the proof of Theorem 1.1 in the next section. Let 2. ProofofTheorem1.1 u = J(u), u H 1 ( ).

A NODAL SOLUTION OF THE SCALAR FIELD EQUATION 1221 Lemma 2.1. The norm is an equivalent norm on H 1 ( ). Proof. Since V L ( ), u C u H1 ( ) for all u H 1 ( ) for some constant C>0. If the reverse inequality does not hold for any C>0, then there exists a sequence u k H 1 ( ) such that u k 2 =1, (2.1) u k 2 0, (2.2) V (x) u 2 k 0. (2.3) By (1.3), there exists R>0 such that V (x) V 2 > 0 x RN \ B R, where B R = {x : x <R}. Then u k L 2 ( \B R) 0by(1.2) and (2.3). By (2.1) and (2.2), u k is bounded in H 1 ( ) and hence converges weakly in H 1 ( )tosomew for a renamed subsequence. By the weak lower semicontinuity of the gradient seminorm, then w 2 =0,so w is a constant function. This constant is necessarily zero since w H 1 ( ). Consequently, u k L 2 (B R) 0 by the compactness of the imbedding H 1 ( ) L 2 (B R ). Thus, u k 2 0, contradicting (2.1). For u H 1 ( ), let K u (v) = u(x) p 2 v 2, v H 1 ( ). Lemma 2.2. The map H 1 ( ) H 1 ( ) R, (u, v) K u (v) is continuous with respect to norm convergence in u and weak convergence in v, that is, K uk (v k ) K u (v) whenever u k u in H 1 ( ) and v k vin H 1 ( ). Proof. It suffices to show that K uk (v k ) K u (v) for a renamed subsequence of (u k,v k ). By the continuity of the Sobolev imbedding H 1 ( ) L p ( ), u k u in L p ( )andv k is bounded in L p ( ). Then u k (x) p 2 vk 2 + u(x) p 2 v 2 u k p 2 L p ( \B v R) k 2 p + u p 2 L p ( \B v 2 R) p, \B R by the Hölder inequality and the right-hand side can be made arbitrarily small by taking R>0 and k sufficiently large, where B R = {x : x <R}. By the compactness of the imbedding H 1 (B R ) L p (B R ), (u k,v k ) (u, v) strongly in L p (B R ) L p (B R ) and almost everywhere in B R B R for a renamed subsequence. Then u k (x) p 2 vk 2 u(x) p 2 v 2, B R B R by the elementary inequality ( a p 2 b 2 1 2 ) a p + 2 p p b p a, b R, and the dominated convergence theorem. Thus, the conclusion follows. For u H 1 ( ) \{0}, let M u = {v H 1 ( ):K u (v) =1}.

1222 KANISHKA PERERA AND CYRIL TINTAREV Lemma 2.3. For u H 1 ( ) \{0}, the infimum μ 1 (u) := inf J(v) (2.4) v M u is attained at a unique v 1 (u) > 0, and the even map H 1 ( ) \{0} H 1 ( ),u v 1 (u) is continuous. Proof. The functional K u is weakly continuous on H 1 ( ) by Lemma 2.2 and J is weakly lower semicontinuous, so the infimum in (2.4) is attained at some v 1 (u) M u. By the strong maximum principle, v 1 (u) > 0. Then the right-hand side of the well-known Jacobi identity ( ) J(v) μ 1 (u)k u (v) = v 1 (u) 2 v 2 R v N 1 (u) vanishes at v M u if and only if v = v 1 (u), so the minimizer is unique. Let u k u 0inH 1 ( ). By Lemma 2.2, K uk (v 1 (u)) K u (v 1 (u)) = 1, so for sufficiently large k, K uk (v 1 (u)) > 0andv 1 (u)/ K uk (v 1 (u)) M uk. Then ( ) v 1 (u) μ J(v 1 (u k )) J 1 (u) = Kuk (v 1 (u)) K uk (v 1 (u)) μ 1(u), so lim sup J(v 1 (u k )) μ 1 (u). (2.5) In particular, v 1 (u k ) is bounded in H 1 ( ) and hence converges weakly in H 1 ( ) to some v for a renamed subsequence. Then 1 = K uk (v 1 (u k )) K u (v) by Lemma 2.2, so K u (v) =1and hence v M u. Since J is weakly lower semicontinuous, then μ 1 (u) J(v) lim inf J(v 1 (u k )). (2.6) Combining (2.5) and (2.6) gives lim J(v 1 (u k )) = J(v) =μ 1 (u), so v 1 (u k ) v and v = v 1 (u) by the uniqueness of the minimizer. Since v 1 (u k ) v, then v 1 (u k ) v 1 (u). For u H 1 ( ) \{0}, let L u (v) = u(x) p 2 v 1 (u)v, v H 1 ( ), and let N u = {v M u : L u (v) =0}. Lemma 2.4. For u H 1 ( ) \{0}, the infimum μ 2 (u) := inf J(v) (2.7) v N u is attained at some v 2 (u). Proof. The functional K u is weakly continuous on H 1 ( ) by Lemma 2.2, L u is a bounded linear functional on H 1 ( ), and J is weakly lower semicontinuous, so the infimum in (2.7) is attained at some v 2 (u) N u. Lemma 2.5. For u H 1 ( ) \{0}, v 1 (u) and v 2 (u) are linearly independent and J(v) μ 2 (u) u(x) p 2 v 2 v span{v 1 (u),v 2 (u)}.

A NODAL SOLUTION OF THE SCALAR FIELD EQUATION 1223 Proof. Since L u (v 2 )=0andK u (v 1 )=1,v 1 and v 2 are linearly independent. We have v 1 w + V (x)v 1 w = μ 1 (u) u(x) p 2 v 1 w w H 1 ( ), and testing with v 2 gives v 1 v 2 + V (x)v 1 v 2 = μ 1 (u) u(x) p 2 v 1 v 2 =0. Then J(c 1 v 1 + c 2 v 2 )=c 2 1J(v 1 )+c 2 2J(v 2 )=c 2 1μ 1 (u)+c 2 2μ 2 (u) (c 1 2 + c 2 2)μ 2 (u) =μ 2 (u) u(x) p 2 (c 1 v 1 + c 2 v 2 ) 2. Let h(u) = u p 2 uv 1 (u), u H 1 ( ) \{0}, and let F = {u M: h(u) =0}. By Lemma 2.3, h is continuous and hence F is closed. Lemma 2.6. For all γ(s 1 ) F for all γ Γ 2. Proof. Since h γ : S 1 R is an odd continuous map by Lemma 2.3, h(u) = 0 for some u γ(s 1 ) by the intermediate value theorem. Lemma 2.7. For all J(u) λ 2 for all u F. Proof. Since K u (u) =I(u) =1andL u (u) =h(u) =0,wehaveu N u and hence J(u) μ 2 (u). (2.8) By Lemma 2.5, v 1 (u) andv 2 (u) are linearly independent and hence we can define an odd continuous map γ u : S 1 M span{v 1 (u),v 2 (u)} by γ u (e iθ )= Take v 0 γ u (S 1 ) such that v 1(u)cosθ + v 2 (u)sinθ v 1 (u)cosθ + v 2 (u)sinθ p, θ [0, 2π]. J(v 0 ) = max J(v) λ 2. (2.9) v γ u(s 1 ) By Lemma 2.5 and the Hölder inequality, J(v 0 ) μ 2 (u) u p 2 p v 0 2 p = μ 2 (u). (2.10) Combining (2.8) (2.10) gives J(u) λ 2. It was shown in Perera and Tintarev [22, Proposition 3.1] that λ 1 <λ 2 < (λ p/(p 2) 1 +(λ 1 ) p/(p 2) ) (p 2)/p,

1224 KANISHKA PERERA AND CYRIL TINTAREV under the hypotheses of Theorem 1.1, so J M satisfies the (PS) condition at the level λ 2 (see, for example, Cerami [7]). Let K λ2 denote the set of critical points of J M at this level. Lemma 2.8. We have K λ2 F Proof. Suppose K λ2 F =. Since K λ2 is compact by the (PS) condition and F is closed, there is a δ>0such that the δ-neighborhood N δ (K λ2 )={u M: dist(u, K λ2 ) δ} does not intersect F. By the first deformation lemma, there are ε>0 and an odd continuous map η : M Msuch that η(j λ2+ε ) J λ2 ε N δ (K λ2 ), where J a = {u M: J(u) a} for a R (see Corvellec et al. [11]). Since J>λ 2 ε on F by Lemma 2.7 and N δ (K λ2 ) F =, then η(j λ2+ε ) F =. Takeγ Γ 2 such that γ(s 1 ) J λ2+ε and let γ = η γ. Then γ Γ 2 and γ(s 1 ) F =, contrary to Lemma 2.6. We are now ready to prove Theorem 1.1. Proof of Theorem 1.1. By Lemma 2.8, there is some u K λ2 F, which then is a solution of (1.1) onm for λ = λ 2 satisfying u p 2 uv 1 (u) =0. (2.11) Since v 1 (u) > 0 by Lemma 2.3, u is nodal. References 1. A. Bahri and P.-L. Lions, On the existence of a positive solution of semilinear elliptic equations in unbounded domains, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997) 365 413. 2. T. Bartsch and Z. Q. Wang, Existence and multiplicity results for some superlinear elliptic problems on, Comm. Partial Differential Equations 20 (1995) 1725 1741. 3. T. Bartsch and M. Willem, Infinitely many nonradial solutions of a Euclidean scalar field equation, J. Funct. Anal. 117 (1993) 447 460. 4. T. Bartsch and M. Willem, Infinitely many radial solutions of a semilinear elliptic problem on, Arch. Ration. Mech. Anal. 124 (1993) 261 276. 5. H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983) 313 345. 6. H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Ration. Mech. Anal. 82 (1983) 347 375. 7. G. Cerami, Some nonlinear elliptic problems in unbounded domains, Milan J. Math. 74 (2006) 47 77. 8. M. Clapp and T. Weth, Multiple solutions of nonlinear scalar field equations, Comm. Partial Differential Equations 29 (2004) 1533 1554. 9. C. V. Coffman, Lyusternik Schnirelman theory: complementary principles and the Morse index, Nonlinear Anal. 12 (1988) 507 529. 10. M. Conti, L. Merizzi and S. Terracini, Radial solutions of superlinear equations on. I. A global variational approach, Arch. Ration. Mech. Anal. 153 (2000) 291 316. 11. J.-N. Corvellec, M. Degiovanni and M. Marzocchi, Deformation properties for continuous functionals and critical point theory, Topol. Methods Nonlinear Anal. 1 (1993) 151 171. 12. G. Devillanova and S. Solimini, Min-max solutions to some scalar field equations, Adv. Nonlinear Stud. 12 (2012) 173 186. 13. B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209 243. 14. M. Grillakis, Existence of nodal solutions of semilinear equations in, J. Differential Equations 85 (1990) 367 400. 15. N. N. Hirano, Multiple existence of sign changing solutions for semilinear elliptic problems on, Nonlinear Anal. 46 (2001) 997 1020. 16. C. Jones and T. Küpper, On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal. 17 (1986) 803 835.

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