On the second minimax level of the scalar field equation and symmetry breaking

We study the second minimax level $$\lambda _2$$λ2 of the eigenvalue problem for the scalar field equation in $$\mathbb{R }^N$$RN. We prove the existence of an eigenfunction at the level $$\lambda _2$$λ2 when the potential near infinity approaches the constant level from below no faster than $$\mathrm{{e}}^{- \varepsilon \, |x|}$$e-ε|x|. We also consider questions about the nodality of eigenfunctions at this level and establish symmetry breaking at the levels $$2,\ldots ,N$$2,…,N.

p ∈ (2, 2 * ), and 2 * = 2N /(N − 2) if N ≥ 3 and 2 * = ∞ if N = 2. Let Then the eigenfunctions of (1.1) on the manifold M = u ∈ H 1 R N : I (u) = 1 and the corresponding eigenvalues coincide with the critical points and the critical values of the constrained functional J | M , respectively. This problem has been studied extensively for more than three decades. Least energy solutions, also called ground states, are well understood. In general, the infimum is not attained. For the autonomous problem at infinity, the corresponding functional at a radial function w ∞ 1 > 0 and this minimizer is unique up to translations (see Berestycki and Lions [1] and Kwong [2]). For the nonautonomous problem, we have λ 1 ≤ λ ∞ 1 by the translation invariance of J ∞ , and λ 1 is attained if the inequality is strict (see Lions [3]).
As for the higher energy solutions, also called bound states, radial solutions have been extensively studied when the potential V is radially symmetric (see, e.g., Berestycki and Lions [4], Grillakis [5], and Jones and Küpper [6]). The subspace H 1 r (R N ) of H 1 (R N ) consisting of radially symmetric functions is compactly imbedded into L p (R N ) for p ∈ (2, 2 * ) by the compactness result of Strauss [7]. Denoting by m, r the class of all odd continuous maps from the unit sphere S m−1 = y ∈ R m : |y| = 1 to M r = M ∩ H 1 r (R N ), increasing and unbounded sequences of critical values of J | M r and J ∞ | M r can therefore be defined by (1.6) respectively. Furthermore, Sobolev imbeddings remain compact for subspaces with any sufficiently robust symmetry (see Bartsch and Wang [8] and Devillanova and Solimini [9]). Clapp and Weth [10] have obtained multiple solutions without any symmetry assumptions, with the number of solutions depending on N , under a robust penalty condition similar to (1.10) below, but their result does not locate the solutions on particular minimax levels. There is also an extensive literature on multiple solutions of scalar field equations in topologically nontrivial unbounded domains, for which we refer the reader to the survey paper of Cerami [11].
In the present paper, we study the second minimax levels where 2 is the class of all odd continuous maps from S 1 = y ∈ R 2 : |y| = 1 to M. It is known that is not critical (see, e.g., Weth [12]). First we give sufficient conditions for λ 2 to be critical. Recall that for some constant C 0 > 0 and that there are constants 0 < a 0 ≤ √ V ∞ and C > 0 such that if λ 1 is attained at w 1 ≥ 0, then Gidas et al. [13] for the case of constant V ; the general case follows from an elementary comparison argument). Write so that W (x) → 0 as |x| → ∞ by (1.2), and write |·| p for the L p -norm. Our main existence result for the nonautonomous problem (1.1) is the following.
for some constants 0 < a < a 0 and c > 0.
The existence of a ground state was initially proved under the penalty condition V (x) < V ∞ by Lions [3], but it was later relaxed by a term of the order e −a |x| by Bahri and Lions [14]. This can be understood in the sense that the existence of the ground state in the autonomous case is somewhat robust. In our case, the same order of correction is involved with the reverse sign, namely while λ ∞ 2 is not a critical level for J ∞ | M , it requires the enhanced penalty V (x) ≤ V ∞ − c e −a |x| to assure that λ 2 is critical for J | M . We believe that careful calculations will show that this correction cannot be removed, which in turn suggests that the nonexistence of the second eigenfunction in the autonomous case is equally robust as the existence of the first eigenfunction.
Next we consider the higher minimax levels where m is the class of all odd continuous maps from S m−1 to M. By (1.2) and the translation invariance of J ∞ , (1.12) In general, λ ∞ m may be different from the more standard minimax values where A m is the family of all nonempty closed symmetric subsets A ⊂ M with genus We prove the following nonexistence result for the autonomous problem (1.4). (1.14) Hence, none of them are critical for J ∞ | M .
Finally, we prove a symmetry breaking result. Recall that the radial minimax levels λ m, r defined in (1.6) are all critical when V is radial.

Preliminaries
We will use the norm · on H 1 (R N ) induced by the inner product which is equivalent to the standard norm.
Proof The function is continuous since the mappings u → u ± on H 1 (R N ) and the imbedding For u 1 , u 2 ∈ M with u 2 = ±u 1 , consider the path in 2 given by which passes through u 1 and u 2 .
and a straightforward calculation yields the conclusion.
For each y ∈ R N , the translation u → u(· − y) is a unitary operator on H 1 (R N ) and an isometry of L p (R N ), in particular, it preserves M.
Proof By density, u 1 and u 2 can be approximated by functions u 1 , u 2 ∈ C ∞ 0 (R N ) ∩ M, respectively. For all y ∈ R N with |y| sufficiently large, u 1 and u 2 (·− y) have disjoint supports, and by (1.2) and the dominated convergence theorem, so the conclusion follows from Lemma 2.2 and the continuity of J and J ∞ .
We can now obtain some bounds for λ 2 .
Proof Every path γ ∈ 2 contains a point u 0 with u ± 0 p = 1/2 1/ p by Lemma 2.1, and hence This proves the lower bounds for λ 2 since λ 2 ≥ λ 1 in general. The upper bounds follow by applying Lemma 2.3 to minimizing sequences for λ 1 and λ ∞ 1 . For Recall that u is called nodal if both u + and u − are nonzero.
in particular, J ( u ± 0 ) have the same sign as J (u 0 ). Since u ± 0 have disjoint supports, then We can now prove the following nonexistence results. This proposition is well known, but we include a short proof here for the convenience of the reader.
Proof Part (i) is immediate from Lemma 2.5. As for part (ii), there is no solution for λ ∞ 1 < λ < λ ∞ 2 by part (i) since ±w ∞ 1 are the only sign definite solutions (see Kwong [2]). If u 0 ∈ M is a solution of (1.4) with λ = λ ∞ 2 , then u 0 is nodal, so as in the proof of Lemma 2.5, By (1.7), equality holds throughout and u ± 0 are minimizers for λ ∞ 1 . Since any nonnodal solution is sign definite by the strong maximum principle, then both u + 0 and u − 0 are positive everywhere, a contradiction.
Recall that a nodal domain of u ∈ H 1 (R N ) is a nonempty connected component of Testing the Eq. (1.1) for u = u 0 with u 0 , u j gives Thus, for some sequence μ k ∈ R. Since J (u k ), u k = 2 J (u k ) and I (u k ), u k = p I (u k ) = p, then μ k → (2/ p) c.

Lemma 2.8
Any sublevel set of J | M is bounded, and λ 1 > −∞. In particular, any critical sequence u k for J | M is bounded.
Proof Let u k ∈ M be any sequence such that J (u k ) ≤ α < ∞. Then Note that the set D = supp (W − 1/2 V ∞ ) + has finite measure and W ∈ L ∞ (R N ), so the second term on the right hand side, by the Hölder inequality on D, is bounded by C |u k | 2 p and hence bounded. Finally, it remains to note that J is a sum of u 2 and a weakly continuous functional, and thus, it is weakly lower semicontinuous. Since its sublevel sets are bounded, it is necessarily bounded from below.
In the absence of a compact Sobolev imbedding, the main technical tool we use here for handling the convergence matters is the concentration compactness principle of Lions [3,15]. This is expressed as the profile decomposition of Benci and Cerami [16] for critical sequences of J | M , which is a particular case of the profile decomposition of Solimini [17] for general sequences in Sobolev spaces.

Proposition 2.9 Let u k ∈ H 1 (R N ) be a bounded sequence and assume that there is a constant
. . , m}, w (n) = 0 for n ≥ 2, such that, on a renumbered subsequence, w on a renumbered subsequence for some y k ∈ R N with |y k | → ∞, then w solves (1.4) with λ = c by (1.2), in particular,
Remark 2.12 Lemma 2.11 is due to Cerami [11]. We have included a proof here merely for the convenience of the reader.
We now have the following existence results for (1.1).
(1.1) has a solution on M for λ = λ 2 in the following cases: Proof Since J | M satisfies the Palais-Smale condition at the level λ 2 by Lemma 2.11, it is a critical level by a standard argument.

Proof of Theorem 1.1
Noting that λ 2 > λ ∞ 1 if 2 ( p−2)/ p λ 1 > λ ∞ 1 by Proposition 2.4, Theorem 1.1 follows from Proposition 2.13, Proposition 3.1 below, and Lemma 2.14.  Under assumption (1.10), so λ 1 is attained at some function w 1 ≥ 0 (see Lions [3]). Our idea of the proof for part (i) of Proposition 3.1 is to show, analogously, that if |y| is sufficiently large, then Lemma 3.2 Let a 0 be as in (1.9). Then, as |y| → ∞, |x| for some C > 0, which together with (1.9) shows that the integral on the left is bounded by a constant multiple of by the triangle inequality.
(ii) Same as part (i) after the change of variable x → x + y.
(iii) We have since w 1 solves (1.1) with λ = λ 1 , and the last term is of the order O(e −a 0 |y| ) by part (i).
(iv) Using the elementary inequality we have by parts (i) and (ii), and the conclusion follows.
by the Hölder inequality.

Proofs ofTheorems 1.2 and 1.3
Proof of Theorem 1.2 An approximation argument as in the proof of Lemma 2.3 shows that given ε > 0, there is a R > 0 such that, for γ R ∈ N given by we have J ∞ (γ R (y)) < 2 ( p−2)/ p λ ∞ 1 + ε ∀y ∈ S N −1 .
If A ∈ A 2 , then A contains a point u 0 with u ± 0 p = 1/2 1/ p since otherwise is an odd continuous map and hence i(A) = 1, so by Proposition 2.4. So λ ∞ 2 ≥ λ ∞ 2 , and the second equality in (1.14) then follows from the first since λ ∞ 2 ≤ λ ∞ N ≤ λ ∞ N by (1.13).  We close with some questions related to the present paper that remain open.
(i) When is every solution of (1.1) at the level λ 2 nodal? Theorem 1.1 gives only a partial answer. What can be said about the geometry of the nodal domains for nodal solutions at this level? (ii) Assuming that V is radial, and taking into account the symmetry analysis of Gladiali et al. [19], is every solution at the level λ 2 foliated Schwarz symmetric? (iii) Can the enhanced penalty condition (1.10) be relaxed? (iv) Is there an analog of Proposition 2.13 for higher minimax levels?