Fourier-extension estimates for symmetric functions and applications to nonlinear Helmholtz equations

We establish weighted $L^p$-Fourier-extension estimates for $O(N-k) \times O(k)$-invariant functions defined on the unit sphere $\mathbb{S}^{N-1}$, allowing for exponents $p$ below the Stein-Tomas critical exponent $\frac{2(N+1)}{N-1}$. Moreover, in the more general setting of an arbitrary closed subgroup $G \subset O(N)$ and $G$-invariant functions, we study the implications of weighted Fourier-extension estimates with regard to boundedness and nonvanishing properties of the corresponding weighted Helmholtz resolvent operator. Finally, we use these properties to derive new existence results for $G$-invariant solutions to the nonlinear Helmholtz equation $$ - \Delta u - u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}(\mathbb{R}^{N}), $$ where $Q$ is a nonnegative bounded and $G$-invariant weight function.


Introduction
Starting with the pioneering work of Stein (cf. [11]), Tomas [15] and Strichartz [13], Fourier restriction and extension estimates have been receiving extensive attention due to their various applications, especially to partial differential equations. For an overview on classical results and recent progress, we refer the reader to e.g. [5,12,14]. In its classical form, the famous Fourier extension theorem of Stein and Tomas (see e.g. [12,§8: Corollary 5.4]) states that the inverse Fourier transformF σ of F ∈ L 2 (S N −1 ), given by belongs to L q (R N ) for N ≥ 2 if q ≥ 2(N +1) N −1 , and that (1.1) with a constant C > 0 depending only on q and N . Here S N −1 denotes the (N − 1)−dimensional sphere in R N and dσ the Lebesgue measure on S N −1 . Due to the Knapp example given by the characteristic function of a small spherical cap, this range of exponents is known to be sharp for arbitrary functions, see e.g [14,Chapter 4]. On the other hand, it is a natural question whether the range of exponents can be improved both by considering weighted L q -norms and by restricting to functions having additional symmetries. A well known and classical observation in this context yields that case (1.1) holds for q > 2N N −1 and radial (and thus constant) functions F ∈ L 2 (S N −1 ), see e.g. [12, §8: Proposition 5.1].
In the present paper, we analyze this question for more general symmetries with respect to closed subgroups of O(N ). For this we introduce the following definition.
Here and in the following, a function F ∈ L 2 (S N −1 ) is called G-invariant if F (Aσ) = F (σ) for every σ ∈ S N −1 , A ∈ G. By the remarks above, ({id}, q, 1) is an admissible extension triple if q ≥ 2(N +1) N −1 and (O(N ), q, 1) is an admissible extension triple if q > 2N N −1 . If we consider weight functions Q ∈ L s (R N ) for suitable s < ∞, then the range of exponents giving rise to admissible extension triples can be readily extended by applying Hölder's inequality to the LHS of (1.2). In particular, this yields that ({id}, q, Q) is an admissible extension triple if Q ∈ L s (R N ) for some s ∈ 2(N +1) N +3 , ∞) and q ≥ 2s(N +1) 2(N +1)+s(N −1) . Moreover, (O(N ), q, Q) is an admissible extension triple if Q ∈ L s (R N ) for some s ∈ 2N N +1 , ∞ and q > 2sN 2N +s(N −1) . In the present paper, we are interested in weight functions Q ∈ L ∞ (R N ), where Hölder's inequality does not yield an extended range of admissible exponents. The main aims of the paper are the following. First, we wish to detect a class of admissible extension triples corresponding to nontrivial subgroups of O(N ) and corresponding to functions Q ∈ L ∞ (R N ) which are not s-integrable for any s < ∞. Second, starting from a range of admissible extension triples (G, q, Q), we wish to derive selfdual (L p ′ , L p )-estimates for the restriction of mappings of the form With regard to our first aim, we focus our attention to the subgroups Moreover, we consider weight functions of the form Q α = 1 Lα for the set where a > 0 is an arbitrary fixed number and α > 0. Since |L α | = ∞, we have Q α ∈ L s (R N ) for any α > 0, s < ∞.
We note that, in Theorem 2.2 below, we shall in fact prove a generalization of this result for characteristic functions of sets of the form L α,β := {x ∈ R N : |x (N −k) | ≤ a max{|x (k) | −α , |x (k) | −β }} with α > β > 0. Regarding Theorem 1.1(ii), we note in particular that λ N,k,α = 0 for α = k N −k , so (G k , q, Q α ) is an admissible extension triple for every q ≥ 1 in this case. More generally, the latter property holds if α ∈ ( k+1 2(N −k) , 2k N −k+1 ), since then we have λ N,k,α < 1. The main part of the proof of this Theorem consists in a detailed asymptotic study of one-dimensional integrals which arise after integrating along the orbits of G k . Here, the well-known bound for the Fourier transform of the standard measure dσ k on S k−1 will play a key role (see e.g. [12, §8: We also remark that, if (G, q, Q) is an admissible extension triple and Q ′ : R N → C is a measurable function with |Q ′ | ≤ |Q| in R N , then, by definition, (G, q, Q ′ ) is also an admissible extension triple. Consequently, the statement of Theorem 1.1 extends to functions Q ∈ L ∞ (R N ) with |Q| ≤ c1 Lα in R N for some c > 0.
Next we state our main result on (L p ′ , L p )-Helmholtz resolvent estimates for G-invariant functions. Here and in the following, for r ∈ [1, ∞], we let L r G (R N ) denotes the closed subspace of G-invariant functions in L r (R N ).
Here and in the following, S G ⊂ S denotes the subspace of G-invariant functions in the Schwartz space S.
Our proof of Theorem 1.2 is based on the strategy used in [7] and [8], see also [3]. We recall that a selfdual estimate of the form (1.7) has been proved for the Helmholtz resolvent R in place of R Q in the range of exponents p ∈ [ 2(N +1) N −1 , 2N N −2 , while non-selfdual estimates were obtained in [7]. Clearly, these already available (L r , L s )-estimates for R extend, by approximation, to the weighted resolvent R Q in the case where Q ∈ L ∞ (R N ). Theorem 1.2 yields an improvement of these estimates, for R Q and G-invariant functions, in the case where (G, q, Q) is an admissible extension triple for some q < 2(N +1) N −1 , which is equivalent to the inequality 2N Under the assumptions of Theorem 1.2, it follows, by density, that the weighted resolvent R Q extends to a bounded linear operator . In our next result we state that, under the same assumptions, a nonvanishing property in the spirit of [3, Theorem 3.1] holds.
exist -after passing to a subsequence -numbers R, ζ > 0 and a sequence of points (x n ) n∈N ⊂ R N with In the special (non-symmetric) case G = {id}, Q ≡ 1, q = 2(N +1) N −1 , this theorem reduces to [3, Theorem 3.1]. Here we note that 2N The general strategy of the proof of Theorem 1.3 is inspired by [3,Theorem 3.1]. However, additional difficulties, related to the fact that the multiplication with Q ∈ L ∞ (R N ) does not map S into itself, lead to a somewhat more involved argument. Theorems 1.2 and 1.3 are useful in the study of real-valued G-invariant solutions of the nonlinear Helmholtz equation (1.4) with a real-valued weight function Q ∈ L ∞ G (R N ), where G ⊂ O(N ) is a given closed subgroup. In the following, we focus on dual bound state solutions, which arise as solutions u ∈ L p (R N ) of the integral equation u = R Q|u| p−2 u , where R is the real part of the resolvent operator R, see Section 5 for details. Our first main result in this context is the following.
be a closed subgroup, and let Q ∈ L ∞ G (R N ) be a real-valued nonnegative function with Q ≡ 0 and with the property that . Finally, we point out that assumption (1.8) holds in particular for functions Q ∈ L ∞ (R N ) satisfying |Q| ≤ c1 Lα for some c, α > 0, where L α is given in (1.6). Using this fact, the following corollary can be deduced from Theorems 1.1 and 1.4.
be a nonnegative function with Q ≡ 0 and satisfying |Q| ≤ c1 Lα for some c > 0 with L α given in (1.6).

Then (1.4) admits a nontrivial G k -invariant dual bound state solution if one of the following holds.
( , We point out that, in contrast to Corollary 1.5 and Theorem 1.6, Corollary 1.7 allows to consider exponents p < 2(N +1) N −1 . To put our existence results for (1.4) into perspective, we recall some previous results. In [7], the existence of small complex solutions has been proved via the use of contraction mappings in dimensions N = 3, 4, p = 3 and Q = ±1. A variant of this technique is developed in [9], where continua of small real-valued solutions of (1.4) are detected for a larger class of nonlinearities. The dual variational approach to (1.4) was introduced in [3], where the existence of nontrivial dual bound state solutions was proved for p ∈ 2(N +1) N −1 , 2N N −2 and for nonnegative weight functions Q ∈ L ∞ (R N ) \ {0} which are either Z N -periodic or satisfy the uniform decay assumptions Q(x) → 0 as |x| → ∞. Under additional restrictions on Q, this approach was extended to the Sobolev critical case p = 2N N −2 in [4]. Moreover, a dual approach in Orlicz spaces was developed in [2] to treat more general nonlinerities in (1.4). The defocusing case Q ≤ 0 in (1.4) and radial solutions are considered in [10]. We are not aware of any previous work where symmetries different from radial symmetry are used to extend the range of admissible exponents to values below the Stein-Tomas exponent 2(N +1) N −1 and to overcome lack of compactness issues in the context of (1.4). . The paper is organized as follows. In Section 2, we derive a Fourier extension estimate for G k -invariant functions, where G k is defined in (1.5). In particular, we prove a generalization of Theorem 1.1. In Section 3, we provide weighted Helmholtz resolvent estimates relative to a given admissible extension triple, thereby giving the proof of Theorem 1.2. In Section 4 we study related nonvanishing properties, and we give the proof of Theorem 1.3. Finally, Section 5 is devoted to our main existence results for dual bound state solutions of (1.4).
We close this introduction by fixing some notation. Throughout the paper we denote by B r (x) the open ball in R N with radius r > 0 and center at x. Moreover, we set B r = B r (0) and S N −1 for the boundary of B 1 =: B. The constant α N represents the volume of the unit ball B 1 in R N . For Moreover by B (k) we denote the unit ball in R k . By 1 L we denote the characteristic function of a measurable set L ⊂ R N . Furthermore, we shall indifferently denote by f or F(f ) the Fourier transform of a function in R N given by and byF σ the inverse Fourier transform of an admissible functions F defined on S N −1 viǎ For 1 ≤ s ≤ ∞, we abbreviate the norm on L s (R N ) by · s . The Schwartz-class of rapidly decreasing functions on R N is denoted by S. For any p ∈ (1, ∞) we always denote by p ′ := p p−1 the Hölder conjugate of p.
Acknowledgement: T. Weth and T. Yesil are supported by the German Science Foundation (DFG) within the project WE-2821/5-2.

Fourier extension estimates for G k -invariant functions
We recall that, for a function F ∈ L 2 (S N −1 ), we define the (inverse) Fourier transform of F dσ byF For F ≡ 1 we use the notationď and will often omit the dimensional index if no confusion is possible. We point out that this function satisfies the key uniform bound with h F given in (2.2). Moreover, for all x ∈ R N .
Proof. By using slice integration (see e.g. [1, A.5]) we havě for all x ∈ R N with h F given in (2.2), as claimed in (2.3). In particular, we get For α > β > 0 and fixed a > 0, we now consider the subset We shall prove the following generalization of Theorem 1.1. (1.6), and let Suppose furthermore that q ≥ 1 and k ∈ {1, . . . , N − 1} satisfy Then there exists a constant C = C(N, k, α, β, a) with the property that Proof. In the following, the letter C stands for positive constants depending only on N, k, α, β and a. Let F ∈ C(S N −1 ) be a G k -invariant function, and assume without loss of generality that Since Q = 1 L α,β , we have, by (2.1), Hence, for these values of β and q we have and the latter integral is finite iff k−2 Here we note that the last inequality is already implied by (2.9). To estimate and therefore which is finite if (2.10) holds. Moreover, integrating over the set B r gives provided that If this holds, then integration over r gives and the latter integral is finite if which is already guaranteed by (2.11). Moreover, (2.13) holds iff We now observe that if k ∈ {2, . . . , N − 1} and (2.6) holds, then all required conditions (2.9), (2.10), (2.11), (2.14) and (2.15) are satisfied. The same is true in the case k = 1 if we assume in addition that q < 2 + 2 β α , which by the first line in (2.6) implies that q < 2 + 2(N − 1)β. So in these cases we conclude, using (2.8) and the fact that H(r) = H 1 (r) + H 1 2 (r) + H 2 2 (r) for r ∈ (0, 1), that QF σ q < ∞. However, since we can always interpolate with the trivial estimate assumptions on upper bounds on q can be removed a posteriori. The proof is thus finished.
We note that Theorem 1.1 is a direct consequence of Theorem 2.2, since the assumptions of Theorem 1.1 imply those of Theorem 2.2 in the case α = β.
Moreover, we have the following duality property.
Here and in the following, S G denotes the subspace of G-invariant functions in S.
In particular, this holds if G = G k and N, k, α, β, q and Q satisfy the assumptions of Theorem 2.2.
Proof. Let f ∈ S G and F := Qf S N−1 ∈ L 2 (S N −1 ). Then we have and therefore F L 2 (S N−1 ) ≤ C f q ′ , as claimed.

Resolvent estimates for G-invariant functions
For N ≥ 3, the radial outgoing fundamental solution of the Helmholtz equation −∆u − u = δ 0 in R N is given by where H , as |x| → ∞. Moreover, it is known (see [6]) that, in the sense of tempered distributions, the Fourier transform of Φ is given by .
In order to prove Theorem 3.1, we adapt the strategy of [7] and [8], see also [3]. Throughout the remainder of this section, we fix a closed subgroup G ⊂ O(N ) and Q ∈ L ∞ G (R N ). We first note the following lemma which is a basic consequence of complex interpolation.
Lemma 3.2. Let 1 ≤ q < ∞ and let ρ ∈ S be a radial function. Suppose furthermore that Proof. Since ρ ∈ S is radial, the convolution with ρ maps G-invariant functions to G-invariant functions. Moreover, by assumption we have for all u, v ∈ S G . By duality, we therefore have Q ρ * (Qu) L q ≤ CD u L 2 for all u ∈ S G . Complex interpolation of this estimate with the assumed (L q ′ , L 2 ) estimate gives q+2 , ∞ . Then we have 1 p = θ s for some θ ∈ (0, 1) and therefore, by complex interpolation of (3.4) with the assumed (L 1 , L ∞ )-estimate, Clearly, A p,q < 0 if and only if p > 2N q+2 .
Next we turn to the operator R 1 Q , and we assume from now on that q ∈ 1, 2(N −1) is given such that (G, q, Q) is an admissible extension triple.
and ϕ ≡ 0 for ||ξ| − 1| ≥ 1 2 . By construction of Φ 1 , we then have Φ 1 = Φ 1 ϕ, which means that Then we have the dyadic composition Using the asymptotics of Φ 1 we deduce where the constant C > 0 is independent of j. Using that Φ j 1 is radial, we get, with Plancharel's theorem and Lemma 2.3, where the constant does not depend on j. Consequently, we thus have , for all j, which implies that Applying Lemma 3.2 to the radial kernel Φ j 1 * ϕ ∈ S(R N ) gives q+2 , ∞ , we have A p,q < 0. Since, as remarked above, Proof of Theorem 3.1. The claim follows readily by combining Lemma 3.3 and Proposition 3.4.

Nonvanishing for G-invariant functions
Our next aim is to deduce a nonvanishing theorem for the operator R Q and G-invariant functions, where again G ⊂ O(N ) is a closed subgroup and Q ∈ L ∞ G (R N ) is a given weight function. We restate Theorem 1.3 for the reader's convenience.
exist -after passing to a subsequence -numbers R, ζ > 0 and a sequence of points (x n ) n∈N ⊂ R N with The remainder of this section is devoted to the proof of this theorem. For this we fix q ∈ 1, 2(N −1) N −3 such that (G, q, Q) is an admissible extension triple. Moreover, we keep using the notation of the previous section, so we write Φ = Φ 1 + Φ 2 and R Q = R 1 Q + R 2 Q . We need to analyze the operators R 1 Q and R 2 Q separately. We start by proving the following variant of Proposition 3.4 for the operator R 1 Q .
Proof. It suffices to prove the assertion for k ≥ 4. Using the given function η, we let Φ j 1 , j ∈ N ∪ {0} be defined as in the proof of Proposition 3.4. As in the proof of this proposition, we see that j and all functions f ∈ S G . Moreover, by construction, we have the dyadic decomposition for all functions f ∈ S G , as claimed.
Proof. Let, as in the assumptions of Proposition 4.2, ϕ ∈ S(R N ) be such that ϕ ∈ C ∞ c (R N ) is radial, 0 ≤ ϕ ≤ 1 with ϕ ≡ 1 for ||ξ| − 1| ≤ 1 4 and ϕ ≡ 0 for ||ξ| − 1| ≥ 1 2 . Moreover, let w n = ϕ * (Qv n ). Then we have (4.2) w n p ′ ≤ ϕ 1 Qv n p ′ ≤ ϕ 1 Q ∞ v n p ′ for all n ∈ N by Young's inequality, so (w n ) n is also a bounded sequence in L p ′ (R N ) by assumption. Since Φ 1 = Φ 1 ϕ, we have (2π) N 2 Φ 1 = Φ 1 * ϕ. Therefore, with η k defined as in Proposition 4.2, we can write by Hölder's inequality and Proposition 4.2. Since A p,q < 0, it follows that For fixed k ∈ N, we now choose R = 2 k+1 , which implies that η k ≡ 0 on R N \ B R . Decomposing R N into disjoint N -cubes {Z l } l∈N of side length R, and considering for each l the N − cube Z ′ l with the same center as Z l but with side length 3R, we find , where we used (4.2) in the last step. By assumption, it now follows that The claim now follows by combining (4.3) and (4.4).
Regarding Φ 2 we make use of the following variant of [4, Theorem 2.5].
Proof. The claim follows from [4,Theorem 2.5] in the case where v n ∈ S for every n ∈ N. If (v n ) n is an arbitrary bounded sequence in L p ′ (R N ), we first recall that, by (3.6), there exists a constant Moreover we choose, by density,ṽ n ∈ S with v n −ṽ n p ′ ≤ 1 n for every n ∈ N. The assumption then implies that also lim n→∞ sup y∈R N Bρ(y) |ṽ n | p ′ dx = 0, for all ρ > 0 and therefore and thus also as claimed.
We are now in position to finish the proof of Theorem 4.1: Proof of Theorem 4.1. Let (v n ) n ⊂ L p ′ G (R N ) be a bounded sequence, and suppose by contradiction that (4.1) does not hold. Then we have lim n→∞ sup y∈R N Bρ(y) |Qv n | p ′ dx = 0, for all ρ > 0.
By density, we may chooseṽ n ∈ S G with v n −ṽ n p ′ ≤ 1 n for every n ∈ N, which implies that Qv n − Qṽ n p ′ ≤ Q ∞ n for all n and therefore also lim n→∞ sup y∈R N Bρ(y) |Qṽ n | p ′ dx = 0, for all¸ρ > 0.
Combining Lemma 4.3 (applied toṽ n ) and Lemma 4.4 (applied to Qṽ n ), we then deduce that Moreover, by Theorem 3.1 we have Consequently, we also have that R N v n R Q v n dx → 0 as n → ∞, contrary to the assumption. The claim thus follows.

Dual variational framework and G−invariant solutions
Let G ⊂ O(N ) be a fixed closed subgroup, and let Q ∈ L ∞ G (R N ) be a nonnegative fixed weight function with Q ≡ 0. We now focus our attention to the equation To prove the existence of nontrivial real-valued solutions of (5.1), we will use the dual variational approach introduced in [3] and consider the operator K Q formally defined as where R denotes the real part of the Helmholtz resolvent operator R, i.e., Rg = Re Φ * g with the fundamental solution Φ defined in (1.3).
To analyze the mapping properties of K Q and to set up a variational framework, we assume, as in Theorem 1.4, that q ∈ 1, 2(N −1) q+2 , 2}, 2N N −2 are chosen such that (G, q, Q 1 p ) is an admissible extension triple. By Theorem 1.2, so this applies in particular to t = p. Moreover, K Q is symmetric in the sense that We also note the following non-selfdual estimates for the operator K Q .
Lemma 5.1. There exist σ 1 < p < σ 2 with the property that K Q is bounded as a map L Then there exists τ 1 > s and τ 2 ∈ (p, s) with 2 N +1 < 1 s ′ − 1 τ i < 2 N for i = 1, 2, which, by the non-selfdual estimates of Gutierrez in [7,Theorem 6], implies that We now fix t ∈ 2N N −1 2q q+2 , p , and we recall that (5.4) K Q is bounded as a map L t ′ G (R N ) → L t G (R N ). Moreover, for i = 1, 2, we let θ i ∈ (0, 1) be defined by Complex interpolation of (5.3) and (5.4) then yields that K Q is bounded as a map L and therefore σ 1 < p < σ 2 . This finishes the proof of the first mapping property, and the second mapping property follows by duality.
Next we note that the functional J has a mountain pass geometry. More precisely, we have: There exists a Palais-Smale sequence for J at the mountain pass level Since p > 2, the parts (i)-(iii) are proved in [3,Lemma 4.2] for G = {id}, and the proof remains the same for general closed subgroups G ⊂ O(N ). Moreover, the positivity of the mountain pass level c defined in (5.5) is a direct consequence of (i) and (ii), which also shows that the set Γ is nonempty. Finally, the proof of the existence of a Palais-Smale sequence for J at level d is exactly the same as the proof of [3, Lemma 6.1]. Here we note that periodicity of Q was assumed in [3,Section 6], but this property is not used in Lemma 6.1.
Moreover, suppose that one of the following conditions hold: (A1) For some R > 0, we have lim Then, after passing to a subsequence, we have Proof. We first note that (v n ) n is bounded by Lemma 5.4. Consequently, since Moreover, by assumption, which implies that Since moreover (G, q, Q 1 p ) is an admissible extension triple by assumption, Theorem 1.3 applies and yields δ, R > 0 and a sequence of points (x n ) n ⊂ R N such that, after passing to a subsequence, We claim that (x n ) n has to be bounded. To see this, we argue by contradiction and assume that, after passing to a subsequence again, |x n | → ∞. We distinguish two cases. Case 1: (A1) holds. In this case we put ϕ n := Q p−1 v n 1 B R (xn) , and we note that (ϕ n ) n is a bounded sequence in L p ′ (R N ). Moreover, we have as n → ∞. By Lemma 5.1, there exists σ > p ′ and C > 0 with the property that whereas, since σ ′ < p ′ and by Hölder's inequality, 8), it thus follows that ϕ n σ ′ → 0 as n → ∞. Here we note that, by an easy covering argument, (1.8) holds for every R > 0 if it holds for one R > 0. Going back to (5.8), we thus deduce that which contradicts (5.7). Case 2: (A2) holds.
In this case it follows from (5.7) and the fact that v n and Q are G-invariant that as n → ∞, which contradicts the boundedness of the sequence (v n ) n in L p ′ (R N ). Since in both cases we have reached a contradiction, we conclude that (x n ) n ist bounded. Therefore, making R larger if necessary, we can assume that (5.7) holds with x n = 0 for all n ∈ N. Now for any fixed G-invariant function ϕ ∈ C ∞ c (R N ), any r > 0 and n, m ∈ N we have So by assumption and the local compactness of K Q , as stated in Lemma 5.2, we get that (|v n | p ′ −2 v n ) n∈N is a Cauchy sequence in L p (B R ). Consequently, |v n | p ′ −2 v n →ṽ strongly in L p (B R ) for somẽ v ∈ L p (B R ), and passing to a subsequence also pointwisely almost everywhere on B R . This clearly implies that v n → |ṽ| p−2ṽ almost everywhere on B r . Now (5.6) and the uniqueness of the weak limit givesṽ = |v| p ′ −2 v and using the local strong convergence of |v n | p ′ −2 v n and the continuity of linear operator K Q : L p ′ G (R N ) → L p G (R N ). By density, it now follows that J ′ (v)w = 0 for every w ∈ L p ′ G (R N ), i.e., v ∈ L p ′ G (R N ) \ {0} is a critical point of J.
We now have all the tools to complete the proofs of our main existence results for nontrivial G-invariant dual ground state solutions as stated in the introduction.
Proof of Theorem 1.4 (completed). By Lemma 5.4(iv), there exists a Palais-Smale sequence (v n ) n in L p ′ G (R N ) for J at the mountain pass level d > 0. By Proposition 5.5, we have v n ⇀ v in L p ′ G (R N ) after passing to a subsequence, where v ∈ L p ′ G (R N ) is a nontrivial critical point of J. Here we note that assumption (A1) of Proposition 5.5 is satisfied by (1.8). The proof is finished by Lemma 5.3.
Proof of Corollary 1.5. Since Q 1 p ∈ L ∞ (R N ), it follows by the classical Stein-Tomas estimate that (G, q, Q 1 p ) is an admissible extension triple for q = 2(N +1) N −1 . Since the assumptions of Theorem 1.4 are satisfied and yield the existence of a nontrivial solution v ∈ L p ′ G (R N ) of (1.4).
Proof of Theorem 1.6. As above, it follows by the classical Stein-Tomas estimate that (G k , q, Q 1 p ) is an admissible extension triple for q = 2(N +1) N −1 , whereas Moreover, since 2 ≤ k ≤ N − 2, we have lim |x|→∞ N G (x, R) = ∞ for R > 0, where N G (x, R) is defined as in Proposition 5.5. Hence assumption (A2) of Proposition 5.5 is satisfied, and thus the proof is completed as the proof of Theorem 1.4 above.
To see the latter, it suffices to consider a sequence (x n ) n = (x (N −k) n , x (k) n ) ⊂ R N −k × R k with x (N −k) n = 0 for all n ∈ N and r n := |x n | = |x (k) n | → ∞ as n → ∞. In this case we have |x (k) −x (k) n | < R for x ∈ B R (x n ) and therefore = C(r n − R) −(N −k)α → 0 as n → ∞ with constants C > 0. Hence (5.9) holds. We now first consider the case where k = 1 and α > 1 N −1 or k = N − 1 and α < N − 1. In this case, (G k , q, 1 Lα ) is an admissible extension triple for every q ≥ 1 by Theorem 1.1. Since moreover, 0 ≤ Q 1 p ≤ c . Thus the claim holds in this case.
Next we consider the case where 2 ≤ k ≤ N − 2 and p ∈ (µ N,k,α , 2N N −2 ), where µ N,k,α is defined in (1.9). A case distinction shows that µ N,k,α = max{ 2N N −1 2λ λ+2 , 2}, where λ := λ N,k,α is given in Theorem 1.1. Consequently, by Theorem 1.1, we may fix q ∈ (µ N,k,α , p) with max{ 2N N −2 and the property that (G k , q, 1 Lα ) is an admissible extension triple. As above, it follows that also (G k , q, Q 1 p ) is an admissible extension triple. Again, Theorem 1.4 applies and yields that (1.4) admits a nontrivial dual bound state solution. Thus the claim holds in this case as well.
Remark 5.6. We note that Corollary 1.7 extends to the case where L α is replaced by the more general class of sets L α,β considered in Theorem 2.2. For this, one has to replace the assumption α > 1 N −1 by β > 1 N −1 in the case k = 1. Moreover, in the case 2 ≤ k ≤ N − 2, the value µ N,k,α needs to be replaced by max{ 2N N −1 2λ λ+2 , 2}, where now λ = λ N,k,α,β is given in (2.5).