Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity

We prove new existence results for a nonlinear Helmholtz equation with sign-changing nonlinearity of the form -Δu-k2u=Q(x)|u|p-2u,u∈W2,pRN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - \Delta u - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}\left( {\mathbb {R}}^{N}\right) \end{aligned}$$\end{document}with k>0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>0,$$\end{document}N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \ge 3$$\end{document}, p∈2(N+1)N-1,2NN-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p \in \left[ \left. \frac{2(N+1)}{N-1},\frac{2N}{N-2}\right) \right. $$\end{document} and Q∈L∞(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q \in L^{\infty }({\mathbb {R}}^{N})$$\end{document}. Due to the sign-changes of Q, our solutions have infinite Morse-Index in the corresponding dual variational formulation.

In the present article, we consider nonlinear Helmholtz equations of the form for p ∈ 2(N +1) N −1 , 2N N −2 and k > 0 with a weight function Q ∈ L ∞ (R N ) that may change sign. One of the main tools for proving the existence of solutions is the dual variational method that, in its classical form, relies on the nonnegativity of the potential Q. In the context of Nonlinear Helmholtz equations this method was first implemented in a paper by Evéquoz and Weth [4]. To highlight the role of the nonnegativity of Q we briefly recapitulate the approach.
Instead of (1.2) one considers a reformulation as the integral equation where R is the real part of a resolvent type operator R, i.e., a right inverse of the Helmholtz operator − − k 2 on R N . For f ∈ S(R N ) the operator R is given by is the fundamental solution of the Helmholtz equation associated with Sommerfeld's outgoing radiation condition , as |x| → ∞. ) stands for the Bessel function of the second kind of order N −2 2 . It is known [7,Theorem 2.3] that R extends as a continuous linear map from L p (R N ) → L p (R N ) precisely for p ∈ 2(N +1) N −1 , 2N N −2 . To explain this briefly notice that the Fourier symbol of R is the tempered distribution (|ξ | 2 − k 2 − i0) −1 . Its behaviour at infinity ∼ |ξ | −2 is responsible for the upper bound 2N N −2 whereas frequencies |ξ | ≈ k give rise to the lower bounds given by the Stein-Tomas exponent p ST := 2(N +1) N −1 . Recall that the Stein-Tomas inequality reads Here, S N −1 denotes the unit sphere in R N and 2(N +1) N +3 = p ST is the dual Stein-Tomas exponent.
Being given these mapping properties of R and hence of R = Re(R) we introduce the dual variableũ := Q 1/ p |u| p−2 u and observe that solutions of (1.3) are precisely the critical points of the (dual) energy functional I : L p (R N ) → L p (R N ) given by see [4,Lemma 4.1]. We point out that the proof of this fact uses that Q is nonnegative. Under the additional assumption that Q vanishes at infinity, one obtains that I is an odd functional of class C 1 that has the Mountain Pass Geometry and satisfies the Palais-Smale Condition. So the existence of an unbounded sequence of solutions to (1.3) follows from the Symmetric Mountain Pass Theorem. Inverting the transformation u →ũ one thus obtains an unbounded sequence of solutions to the nonlinear Helmholtz equation (1.2). This is the strategy proposed by Evéquoz and Weth [4] for the focusing nonlinear Helmholtz equation where Q ≥ 0. We refer to [9] for analogous results in the defocusing case Q ≤ 0, where the dual variational approach was implemented for the dual variableũ := |Q| 1/ p |u| p−2 u. In view of these two results it is natural to ask for a dual variational approach work in the intermediate case of sign-changing Q. In this paper, we provide a solution for this problem.
To treat sign-changing coefficients Q ∈ L ∞ (R N ) we have to come up with a new idea to make the dual variational approach work. We write Q = Q + − Q − where Q ± := |Q|1 A ± and In fact we will consider Q λ := λQ + − Q − for λ > 0 in the following. Our main idea is to introduce the dual variables (φ, ψ) and to derive a coupled system of nonlinear integral equations the solutions of which are precisely the critical points of an associated strongly indefinite dual energy functional. We will see that the indefiniteness comes from the presence of Q − and thus vanishes in the case of a nonnegative function Q ≥ 0. In particular, the critical points of this dual energy functional will have infinite Morse index, which clearly distinguishes these solutions from the dual bound and ground states obtained in [4] in the case Q ≥ 0. We will explain the dual variational framework in detail in Sect. 2. Our conditions for the existence of critical points involve the linear operator K : p f ) as well as the numbers Since we will assume K to be compact, both values are indeed attained. Moreover we have β ≥ 0 and [4, Lemma 4.2(ii)] gives α > 0 once we assume that A + has positive measure, i.e., Q + ≡ 0. Our main result reads as follows.
Then for almost all λ > λ 0 : (1.10) The proof relies on a combination of a saddle-point reduction and the abstract monotonicity trick by Jeanjean-Toland [6], which provides bounded Palais-Smale sequences (only) for almost all λ > λ 0 . It would clearly be desirable to extend our result to all λ > λ 0 , but related a priori bounds seem to be out of reach. Notice also that [9, Theorem 1.4] suggests the existence of nontrivial solutions also for small λ > 0, possibly assuming the set {Q > 0} to be small enough and following a different variational approach. Let us point out that λ 0 is small provided that the number β, which is the same as the operator norm In the following Corollary, we show that the abstract conditions (1.9) hold for a reasonable class of sign-changing functions Q ∈ L ∞ (R N ). If for instance Q vanishes at infinity, then It is less immediate to verify the non-negativity assumption on the bilinear form (1.9). From [2, Corollary 5.4] we infer that this condition holds for measurable sets A − with small enough diameter. To be more precise, if y N −2 2 denotes the first (positive) zero of the Bessel π t cos t, thus y 1/2 = π/2 and y N −2 2 > y 1/2 for N > 3 (see [1], [Section 9.5]). We thus conclude as follows.
The regularity results in Theorem 1.1 and Corollary 1.2 are direct consequences of [4, Lemma 4.3] and of the iteration procedure from Step 3 in the proof of [8, Theorem 1]. Notice that [8, Theorem 1] provides solutions to far more general Nonlinear Helmholtz equations than (1.2) regardless of whether sign-changes occur or not, but the constructed solutions are small. This result relies on a fixed point approach. Let us also mention [3] where nontrivial solutions of Nonlinear Helmholtz equations are constructed for rather general and possibly sign-changing nonlinearities vanishing identically outside some compact subset of R N . Our method is entirely different from any of these approaches. This paper is organzied as follows: In Sect. 2 we introduce our basic tools and develop the dual variational framework by reformulating the indefinite Nonlinear Helmholtz equation as a coupled system of integral equations. Then we prove that nontrivial critical points of the associated energy functional J λ are indeed nontrivial solutions u ∈ L p (R N ) of the integral equation u = R(Q λ |u| p−2 u). This motivates the search for critical points of J λ . In Sect. 3 we perform the saddle-point reduction of (φ, ψ) → J λ (φ, ψ) leading to a reduced functioñ J λ that depends on φ only. In Sect. 4 we establish the existence of bounded Palais-Smale sequences for these reduced functionals for almost all λ > λ 0 . As mentioned above, this step entirely relies on the monotonicity trick by Jeanjean and Toland [6]. Finally, we combine all the auxiliary results to prove Theorem 1.1 and Corollary 1.2 in Sect. 5.
Let us 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 ∈ R N . Moreover, we set B r = B r (0) for any r > 0. For 1 ≤ s ≤ ∞, we abbreviate the standard norm on L s (R N ) by · s . The Schwartz-class of rapidly decreasing functions on R N is denoted by S(R N ). For any p ∈ (1, ∞) we always denote by p := p p−1 the Hölder conjugate of p. The indicator function of a measurable set B ⊂ R N is 1 B . By diam (B) we always denote the diameter of a set. We will always use the symbols φ, ψ to denote L p (A + )− and L p (A − )-functions that are continued trivially to the whole of R N .

Dual variational formulation
In this section we will formulate a variational framework to the Eq. (1.10). We recall from the introduction that solutions of our problem are obtained as solutions of the integral equation where R( f ) = * f for the function introduced in (1.5) and for all p ∈ 2(N +1) N −1 , 2N N −2 and some constant C > 0. To obtain the dual variational formulation of (2.1) we introduce v := 1 A + u and w := 1 A − u.
In terms of the Birman-Schwinger operator K : f → |Q| 1 p R |Q| 1 p f introduced above this can be reformulated as and therefore carries a variational structure through the (dual) energy functional J λ on L p (A + ) × L p (A − ) given by This functional is of class C 1 with Here ∂ 1 , ∂ 2 stand for partial derivatives with respect to φ and ψ. For this reason we will look for critical points of J λ . These solve the integral equation ( as well as So we conclude that it remains to find nontrivial critical points of the functionals J λ for as many λ > 0 as possible. This will be achieved with the Mountain Pass Theorem for families of C 1 -functionals by Jeanjean and Toland [6].

Saddle-point reduction
In this section we perform the saddle-point reduction of J λ with respect to the ψ-variable. To this end, we prove that for any fixed φ ∈ L p (A + ) the functional ψ → J λ (φ, ψ) attains its maximum at some uniquely defined function in L p (A − ) that we will call Z (φ) in the following. We shall see that the positivity assumption ensures that the functional ψ → J λ (φ, ψ) is strictly concave so that the global maximization with respect to ψ is the only reasonable approach to perform a saddle point reduction. We introduce the reduced functionalJ λ : such that for all λ > 0 we haveJ Moreover: Proof We first establish the existence of a maximizer. So fix φ ∈ L p (A + ) and consider a maximizing sequence (ψ n ) n ⊂ L p (A − ). Using J λ (φ, 0) ≤ sup Here we used the nonnegativity assumption on K as well as (1.8). Hence, so (ψ n ) n is bounded. Passing to a subsequence we find ψ * ∈ L p (A − ) such that ψ n ψ * in L p (A − ) as n → ∞. Using the compactness of K and the weak lower semicontinuity of the norm we find Hence the supremum is attained at ψ * . Since equality must hold in the above estimate we conclude ψ n p → ψ * p , whence ψ n → ψ * in L p (A − ) as n → ∞. This shows the existence of a maximizer satisfying the estimate stated in (i). So (i) is proved once we have established the uniqueness of the maximizer.
To this end assume that ψ * , ψ † ∈ L p (A − ) are maximizers. Then we have where we have used the nonnegativity condition in the second last step and the convexity of z → |z| p in the last step. So we have equality in each estimate and conclude ψ * = ψ † . Note that the maximizer does not depend on λ since the only λ-dependent term in J λ (φ, ψ) is λ 1− p p φ p p , which is independent of ψ. We now prove (ii), i.e., the continuity of the map Z : Assume φ n → φ in L p (A + ) and let (ψ n ) n := (Z (φ n )) n ⊂ L p (A − ) be the associated maximizers. By (3.3), the sequence (ψ n ) n is bounded and after passing to a subsequence we may assume ψ n ψ 0 in L p (A − ) as n → ∞. Arguing as above we deduce using weak lower semicontinuity and lim inf n→∞ ψ n p ≥ ψ 0 p . On the other hand, with the (3.5) Combining both estimates gives ψ n p → ψ 0 p as well asJ λ (φ n ) →J λ (φ) as n → ∞.
Thus we have equality in (3.4), (3.5). Since maximizers are unique, we obtain ψ 0 = Z (φ) and in particular Z We are left to prove (iii). Let h ∈ L p (A + ) be arbitrary. We can estimate the difference quotients as follows: Here we used that Z is continuous and that ∂ 1 J λ is continuous, see [10,Proposition 9] for a similar computation. We conclude thatJ λ is Gâteaux-differentiable with continuous as claimed.
Notice that the condition R N ψKψ dx ≥ 0 is also necessary for the existence of a global maximizer of ψ → J λ (φ, ψ) because otherwise this functional is unbounded from above.

Palais-Smale sequences for the reduced functional
In view of the results of the previous sections, we obtain a solution to our problem by proving the existence of a nontrivial critical point of the reduced functionalJ λ : L p (A + ) → R introduced in (3.1). This will be done via Mountain-pass techniques for monotone families of functionals originating from the work of Jeanjean and Toland [6].  For every sequence (ν n , φ n ) ∈ M × X with ν n ν * ∈ M and with (H) We may thus assume w.l.o.g. φ j φ * . Moreover, Proposition 3.1 (i) implies the boundedness of (ψ j ) j := (Z (φ j )) j and hence w.l.o.g. also weak convergence. For all h ∈ L p (A + ) we then have, in view of the formula forJ λ from Proposition 3.1 (iii), We infer that (|φ j | p −2 φ j ) j converges strongly in L p (A + ). By uniqueness of weak limits, we infer |φ j | p −2 φ j → |φ * | p −2 φ * strongly in L p (A + ) and hence in particular φ j p → φ * p . This finally implies φ j → φ * strongly in L p (A + ). A standard computation finally showsJ λ (φ * ) = c λ as well asJ λ (φ * ) = 0.

Proof of Theorem 1.1 and Corollary 1.2
We finally combine all auxiliary results to prove Theorem 1.1.
Proof of Theorem 1.1 From Proposition 2.1 and Proposition 3.1 we infer that for almost all λ ∈ (λ 0 , ∞) a nontrivial solution u ∈ L p (R N ) of the nonlinear Helmholtz equation (1.10) is found once we have proved the existence of nontrivial critical points of the reduced functional J λ for almost all λ ∈ M where M is an arbitrary compact subinterval of (λ 0 , ∞). From Proposition 4.3 we infer that the family (J λ ) λ∈M has the Mountain Pass Geometry. Moreover, by Proposition 4.4, condition (H) holds. So Theorem 4.2 yields for almost all λ ∈ M a BPS sequence forJ λ at the corresponding Mountain Pass level. By Proposition 4.5 each of these BPS sequences converges to a critical point φ λ ofJ λ at the Mountain Pass level. Since this critical point is necessarily nontrivial, we have thus obtained the desired claim for L p (R N )solutions of (1.10). From [4,Lemma 4.3] we infer that each of these solutions belongs to W 2,q (R N ) ∩ C 1,α (R N ) for all p ≤ q < ∞ and α ∈ (0, 1). Arguing as in Step 3 and Step 4 [8, p.13] one even obtains that these solutions belong to W 2,q (R N ) for all q ∈ ( 2N  N −1 , p). In particular, these solutions are strong solutions of (1.10), which finishes the proof.
Proof of Collorary 1.2 In order to apply Theorem 1.1 we show that (1.11) implies (1.9). In the special case k = 1 the compactness of K was shown in Lemma 4.2 in [4]. So the general case follows by rescaling. It therefore remains to show that δ := diam (A − ) ≤ k −1 y N −2 2 implies R N ψKψ dx ≥ 0 for all ψ ∈ L p (A − ). Due to (2.2), K = |Q| 1/ p R(|Q| 1/ p ·) and Q ∈ L ∞ (R N ) it suffices to prove R N ψRψ dx ≥ 0, for all ψ ∈ S( A − ). (5.1) Using that x, y ∈ A − implies x − y ∈ B δ we infer from Corollary 5.4 in [2] R which proves (5.1) and hence the Corollary.