Radial sign-changing solutions to biharmonic nonlinear Schr\"odinger equations

In this work we obtain three radial solutions of a biharmonic stationary Schr\"odinger equation, being one positive, one negative and one that changes sign. The Dual Decompostion Method is used to split the natural second order Sobolev space considered in order to apply the appropriate variational approach.


Introduction
In the last two decades, the stationary nonlinear Schrödinger equation given by −ǫ 2 ∆u + V (x)u = f (x, u) in R N u ∈ H 1 (R N ), (1.1) has been widely studied by several authors in works dealing with existence, multiplicity and concentration of solutions when the parameter ǫ → 0. In order to highlight some of the most influent works about this subject, we could quote the pioneering work of Floer and Weinstein [10], in which they used the Lyapunov-Schmidt reduction to obtain positive solutions exhibiting concentration behavior for the unidimensional case. In [17], Rabinowitz used a variational approach to obtain under global hypotheses in the potential V , positive solutions to (1.1). Later, Wang in [18] proved that the solutions obtained by Rabinowitz concentrate around the global infimum of the potential. In the celebrated paper [7], del Pino and Felmer used the so called penalization technique to prove the same kind of concentration behavior of solutions of (1.1), considering the potential V under a local version of the Rabinowitz condition. In all of the above mentioned works, the authors worked with potentials V bounded away from zero. Stationary NLS problems with vanishing potentials were trated for instance by Bonheure et al in [3,4], where the authors obtained concentration of positive solutions around global minimum points of an auxiliary function and even around some lower dimensional spheres in R N .
Although this amount of results treating with the second order case, just few works can be found dealing with similar questions involving the fourthorder equation In [13,14], the author and Soares proved the existence of a concentrating sequence of solutions of the singular perturbed version of (1.2), considering, respectively, a global and a local condition in the potential V , and a subcritical power-type nonlinearity f . The main arguments of these works were strongly inspired by Rabinowitz [17], Wang [18] and Del Pino and Felmer [7], in which other arguments to overcome the lack of a maximum principle of the biharmonic operator were required. In [9] the author and Figueiredo using Ljusternik-Schnirelmann theory, deal with the same problem that in [14], obtaining multiple solutions exhibiting the concentration phenomenon.
Another relevant question about both problems (1.1) and (1.2) is the existence of sign-changing solutions, sometimes called nodal solutions. For the second-order problem there are several papers dealing with this subject.
In a pioneering work [5], Castro, Cossio and Neuberger have obtained three solutions, including a nodal one, for a second order problem in a bounded domain and with Dirichlet boundary conditions. In [1,2], Alves and Soares use the penalization technique to get nodal solutions concentrating around extremal points of the potential V . In this approach, they use arguments based in minimization of the energy functional in some Nehari sets, considering u + = max{u, 0} and u − = min{u, 0}, respectively, the positive and negative parts of a function u ∈ H 1 (R N ). At a first sight, one could think that these arguments are trivially adaptable to the fourth order case, however, as long in H 1 (R N ) the decomposition u = u + + u − is trivially allowed, in H 2 (R N ) this factoration is no longer to be available.
In this work, we consider the problem (1.2) where N ≥ 5 and f and V satisfies the following assumption set: such that |f (x, s)−f (x, t)| ≤ (c 1 +c 2 (|s| p +|t| p ))|s−t| for a.e. x ∈ R N and s, t ∈ R; is nondecreasing for s > 0 and nonincreasing for s < 0, for a.e.
x ∈ R N .
is also nondecreasing for s > 0 and nonincreasing for s < 0, for a.e. x ∈ R N .
In order to overcome the lack of the decomposition of H 2 (R N ) in terms of positive and negative parts of theirs functions, we use an alternative method developed by Moreau in [12] called Dual Cones Decomposition Method, which consists in split a function u in a Hilbert space H as u = u 1 + u 2 , in such a way that u ∈ K and u 2 ∈ K * , where K is a cone in H and K * is called his dual cone. Considering H = H 2 (R N ) and K = {u ∈ H 2 (R N ); u ≥ 0}, if it is possible to prove that K * ⊂ −K, then we have a decomposition of a function u ∈ H 2 (R N ) in term of a non-negative and a non-posive functions such that, in many times, can substitute the trivial decomposition not-available in our case. Our approach follows closely the work of Weth [19], in which the author takes advantage of the Moreau's method to obtain signed and sign-changing solutions of the problem under Dirichlet or Navier boundary conditions in a bounded domain Ω. It is worth pointing out that to prove that K * ⊂ −K, it is necessary some kind of a maximum principle to ∆ 2 . In fact, this is the main reason why this method becomes so restrictive when dealing with fourth order problems. In [19], Weth uses the fact that under Navier boundary conditions, one can use twice the strong maximum principle to −∆ in Ω, in order to obtain a version of this result to ∆ 2 . Considering Dirichlet boundary conditions, the same is true at least to some domains like balls and limaçons. In this sense, we use some arguments of Chabrowski and Yang in [6], to prove that there exists positive solutions of a linear version of (1.2), which in particular implies that K * ⊂ −K (see Lemma 2.10). Another difficulty that deserves to be highlighted is the lack of compactness, since the problem is in R N . In order to overcome this difficulty, we consider the problem restricted to H 2 rad (R N ) consisting in the radial functions belonging to H 2 (R N ). This is interesting because of a version of the Strauss Lemma to higher order Sobolev spaces proved by Ebihara and Schonbek in [8]. At the end, once critical points of the restricted energy functional are in our hand we obtain critical points of the functional using the principle of symmetric criticality of Palais.
Our main result is the following. The proof involves variational arguments consisting in searching for critical points of the energy functional, looking for stationary points of a Cauchy problem in Banach spaces. In this sense, some results of Liu and Sun [11] about the invariance of some sets will be necessary.
Finally, we note that the nonlinearity f do not satisfies the well known Ambrosetti-Rabinowitz super-linearity condition, given by and a.e. in R N .
Instead of this condition, in order to increase the range of admissible nonlinearities, we consider the weaker assumption (f 4 ). This requires some arguments of Miyagaki and Souto in [15] to prove the boundedness of a certain sequence.
In the first section we describe the variational framework. In the second one we introduce the Cauchy problem and prove the invariance of some sets. The last section is left for the proof of the main result.
To save notation in all of this paper we denote R N gdx just by g. The norm · L p (R N ) will be simple denoted by · p .

The variational framework
As mentioned in the introduction, in order to overcome the lack of compactness, let us consider H = H 2 rad (R N ) which is a Hilbert space when endowed with the following inner product which gives rise to the following norm By (V 1 ), it follows easily that · is equivalent to the usual norm in H 2 (R N ). Before introducing the energy functional associated to (1.2), let us remember some results proved by Ebihara in [8] that will be used along this text. Lemma 2.1 (Corollary 2 in [8]). The following embeddings are compact Let us consider the restriction to H of the energy functional whose Euler-Lagrange equation is (1.2), I : H → R, given by Note that by (f 3 ) and Sobolev embeddings, I is well defined.
The following result states some interesting properties of the operator A : H → H defined above.
Proof. Let us prove the estimate. Note that from (f 2 ) and (f 3 ) it follows that for each ǫ < 0, there exists A(ǫ) > 0 such that Using Hölder with the conjugated exponents 2 * p+1 and Just by definition and Lebesgue Dominated Convergence Theorem, it follows that A = ∇Ψ, where Ψ(u) = F (u). On the other hand, (f 2 ) implies that A is a locally Lipschitz operator.
What is left to show is that A is a compact operator. Although this follows by straightforward calculations, we describe all the details, since this was the reason why we had to consider the space it follows by (f 3 ) and Hölder inequality with 2 + ǫ and (2 + ǫ) ′ , for small enough ǫ > 0, that |u| p |u n − u| .
By taking 2 < r < min{2, 2 * /p}, using Hölder inequality with r and r ′ and by (2.1), we obtain It is straightforward to prove that critical points of I correspond to fixed points of A.
Let us consider the following Cauchy problem in the Hilbert space Note that since A is a Lipschitz continuous operator, the flow ϕ is well defined.
The follow is a key point in our approach.
is a non-empty set formed by critical points of I.

Proof.
i) Note that where we have used Hölder and the equation (2.3). Suppose the assertion of the item is false. Then by last estimate, the trajectory {ϕ(t, u); t ∈ [0, T (u))} would be bounded, which implies that T (u) = ∞, which gives rise to a contradiction.
ii) First of all, let us note that there exists a sequence t n → ∞ such that ∇I(ϕ(t n , u)) → 0, as n → ∞. This follows just by noting that We claim that {ϕ(t n , u)} n∈N is uniformly bounded in H with respect to n ∈ N.
Since (w n ) is a bounded sequence in H, it follows that there exists w ∈ H such that w n ⇀ w em H, up to a subsequence. Then Lemma 2.1 implies that w n → w in L q (R N ), 2 < q < 2 * and also w n → w a.e. in R N .
In order to prove that w = 0, let us consider Γ = {x ∈ R N ; w(x) = 0} and prove that Γ has zero Lebesgue measure. For all x ∈ Γ, we have that lim n→∞ ϕ(t n , u)(x) = ∞. By (f 4 ), for each M > 0, there exists r > 0 such that F (x, s) ≥ Ms 2 , for all s ≥ r and for a.e. x ∈ R N .
In order to show that ω(u) = ∅, let us consider the bounded sequence {ϕ(t n , u)}. Since A is compact, there exists u 0 ∈ H such that A(ϕ(t n , u)) → u 0 along a subsequence. Hence Then lim n→∞ ϕ(t n , u) = u 0 and u 0 ∈ ω(u).
The following is a key result in our argument and in particular imply that ∂A 0 is a great place to look for nontrivial critical points of I. The proof of the positively invariance can be found in [11] while the other results are straightforward to see.
Although ∂A 0 is a great set to look for nontrivial critical points of I, once found, nothing can be sad about its signal. Let us introduce the concept of dual cone and state the Dual Cone Decompostions Theorem which is given by Moreau in [12].
Definition 2.8. Given a cone K in a Hilbert space H, its dual cone is defined by Theorem 2.9. Let K ⊂ H a closed convex cone. Then for all x ∈ H, there exist y ∈ K and z ∈ K * such that x = y + z and y, z = 0.
Let us define the following cones and afterwards prove some invariance properties of them. Let which are closed convex cones.
Let us denote by P and Q the orthogonal projections of H in K and −K, respectively. Denoting by P * = Id − P and Q * = Id − Q, note that P u, P * u = 0, for all u ∈ H and where K * is the dual cone associated to K. We have analogous results involving Q, −K and (−K) * . In order to prove the invariance of K and −K, as we will see, it will be necessary to prove that K * ⊂ −K and (−K) * ⊂ K . In the classical argument developed by Weth in [19], the maximum principle to the operator ∆ 2 under certain boundary conditions and in certain domains is absolutely useful. Since this is not an option for us, let us prove some result that in some sense will substitute the lack of this result.
Proof. The existence of a solution v ∈ H 2 rad (R N ) follows straightforwardly just by applying Riesz Theorem. Regularity is a simple matter just by calling Proposition 2.5 in [16].
To the positiveness we apply some arguments of Chabrowski and Yang in [6] which we describe below.
Using the last lemma it is possible to prove the following claim.
. For each n ∈ N, let v n be the positive solution of the linear problem (2.8) with h = h n , given by Lemma 2.10. Then v n ∈ K and, 0 ≥ u, v n = (∆u∆v n + V (x)uv n ) = uh n . (2.11) Hence, by (2.11) and Fatou's Lemma, Then it follows that u ≤ 0 a.e. in R N and therefore u ∈ −K.
Remark 2.11. It is worth pointing out that if u ∈ H, then u = P u + P * u where P u ≥ 0 and P * u ≤ 0 a.e. in R N . Then u ≤ P u, and consequently, u + ≤ P u a.e. in R N . In the same way one can prove that P * u ≤ u − , Qu ≤ u − and u + ≤ Q * u a.e. in R N .
The following is a very important result to prove the invariance of K and −K under the flux ϕ. ii) A(u), v ≤ A(Q * u), v , for all u ∈ H and v ∈ (−K) * .
Proof. Since ii) can be proved in the same way, we just prove i). Let u ∈ H and v ∈ K * . By Remark 2.11, P * u ≤ u − and by (f 5 ) f (x, P * u(x)) ≤ f (x, u − (x)).
Since v ≤ 0, once more by (f 5 ) it follows that ii) For sufficiently small α > 0, the α-neighborhood of K, B α (K) is positively invariant under ϕ. Moreover, all critical point of I in B α (K) belong to K. The same holds to the cone −K.
iii) K and −K are positively invariant under ϕ.
Proof. Let us prove the results just to K, since for −K the arguments are the same.
From now on let us consider α > 0 such that the statement of Lemma 2.13 holds for K and −K.
To obtain the signed solutions we will use the following result.
Proposition 2.14. Assume that there exists u 0 ∈ K such that I(u 0 ) < 0, then there exists a nontrivial critical point of I in K. The same holds for −K.
Proof. First note that by definition of A 0 , I(u) ≥ 0 for all u ∈ A 0 . Then by continuity, I(u) ≥ 0 for all u ∈ A 0 . Since I(u 0 ) < 0 then u 0 ∈ A 0 . As A 0 is an open neighborhood of the origin, there exists s ∈ (0, 1) such that su 0 ∈ ∂A 0 ∩ K. Since ∂A 0 ∩ K is a closed positively invariant set, by Proposition 2.4 ω(su 0 ) ⊂ ∂A 0 ∩ K is nonempty and any of its points are critical points of I.
Proof. Proposition 2.14 gives the existence of the signed critical points u 1 ∈ K and u 2 ∈ −K.
The next result will be useful to put the energy functional I in the context of the last proposition. Proof. Let (u n ) ⊂S be a sequence such that u n → ∞. Then there exist a sequence in R + (t n ) such that u n = t n v n and (v n ) ⊂ S. Since S is compact, we can suppose that along a subsequence v n → v, as n → ∞, for some v ∈ S. As u n → ∞, then one trivially see that t n → ∞. For each s > 0, • h s (1) = su ∈ K\{0}, • h s (0) = sv ∈ K * \{0}.
By Lemma 2.17 applied to the compact set S = {tu + (1 − t)v; t ∈ [0, 1]}, we see that if s is large enough, then I(h s (t)) < 0, ∀t ∈ [0, 1]. Hence, Proposition 2.16 give us the existence of u 1 , u 2 and u 3 , respectively, a positive, a negative and a nodal critical point of I restricted to H. The existence of the critical points of I in all space H 2 (R N ) follows just by applying the Principle of symmetric criticality of Palais to the functional I, once observed that I is invariant by the action of the group O(N), and H 2 rad (R N ) = {u ∈ H 2 (R N ); u(g(x)) = u(x), ∀g ∈ O(N)}. Hence the theorem follows.