Radial signchanging solutions to biharmonic nonlinear Schrödinger equations
 Marcos TO Pimenta^{1}Email author
https://doi.org/10.1186/s136610150282z
© Pimenta; licensee Springer 2015
Received: 20 August 2014
Accepted: 4 January 2015
Published: 31 January 2015
Abstract
In this work we obtain three radial solutions of a biharmonic stationary Schrödinger equation, one being positive, one negative, and one sign changing. The dual decomposition method is used to split the natural secondorder Sobolev space considered in order to apply the appropriate variational approach.
Keywords
MSC
1 Introduction
In all of the above mentioned works, the authors worked with potentials V bounded away from zero. Stationary NLS problems with vanishing potentials were treated for instance by Bonheure et al. in [5, 6], where the authors obtained concentration of positive solutions around global minimum points of an auxiliary function and even around some lower dimensional spheres in \(\mathbb{R}^{N}\).
Another relevant question about both problems (1.1) and (1.2) is the existence of signchanging solutions, sometimes called nodal solutions. For the secondorder problem there are several papers dealing with this subject. In a pioneering work [10], Castro et al. have obtained three solutions, including a nodal one, for a secondorder problem in a bounded domain and with Dirichlet boundary conditions. In [11, 12], 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 on the 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 \(u \in H^{1}(\mathbb{R}^{N})\). At first sight, one could think that these arguments are trivially adaptable to the fourthorder case, however, as long as in \(H^{1}(\mathbb{R}^{N})\) the decomposition \(u = u^{+} + u^{}\) is trivially allowed; in \(H^{2}(\mathbb{R}^{N})\) this factorization is no longer to be available.
 (V_{1}):

\(0 < V_{0}:=\inf_{\mathbb{R}^{N}}V\) and \(V(x) = V(x)\), for all \(x \in\mathbb{R}^{N}\);
 (V_{2}):

there exist \(\Omega\subset\mathbb{R}^{N}\), \(0 < V_{0}:=\inf_{\mathbb{R}^{N}}V\), and \(V(x) = V(x)\), for all \(x \in \mathbb{R}^{N}\);
 (f_{1}):

\(f:\mathbb{R}^{N}\times\mathbb{R} \to\mathbb{R}\) is a Carathéodory function;
 (f_{2}):

\(f(x,s) = o(s)\) as \(s\to0\) a.e. in \(\mathbb{R}^{N}\);
 (f_{3}):

there are constants \(c_{1},c_{2} > 0\), and \(0 < p < 2_{*}2\), where \(2_{*} = 2N/(N4)\), such that$$\bigl\vert f(x,s)  f(x,t)\bigr\vert \leq\bigl(c_{1} + c_{2}\bigl(s^{p} + t^{p}\bigr)\bigr)st, \quad \mbox{for a.e. }x\in\mathbb{R}^{N}\mbox{ and }s,t \in\mathbb{R}; $$
 (f_{4}):

\(\lim_{s\rightarrow\infty}\frac {F(x,s)}{s^{2}} = +\infty\) a.e. in \(\mathbb{R}^{N}\), where \(F(x,s) = \int_{0}^{s} f(x,t)\, dt\);
 (f_{5}):

\(\frac{f(x,s)}{s}\) is nondecreasing for \(s > 0\) and nonincreasing for \(s<0\), for a.e. \(x\in\mathbb{R}^{N}\).
Remark 1.1
Note that (f_{5}) implies that \(f(x,s)\) is nondecreasing in ℝ and that \((x,s)\mapsto f(x,s)s  2F(x,s)\) is also nondecreasing for \(s > 0\) and nonincreasing for \(s<0\), for a.e. \(x\in\mathbb{R}^{N}\).
Remark 1.2
An example of nonlinearity satisfying (f_{1})(f_{5}) is \(f(x,s) = \sum_{i=1}^{k} a_{i}(x)s^{p_{i}}s\), where \(a_{i} \in L^{\infty}(\mathbb{R}^{N})\), \(a_{i} \geq0\) in a positive measure set of \(\mathbb{R}^{N}\), and \(0 < p_{i} < 2_{*}2\), for all \(i \in\{1,\ldots,k\}\).
Another difficulty that deserves to be highlighted is the lack of compactness, since the problem is in \(\mathbb{R}^{N}\). In order to overcome this difficulty, we consider the problem restricted to \(H^{2}_{\mathrm{rad}}(\mathbb{R}^{N})\) consisting in the radial functions belonging to \(H^{2}(\mathbb{R}^{N})\). This is interesting because of a version of the Strauss lemma as regards higherorder Sobolev spaces proved by Ebihara and Schonbek in [16]. At the end, once critical points of the restricted energy functional are at hand we obtain critical points of the functional using the principle of symmetric criticality of Palais.
Huang and Liu, in very recent papers [17, 18], have applied similar arguments to prove the existence of signchanging solutions to, respectively, a biharmonic and a pbiharmonic problem. In the first one the authors have studied a biharmonic problem with an asymptotic linear nonlinearity and in the second, a pbiharmonic problem with a Hardy type potential.
Our main result is the following.
Theorem 1.3
Assume that conditions (V_{1}) and (f_{1})(f_{5}) hold. Then there exist at least one positive, one negative and one nodal classical radial solution of (1.2).
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 [19] about the invariance of some sets will be necessary.
 (AR)
there exists \(\mu> 2\) such that \(0 < \mu F(x,s) \leq f(x,s)s\), for all \(s \neq0\) and a.e. in \(\mathbb{R}^{N}\).
When proving Theorem 1.3 our main contribution is in providing a result on the existence of nodal solutions to the BNLS equation, which seems to be difficult to obtain by other methods like Nehari analysis. In fact the real difficulty here is in overcoming the lack of the strong maximum principle in applying a method which works very well in those situations where this principle is available.
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 \(\int_{\mathbb{R}^{N}} g\, dx\) just by ∫g. The norm \(\\cdot\_{L^{p}(\mathbb{R}^{N})}\) will be simple denoted by \(\\cdot\_{p}\).
2 The variational framework
Before introducing the energy functional associated to (1.2), let us remember some results proved by Ebihara and Schonbek in [16] that will be used along this text.
Lemma 2.1
(Corollary 2 in [16])
Lemma 2.2
(Theorem 2.1 in [16])
\(H^{2}_{\mathrm{rad}}(\mathbb{R}^{N})\) is not compactly embedded into \(L^{2}(\mathbb{R}^{N})\).
Note that by (f_{3}) and Sobolev embeddings, I is well defined.
The following result states some interesting properties of the operator \(A: H \to H\) defined above.
Lemma 2.3
Proof
Just by definition and Lebesgue dominated convergence theorem, it follows that \(A = \nabla\Psi\), where \(\Psi(u) = \int F(u)\). 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 \(H^{2}_{\mathrm{rad}}(\mathbb{R}^{N})\) rather than \(H^{2}(\mathbb{R}^{N})\).
It is straightforward to prove that critical points of I correspond to fixed points of A.
The following is a key point in our approach.
Proposition 2.4
 (i)
\(T(u)=\infty\),
 (ii)there exists \(t_{n}\rightarrow\infty\) such that \(\{\varphi(t_{n},u), n\in\mathbb{N} \}\) is bounded in H and the ωlimit set of u,is a nonempty set formed by critical points of I.$$\omega(u) = \bigcap_{0\leq t<\infty} \overline{\bigcup _{t \leq s<\infty}\varphi(s,u)} $$
Proof
We claim that \(\{\varphi(t_{n},u)\}_{n\in\mathbb{N}}\) is uniformly bounded in H with respect to \(n\in\mathbb{N}\).
Since (2.4) contradicts (2.5), we find that \(\{\varphi (t_{n},u)\}_{n\in\mathbb{N}}\) is uniformly bounded in H with respect to \(n\in\mathbb{N}\) and the claim is proved.
Then \(\lim_{n\to\infty}\varphi(t_{n},u) = u_{0}\) and \(u_{0} \in\omega(u)\).
Definition 2.5
Lemma 2.6
\(\mathcal{A}_{0}\) is an open subset of H and there exists \(r > 0\) such that \(B_{r}(0) \subset\mathcal{A}_{0}\).
Proof
The following is a key result in our argument and in particular implies that \(\partial\mathcal{A}_{0}\) is a great place to look for nontrivial critical points of I.
Proposition 2.7
\(\partial\mathcal{A}_{0}\) is a closed positively invariant set of H and \(\inf_{u \in\partial\mathcal{A}_{0}}I(u) \geq0\). In particular, for all \(u \in\partial\mathcal{A}_{0}\), \(\omega(u)\) is a nonempty set consisting in nontrivial critical points of I.
The proof of the positively invariance can be found in [19] while the other results are straightforward to see.
Although \(\partial\mathcal{A}_{0}\) is a great set to look for nontrivial critical points of I, once found, nothing can be said about its signal. Let us introduce the concept of dual cone and state the dual cone decomposition theorem, which is given by Moreau in [13].
Definition 2.8
Theorem 2.9
Let us define the following cones and afterwards prove some invariance properties of them.
In order to prove the invariance of \(\mathcal{K}\) and \(\mathcal{K}\), as we will see, it will be necessary to prove that \(\mathcal{K}^{*} \subset\mathcal{K}\) and \((\mathcal{K})^{*} \subset\mathcal{K}\). In the classical argument developed by Weth in [14], the maximum principle to the operator \(\Delta^{2}\) 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.
Lemma 2.10
Proof
The existence of a solution \(v \in H^{2}_{\mathrm{rad}}(\mathbb{R}^{N})\) follows straightforwardly just by applying the Riesz theorem. Regularity is a simple matter just by calling Proposition 2.5 in [21].
To the positiveness we apply some arguments of Chabrowski and Yang in [15] which we describe below.
Using the last lemma it is possible to prove the following claim.
Claim
\(\mathcal{K}^{*} \subset\mathcal{K}\).
Proof
Then it follows that \(u \leq0\) a.e. in \(\mathbb{R}^{N}\) and therefore \(u \in\mathcal{K}\). □
Remark 2.11
It is worth pointing out that if \(u \in H\), then \(u = Pu + P^{*}u\) where \(Pu \geq0\) and \(P^{*}u \leq0\) a.e. in \(\mathbb{R}^{N}\). Then \(u \leq Pu\), and consequently, \(u^{+} \leq Pu\) a.e. in \(\mathbb{R}^{N}\). In the same way one can prove that \(P^{*} u \leq u^{}\), \(Qu \leq u^{}\), and \(u^{+} \leq Q^{*} u\) a.e. in \(\mathbb{R}^{N}\).
The following is a very important result to prove the invariance of \(\mathcal{K}\) and \(\mathcal{K}\) under the flow φ.
Lemma 2.12
 (i)
\(\langle A(u),v \rangle\leq\langle A(P^{*}u),v \rangle \), for all \(u \in H\) and \(v\in\mathcal{K}^{*}\);
 (ii)
\(\langle A(u),v \rangle\leq\langle A(Q^{*}u),v \rangle\), for all \(u \in H\) and \(v\in(\mathcal{K})^{*}\).
Proof
Lemma 2.13
 (i)
\(A(\mathcal{K}) \subset\mathcal{K}\) and \(A(\mathcal{K}) \subset\mathcal{K}\).
 (ii)
For sufficiently small \(\alpha> 0\), the αneighborhood of \(\mathcal{K}\), \(B_{\alpha}(\mathcal{K})\) is positively invariant under φ. Moreover, all critical points of I in \(\overline{B_{\alpha}(\mathcal{K})}\) belong to \(\mathcal{K}\). The same holds for the cone \(\mathcal{K}\).
 (iii)
\(\mathcal{K}\) and \(\mathcal{K}\) are positively invariant under φ.
Proof
Let us prove the results just for \(\mathcal{K}\), since for \(\mathcal {K}\) the arguments are the same.
Now let us prove that \(B_{\alpha}(\mathcal{K})\) is positively invariant.
Suppose, contrary to our claim, that there exists \(u_{0}\in B_{\alpha}(\mathcal{K})\) such that \(\varphi(t_{0},u_{0}) \in\partial B_{\alpha}(\mathcal{K})\) where \(t_{0}\in[0,T(u_{0}))\) is the least positive real with this property. As \(B_{\alpha}(\mathcal{K})\) is a convex open set and \(\{\varphi(t_{0},u_{0})\}\) is compact, by Mazur’s separation theorem, there exist a linear functional \(\rho\in H^{*}\) and a real number β such that \(\rho(\varphi(t_{0},u_{0})) = \beta\) and \(\rho(u) > \beta\) for all \(u \in B_{\alpha}(\mathcal{K})\).
(iii) This item follows straightforwardly observing that \(\mathcal{K} = \bigcap_{\alpha>0} B_{\alpha}(\mathcal{K})\). □
From now on let us consider \(\alpha> 0\) such that the statement of Lemma 2.13 holds for \(\mathcal{K}\) and \(\mathcal{K}\).
To obtain the signed solutions we will use the following result.
Proposition 2.14
Assume that there exists \(u_{0}\in\mathcal{K}\) such that \(I(u_{0}) < 0\), then there exists a nontrivial critical point of I in \(\mathcal{K}\). The same holds for \(\mathcal{K}\).
Proof
First note that by definition of \(\mathcal{A}_{0}\), \(I(u) \geq0\) for all \(u\in\mathcal{A}_{0}\). Then by continuity, \(I(u) \geq0\) for all \(u\in\overline{\mathcal{A}}_{0}\).
Since \(I(u_{0}) < 0\) then \(u_{0} \notin\overline{\mathcal{A}}_{0}\). As \(\mathcal{A}_{0}\) is an open neighborhood of the origin, there exists \(s\in(0,1)\) such that \(su_{0} \in\partial\mathcal{A}_{0}\cap\mathcal {K}\). Since \(\partial\mathcal{A}_{0}\cap\mathcal{K}\) is a closed positively invariant set, by Proposition 2.4, \(\omega(su_{0}) \subset\partial\mathcal{A}_{0}\cap\mathcal{K}\) is nonempty and any of its points are critical points of I. □
Lemma 2.15
\(\mathcal{A}_{+}\) and \(\mathcal{A}_{}\) are disjoint relatively open sets of \(\partial\mathcal{A}_{0}\).
Proof
Since \(B_{\alpha}(\mathcal{K})\) and \(B_{\alpha}(\mathcal{K})\) are open sets, then so are \(\mathcal{A}(B_{\alpha}(\mathcal{K}))\) and \(\mathcal {A}(B_{\alpha}(\mathcal{K}))\).
Suppose, contrary to our claim, that there exists \(u\in\mathcal {A}_{+}\cap\mathcal{A}_{}\). Since \(u\in\partial\mathcal{A}_{0}\), then \(T(u)=\infty\) and \(\omega(u)\neq\emptyset\). Further, since \(u\in \mathcal{A}(B_{\alpha}(\mathcal{K})) \cap\mathcal{A}(B_{\alpha}(\mathcal{K}))\), then \(\omega(u)\subset\overline{B_{\alpha}(\mathcal{K})} \cap\overline{B_{\alpha}(\mathcal{K})}\). But since \(\omega(u)\) consists of critical points of I, by Lemma 2.13, \(\omega(u) \in\mathcal{K} \cap\mathcal{K} = \{0\}\), which contradicts the fact that \(u\in\partial\mathcal{A}_{0}\). □
Proposition 2.16
Suppose that there exists a continuous path \(h:[0,1] \rightarrow H\), such that \(h(0)\in\mathcal{K}\), \(h(1)\in \mathcal{K}\), and \(I(h(t)) < 0\) for all \(t\in[0,1]\). Then I has at least three nontrivial critical points, these being \(u_{1}\in\mathcal{K}\), \(u_{2}\in  \mathcal{K}\), and \(u_{3}\in H\backslash(\mathcal{K}\cup\mathcal{K})\).
Proof
Proposition 2.14 gives the existence of the signed critical points \(u_{1} \in\mathcal{K}\) and \(u_{2} \in\mathcal{K}\).
To get the nodal one, let us first highlight that \(h([0,1]) \cap \overline{\mathcal{A}}_{0} = \emptyset\).

\(\{0\} \times[0,1] \subset\mathcal{B}\), since \(0h(s) = 0\in \mathcal{A}_{0}\), \(\forall s\in[0,1]\);

\(\{1\} \times[0,1] \cap\overline{\mathcal{B}} = \emptyset\), since \(I(h(s))<0\), \(\forall s\in[0,1]\).
Now, let us consider Σ the closure of the connected component of \(\Gamma\backslash\partial Q\) that intersects \([0,1]\times\{0\}\) and \([0,1]\times\{1\}\).
Denoting by \(\Gamma_{0} = \{s_{1}h(s_{2}); (s_{1},s_{2})\in\Sigma\}\), we find that \(\Gamma_{0}\) is a connected subset of \(\partial\mathcal {A}_{0}\) such that \(\Gamma_{0} \cap\pm\mathcal{K} \neq\emptyset\). Since by Lemma 2.15, \(\mathcal{A}_{\pm}\) are disjoint open subsets of \(\partial\mathcal{A}_{0}\), \(\Gamma_{0} \cap\mathcal{A}_{\pm}\) are open disjoints subsets of \(\Gamma_{0}\). By connectedness of \(\Gamma_{0}\), there exists \(u\in\Gamma_{0}\backslash(\mathcal{A}_{+} \cup\mathcal{A}_{})\) and once \(\partial\mathcal{A}_{0} \backslash (\mathcal{A}_{+} \cup\mathcal{A}_{})\) is positively invariant, then \(\{ \varphi(t,u); t \geq0\} \subset\partial\mathcal{A}_{0} \backslash (\mathcal{A}_{+} \cup\mathcal{A}_{})\). Using the fact that this is a closed subset in \(\partial\mathcal{A}_{0}\), it follows that \(\omega (u)\subset\partial\mathcal{A}_{0} \backslash(\mathcal{A}_{+} \cup \mathcal{A}_{})\). In particular, \(\omega(u) \cap(\mathcal{K} \cup \mathcal{K}) \neq\emptyset\) and any of his points are nodal critical points of I. □
The next result will be useful to put the energy functional I in the context of the last proposition.
Lemma 2.17
Proof
Let \((u_{n})\subset\tilde{S}\) be a sequence such that \(\u_{n}\ \rightarrow\infty\). Then there exists a sequence in \(\mathbb{R}_{+}\) \((t_{n})\) such that \(u_{n} = t_{n} v_{n}\) and \((v_{n})\subset S\). Since S is compact, we can suppose that along a subsequence \(v_{n} \rightarrow v\), as \(n\rightarrow\infty\), for some \(v\in S\). As \(\u_{n}\\rightarrow \infty\), then one trivially sees that \(t_{n}\rightarrow\infty\).
Now, let us prove the main result of this work
Proof of Theorem 1.3
Let \(u\in\mathcal{K}\) and \(v\in\mathcal{K}^{*}\), such that \(u,v\neq 0\) and u and v are linearly independent in H. For each \(s>0\), let us define \(h_{s}:[0,1] \rightarrow H\) as \(h_{s}(t) = s(tu + (1t)v)\).

\(h_{s}(1) = su \in\mathcal{K}\backslash\{0\}\),

\(h_{s}(0) = sv \in\mathcal{K}^{*}\backslash\{0\}\).
Declarations
Acknowledgements
The author would like to thank PropeUnesp and Fundunesp for the support and Prof. Sérgio Monari Soares and Antonio Suárez for useful suggestions concerning the paper. This work was supported by FAPESP  2012/201600  2014/161361 and CNPq 442520/20140
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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