Nodal solutions of an NLS equation concentrating on lower dimensional spheres
 Giovany M Figueiredo^{1} and
 Marcos TO Pimenta^{2}Email author
https://doi.org/10.1186/s1366101504118
© Figueiredo and Pimenta 2015
Received: 30 March 2015
Accepted: 10 August 2015
Published: 18 September 2015
Abstract
Keywords
MSC
1 Introduction
In the last ten years, solutions which concentrate on higher dimensional sets have received more and more attention. The first work which seems to show this kind of result is [5] in which the authors study an NLS equation on a bounded domain with Neumann boundary condition and prove the existence of a sequence of solutions which concentrate on some component of the boundary. One of the first works dealing with solutions concentrating around a sphere is [6] in which Ambrosetti, Malchiodi and Ni give necessary and sufficient conditions under which (1.1) exhibits solutions concentrating around a sphere. The radius of such a sphere is given by a minimum point of a function \(\mathcal{M}\), which takes into account the value of the radial potential \(V(\vert x\vert )\). The role played by \(\mathcal {M}\) is in order to balance the potential energy (coming from V) and the volume energy which arise from the other terms of the energy functional (see the introduction of [6] for more details). In fact, sphereconcentrating solutions show a rather different behavior when compared with spikelayered ones. To be more specific, in [6], the authors prove the existence of sphereconcentrating solutions to (1.1) even for critical or supercritical exponent p. This is in a strike contrast with the fact that, as showed in [7], no spikelayered solution exists to (1.1) for \(p = 2^{*}1\). Other significant difference is that the energy of the sphereconcentrating solutions tends to zero in contrast with that of spikelayered solutions which converges to the mountainpass level of the energy functional. In these and so many other works ([8–11] for example), LyapunovSchmidt reduction methods have been used in order to construct the sphereconcentrating solutions for Schrödinger equations, SchrödingerPoisson systems and other related problems.
In the spikelayered solutions setting, the existence of signchanging (or nodal) solutions was investigated by some authors. In [14] and [15], Alves and Soares study problem (1.2), with K to be a constant, and prove the existence of nodal solutions which concentrate on minima of the potential V. They consider f as a subcritical powertype nonlinearity in their first work and as presenting a critical exponential growth at infinity in the second. In both they employ the penalization technique together with a careful analysis of the profile of the solutions. In [16], Sato proposes a different kind of penalization in order to show the existence of multipeak nodal solutions to a Schrödinger equation with a vanishing potential.
 \((f_{1})\) :

There exists \(\nu> 1\) such that \(f(\vert s\vert ) = o(\vert s\vert ^{\nu})\) as \(s \to0\);
 \((f_{2})\) :

There exist \(c_{1}, c_{2} > 0\) such that \(\vert f'(s)\vert \leq c_{1} + c_{2}\vert s\vert ^{p1}\), where \(0 < p < \frac{2N}{N2}  2\);
 \((f_{3})\) :

There exists \(\theta> 2 \) such thatwhere \(F(s) = \int_{0}^{s} f(t)\,dt\);$$0 < \theta F(s) \leq f(s)s\quad \mbox{for }s \neq0, $$
 \((f_{4})\) :

\(s \mapsto f(s)/s\) is increasing in \(s > 0\) and decreasing for \(s < 0\).
The potential V will be assumed to satisfy a symmetry condition which we explain in the next section.
1.1 Statement of the main result
Let \(1 \leq k \leq N1\) be an integer which determines the dimension of the sphere in which the solutions obtained are going to concentrate. Consider \(\mathcal{H}\) to be an \((Nk1)\)dimensional linear subspace of \(\mathbb{R}^{N}\) and note that \(\mathcal{H}^{\perp}\) is a \((k+1)\)dimensional subspace. All along the paper we use the notation for \(x\in\mathbb{R}^{N}\) as \(x = (x',x'')\), in which \(x'\in\mathcal {H}\), \(x''\in\mathcal{H}^{\perp}\) are such that \(x = x' + x''\).
From now on, if \(h:\mathbb{R}^{N} \to\mathbb{R}\) is a function, by saying that \(h(x',x'') = h(x',\vert x''\vert )\) (which rigorously does not make sense), we mean that \(h(x',y) = h(x',z)\) for all \(y,z \in\mathcal {H}^{\perp}\) such that \(\vert y\vert = \vert z\vert \).
 \((V_{1})\) :

There exists \(V_{0} > 0\) such that \(V_{0} \leq V(x)\) and, for all \(x \in\mathbb{R}^{N}\), \(V(x) = V(x',x'') = V(x',\vert x''\vert )\).
 \((\mathcal{M}_{1})\) :

There exists an open bounded set \(\Omega \subset\mathbb{R}^{N}\) such that if \((x',x'') \in\Omega\) then \((x',y'') \in\Omega\) for all \(y''\in\mathcal{H}^{\perp}\), \(\vert x''\vert =\vert y''\vert \). Moreover,$$0 < \mathcal{M}_{0} : = \inf_{x \in\Omega} \mathcal{M}(x) < \inf_{x\in \partial\Omega} \mathcal{M}(x). $$
Theorem 1.1
The arguments in proving the existence of solutions were strongly influenced by the works of Alves and Souto [17], in which they prove the existence of nodal solutions to a SchrödingerPoisson system. In the concentration, we follow closely the arguments in [14] and [12, 13].
After our work has been finished, we found a very recent paper [18] in which the author uses a similar argumentation in order to prove the existence of a sequence of nodal multipeak solutions which concentrate around the minimum points of a modified potential, associated to a vanishing potential. The existence arguments in both works rely on a minimization of the penalized energy functional on the nodal Nehari set, and the concentration arguments follow the same general lines. Nevertheless, it is worth pointing out that in our work, since we get sphereconcentrating solutions, several technical difficulties arise. Moreover, in our work proving that the solution of the modified problem is in fact a solution of the original one involves different comparison functions since our penalization is slightly different.
In Section 2 we present the penalization scheme and the variational framework. In Section 3 we prove the existence of nodal solutions of the modified problem. In Section 4 we exhibit the concentration arguments in order to prove that the solutions of the modified problem concentrate around a kdimensional sphere; and in the last section we complete the proof of Theorem 1.1 by showing that the solutions of the modified problem satisfy the original one.
2 The penalized nonlinearity and the variational framework
 \((g_{1})\) :

\(g_{\epsilon}(x,s) = o(\vert s\vert ^{\nu})\), as \(s \to0\), uniformly in compact sets of \(\mathbb{R}^{N}\).
 \((g_{2})\) :

There exist \(c_{1}, c_{2} > 0\) such that \(\vert g_{\epsilon}(x,s)\vert \leq c_{1}\vert s\vert + c_{2}\vert s\vert ^{p}\), where \(1 < p < \frac{N+2}{N2}\);
 \((g_{3})\) :

There exists \(\theta> 2\) such that:
 (i)
\(0 < \theta G_{\epsilon}(x,s) \leq g_{\epsilon}(x,s)s\) for \(x \in\Omega\) and \(s \neq0\),
 (ii)
\(0 < 2 G_{\epsilon}(x,s) \leq g_{\epsilon}(x,s)s\) for \(x\in \mathbb{R}^{N}\backslash\Omega\) and \(s \neq0\),
 (i)
 \((g_{4})\) :

\(s \mapsto\frac{g_{\epsilon}(x,s)}{s}\) is a nondecreasing function for \(s > 0\) and nonincreasing for \(s < 0\), for all \(x\in \mathbb{R}^{N}\).
Remark 2.1
In this section and throughout the rest of the paper, we omit the dx in all the integrals and, when the domain over which the integral is calculated is \(\mathbb{R}^{N}\), we write ∫ rather than \(\int_{\mathbb{R}^{N}}\).
3 Existence results
In the next result we try to infer information of \(I_{\epsilon}\) with respect to \(\mathcal{N}_{\epsilon}^{\pm}\) in the same way that one is used to do with \(\mathcal{N}_{\epsilon}\).
Lemma 3.1
Let \(v \in\tilde{H}\) such that \(v^{\pm}\neq0\). Then there exist \(t,s > 0\) such that \(t v^{+} + s v^{} \in\mathcal{N}_{\epsilon}^{\pm}\).
Proof
Lemma 3.2
Let \(v \in\mathcal{N}_{\epsilon}^{\pm}\), then \((t,s) = (1,1)\) is a strict global maximum point of \(\psi_{v}\).
Proof
Since \(D^{2}\psi_{v}(1,1)\) is a positive definite form, we just have to verify that \(\frac{\partial^{2} \psi_{v}}{\partial t^{2}} = I_{\epsilon}''(tv^{+}){v^{+}}^{2} < 0\). But this follows since 1 is a maximum point of \(t \mapsto I_{\epsilon}(tv^{+})\). □
Still, as a consequence of the arguments employed in the construction of the Nehari manifold as in [2], we have the following result.
Lemma 3.3
Proof
Now we are going to state and prove the main result of this section.
Theorem 3.4
For sufficiently small \(\epsilon> 0\), there exists a nodal solution of (2.5), \(v_{\epsilon}\in\tilde{H}\) such that \(I_{\epsilon}(v_{\epsilon}) = d_{\epsilon}\).
Before proceeding with the proof of Theorem 3.4, let us state some technical result about \(\mathcal{N}_{\epsilon}^{\pm}\).
Lemma 3.5
 (i)
There exists \(\rho> 0\) such that \(\Vert v\Vert _{\epsilon}\geq \rho \) for all \(v \in\mathcal{N}_{\epsilon}\).
 (ii)
There exists a constant \(C > 0\) such that, for all \(v \in \mathcal{N}_{\epsilon}\), \(I_{\epsilon}(v) \geq C\Vert v\Vert _{\epsilon}^{2}\).
Proof
Proof of Theorem 3.4
The proof will be carried out in two steps. In the first one we prove that \(d_{\epsilon}\) is attained by a function \(u_{\epsilon}\in\tilde{H}\).
Hence \(I_{\epsilon}(t_{\epsilon}w_{\epsilon}^{+} + s_{\epsilon}w_{\epsilon}^{}) = b_{\epsilon}\).
The second step is proving that \(v_{\epsilon}\in\tilde{H}\), which minimizes \(I_{\epsilon}\) on \(\mathcal{N}_{\epsilon}^{\pm}\) is a critical point of \(I_{\epsilon}\) in H̃. This can be done by employing the same arguments as those of Section 3 in [20]. For the sake of completeness, we include all the details of this proof.
 (i)
\(\eta(1,v) = v\) for all \(v \notin I_{\epsilon}^{1}([d_{\epsilon}2\rho,d_{\epsilon}+2\rho])\),
 (ii)
\(\eta(1,I_{\epsilon}^{d_{\epsilon}+\rho}\cap S) \subset I_{\epsilon}^{d_{\epsilon}\rho}\),
 (iii)
\(I_{\epsilon}(\eta(1,v)) \leq I_{\epsilon}(v)\) for all \(u \in\tilde{H}\).
Now, let us prove that there exists \((t,s) \in D\) such that \(\eta (1,tv_{\epsilon}^{+} + sv_{\epsilon}^{}) \in\mathcal{N}_{\epsilon}^{\pm}\), which together with the claim contradicts the definition of \(d_{\epsilon}\).
Finally, this contradiction proves the theorem. □
4 Concentration results
Let us introduce a sequence \(\epsilon_{n} \to0\) as \(n \to\infty\) and, for each \(n \in\mathbb{N}\), let us denote by \(v_{n}\) the solution \(v_{\epsilon_{n}}\) given by Theorem 3.4 and consider \(d_{n}:= d_{\epsilon_{n}}\), \(\Vert \cdot \Vert _{n}:=\Vert \cdot \Vert _{\epsilon_{n}}\) and \(I_{n}: = I_{\epsilon_{n}}\).
The following result provides an upper estimate for the sequence of values \(d_{n}\). Its proof is inspired by the arguments of Alves and Soares in [14].
Lemma 4.1
Proof
Let \(z_{0}=(x_{0},y_{0}) \in\Omega\) be such that \(\mathcal{M}(z_{0}) = \inf_{x \in\Omega} \mathcal{M}(x)\). Since Ω is an open set, there exists \(R > 0\) such that \(B_{2R}(z_{0}) \subset\Omega\). Let us choose points \(z_{1},z_{2} \in\partial B_{R}(z_{0})\) such that, if \(z_{i}=(Q_{i}',Q_{i}'')\), then \(\vert Q_{1}''  Q_{2}''\vert = 2R\). Note that \(B_{R}(z_{i}) \subset\Omega\) for \(i = 1,2\). In the rest of this proof, \(i \in\{1,2\}\). Let us choose smooth cutoff functions \(\psi_{i}:\mathbb{R}^{Nk} \to \mathbb{R}\) such that \(\psi_{i} = 1\) in \(B_{\mathbb {R}^{Nk}}((Q_{i}',\vert Q_{i}''\vert ),R/4)\) and \(\psi_{i} = 0\) in \(\mathbb{R}^{Nk} \backslash B_{\mathbb{R}^{Nk}}((Q_{i}',\vert Q_{i}''\vert ),R/2)\).
The next lemma implies that solutions found in Theorem 3.4 do not vanish when \(n \to\infty\).
Lemma 4.2
Proof
The same argument of [14] with short modifications can be used to prove the following result.
Lemma 4.3
Lemma 4.4
Proof
The proof is analogous to [12][Lemma 4.3], but we sketch it here for the sake of completeness.
By choosing a sequence \(R_{n} \to\infty\) such that \(\epsilon_{n} R_{n} \to 0\) and considering a smooth cutoff function in \(\mathbb{R}^{Nk}\), \(\eta_{R}\) such that \(0 \leq\eta_{R} \leq1\), \(\eta_{R}(z) = 1\) if \(\vert z\vert \leq \frac{R}{2}\) and \(\eta_{R}(z) = 0\) if \(\vert z\vert > R\), and \(\Vert \nabla\eta_{R}\Vert _{\infty}\leq\frac{C}{R}\), it can be proved using (4.8) that \(\overline{w}_{n}(z) := \eta_{R_{n}}(z)\tilde {w}_{n}(z)\) is bounded in \(H^{1}(\mathbb{R}^{Nk})\), uniformly in n.
By (4.9) and (4.10) it follows that \(\tilde{w}_{n} \to\tilde{w}\) in \(C^{2}_{loc}(\mathbb{R}^{Nk})\) and that w̃ satisfies (4.6). □
The following is the main step in proving the concentration result.
Proposition 4.5
 (i)
\(\lim_{n \to\infty} \epsilon_{n}^{k}d_{n} = 2\omega_{k}\inf_{\Omega}\mathcal{M}\);
 (ii)
\(\lim_{n \to\infty} \mathcal {M}(\epsilon_{n} P_{n}^{i}) = \inf_{\Omega}\mathcal{M}\), \(i \in\{1,2\}\).
Proof
By Lemmas 4.2 and 4.4 it follows that \(\tilde {w}^{i}_{n}(x',r):= \tilde{v}_{n}(x',r + P_{n}^{i}) \to\tilde{w}_{i}\) in \(C^{2}_{loc}(\mathbb{R}^{Nk})\), where \(\tilde{w}_{i} \neq0\) and satisfies (4.6) with \((\bar{y}',\bar{y}'') = P^{i}\).
Remark 4.6
Note that by (ii) of the last result it follows that \(P^{i} \in\Omega\). In fact, if \(P^{i} \in\partial\Omega\), by \((\mathcal {M}_{1})\), \(\mathcal{M}(P^{i}) > \inf_{\Omega}\mathcal{M}\), which together with the continuity of \(\mathcal{M}\) (see Lemma 2.1 in [12]) leads to a contradiction.
5 Proof of Theorem 1.1
Let \(v_{n}\) be as at the beginning of Section 4 and \(u_{n}(x):=v_{n}(\epsilon _{n}^{1}x)\).
Proposition 5.1
Proof
Declarations
Acknowledgements
We would like to express our gratitude to Prof. Antonio Suárez for some discussions on this subject and to the anonymous referee for comments and careful reading of the manuscript.
GMF is supported by PROCAD/CASADINHO 552101/20117, CNPq/PQ 301242/20119 and CNPq/CSF 200237/20128. MTOP is supported by FAPESP 2014/161361, CNPq 442520/20140 and UNESP/PROPe.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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