Infinite solutions having a prescribed number of nodes for a Schrödinger problem
 Jing Zeng^{1}Email author
Received: 23 March 2017
Accepted: 10 August 2017
Published: 18 September 2017
Abstract
In this paper, we are concerned with the multiplicity of solutions for a Schrödinger problem. A weaker superquadratic assumption is required for the nonlinearity. Then we give a new proof for the infinite solutions to the problem, having a prescribed number of nodes. It turns out that the weaker condition of the nonlinearity suffices to guarantee the infinitely many solutions. At the same time, a global characterization of the critical values of the nonradial nodal solutions are given.
Keywords
1 Introduction
Since (1.1) is invariant under rotations, it is natural to search for spherically symmetric solutions. The radial solutions of (1.1) are proved by BartschWillem [3] and LiuWang [4]. The existence question of nonradial solutions to (1.1) or (1.2) was open for a long time [5], until it was proved by BartschWillem [6] and LiuWang [4]. Fan [7] considered \(p(x)\)Laplacian equations in \(\mathbb{R}^{N}\) with periodic data and nonperiodic perturbations being stationary at infinity, where the perturbations are carried out not only on the coefficients but also on the exponents. Using the concentrationcompactness principle, Fan proved the existence of ground state solutions vanishing at infinity under appropriate assumptions. Later AlvesLiu [8] improved the result of Fan in [7], and obtained ground states of \(p(x)\)Laplacian equations in \(\mathbb{R}^{N}\). They also established a BartschWang type compact embedding theorem for variable exponent spaces. Ayoujil [9] was concerned with the existence and multiplicity of solutions to the \(p(x)\)Laplacian Steklov problem without the wellknown AmbrosettiRabinowitz type growth conditions. By means of critical point theorems with Cerami condition, he proved the existence and multiplicity results of the solutions under weaker conditions.
 \((f_{1})\) :

\(f(x, 0)=0\), \(f(x, t)=o(\vert t\vert ^{p2}t)\), as \(\vert t\vert \rightarrow 0\), uniformly in x.
 \((f_{2})\) :

\(f\in C(\mathbb{R}^{N}, \mathbb{R})\) and there exist \(C>0\) and \(q\in (p, p^{*})\) such thatwhere \(p^{*}=Np/(Np)\) if \(N>p\), and \(p^{*}=\infty \) if \(N\leq p\).$$\bigl\vert f(x, t)\bigr\vert \leq C\bigl(1+\vert t\vert ^{q1} \bigr), $$
 \((f_{3})\) :

\(\lim_{\vert t\vert \rightarrow \infty }\frac{F(x, t)}{\vert t\vert ^{p}}=+\infty \), where \(F(x, t)= \int^{t}_{0}f(x, s)\,ds\).
 \((f_{4})\) :

There exists \(R>0\) such that, for any x, \(\frac{f(x, t)}{\vert t\vert ^{p2}t}\) is increasing in \(t\geq R\), and decreasing in \(t\leq R\).
Remark 1.1
Remark 1.2
In recent years there were some articles trying to drop the AR condition in the study of the superlinear problems. For equation (1.1), LiuWang [4] first posed the SQ condition to get the bounds of a minimizing sequence on the Nehari manifold. Furthermore, under coercive conditions of a potential function \(V(x)\), they proved the existence of three solutions of equation \(\Delta u+V(x)u=f(x,u)\) (\(u\in H^{1}(\mathbb{R}^{N})\)), one positive, one negative and one signchanging solution. Later LiWangZeng [14] gave a natural generalization of LiuWang’s results [4] to two noncompact cases, which do not have compact embedding. We made use of a combination of the techniques in [4] and the concentrationcompactness principle of Lions [15, 16]. Then we gave general conditions which ensure the existence of ground state solutions. MiyagakiSouto [17] established the existence of a nontrivial solution of (1.1) by combining some arguments of StruweTarantello in [18]. Then Liu [19] extended the results of MiyagakiSouto [17], and obtained the existence and multiplicity results for superlinear pLaplacian equations (1.2) without the AR condition. To overcome the difficulty that the PalaisSmale sequences of the EulerLagrange functional may be unbounded, they consider the Cerami sequences.
TanFang [20] considered the \(p(x)\)Laplacian equations on the bounded domain and expanded a recent result [21] of GasinskiPapageorgiou. The nonlinearity is superlinear but does not satisfy the usual AR condition near infinity, or its dual version near zero. They obtained the existence and multiplicity results via Morse theory and modified functional methods. Ge [22] dealt with the superlinear elliptic problem without AmbrosettiRabinowitz type growth condition in a bounded domain with smooth boundaries. He obtained the existence results of nontrivial solutions for every parameter. Using variational arguments, CarvalhoGoncalvesSilva [23] established the existence of multiple solutions for quasilinear elliptic problems driven by the ΦLaplacian operator.
The main result of this paper is as follows.
Theorem 1.3
Under the assumptions \((f_{1})\)\((f_{4})\), for every integer \(k>0\), there exist a pair \(u^{+}_{k}\) and \(u_{k}^{}\) of radial solutions of (1.2) with \(u_{k}^{}(0)<0<u_{k}^{+}(0)\), having exactly k nodes; \(0<\rho_{1}^{\pm }<\cdots <\rho_{k}^{\pm }<\infty \).
Here a node \(\rho >0\) is defined such that \(u(\rho)=0\).
Theorem 1.4
Under the assumptions \((f_{1})\)\((f_{4})\) and if \(f(x, u)\) is odd in u, there exist infinitely many nonradial nodal solutions of (1.2).
Remark 1.5
It is also possible to replace the oddness of \(f(x, u)\) in Theorem 1.4 by other conditions, we refer the reader to the work of JonesKüpper [24].
 \((f_{5})\) :

\(f\in C(\mathbb{R}^{N}, \mathbb{R})\) and for some \(C>0\),where \(q=p^{*}\) if \(N\geq 3\) and \(q\in (p, p^{*})\) if \(N=2\).$$\bigl\vert f'_{u}(x, t)\bigr\vert \leq C\bigl(1+ \vert t\vert ^{q2}\bigr), $$
Corollary 1.6
Assume \(N=4\) or \(N\geq 6\), the assumptions \((f_{2})\)\((f_{5})\) hold, and f is odd in u. Then equation (1.2) has an unbounded sequence of nonradial signchanging solutions.
In the present paper, we give a new proof for the infinite solutions to problem (1.2) having a prescribed number of nodes, and the results are proved under the weaker SQ condition. It turns out that the SQ condition on \(f(x, u)\) suffice to guarantee infinitely many solutions. Our theorems generalize the results in [4] to the case of \(p\neq 2\). At the same time, a global characterization of the critical values of the nodal radial solutions are given.
2 Preliminaries
In this section, we give some notations and some preliminary lemmas, which will be adopted in Section 3.
Notation 2.1
The letters C will always denote various universal constants.
Lemma 2.2
Proof
First we want to prove that \(\{u_{n}\}\) is bounded. If not, consider \(v_{n}:=u_{n}/\Vert u_{n}\Vert \), then \(\Vert v_{n}\Vert =1\). By passing to a subsequence, we may assume \(v_{n}\rightarrow v\) weakly in \(X_{\rho, \sigma }\) and strongly in \(L^{r}(X_{\rho, \sigma })\) for any \(r\in [p, p^{*}]\). Note that \((f_{1})\) and \((f_{2})\) imply \(\int_{X_{\rho, \sigma }}F(x, u)\) is weakly continuous on \(X_{\rho, \sigma }\).
\((f_{4})\) implies that \(\max_{t>0} J(tu)\) is achieved at only one point \(t=s\). It is also the unique one such that \(\langle J'(tu), u\rangle =0\).
Next we claim that su is a critical point of J. Without loss of generality, we assume \(s=1\). If u is not a critical point, there is \(v\in C_{0}^{\infty }(\Omega)\) such that \(\langle J'(u), v\rangle =2\). There is \(\varepsilon_{0}>0\) such that, for \(\vert t1\vert +\vert \varepsilon \vert \leq \varepsilon_{0}\), \(\langle J'(tu+\varepsilon v)\), \(v\rangle \leq 1\).
Lemma 2.3
Under assumptions \((f_{2})\)\((f_{5})\), if f is odd in u, equation (2.1) has infinitely many pairs of solutions.
Proof
It is clear that the solutions occur in pairs due to the oddness of \(f(x, u)\). Under the assumptions, any critical point of J restricted on \(\mathcal{N}_{2}\) is a critical point of J in \(X_{2}\). To verify the PS condition it suffices to show that any PS sequence is bounded. This is similar to the proof of Lemma 2.2. We omit the details.
If the PS condition is satisfied on \(\mathcal{N}_{2}\), then the standard LjusternikSchnirelmann theory gives rise to an unbounded sequence of critical values of J; see the details in [25]. □
3 Proof of theorems
In this section, we prove Theorem 1.3 and Theorem 1.4.
Proof of Theorem 1.3
Let \(\{u_{n}\}\) be a minimizing sequence of \(c^{+}_{k}\). As the same arguments hold in the proof of Lemma 2.2, \(\{u_{n}\}\) is bounded.
Since \(u_{n}\in \mathcal{N}^{+}_{k}\), there exist \(0=\rho_{0}^{n}<\rho _{1}^{n}<\cdots <\rho_{k}^{n}<\rho^{n}_{k+1}=\infty \) such that \((1)^{j}u_{n}\vert _{\Omega (\rho_{j}^{n}, \rho_{j+1}^{n})}\geq 0\) and \(u_{n}\vert _{\Omega (\rho_{j}^{n}, \rho_{j+1}^{n})}\in \mathcal{N}_{\rho_{j}^{n}, \rho_{j+1}^{n}}\) for \(j=0, \ldots,k\).
Along a subsequence of \(\{u_{n}\}\), we may assume that \(u_{n}\rightarrow u\) weakly in \(X_{1}\), and strongly in \(L^{r}(X_{1})\) for any \(r\in [p, p^{*}]\). It follows that \(u_{n}\vert _{\Omega (\rho_{j}^{n}, \rho_{j+1}^{n})}\rightarrow u\vert _{\Omega (\rho_{j}, \rho_{j+1})}\) weakly in \(X_{1}\), and strongly in \(L^{r}(X_{1})\) (\(r\in [p, p^{*})\)). And \((1)^{j}u\vert _{\Omega (\rho_{j}, \rho_{j+1})}\geq 0\), for \(u\in \mathcal{N}_{\rho_{j}^{n}, \rho_{j+1}^{n}}\).
 1.
\(c_{k}^{+}\) is archived by \(u_{k}^{+}\), that is, \(J(u_{k}^{+})=c_{k} ^{+}\),
 2.
\(u_{k}^{+}\) is a radial function having nodes \(0<\rho_{1}<\cdots <\rho _{k}<\infty \),
 3.
\(u_{k}^{+}\) is a solution of (1.2).
Proof of Theorem 1.4
Using a result of Lions in [26], it is possible to find a subspace E of \(X_{2}\) consisting of functions which are not radial and such that the inclusion \(E\hookrightarrow L^{s}\) is compact for \(p< s< p^{*}\); see the details in Theorem IV.1 of [26] or the proof of Theorem 2.1 in [6]. By Proposition 3.2 in [6], the subspace E should be chosen to satisfy the compactness. Here we describe E briefly. Let G be a group acting on \(X_{2}\) via orthogonal maps \(\varrho (g):X_{2}\rightarrow X_{2}\), such that the functional J is Ginvariant, and the inclusion \(X_{2}^{G}\hookrightarrow L^{s}\) is compact for \(p< s< p^{*}\), where \(X_{2}^{G}:=\{u\in X_{2}, \varrho (g)u=u,\mbox{ for all }g\in G\}\). We set \(E:=X_{2}^{G}\). Then we follow the same steps in Lemma 2.2, and combine with Lemma 2.3 to get the infinitely many nonradial nodal solutions of (1.2). □
4 Conclusion
We are concerned with the multiplicity of solutions for a Schrödinger problem. Based on our work [14] about (1.1), which gave a natural generalization of LiuWang’s results [4] to two noncompact cases, we give a new proof for the infinite solutions having a prescribed number of nodes to problem (1.2) in the present paper. It turns out that the weaker condition of the nonlinearity suffices to guarantee the infinitely many solutions.
Declarations
Acknowledgements
The author is supported by the National Natural Science Foundation of China (NSFC, 11501110). She thanks the anonymous referee for a careful and thorough reading of the manuscript.
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.
Authors’ Affiliations
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