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Infinite solutions having a prescribed number of nodes for a Schrödinger problem
Boundary Value Problems volume 2017, Article number: 135 (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.
1 Introduction
The semilinear equation
originates from various problems in physics and mathematical physics, and is called Euclidean field equation in cosmology [1], and nonlinear KleinGordon or Schrödinger equations when one is looking for certain types of solitary waves [2]. More generally, (1.1) can be explained as the case of \(p=2\) in a more general problem:
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.
For the equation
where Ω is a bounded domain in \(\mathbb{R}^{N}\), DincaJebeleanMawhin [10] obtained the existence results under Dirichlet boundary condition. BartschLiu [11] proved the existence of several solutions of (1.3), that is, a pair of subsolution and supersolution, a positive and a negative solution, and a signchanging solution. BonannoCandito [12] established the existence of three solutions for the Neumann boundary condition of (1.3).
In this paper, we are concerned with the multiple solutions of (1.2), and require the following assumptions on the nonlinearity \(f(x, u)\):
 \((f_{1})\) :

\(f(x, 0)=0\), \(f(x, t)=o(\vert t\vert ^{p2}t)\), 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 that
$$\bigl\vert f(x, t)\bigr\vert \leq C\bigl(1+\vert t\vert ^{q1} \bigr), $$where \(p^{*}=Np/(Np)\) if \(N>p\), and \(p^{*}=\infty \) if \(N\leq p\).
 \((f_{3})\) :

\(\lim_{\vert t\vert \rightarrow \infty }\frac{F(x, t)}{\vert t\vert ^{p}}=+\infty \), 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
The assumption \((f_{3})\) comes from the following condition:
In the case \(p=2\), \((f_{3})\) characterizes problem (1.2) as superlinear at infinity. It is an extension of a very natural superquadratic condition (SQ condition for short), SQ condition: \(\lim_{\vert t\vert \rightarrow \infty }\frac{F(x, t)}{t^{2}}=\infty\).
Remark 1.2
The SQ condition is weaker than the famous AmbrosettiRabinowitz growth condition (AR condition for short). Since the work of AmbrosettiRabinowitz [13], the AR condition has most frequently appeared in the superlinear elliptic boundary value problem. AR condition: There exist \(\mu >p\) and \(R>0\) such that
It is important not only in establishing the mountainpass geometry of the functional, but also in obtaining the bounds of PS sequences. In fact, the AR condition implies that, for some \(C>0\),
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].
We further assume that:
 \((f_{5})\) :

\(f\in C(\mathbb{R}^{N}, \mathbb{R})\) and for some \(C>0\),
$$\bigl\vert f'_{u}(x, t)\bigr\vert \leq C\bigl(1+ \vert t\vert ^{q2}\bigr), $$where \(q=p^{*}\) if \(N\geq 3\) and \(q\in (p, p^{*})\) if \(N=2\).
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.
Solutions of (1.2) correspond to the critical points of the functional
where \(W^{1,p}(\mathbb{R}^{N})\) is endowed with the norm \(\Vert u\Vert = (\int_{\mathbb{R}^{N}}(\vert \nabla u\vert ^{p}+\vert u\vert ^{p}))^{\frac{1}{p}}\).
Notation 2.1
We define the Nehari manifold
where \(X_{1}:=\{u\in W^{1,p}(\mathbb{R}^{N}):u(x)=u(\vert x\vert )\}\). And
where \(X_{2}:=W^{1,p}(\mathbb{R}^{N})\). For \(0\leq \rho <\sigma \leq \infty \), we define
Define \(u(x)=0\) for \(x \notin \Omega (\rho, \sigma)\) if \(u\in X_{1}\). Obviously \(X_{\rho, \sigma }\subset X_{1}\) and \(\mathcal{N}_{\rho, \sigma }\subset \mathcal{N}_{1}\).
We fix k, and we define
On \([0, \infty)\times \mathbb{R}\), we define
and \(F^{+}(r, u):= \int_{0}^{u}f^{+}(r, s)\,ds\),
Similarly, we define
and \(F^{}(r, u)\), \(J^{}(u)\).
The letters C will always denote various universal constants.
Lemma 2.2
Under assumptions \((f_{1})\)\((f_{4})\), the equation
has a weak solution u such that
Proof
By the assumptions \((f_{1})\) and \((f_{2})\), J has a strict local minimum at 0. For any \(u\neq 0\), \(J(tu)\rightarrow  \infty \) as \(t\rightarrow \infty \). Thus
is well defined.
Let \(\{u_{n}\}\) be a minimizing sequence of c such that
as \(n\rightarrow \infty \).
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 \(X_{\rho, \sigma }\).
If \(v\neq 0\), we have
By (2.2),
Then by \((f_{3})\) and Fadou’s lemma, passing to the limit on both sides,
This gives a contradiction.
If \(v=0\), fixing an \(R>\sqrt[p]{pc}\), by \(\Vert v_{n}\Vert =1\), we have
\(J(u_{n})\) converges towards c, but \(R^{p}/p\int_{X_{\rho, \sigma }}F(x, Rv_{n})\) tends to \(R^{p}/p>c\), a contradiction. Thus \(\{u_{n}\}\) is bounded.
Assume \(u_{n}\) weakly converges to u. As \(n\rightarrow \infty \),
Since, for some \(\alpha >0\), \(\Vert u_{n}\Vert ^{p}>\alpha \), and
so \(u\neq 0\).
There is \(s>0\) such that \(J(su)= \max_{t>0} J(tu)\). Then
\((f_{4})\) 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 \(\vert t1\vert +\vert \varepsilon \vert \leq \varepsilon_{0}\), \(\langle J'(tu+\varepsilon v)\), \(v\rangle \leq 1\).
If \(\varepsilon >0\) is small, let \(t_{\varepsilon }>0\) be the unique number such that
Then \(t_{\varepsilon }\rightarrow 1\) as \(\varepsilon \rightarrow 0\).
If ε is small such that \(\vert t_{\varepsilon }1\vert +\varepsilon \leq \varepsilon_{0}\), then \(J(t_{\varepsilon }u+\varepsilon v) \geq c\), but by the assumption that \(\langle J'(tu+\varepsilon v)\), \(v\rangle \leq 1\), so
This is a contradiction. □
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
First by Lemma 2.2, the infimum
is achieved. Since \(\vert u\vert \) is also a minimizer, we assume the minimizer u is a positive solution of the problem
Similarly, the infimum
is also achieved by negative minimizers which are negative solutions of (3.1).
Then we work on the Nehari manifold \(\mathcal{N}_{k}^{+}\), and construct a \(u^{+}_{k}\in \mathcal{N}^{+}_{k}\) such that
is achieved by some \(u^{+}_{k}\), which gives the desired solutions in 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 \(j=0, \ldots,k\).
Note that
By \((f_{1})\)\((f_{2})\), 0 is a strict local minimizer of J, thus there is a \(\delta >0\) such that \(\Vert u\Vert \geq \delta \) for \(u\in \mathcal{N} _{\rho_{j}^{n}, \rho_{j+1}^{n}}\). Fix \(q\in (p, p^{*})\), and for any \(\varepsilon >0\), there is a constant \(C>0\) such that
where \(q\in (p, p^{*})\). Therefore, by choosing \(\varepsilon >0\) small we find a \(C>0\) such that
Using (3.2), in a similar way as in [3], we see that \(\{\rho_{k+1}^{n}\}_{n}\) is bounded away from ∞, \(\{\rho_{j+1} ^{n}\rho_{j}^{n}\}_{n}\) is bounded away from 0 for each j, and there are \(0=\rho_{0}<\rho_{1}<\cdots <\rho_{k}<\rho_{k+1}=\infty \) such that \(\rho_{j}^{n}\rightarrow \rho_{j}\) as \(n\rightarrow \infty \), for \(j=1, \ldots, k\).
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 \(u\in \mathcal{N}_{\rho_{j}^{n}, \rho_{j+1}^{n}}\).
Let \(n\rightarrow \infty \) in (3.2). It implies that \(u\vert _{\Omega (\rho _{j}, \rho_{j+1})}\neq 0\). Thus we can choose an \(\alpha_{j}>0\) such that \(\alpha_{j}u\vert _{\Omega (\rho_{j}, \rho_{j+1})}\in \mathcal{N}_{( \rho_{j}, \rho_{j+1})}\) for \(j=1, \ldots, k\). Define
By the definition of \(u_{k}^{+}\), we observe that \(u_{k}^{+}\in \mathcal{N}_{k}^{+}\).
Next we want to show

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).
The weak convergence of \(u_{n}\vert _{\Omega_{(\rho_{j}^{n}, \rho^{n}_{j+1})}}\) in \(X_{1}\) and strong convergence in \(L^{r}(X_{1})\) (\(p< r< p ^{*}\)) imply
Also
So \(J(u_{k}^{+})=c_{k}^{+}\).
Then the equality in (3.3) implies that \(\alpha_{j}u\vert _{\Omega (\rho_{j}^{n}, \rho^{n}_{j+1})}\) is a minimizer of
and a minimizer of
where \(P^{\pm }:=\{u\in X_{1}:\pm u\geq 0\}\). At the same time, \(\alpha_{j}u\vert _{\Omega (\rho_{j}, \rho_{j+1})}\) is a minimizer of \(\inf_{\mathcal{N}_{\rho_{j}, \rho_{j+1}}} J^{\pm }\). When j is even, \(\alpha_{j}u\vert _{\Omega (\rho_{j}^{n}, \rho^{n}_{j+1})}\) is a positive solution of (1.2), and when j is odd, \(\alpha_{j}u\vert _{\Omega (\rho _{j}^{n}, \rho^{n}_{j+1})}\) is a negative solution. Then the strong maximum principle implies that \(u_{k}^{+}(0)>0\) and \((1)^{j}u_{k} ^{+}(x)>0\), for \(\rho_{j}<\vert x\vert <\rho_{j+1}\) (\(j=0, 1, \ldots, k\)), and
So \(u_{k}^{+}\) has exactly k nodes.
In order to prove \(u_{k}^{+}\) is a solution of (1.2), for simplicity we assume \(\alpha_{j}=1\) for all j. If \(u_{k}^{+}\) is not a critical point of J, then there is a \(\varphi \in C_{0}^{\infty }(\mathbb{R}^{N})\) such that
Observe that there is a \(\tau >0\) such that, if \(\vert s_{j}1\vert \leq \tau \) (\(j=0, \ldots, k\)) and \(0\leq \varepsilon \leq \tau \), the function
where \(s=(s_{1}, \ldots, s_{k})\), has exactly k nodes \(0<\rho_{1}(s, \varepsilon)<\cdots <\rho_{k}(s, \varepsilon)<\infty\). And \(\rho_{j}(s, \varepsilon)\) is continuous in \((s, \varepsilon)\in D \times [0, \tau ]\), where \(D:=\{(s_{1}, \ldots, s_{k})\in \mathbb{R}^{k}:\vert s _{j}1\vert \leq \tau \}\), and
In order to deduce a contradiction, we set \(s\in D\), and
where \(\eta (s): D\rightarrow [0, 1]\) (\(s=(s_{1}, \ldots, s_{k})\)) is a cutoff function such that
Then, for each \(s\in D\), \(g_{1}(s)\in C(D, X)\) and \(g_{1}(s)\) has exactly k nodes \(0<\rho_{1}(s)<\cdots <\rho_{k}(s)<\infty \), where \(\rho_{j}(s)\) is continuous.
Further, we define for \(j=1, \ldots, k\),
And we define \(h: D\rightarrow \mathbb{R}^{k}\) as \(h(s):=(h_{1}(s), \ldots, h_{k}(s))\). Then \(h(s)\in C(D, \mathbb{R}^{k})\).
For a fixed j, if \(\vert s_{j}1\vert =\tau \) then \(\eta (s)=0\) and \(\rho_{i}(s)=\rho_{i}\) for all \(i=1, \ldots, k\). So by the definition of \(g_{1}(s)\),
Therefore, the degree \(\operatorname{deg}(h, \operatorname{int}(D), 0)\) is well defined and \(\operatorname{deg}(h, \operatorname{int}(D), 0)=(1)^{k}\). Thus there is an \(s\in \operatorname{int}(D)\) such that \(h(s)=0\), that is, \(g_{1}(s)\in \mathcal{N}_{k}^{+}\).
It is obvious that
On the other hand, by (3.5),
If \(\vert s_{j}1\vert \leq \tau /2\) for each j, then by (3.4)
which contradicts (3.6).
If \(\vert s_{j}1\vert >\tau /2\) for at least one j, by (3.4)
which is a contradiction with (3.6) too. The proof is finished. □
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.
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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.
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Zeng, J. Infinite solutions having a prescribed number of nodes for a Schrödinger problem. Bound Value Probl 2017, 135 (2017). https://doi.org/10.1186/s1366101708612
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DOI: https://doi.org/10.1186/s1366101708612