Advertisement

Annali di Matematica Pura ed Applicata (1923 -)

, Volume 197, Issue 6, pp 1875–1883 | Cite as

Foliated Schwarz symmetry for the nodal solution at the second minimax level

  • Kui Li
  • Zhitao Zhang
Article

Abstract

We study the second minimax level \(\lambda _2\) of the eigenvalue problem for the scalar field equation in \({\mathbb {R}}^N\). By using the tool of polarization, we prove that every nodal solution at the second minimax level is foliated Schwarz symmetric. As a consequence, we prove an open problem of Perera and Tintarev (Annali di Matematica Pura ed Applicata (4) 194(1):131–144, 2015).

Keywords

Second minimax level Nodal solution Foliated Schwarz symmetry Maximum principle 

Mathematics Subject Classification

35J20 35J61 35P30 

1 Introduction

In this paper, we consider variational problems of the form
$$\begin{aligned} \inf \limits _{\gamma \in \Gamma _2} \max \limits _{u \in \gamma (S^1)}J(u), \end{aligned}$$
where \(\Gamma _2 :=\{ \gamma \in C(S^1,{\mathcal {M}}): \gamma (-\theta )=-\gamma (\theta ), \forall \theta \in S^1\}\), \(S^1:=\{y \in {{\mathbb {R}}}^2: |y|=1\}\), \({\mathcal {M}}\) is a \(C^2\) connected sub-manifold of a Hilbert space H, \({\mathcal {M}}=-{\mathcal {M}}\) and \(J: H\rightarrow {{\mathbb {R}}}\) is a \(C^1\) functional. For some differential equations, we want to show that any nodal solution which attains the above minimax level is foliated Schwarz symmetric with respect to some point \(P\in S^{N-1}\).

Let \(\Omega \subseteq {{\mathbb {R}}}^N\) be a symmetric domain, a continuous function \(u(x): \Omega \rightarrow {{\mathbb {R}}}\) is called to be foliated Schwarz symmetric with respect to some point \(P \in S^{N-1}=\{y \in {\mathbb {R}}^N: |y|=1 \}\) if u depends on \(r=|x|>0\) and \(\theta = \arccos (\frac{x}{|x|}\cdot P)\) and u is nonincreasing in \(\theta \) for any fixed \(r>0\).

We focus on the topic of the eigenvalue problem for the scalar field equation
$$\begin{aligned} -\Delta u + V(x) u = \lambda |u|^{p-2} u,~~~u \in H^1({\mathbb {R}}^N), \end{aligned}$$
(1.1)
where \(N \ge 3\), \(p \in (2,2^*]\) (\(2^*=\frac{2N}{N-2}\)), \(V \in L^\infty ({\mathbb {R}}^N)\).
For \(u \in H^1(\mathbb {{\mathbb {R}}}^N)\), let
$$\begin{aligned} J(u)=\int _{{\mathbb {R}}^N}[|\nabla u|^2+ V(x)u^2]\mathrm{d}x,~~I(u)=\int _{{\mathbb {R}}^N}|u|^p \mathrm{d}x. \end{aligned}$$
Thus, we define
$$\begin{aligned} {\mathcal {M}}= \{u \in H^1(\mathbb {{\mathbb {R}}}^N): I(u)=1\}, ~~\lambda _2:= \inf \limits _{\gamma \in \Gamma _2} \max \limits _{u \in \gamma (S^1)}J(u), \end{aligned}$$
where \(\Gamma _2 :=\{ \gamma \in C(S^1,{\mathcal {M}}): \gamma (-\theta )=-\gamma (\theta ), \forall \theta \in S^1\}\).

We use the tool of polarization (see [2, 3, 8, 9, 14]) to prove that any sign changing solution of (1.1) on \({\mathcal {M}}\) corresponding to \(\lambda =\lambda _2\) is foliated Schwarz symmetric.

Theorem 1.1

Suppose that \(p \in (2, 2^*]\), \(\lambda _2>0\) and that \(V\in L^{\infty }({\mathbb {R}}^N)\) is radially symmetric. If u is a nodal solution of (1.1) on \({\mathcal {M}}\) corresponding to \(\lambda =\lambda _2\), then u is foliated Schwarz symmetric with respect to some point \(P \in S^{N-1}\), where \(S^{N-1}=\{y \in {\mathbb {R}}^N: |y|=1\}\).

Remark 1.2

As \(N=2\), \(2^*=+\infty \), then for \(2<p<+\infty \), Theorem 1.1 still holds.

Remark 1.3

Let \(V \in L^\infty ({\mathbb {R}}^N)\), \(\lambda _2>0\) and u be a nodal solution of (1.1) on \({\mathcal {M}}\) corresponding to \(\lambda =\lambda _2\). Assume that \(\Omega ={\mathbb {R}}^N\) and \(\beta \) is the number defined for the equation (1.1) with \(\lambda =1\) in [2]. Then, by scaling methods, we have
$$\begin{aligned} \beta \le (\frac{1}{2}-\frac{1}{p})\lambda _2^{\frac{p}{p-2}}. \end{aligned}$$

Perera and Tintarev [9] proposed an open problem which states that if V is radially symmetric, whether every solution of (1.1) at level \(\lambda _2\) is foliated Schwarz symmetry or not. Here, we answer this open problem when the solution is nodal. In [10], they obtain the existence of nonradial nodal solution of (1.1) under some conditions on V(x). In this paper, we can show that this solution is foliated Schwarz symmetric with respect to some point \(P \in S^{N-1}\).

Corollary 1.4

Suppose that \(p \in (2, 2^*)\), \(V\in L^{\infty }({\mathbb {R}}^N)\) is radially symmetric and satisfies
$$\begin{aligned} V(x)\ge 0~\forall x \in {\mathbb {R}}^N,~~\lim \limits _{|x| \rightarrow \infty } {V(x)}= V^{\infty }>0. \end{aligned}$$
Suppose that \(W(x):=V^{\infty }-V(x)\in L^{\frac{p}{p-2}}({\mathbb {R}}^N)\) satisfies
$$\begin{aligned} W(x)\ge c_0 e^{-a|x|}, \forall x \in {\mathbb {R}}^N,~~|W|_{\frac{p}{p-2}}<(2^{\frac{p}{p-2}}-1)\lambda _1^{\infty }, \end{aligned}$$
where \(c_0, a\) are some positive constants and
$$\begin{aligned} \lambda _1^{\infty }:= \inf \limits _{u \in {\mathcal {M}}}\int _{{\mathbb {R}}^N}[|\nabla u|^2+ V^{\infty }u^2]\mathrm{d}x. \end{aligned}$$
Then, (1.1) has a nodal solution on \({\mathcal {M}}\) corresponding to \(\lambda =\lambda _2\), which is foliated Schwarz symmetric with respect to some point \(P \in S^{N-1}\).
For \(u, v \in H^1({\mathbb {R}}^N)\), we define the inner product as follows
$$\begin{aligned} (u,v)={\int }_{{\mathbb {R}}^N} \nabla u \nabla v \mathrm{dx+ {}\int }_{{\mathbb {R}}^N}uv \mathrm{d}x, ~~\forall u,v \in H^1({\mathbb {R}}^N), \end{aligned}$$
and the corresponding norm is denoted by \(\Vert u\Vert \). For \(1\le p \le +\infty \) and \(f\in L^p({\mathbb {R}}^N)\), we denote by \(|f|_p\) the usual \(L^p\)-norm of f, \(f^+=\max \{f,0\}\) and \(f^-=\max \{-f,0\}\).

This paper is organized as follows: in Sect. 2, we give some preliminaries; in Sect. 3, we prove Theorem 1.1.

2 Preliminaries

In this section, we give some preliminaries. Let \({\mathcal {H}}\) be all the closed affine halfspaces in \({\mathbb {R}}^N\) and \({\mathcal {H}}_0=\{H:H\in {\mathcal {H}},0\in \partial H\}\). For \(H \in {\mathcal {H}}\), let \(\sigma _H : {\mathbb {R}}^N \rightarrow {\mathbb {R}}^N\) be the reflection with respect to the boundary of H, and for a measurable function \(u:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\), let
$$\begin{aligned} u_H(x)= {\left\{ \begin{array}{ll} \max \{u(x), u(\sigma _H(x))\}, &{} x \in H,\\ \min \{u(x), u(\sigma _H(x))\}, &{} x \in {\mathbb {R}}^N \backslash H \end{array}\right. } \end{aligned}$$
be the polarization of u with respect to H (see [2]).

Remark 2.1

From the definition of the polarization, it is easy to check that \(u_H\) is measurable.

The following definition can be found in [4] or [12].

Definition 2.2

Let \(\Omega \subseteq {\mathbb {R}}^N\) be a symmetric domain, a continuous function \(u(x): \Omega \rightarrow {\mathbb {R}}\) is said to be foliated Schwarz symmetric with respect to some point \(P \in S^{N-1}=\{y \in {\mathbb {R}}^N: |y|=1 \}\) if u depends on \(r=|x|>0\) and \(\theta = \arccos (\frac{x}{|x|}\cdot P)\) and u is nonincreasing in \(\theta \) for any fixed \(r>0\).

Remark 2.3

Let \({\mathcal {H}}_P:=\{H :H\in {\mathcal {H}}_0~\text{ and }~P \in \mathring{H}\}\). The symmetrization \(A^P\) of a set \(A \subseteq S_r^{N-1}(0)= \partial B_r(0)\) with respect to P is defined as the closed geodesic ball in \(S_r^{N-1}(0)\) centered at rP which satisfies \( H^{N-1}(A^P)=H^{N-1}(A)\), where \(H^{N-1}\) is the \({N-1}\)-dimensional Hausdorff measure (see [1]). For a continuous function \(u(x): \Omega \rightarrow {\mathbb {R}}\), the foliated Schwarz symmetrization \(u_P: \Omega \rightarrow {\mathbb {R}}\) of u with respect to P is defined by the condition
$$\begin{aligned} \{u_P\ge t\} \cap S_r^{N-1}(0)=[\{u \ge t\}\cap S_r^{N-1}(0)]^P, \quad \forall ~r>0,~t \in {\mathbb {R}}. \end{aligned}$$
One can check that \(u_P\) is measurable and that u is foliated Schwarz symmetric with respect to P if and only if \(u=u_P\) (See [14], p. 214).

The next two Lemmata can be found in [2], we omit their proofs.

Lemma 2.4

For every measurable function \(u:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) and for every \(H \in {\mathcal {H}}\), there holds
$$\begin{aligned} (u_H)^+=(u^+)_H, (u_H)^-=(u^-)_H. \end{aligned}$$

In the following, we may simply write \(u_H^{\pm }\) due to Lemma 2.4.

Lemma 2.5

If \(u \in W^{1,p}({\mathbb {R}}^N)\) and \(v \in L^p({\mathbb {R}}^N)\) for some \(1\le p < +\infty \), then for any \(H \in {\mathcal {H}}\), we have
$$\begin{aligned} u_H \in W^{1,p}({\mathbb {R}}^N),~~ v_H \in L^p({\mathbb {R}}^N), \end{aligned}$$
and
$$\begin{aligned} \int _{{\mathbb {R}}^N}|\nabla u^{\pm }|^p\mathrm{d}x=\int _{{\mathbb {R}}^N}|\nabla u_H^{\pm }|^p\mathrm{d}x,~~ \int _{{\mathbb {R}}^N}|v^{\pm }|^p\mathrm{d}x= \int _{{\mathbb {R}}^N}|v_H^{\pm }|^p\mathrm{d}x. \end{aligned}$$
Moreover, if \(U:{\mathbb {R}}^N\rightarrow {\mathbb {R}}\) is measurable and radially symmetric, and if
$$\begin{aligned} \int _{{\mathbb {R}}^N}|U||v^{\pm }|^p\mathrm{d}x <+\infty , \end{aligned}$$
then for any \(H \in {\mathcal {H}}_0\), we have
$$\begin{aligned} \int _{{\mathbb {R}}^N}U|v^{\pm }|^p\mathrm{d}x= \int _{{\mathbb {R}}^N}U|v_H^{\pm }|^p\mathrm{d}x. \end{aligned}$$

Lemma 2.6

Suppose that \(P \in S^{N-1}\) and \(u \in C({\mathbb {R}}^N)\). If \(u_H(x)=u(x)\) for any \(H \in {\mathcal {H}}_P\), then u is foliated Schwarz symmetric with respect to P.

Proof

According to Definition 2.2, we only need to prove that if \(|x_1|=|x_2|\) and \(\theta _1 = \arccos (\frac{x_1}{|x_1|}\cdot P) \le \theta _2 = \arccos (\frac{x_2}{|x_2|}\cdot P)\), then \(u(x_1)\ge u(x_2)\).

If \(\theta _1 < \theta _2\), then \(x_1\) is closer to P than \(x_2\); hence, there exists \(H \in {\mathcal {H}}_P \) such that \(x_1 \in H\) and \(\sigma _H(x_1)=x_2\). Since \(u_H(x_i)=u(x_i)\) for \(i=1,2\), we have \(u(x_1)\ge u\circ \sigma _H (x_1)=u(x_2)\) by the definition of \(u_H\).

If \(\theta _1 = \theta _2\), then there exists a sequence \(\{\tilde{H}_n\}_{n=1}^{\infty } \subseteq {\mathcal {H}}_P\) with \(x_1 \in \tilde{H}_n \) and \(\sigma _{\tilde{H}_n}(x_1)\rightarrow x_2\) as \(n \rightarrow \infty \). Since \(u(x_1)\ge u\circ \sigma _{\tilde{H}_n} (x_1)\), we have \(u(x_1)\ge \lim \limits _{n \rightarrow \infty } u\circ \sigma _{\tilde{H}_n} (x_1)=u(x_2)\). \(\square \)

3 Proof of Theorem 1.1

For \(\lambda \in {\mathbb {R}}\), let \(K_\lambda =\{u\in {\mathcal {M}}:J|_{\mathcal {M}}'(u)=0, J(u)=\lambda \}\).

Lemma 3.1

If \(\gamma _0 \in \Gamma _2\) satisfying \(\max \limits _{\theta \in S^1}J(\gamma _0(\theta ))= \lambda _2\), then there exists \(\theta _0 \in S^1\) such that \(J|_{\mathcal {M}}'(\gamma _0(\theta _0))=0\) and \(J(\gamma _0(\theta _0))=\lambda _2\).

Proof

Let \(\Sigma =\{\theta \in S^1:J(\gamma _0(\theta ))=\lambda _2\}\). Then, \(\Sigma \) and \(\gamma _0(\Sigma )\) are compact. If \(J|_{\mathcal {M}}'(\gamma _0(\theta ))\ne 0\) for any \(\theta \in \Sigma \), then there exist \(\varepsilon ,\delta >0\) such that
$$\begin{aligned} \forall u \in J^{-1}(\left[ \lambda _2-2\varepsilon ,\lambda _2+2\varepsilon ]\right) \cap (\gamma _0(\Sigma ))_{3\delta }:\Vert J|_{\mathcal {M}}'(u)\Vert \ge \frac{8\varepsilon }{\delta }, \end{aligned}$$
where \((\gamma _0(\Sigma ))_{3\delta }\) is the \(3\delta \) neighbor of \(\gamma _0(\Sigma )\) in \({\mathcal {M}}\). Therefore, by Deformation Lemma (see [11] or [15]), there exists \(\eta : C([0,1]\times {\mathcal {M}}, {\mathcal {M}})\) such that
  1. (i)

    \(\eta (1,J^{\lambda _2+\varepsilon })\cap J((\gamma _0(\Sigma ))_\delta ) \subset J^{\lambda _2-\varepsilon }\),

     
  2. (ii)

    \(\eta (t,-u)=-\eta (t,u),\forall t\in [0,1], u\in {\mathcal {M}}\),

     
  3. (iii)

    \(J(\eta (\cdot ,u))\) is nonincreasing in \({\mathbb {R}}\) for any \(u\in {\mathcal {M}}\).

     
Let \(\tilde{\gamma _0}(\theta )=\eta (1,\gamma _0(\theta ))\), then \(\tilde{\gamma _0}\in \Gamma _2\) by (ii). Since \(\gamma _0(S^1) \subseteq J^{\lambda _2}\), we have \(\sup \limits _{\theta \in S^1}J(\tilde{\gamma _0}(S^1)) < \lambda _2\) by (i) and (iii), this contradicts to the definition of \(\lambda _2\). \(\square \)
For \(u,v\in H^1({\mathbb {R}}^N)\backslash \{0\}\),\( v^{\pm }\ne 0\), define
$$\begin{aligned} \widehat{u}=\frac{u}{|u|_p} ~~\text{ and }~~\gamma _v(\cos (t),\sin (t))=\frac{\cos (t)\widehat{v^+} +\sin (t)\widehat{v^-}}{\left( |\cos (t)|^p+|\sin (t)|^p\right) ^\frac{1}{p}}, \end{aligned}$$
then
$$\begin{aligned} \gamma _v \in \Gamma _2. \end{aligned}$$
Now, we prove another Lemma which is similar to Proposition 4.2 of [2].

Lemma 3.2

Suppose that \(\lambda _2 > 0\) and (1.1) has a nodal solution on \({\mathcal {M}}\) corresponding to \(\lambda =\lambda _2\). Then,
$$\begin{aligned} \lambda _2=\inf \limits _{u \in {\mathcal {M}}, u^{\pm }\ne 0}\max \limits _{t \in [0, 2\pi ]} \frac{J(\widehat{u^+})\cos ^2(t)+ J(\widehat{u^-})\sin ^2(t)}{(|\cos (t)|^p+|\sin (t)|^p)^\frac{2}{p}}. \end{aligned}$$
Moreover, if \(u\in {\mathcal {M}}\) satisfies \(u^{\pm }\ne 0\) and
$$\begin{aligned} \max \limits _{t \in [0, 2\pi ]} \frac{J(\widehat{u^+})\cos ^2(t)+ J(\widehat{u^-})\sin ^2(t)}{(|\cos (t)|^p+|\sin (t)|^p)^\frac{2}{p}}=\lambda _2, \end{aligned}$$
then there exits \(t_0 \in [0,2\pi ]\) such that
$$\begin{aligned} J|_{\mathcal {M}}'(\gamma _u(\theta _0))=0~ \text{ and }~J(\gamma _u(\theta _0))=\lambda _2. \end{aligned}$$

Proof

Let
$$\begin{aligned} \begin{aligned} \lambda '_2&=\inf \limits _{u \in {\mathcal {M}}, u^{\pm }\ne 0} ~\max \limits _{t \in (0, 2\pi )} \frac{J(\widehat{u^+})\cos ^2(t)+ J(\widehat{u^-})\sin ^2(t)}{(|\cos (t)|^p+|\sin (t)|^p)^\frac{2}{p}}\\&= \inf \limits _{u \in {\mathcal {M}}, u^{\pm }\ne 0}\max \limits _{S^1}J(\gamma _u). \end{aligned} \end{aligned}$$
Hence,
$$\begin{aligned} \lambda '_2 \ge \lambda _2. \end{aligned}$$
On the other hand, let \(u_0\) be a nodal solution of (1.1) on \({\mathcal {M}}\) corresponding to \(\lambda =\lambda _2\), and then
$$\begin{aligned} -\Delta u_0 + V(x) u_0 = \lambda _2|u_0|^{p-2} u_0. \end{aligned}$$
Testing above equation with \(u_0^{+}\) and \(u_0^{-}\) and integrating over \({\mathbb {R}}^N\) give
$$\begin{aligned} J(u_0^{\pm })=\lambda _2|u_0^{\pm }|_p^p,~~~J(\widehat{u_0^{\pm }})=\frac{J(u_0^{\pm })}{|u_0^{\pm }|_p^2}=\lambda _2|u_0^{\pm }|_p^{p-2}. \end{aligned}$$
Therefore, according to the Hölder inequality and the positivity of \(\lambda _2\), we have
$$\begin{aligned} \begin{aligned} \lambda '_2&\le J(\gamma _{u_0}(\cos (t),\sin (t)))=\frac{J(\widehat{u_0^+})\cos ^2(t)+ J(\widehat{u_0^-})\sin ^2(t)}{(|\cos (t)|^p+|\sin (t)|^p)^\frac{2}{p}}\\&=\lambda _2 \frac{|u_0^+|_p^{p-2}\cos ^2(t)+ |u_0^-|_p^{p-2}\sin ^2(t)}{(|\cos (t)|^p+|\sin (t)|^p)^\frac{2}{p}}\le \lambda _2(|u_0^+|_p^p+|u_0^-|_p^p)^{\frac{p-2}{p}}=\lambda _2. \end{aligned} \end{aligned}$$
Hence,
$$\begin{aligned} \lambda _2=\lambda '_2. \end{aligned}$$
If
$$\begin{aligned} \max \limits _{t \in (0, 2\pi )} \frac{J(\widehat{u^+})\cos ^2(t)+ J(\widehat{u^-})\sin ^2(t)}{(|\cos (t)|^p+|\sin (t)|^p)^\frac{2}{p}}=\max \limits _{S^1}J(\gamma _u)=\lambda _2, \end{aligned}$$
then Lemma 3.1 implies that there exits \(t_0 \in [0,2\pi ]\) such that
$$\begin{aligned} J|_{\mathcal {M}}'(\gamma _u(\theta _0))=0~ \text{ and }~J(\gamma _u(\theta _0))=\lambda _2. \end{aligned}$$
\(\square \)

Lemma 3.3

If \(\lambda _2>0\), \(V \in L^\infty ({\mathbb {R}}^N)\) is symmetric and \(u_0 \in K_{\lambda _2}\) is nodal, then \(u_{0H}\) is also nodal and \(u_{0H}\) belongs to \(K_{\lambda _2}\) for any \(H \in {\mathcal {H}}_0\).

Proof

In the following, we fixed a \(H \in {\mathcal {H}}_0\). Since \(u_0\) satisfies (1.1) with \(\lambda = \lambda _2\) and \(V \in L^\infty ({\mathbb {R}}^N)\), we have \(u_0 \in C^{1,\alpha }({\mathbb {R}}^N)\) for any \(\alpha \in (0,1)\) by elliptic regularity (see [6]). Hence,
$$\begin{aligned} u_{0H}=\frac{1}{2}u_0+\frac{1}{2}u_0\circ \sigma _H+\frac{1}{2}\mathcal {\chi }_H|u_0-u_0\circ \sigma _H|-\frac{1}{2} (1-\mathcal {\chi }_H)|u_0-u_0\circ \sigma _H| \in C({\mathbb {R}}^N). \end{aligned}$$
Since \(u_0\) is nodal, we have that \(u_{0H}\) is nodal and \(\{y \in {\mathbb {R}}^N: |u_{0H}(y)|=0\} \ne \emptyset \).
Note that
$$\begin{aligned} \begin{aligned}&J(\gamma _{u_0}(\cos (t),\sin (t)))=\frac{J(\widehat{u_0^+})\cos ^2(t)+ J(\widehat{u_0^-})\sin ^2(t)}{(|\cos (t)|^p+|\sin (t)|^p)^\frac{2}{p}},\\&J(\gamma _{u_{0H}}(\cos (t),\sin (t)))=\frac{J(\widehat{u_{0H}^+})\cos ^2(t)+ J(\widehat{u_{0H}^-})\sin ^2(t)}{(|\cos (t)|^p+|\sin (t)|^p)^\frac{2}{p}}. \end{aligned} \end{aligned}$$
Lemmas 2.4 and 2.5 give
$$\begin{aligned} J(\widehat{u_0^+})=J(\widehat{u_{0H}^+}), J(\widehat{u_0^-})=J(\widehat{u_{0H}^-}), \end{aligned}$$
hence
$$\begin{aligned} J(\gamma _{u_0}(\cos (t),\sin (t)))=J(\gamma _{u_{0H}}(\cos (t),\sin (t))), \end{aligned}$$
$$\begin{aligned} \begin{aligned} \max \limits _{t \in [0, 2\pi ]}J(\gamma _{u_0}(\cos (t),\sin (t)))=\max \limits _{t \in [0, 2\pi ]}J(\gamma _{u_{0H}}(\cos (t),\sin (t)))=\lambda _2. \end{aligned} \end{aligned}$$
On the other hand,
$$\begin{aligned} \begin{aligned}&J(\gamma _{u_{0H}}(\cos (t),\sin (t)))\\&=J(\gamma _{u_0}(\cos (t),\sin (t)))\\&=\lambda _2 \frac{|u_0^+|_p^{p-2}\cos ^2(t)+ |u_0^-|_p^{p-2}\sin ^2(t)}{(|\cos (t)|^p+|\sin (t)|^p)^\frac{2}{p}}\\&=\lambda _2 \frac{|u_{0H}^+|_p^{p-2}\cos ^2(t)+ |u_{0H}^-|_p^{p-2}\sin ^2(t)}{(|\cos (t)|^p+|\sin (t)|^p)^\frac{2}{p}}\\&\le \lambda _2 \end{aligned} \end{aligned}$$
with equality holds if and only if \(|\tan (t)|=\frac{|u_0^+|_p}{|u_0^-|_p}=\frac{|u_{0H}^+|_p}{|u_{0H}^-|_p}\). Thus, Lemma 3.2 implies that there is a \(t_0 \in [0,2\pi ]\) such that
$$\begin{aligned} |\tan (t_0)|=\frac{|u_0^+|_p}{|u_0^-|_p}=\frac{|u_{0H}^+|_p}{|u_{0H}^-|_p}~\text{ and } ~ \frac{\cos (t_0)\widehat{u_{0H}^+}+\sin (t_0)\widehat{u_{0H}^-}}{(|\cos (t_0)|^p+|\sin (t_0)|^p)^\frac{1}{p}}\in K_{\lambda _2}, \end{aligned}$$
hence
$$\begin{aligned} u_{0H}=u_{0H}^+-u_{0H}^-~\text{ or }~|u_{0H}|=u_{0H}^++u_{0H}^- \in K_{\lambda _2}. \end{aligned}$$
If \(|u_{0H}| \in K_{\lambda _2}\), then
$$\begin{aligned} -\Delta |u_{0H}| + V(x) |u_{0H}| = \lambda _2 |u_{0H}|^{p-1}\ge 0. \end{aligned}$$
Since \(V \in L^\infty ({\mathbb {R}}^N)\), elliptic regularities ([6]) give \(|u_{0H}| \in C^{1,\alpha }({\mathbb {R}}^N)\) for any \(\alpha \in (0,1)\). Maximal principle (see [5] or [7]) and the fact that \(\{y \in {\mathbb {R}}^N: |u_{0H}(y)|=0\} \ne \emptyset \) imply that \(|u_{0H}|\equiv 0\), and this contradicts with the fact that \(|u_{0H}|_p=|u|_p=1\).
Therefore,
$$\begin{aligned} u_{0H}\in K_{\lambda _2}. \end{aligned}$$
\(\square \)

Now, we turn to the proof of Theorem 1.1.

Proof

If u is nodal and solves (1.1) with \(\lambda = \lambda _2\), then \(u_H\) also solves (1.1) with \(\lambda = \lambda _2\) for any \(H\in {\mathcal {H}}_0\) by Lemma 3.3. Since \(V\in L^{\infty }({\mathbb {R}}^N)\), elliptic regularities (see [6]) give \(u, u_H \in C^{1,\alpha }({\mathbb {R}}^N)\) for any \(\alpha \in (0,1)\). Choose \(P \in S^{N-1}\) such that \(u(P)= \max \{u(x):x \in S^{N-1}\}\) and we will show that u is foliated Schwarz symmetric with respect to P.

Fix any \(H \in {\mathcal {H}}_P\). In \(\mathring{H}:=H\backslash \partial H\), we have
$$\begin{aligned} |u-u\circ \sigma _H|=2u_H-(u+u\circ \sigma _H), \end{aligned}$$
therefore,
$$\begin{aligned} \begin{aligned}&-\Delta |u-u\circ \sigma _H| + V(x) |u-u\circ \sigma _H|\\&\quad =2(-\Delta u_H + V(x)u_H)-(-\Delta u + V(x)u-\Delta u\circ \sigma _H + V(x)u\circ \sigma _H))\\&\quad =2\lambda _2 |u_H|^{p-2} u_H-(\lambda _2 |u|^{p-2} u + \lambda _2 |u\circ \sigma _H|^{p-2} u\circ \sigma _H)\\&\quad =\lambda _2 [(|u_H|^{p-2} u_H-|u|^{p-2} u)+(|u_H|^{p-2} u_H-|u\circ \sigma _H|^{p-2} u\circ \sigma _H))]\ge 0, \end{aligned} \end{aligned}$$
hence
$$\begin{aligned} |u-u\circ \sigma _H| > 0~\text{ in } ~\mathring{H}~\text{ or } ~|u-u\circ \sigma _H|=0~\text{ in } ~\mathring{H}. \end{aligned}$$
Since \(P \in \mathring{H}\), we have
$$\begin{aligned} u(P) \ge u\circ \sigma _H(P), \end{aligned}$$
hence
$$\begin{aligned} u > u\circ \sigma _H ~\text{ in } ~\mathring{H}~\text{ or } ~ u=u\circ \sigma _H ~\text{ in } ~\mathring{H}. \end{aligned}$$
Therefore,
$$\begin{aligned} u_H=u. \end{aligned}$$
Thus, for any \(H \in {\mathcal {H}}_P\), we have
$$\begin{aligned} u_H=u. \end{aligned}$$
Lemma 2.6 implies u is foliated Schwarz symmetric with respect to P, and hence we get Theorem 1.1. \(\square \)

Remark 3.4

The condition on V(x) in Theorem 1.1 can be relaxed to “\(V\in L_{loc}^\infty \)”. Replacing \(H^1({\mathbb {R}}^N)\) by
$$\begin{aligned} H=\{u\in H^1(\mathbb {{\mathbb {R}}}^N):\int _{{\mathbb {R}}^N}[|\nabla u|^2+ (1+|V(x)|)u^2]\mathrm{d}x<\infty \}, \end{aligned}$$
the inner product on H can be defined as follows
$$\begin{aligned} (u,v)={\int }_{{\mathbb {R}}^N} \nabla u \nabla v \mathrm{d}x+ {\int }_{{\mathbb {R}}^N}(1+|V(x)|)uv\mathrm{d}x, ~~\forall u,v \in H. \end{aligned}$$
It is clear that H is a Hilbert space (see [13]).
Then, we define
$$\begin{aligned} {\mathcal {M}}= \{u \in H: I(u)=1\}, ~~\lambda _2:= \inf \limits _{\gamma \in \Gamma _2} \max \limits _{u \in \gamma (S^1)}J(u), \end{aligned}$$
where \(\Gamma _2 :=\left\{ \gamma \in C(S^1,{\mathcal {M}}): \gamma (-\theta )=-\gamma (\theta ), \forall \theta \in S^1\right\} \).

Using the similar methods, we can prove Theorem 1.1 under the assumption “\(V\in L_{loc}^\infty \)”.

Notes

Acknowledgements

The authors would like to thank the referees very much for their careful reading and useful comments on the previous version of this paper.

References

  1. 1.
    Albert Baernstein, I.I., Taylor, B.A.: Spherical rearrangements, subharmonic functions, and \(\ast \)-functions in \(n\)-space. Duke Math. J. 43(2), 217–439 (1976)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bartsch, T., Weth, T., Willem, M.: Partial symmetry of least energy nodal solutions to some variational problems. J. d’Anal. Mathmat. 96(1), 1–18 (2005)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bartsch, T., Wang, Z.Q., Zhang, Z.T.: On the Fučik point spectrum for Schrödinger operators on \(\mathbb{R}^N\). J. Fixed Point Theory Appl. 5(2), 305–317 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Damascelli, L., Pacella, F.: Symmetry results for cooperative elliptic systems via linearization. SIAM J. Math. Anal. 45(3), 1003–1026 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Evans, L.C.: Partial Differential Equations, Graduate Students in Mathematics, vol. 19. AMS, Providence (1998)Google Scholar
  6. 6.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 1998th edn. Springer, Berlin (2001). (Reprint of the 1998 ed.) zbMATHGoogle Scholar
  7. 7.
    Han, Q., Lin, F.: Equations, Elliptic Partial Differential. Lecture Notes, Courant Institute of Mathematical Sciences, New York (1997)Google Scholar
  8. 8.
    Pacella, F.: Symmetry results for solutions of semilinear elliptic equations with convex nonlinearities. J. Funct. Anal. 192(1), 271–282 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Perera, K., Tintarev, C.: On the second minimax level of the scalar field equation and symmetry breaking. Annali di Matematica Pura ed Applicata (4) 194(1), 131–144 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Perera, K., Tintarev, C.: A nodal solution of the scalar field equation at the second minimax level. Bull. Lond. Math. Soc. 46(6), 1218–1225 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Struwe, M.: Variational Methods. Springer, Berlin (1990)CrossRefGoogle Scholar
  12. 12.
    Weth, T.: Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods. Jahresbericht der Deutschen Mathematiker-Vereinigung 112(3), 119–158 (2010)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Willem, M.: Minimax Theorems. Springer, Berlin (1996)CrossRefGoogle Scholar
  14. 14.
    Zhang, Z.: Variational, Topological, and Partial Order Methods with Their Applications. Developments in Mathematics, vol. 29. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  15. 15.
    Zhong, C.K., Fan, X.L.: Introduction to Nonlinear Functional Analysis. Lanzhou University Press, Lanzhou (2004). (in Chinese) Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsZhengzhou UniversityZhengzhouPeople’s Republic of China
  2. 2.HLM, CEMS, HCMS, Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China
  3. 3.School of Mathematical SciencesUniversity of Chinese Academy of SciencesBeijingPeople’s Republic of China

Personalised recommendations