Localized nodal solutions of higher topological type for semiclassical nonlinear Schrödinger equations

Abstract

We investigate the existence of localized sign-changing solutions for the semiclassical nonlinear Schrödinger equation

$$\begin{aligned} -\epsilon ^2\Delta v+V(x)v=|v|^{p-2}v,\ v\in H^1(\mathbb {R}^N) \end{aligned}$$

where $N\ge 2,$ $2<p<2^*$, $\epsilon >0$ is a small parameter, and V is assumed to be bounded and bounded away from zero. When V has a local minimum point P, as $\epsilon \rightarrow 0$, we construct an infinite sequence of localized sign-changing solutions clustered at P and these solutions are of higher topological type in the sense that they are obtained from a minimax characterization of higher dimensional symmetric linking structure via the symmetric mountain pass theorem. It has been an open question whether the sign-changing solutions of higher topological type can be localized and our result gives an affirmative answer. The existing results in the literature have been subject to some geometrical or topological constraints that limit the number of localized sign-changing solutions. At a local minimum point of V, Bartsch et al. (Math Ann 338:147–185, 2007) proved the existence of N pairs of localized sign-changing solutions and D’Aprile and Pistoia (Ann Inst Hénri Poincare Anal Non Linéaire 26:1423–1451, 2009) constructed 9 pairs of localized sign-changing solutions for $N\ge 3$. Our result gives an unbounded sequence of such solutions. Our method combines the Byeon and Wang’s penalization approach and minimax method via a variant of the classical symmetric mountain pass theorem, and is rather robust without using any non-degeneracy conditions.

Appendix

The goal of this appendix is to give a proof for Lemma 3.3. We first define an operator A on $H^1(\mathbb {R}^N)$. For $u\in H^1(\mathbb {R}^N),$ defined $w=A(u)$ by the unique solution to

$$\begin{aligned} -\Delta w+V(\epsilon x)w+2\beta \left( \int _{\mathbb {R}^N}\chi _\epsilon u^2dx-1\right) ^{\beta -1}_+\chi _\epsilon w=|u|^{p-2}u,\quad w\in H^1(\mathbb {R}^N).\nonumber \\ \end{aligned}$$

(5.3)

We note that A is odd on $H^1(\mathbb {R}^N)$.

Lemma 5.3

A is well defined and continuous on $H^1(\mathbb {R}^N).$

Proof

It is easy to see A is well defined as

$$\begin{aligned} \xi (u):=2\beta \left( \int _{\mathbb {R}^N}\chi _\epsilon u^2dx-1\right) ^{\beta -1}_+ \end{aligned}$$

(5.4)

is non-negative. Now, if $u_n\rightarrow u$ in $H^1(\mathbb {R}^N)$, we have

$$\begin{aligned}&\min \{1,a\}||A(u_n)-A(u)||^2\nonumber \\&\quad \le \int _{\mathbb {R}^N}\Big ||u_n|^{p-2} u_n-|u|^{p-2}u\Big |\cdot |A(u_n)-A(u)|dx\\&\qquad +\,|\xi (u_n)-\xi (u)|\int _{\mathbb {R}^N}\chi _\epsilon |A(u_n)-A(u)|\cdot |A(u)|dx. \end{aligned}$$

By Sobolev embedding, it suffices to show $|\xi (u_n)-\xi (u)|\rightarrow 0,$ which is obvious. $\square$

Lemma 5.4

For any $u\in H^1(\mathbb {R}^N),$

$$\begin{aligned} \langle \Gamma '_\epsilon (u), u-A(u)\rangle =||u-A(u)||^2_V+\xi (u)\int _{\mathbb {R}^N}\chi _\epsilon \cdot (u-A(u))^2dx, \end{aligned}$$

(5.5)

where $||u||_V=(\int _{\mathbb {R}^N}(|\nabla u|^2+V(\epsilon x)u^2)dx)^{1/2}.$ And there exists a positive constant C such that

$$\begin{aligned} ||\Gamma '_\epsilon (u)||\le ||u-A(u)|| (1+C||u||^{2\beta -2}),\quad \forall u\in H^1(\mathbb {R}^N). \end{aligned}$$

(5.6)

Proof

Direct computations show (5.5) and for any $\varphi \in H^1(\mathbb {R}^N)$,

$$\begin{aligned} \langle \Gamma '_\epsilon (u), \varphi \rangle =(u-A(u),\varphi )_V+\xi (u)\int _{\mathbb {R}^N}\chi _\epsilon \cdot (u-A(u))\cdot \varphi dx, \end{aligned}$$

where

$$\begin{aligned} (u,v)_V=\int _{\mathbb {R}^N}(\nabla u\nabla v+V(\epsilon x)uv)dx. \end{aligned}$$

Here we used the fact that A(u) solves Eq. (5.3). Then using

$$\begin{aligned} |\xi (u)\int _{\mathbb {R}^N}\chi _\epsilon \cdot (u-A(u))\varphi dx|\le C||u||^{2\beta -2}||u-A(u)||\cdot ||\varphi ||, \end{aligned}$$

we see the result (5.6) follows. $\square$

Lemma 5.5

There exists $\sigma _0>0$ such that for $\sigma \in (0,\sigma _0)$,

$$\begin{aligned} A(\partial (P^\sigma _-))\subset P^\sigma _-,\ A(\partial (P^\sigma _+))\subset P^\sigma _+. \end{aligned}$$

Proof

We only prove $A(\partial (P^\sigma _-))\subset P^\sigma _-.$ For $u\in H^1(\mathbb {R}^N)$, let $w=A(u)$. We have

$$\begin{aligned} \text{ dist }_{H^1}(w,P_-)||w^+||\le & {} ||w^+||^2\le (\min \{1,a\})^{-1}||w^+||^2_V=(\min \{1,a\})^{-1}(w, w^+)_V\nonumber \\= & {} -(\min \{1,a\})^{-1}\xi (u)\int _{\mathbb {R}^N}\chi _\epsilon \cdot w\cdot w^+dx\nonumber \\&+\;(\min \{1,a\})^{-1}\int _{\mathbb {R}^N}|u|^{p-2}uw^+dx\nonumber \\\le & {} (\min \{1,a\})^{-1}\int _{\mathbb {R}^N}|u|^{p-2}u^+w^+dx\nonumber \\\le & {} (\min \{1,a\})^{-1}||u^+||^{p-1}_{L^p}||w^+||_{L^p}\nonumber \\= & {} (\min \{1,a\})^{-1}(\text{ dist }_{L^p}(u, P_-))^{p-1}||w^+||_{L^p}\nonumber \\\le & {} C(\text{ dist }_{H^1}(u, P_-))^{p-1}||w^+||. \end{aligned}$$

(5.7)

Here we used $\text{ dist }_{L^p}(v, P_-)=||v^+||_{L^p}$ for any $v\in L^p(\mathbb {R}^N)$ in the last equality, where

$$\begin{aligned} \text{ dist }_{L^p}(u, B)=\inf _{v\in B}||u-v||_{L^p} \end{aligned}$$

for any $u\in L^p(\mathbb {R}^N)$ and any $B\subset L^p(\mathbb {R}^N).$ Thus we have

$$\begin{aligned} \text{ dist }_{H^1}(w,P_-)\le C\sigma ^{p-1}. \end{aligned}$$

For $\sigma >0$ small, the conclusion follows. $\square$

Since A may be only continuous, we need to have a locally Lipschitz perturbation of A. Set $E_0=H^1(\mathbb {R}^N){\setminus }\mathcal {K}$, where $\mathcal {K}$ is the set of fixed points of A,  i.e., the set of critical points of $\Gamma _\epsilon$.

Lemma 5.6

There exists a locally Lipschitz continuous operator $B:E_0\rightarrow H^1(\mathbb {R}^N)$ such that

• (1) $B(\partial ( P^\sigma _+))\subset P^\sigma _+$ and $B(\partial (P^\sigma _-))\subset P^\sigma _-$ for $\sigma \in (0,\sigma _0).$

• (2) $\frac{1}{2}||u-B(u)||\le ||u-A(u)||\le 2||u-B(u)||,$ $\forall u\in E_0.$

• (3) $\langle \Gamma '_\epsilon (u), u-B(u)\rangle \ge \frac{1}{2}||u-A(u)||^2$, $\forall u\in E_0.$

• (4) B is odd.

Proof

The proof is similar to the proofs of [6, Lemma 4.1] and [7, Lemma 2.1].We omit the details. $\square$

Since $\Gamma _\epsilon$ satisfies $(PS)_c$ condition for $c<L$ if $0<\epsilon <\epsilon _L$ (see Lemma 2.2), using the map B and by the same argument as [26, Lemma 3.5] or [28, Lemma 3.6], we have the following lemma

Lemma 5.7

Let $0<\epsilon <\varepsilon _L$ and $c<L$. Let $\mathcal {N}$ be a symmetric closed neighborhood of $K_c.$ Then there exists a positive constant $\tau _0$ such that for $0<\tau<\tau '<\tau _0,$ there exists a continuous map $\zeta :[0,1]\times H^1(\mathbb {R}^N)\rightarrow H^1(\mathbb {R}^N)$ satisfying

• (1) $\zeta (0,u)=u$ for all $u\in H^1(\mathbb {R}^N)$;

• (2) $\zeta (t,u)=u$ for $t\in [0,1]$, $\Gamma _\epsilon (u)\not \in [c-\tau ',c+\tau ']$;

• (3) $\zeta (t,-u)=-\zeta (t,u)$ for all $t\in [0,1]$ and $u\in H^1(\mathbb {R}^N)$;

• (4) $\zeta (1,(\Gamma _\epsilon )^{c+\tau }{\setminus }\mathcal {N})\subset (\Gamma _\epsilon )^{c-\tau }$;

• (5) $\zeta (t, \partial (P^\sigma _+))\subset P^\sigma _+,$ $\zeta (t,\partial (P^\sigma _-))\subset P^\sigma _-$, $\zeta (t, P^\sigma _+)\subset P^\sigma _+,$ $\zeta (t, P^\sigma _-)\subset P^\sigma _-$, $t\in [0,1]$.

Proof of Lemma 3.3

Let $\mathcal {D}$ be a closed symmetric neighborhood of $K_c{\setminus } W.$ Note that $\mathcal {N}=\mathcal {D}\cup \overline{P^\sigma _+}\cup \overline{P^\sigma _-}$ is a closed symmetric neighborhood of $K_c$. According to Lemma 5.7, we can choose $\eta = \zeta (1, \cdot )$ in Definition 3.1. $\square$