Existence and orbital stability/instability of standing waves with prescribed mass for the $L^{2}$-supercritical NLS in bounded domains and exterior domains

Abstract

Relying on bifurcation type arguments, we obtain threshold results for the existence, non-existence and multiplicity of positive solutions with prescribed $L^{2}$ norm for the semi-linear elliptic equation in bounded domains:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u - f(x,u) = \lambda u, x \in \Omega , \\ u_{|\partial \Omega } = 0, \end{array} \right. \end{aligned}$$

and we do not assume that f is autonomous. Moreover, we provide a lower bound for the threshold. For almost every $L^{2}$ mass in the existence range, there exist one orbitally stable standing wave and one unstable standing wave associated to these positive solutions. We also study the existence of prescribed norm solutions in exterior domains and orbitally unstable standing waves for almost every $L^{2}$ mass in the existence range.

A Appendix

In this appendix, we show some details of the calculations in Remark 6.3 (ii). As a byproduct, we prove that $u_{\lambda }(0) > (-\frac{2p}{2N - (N-2)p}\lambda )^{\frac{1}{p-2}}$ in this case. Applying [4, Theorem 1.5] on (5.1) where $\lambda \le 0$, $N \ge 2$, then any solution of (5.1) is nondegenerate if the following assumption holds:

(A) G(r) is either nonnegative in (0, R) or positive in $(0,r_{0})$ and negative in $(r_{0},R)$ for some $r_{0} \in (0, R)$ where $u(r) = u(|x|)$ is a positive solution of

$$\begin{aligned} -u''(r)-\frac{N-1}{r}u'(r) = \lambda u(r) + \frac{1}{(1 + r^{s})^{k}}u(r)^{p-1}, u'(0) = 0, u(R) = 0 \end{aligned}$$

and

$$\begin{aligned} G(r) = \lambda u(r)^{2} + \left[ \frac{2N-(N-2)p-2ks}{2p}\frac{1}{(1 + r^{s})^{k}} + \frac{ks}{p(1 + r^{s})^{k+1}}\right] u(r)^{p}. \end{aligned}$$

Theorem A.1

If $\lambda \le 0$, $N \ge 2$, $(N-2)p - 2(N-ks) \ge 0$, then any solution of (5.1) is nondegenerate.

Proof

We verify the assumption (A). Rewrite $G(r) = u(r)^{2}\tilde{G}(r)$ where

$$\begin{aligned} \tilde{G}(r)= & {} g(r)h(r) + \lambda , \\ g(r)= & {} \frac{2N-(N-2)p-2ks}{2p}r^{s} + \frac{2N-(N-2)p}{2p}, \\ h(r)= & {} \frac{u(r)^{p-2}}{(1 + r^{s})^{k+1}}. \end{aligned}$$

$u(r) > 0$ and $u'(r) < 0$ in (0, R), thus $h(r) > 0$ is decreasing in (0, R). Since $(N-2)p - 2(N-ks) \ge 0$ and $(N-2)p < 2N$, g(r) is nonincreasing in (0, R) and $g(0) > 0$.

Case 1: $g(R) \ge 0$.

In this case, $g(r) > 0$ in (0, R) and $\tilde{G}(r)$ is decreasing in (0, R). Since $\tilde{G}(R) = \lambda \le 0$, if $\tilde{G}(0) > 0$, $\tilde{G}(r)$ is positive in (0, R) when $\lambda = 0$ or positive in $(0,r_{0})$ and negative in $(r_{0},R)$ for some $r_{0} < R$ when $\lambda < 0$ and if $\tilde{G}(0) \le 0$, $\tilde{G}(r) \le 0$. Since $u(r)^{2} > 0$, G(r) has the same sign as $\tilde{G}(r)$ in (0, R).

Case 2: $g(R) < 0$.

In this case, there exists a unique $r^{*} \in (0, R)$ such that $g(r^{*}) = 0$, $g(r) > 0$ in $(0, r^{*})$ and $g(r) < 0$ in $(r^{*}, R)$. $\tilde{G}(r)$ is decreasing in $(0,r^{*})$. Since $\tilde{G}(r^{*}) = \lambda \le 0$, if $\tilde{G}(0) > 0$, $\tilde{G}(r)$ is positive in $(0, r^{*})$ when $\lambda = 0$ or positive in $(0, r_{0})$ and negative in $(r_{0}, r^{*})$ for some $r_{0} < r^{*}$ when $\lambda < 0$ and if $\tilde{G}(0) \le 0$, $\tilde{G}(r) \le 0$ in $(0,r^{*})$. Moreover, $\tilde{G}(r) \le \lambda \le 0$ in $[r^{*}, R)$. Thus $\tilde{G}(r)$ is either positive in $(0,r_{0})$ and negative in $(r_{0},R)$ for some $r_{0} < R$ or $\tilde{G}(r) \le 0$ in (0, R). Since $u(r)^{2} > 0$, G(r) has the same sign as $\tilde{G}(r)$.

Finally, we show that it is impossible that $G(r) \le 0$ in (0, R). Arguing by contradiction, we assume that $G(r) \le 0$ in (0, R). Define

$$\begin{aligned} \phi (r) = \frac{N-2}{2}r^{N-1}u(r)u'(r) + \frac{1}{2}r^{N}u'(r)^{2} + \frac{\lambda }{2}r^{N}u(r)^{2} + \frac{r^{N}}{p(1 + r^{s})^{k}}u(r)^{p}. \end{aligned}$$

Since $-u''(r)-\frac{N-1}{r}u'(r) = \lambda u(r) + \frac{1}{(1 + r^{s})^{k}}u(r)^{p}$,

$$\begin{aligned} \phi '(r)= & {} \frac{(N-1)(N-2)}{2}r^{N-2}u(r)u'(r) + \frac{N-2}{2}r^{N-1}u'(r)^{2} + \frac{N-2}{2}r^{N-1}u'(r)u''(r) \nonumber \\{} & {} + \frac{N}{2}r^{N-1}u'(r)^{2} + r^{N}u'(r)u''(r) + \frac{N\lambda }{2}r^{N-1}u(r)^{2} + \lambda r^{N}u(r)u'(r) \nonumber \\{} & {} + \frac{Nr^{N-1}}{p(1 + r^{s})^{k}}u(r)^{p} - \frac{ksr^{N+s-1}}{p(1 + r^{s})^{k+1}}u(r)^{p} + \frac{r^{N}}{(1 + r^{s})^{k}}u(r)^{p-1}u'(r) \nonumber \\= & {} r^{N-1}G(r). \end{aligned}$$

(A.1)

Thus $\phi (r)$ is nonincreasing in (0, R), contradicting $\phi (0) = 0$ and $\phi (R) = \frac{R^{N}}{2}u'(R)^{2} > 0$. The proof is completed. $\square$

Corollary A.2

If $N \ge 2$, $(N-2)p - 2(N-ks) \ge 0$, $u_{\lambda }$ is a positive solution of (5.1) for some $\lambda \le 0$, then $u_{\lambda }(0) > (-\frac{2p}{2N - (N-2)p}\lambda )^{\frac{1}{p-2}}$.

Proof

From the proof of Theorem A.1, $\tilde{G}(0) > 0$ since it is impossible that $\tilde{G}(r) \le 0$ in (0, R). Notice that

$$\begin{aligned} \tilde{G}(0) = \frac{2N-(N-2)p}{2p}u_{\lambda }(0)^{p-2} + \lambda > 0, \end{aligned}$$

implying that

$$\begin{aligned} u_{\lambda }(0) > \left( -\frac{2p}{2N - (N-2)p}\lambda \right) ^{\frac{1}{p-2}}. \end{aligned}$$

$\square$