On a stationary Schrödinger equation with periodic magnetic potential

Abstract

We prove existence results for a stationary Schrödinger equation with periodic magnetic potential satisfying a local integrability condition on the whole space using a critical value function.

Appendices

Appendix

A Some proofs

In this appendix, we adapt the proof of [6, Theorem 1.1] to our family of functionals $(G_{\rho })_{\rho \in I_{\gamma }},$ where the original idea is due to [12]. We also give the proof of Proposition 5.1.

In [6], the family of functionals is of the form

$$\begin{aligned} \forall \lambda >0, \; I_{\lambda }(u)=A(u)-\lambda B(u), \end{aligned}$$

where $A(u)\xrightarrow {\Vert u\Vert \rightarrow \infty }\infty$ or $B(u)\xrightarrow {\Vert u\Vert \rightarrow \infty }\infty ,$ and with $B\geqslant 0$ everywhere. Unfortunately, in our case,

$$\begin{aligned} \forall \rho >0, \; G_{\rho }(u)=\frac{1}{\lambda }I_{\lambda }(u)=\frac{1}{\lambda }\left( \frac{\Vert u\Vert ^2}{2}-\lambda \psi (u)\right) , \;\; \lambda =\frac{1}{\rho }, \end{aligned}$$

and we do not have $B=\psi \geqslant 0,$ everywhere, but only somewhere. So we have, in some sense, to reverse the role of $A=\Vert u\Vert ^2$ and $B=\psi .$ The following theorem is an easy adaptation of [6, Theorem 1.1], but for the convenience of the reader, we give its proof.

Theorem A.1

[6, Theorem 1.1] Let $(X,\Vert \,\cdot \,\Vert )$ be a Banach space, let $I\subset (0,\infty )$ be a nonempty open interval and let $(G_{\rho })_{\rho \in I}\subset C^1\big (X;{\mathbb {R}}\big )$ be a family of functionals of the form,

$$\begin{aligned} \forall \rho \in I, \; G_{\rho }(u)=\rho A(u)-B(u), \end{aligned}$$

(A.1)

where $A\not \equiv 0$ and for any $u\in X$, $A(u)\geqslant 0.$ Assume that either $A(u)\xrightarrow {\Vert u\Vert \rightarrow \infty }\infty$ or $B(u)\xrightarrow {\Vert u\Vert \rightarrow \infty }\infty .$ We also assume that $(G_{\rho })_{\rho \in I}$ has mountain pass geometry: there exist $u_1\in X$ and $u_2\in X$ such that, denoting by

$$\begin{aligned} \Gamma {\mathop {=}\limits ^{\mathrm{def}}}\left\{ \xi \in C\left( [0,1];X\right) ; \; \xi (0)=u_1 \text { and } \xi (1)=u_2\right\} , \end{aligned}$$

the set of continuous paths joining $u_1$ to $u_2,$ we have for any $\rho \in I,$

$$\begin{aligned} c(\rho ){\mathop {=}\limits ^{\mathrm{def}}}\inf _{\xi \in \Gamma }\max _{t\in [0,1]}G_{\rho }\big (\xi (t)\big )>\max \big \{G_{\rho }(u_1),G_{\rho }(u_2)\big \}. \end{aligned}$$

(A.2)

Then for almost every $\rho \in I$, $G_{\rho }$ admits a bounded Palais–Smale sequence: there exists a sequence $(u_n)_{n\in {\mathbb {N}}}\subset X$ satisfying,

$$\begin{aligned}&(u_n)_{n\in {\mathbb {N}}}\subset X \text { is bounded,} \end{aligned}$$

(A.3)

$$\begin{aligned}&{\left\{ \begin{array}{ll} G_{\rho } (u_n) \xrightarrow [n\rightarrow \infty ]{} c(\rho ),\\ G_{\rho }^{\prime }(u_n) \xrightarrow [n\rightarrow \infty ]{X^{\star }}0, \end{array}\right. } \end{aligned}$$

(A.4)

where $X^{\star }$ denotes the topological space of X.

Remark A.2

Here are some comments of Theorem A.1.

• 1) If there exist $\rho \in I$ and $(u_1,u_2)\in X\times X$ satisfying (A.2) then it is well-known, by the Mountain Pass Theorem, that there exists a Palais–Smale sequence $(u_n)_{n\in {\mathbb {N}}}\subset X$ satisfying (A.4) (see, for instance, [14, Theorem 6.2, p. 144]). The difficulty is to find such a bounded sequence.

• 2) The proof of Theorem A.1 relies on the existence of the derivative $c^{\prime }(\rho )$ of $c(\rho ).$ Since $A\geqslant 0,$ we have by (A.2) that the mapping $c:\rho \longmapsto c(\rho )$ is nondecreasing over I. It follows that c has a derivative $c^{\prime }$ almost everywhere on I. In the original proof, the existence almost everywhere on I of $c^{\prime }$ is ensured by the fact that the mapping $c:\rho \longmapsto c(\rho )$ is nonincreasing over I.

Before proceeding to the proof of Theorem A.1, let us pick any $\rho \in I$ such that the derivative $c^{\prime }(\rho )$ exists (see the item 2) in the above remark). Let then $\rho _0\in (0,\rho )$ be small enough to have $(\rho -\rho _0,\rho +\rho _0)\subset I$ and

$$\begin{aligned} \forall \widetilde{\rho }\in (\rho -\rho _0,\rho +\rho _0), \; \left| \frac{c(\widetilde{\rho })-c(\rho )}{\widetilde{\rho }-\rho }-c^{\prime }(\rho )\right| \leqslant 1. \end{aligned}$$

(A.5)

Now, let us choose $(\rho _n)_{n\in {\mathbb {N}}}\subset (\rho ,\rho +\rho _0)$ be a decreasing sequence such that $\rho _n\xrightarrow {n\rightarrow \infty }\rho .$ Finally, since $A(u)\xrightarrow {\Vert u\Vert \rightarrow \infty }\infty$ or $B(u)\xrightarrow {\Vert u\Vert \rightarrow \infty }\infty$ there exists $M>10$ such that for any $u\in X,$

$$\begin{aligned} \Vert u\Vert>M \implies \max \big \{A(u),B(u)\big \}>\max \big \{c^{\prime } (\rho )+3,2\rho \big (c^{\prime }(\rho )+4\big )-c(\rho )\big \}. \end{aligned}$$

(A.6)

We shall need of the two following lemmas.

Lemma A.3

There exists $(\xi _n)_{n\in {\mathbb {N}}}\subset \Gamma$ satisfying the following properties.

• 1) Let $t\in [0,1].$ If $n\in {\mathbb {N}}$ is such that $G_{\rho }\big (\xi _n(t)\big )\geqslant c(\rho )-(\rho _n-\rho )$ then $\Vert \xi _n(t)\Vert \leqslant M.$

• 2) $\forall n\in {\mathbb {N}}$, $\max \limits \nolimits _{t\in [0,1]}G_{\rho }\big (\xi _n(t)\big )\leqslant c(\rho )+(c^{\prime }(\rho )+2)(\rho _n-\rho ).$

Proof

Let $(\xi _n)_{n\in {\mathbb {N}}}\subset \Gamma$ be such that for any $n\in {\mathbb {N}},$

$$\begin{aligned} \max \limits _{t\in [0,1]}G_{\rho _n}\big (\xi _n(t)\big )\leqslant c(\rho _n)+(\rho _n-\rho ). \end{aligned}$$

(A.7)

Let $t\in [0,1].$ Let $n\in {\mathbb {N}}.$ We have by the hypothesis in 1), (A.7) and (A.5),

$$\begin{aligned} A\big (\xi _n(t)\big )=\frac{G_{\rho _n}\big (\xi _n(t)\big )-G_{\rho }\big (\xi _n(t)\big )}{\rho _n-\rho } \leqslant \frac{c(\rho _n)-c(\rho )}{\rho _n-\rho }+2\leqslant c^{\prime }(\rho )+3. \end{aligned}$$

(A.8)

In addition, since for any $u\in X,$ the mapping $\rho \longmapsto G_{\rho }(u)$ is nondecreasing, it follows from (A.8) and the hypothesis in 1),

$$\begin{aligned} B\big (\xi _n(t)\big )=\rho _nA\big (\xi _n(t)\big )-G_{\rho _n}\big (\xi _n(t)\big )\leqslant 2\rho \big (c^{\prime }(\rho )+4\big )-c(\rho ). \end{aligned}$$

(A.9)

Hence $\Vert \xi _n(t)\Vert \leqslant M,$ by (A.6), (A.8) and (A.9). To prove the second part of the lemma, we see that (A.5) implies,

$$\begin{aligned} c(\rho _n)\leqslant c(\rho )+\big (c^{\prime }(\rho )+1\big )(\rho _n-\rho ). \end{aligned}$$

(A.10)

Finally, (A.7) and (A.10) yield,

$$\begin{aligned} \max _{t\in [0,1]}G_{\rho }\big (\xi _n(t)\big )\leqslant \max _{t\in [0,1]}G_{\rho _n} \big (\xi _n(t)\big )\leqslant c(\rho )+\big (c^{\prime }(\rho )+2\big )(\rho _n-\rho ). \end{aligned}$$

This ends the proof of the lemma. $\square$

Lemma A.4

Define for any $\varepsilon >0,$

$$\begin{aligned} F_{\varepsilon }{\mathop {=}\limits ^{\mathrm{def}}}\Big \{u\in X; \; \Vert u\Vert \leqslant 2M \text { and } |G_{\rho }(u)-c(\rho )|\leqslant \varepsilon \Big \}. \end{aligned}$$

Then for any $\varepsilon >0$, $F_{\varepsilon }\ne \emptyset$ and $\inf \limits _{u\in F_{\varepsilon }}\Vert G_{\rho }^{\prime }(u)\Vert _{X^{\star }}=0$.

Proof

Let $(\xi _n)_{n\in {\mathbb {N}}}\subset \Gamma$ be given by Lemma A.3. Then for each $n\in {\mathbb {N}},$ there exists $t_n\in [0,1]$ such that $0\leqslant G_{\rho }\big (\xi _n(t_n)\big ) -c(\rho )\leqslant (c^{\prime }(\rho )+2)(\rho _n-\rho )\xrightarrow {n\rightarrow \infty }0$ and $\Vert \xi (t_n)\Vert \leqslant M.$ We infer that for any $\varepsilon >0,$ there exists $n_0\in {\mathbb {N}}$ large enough such that $\xi (t_{n_0})\in F_{\varepsilon }.$ Now, we note that it is sufficient to show the result for any $\varepsilon >0$ small enough. If the result does not hold then there exists $0<\varepsilon _0<\frac{c(\rho )-\max \{G_{\rho }(u_1),G_{\rho }(u_2)\}}{2}$ such that $\inf \limits \nolimits _{u\in F_{2\varepsilon _0}}\Vert G_{\rho }^{\prime }(u)\Vert _{X^{\star }}\geqslant 2\varepsilon _0.$ We then may apply a deformation lemma to affirm that there exists a homeomorphism $\eta :X\longrightarrow X$ satisfying the following properties.

$$\begin{aligned}&\text {If } |G_{\rho }(u)-c(\rho )|>2\varepsilon _0 \text { then } \eta (u)=u. \end{aligned}$$

(A.11)

$$\begin{aligned}&\forall u\in X, \; G_{\rho }\big (\eta (u)\big )\leqslant G_{\rho }(u). \end{aligned}$$

(A.12)

$$\begin{aligned}&\text {If } \Vert u\Vert \leqslant M \text { and } G_{\rho }(u)<c(\rho )+\varepsilon _0 \text { then } G_{\rho }\big (\eta (u)\big )<c(\rho )-\varepsilon _0. \end{aligned}$$

(A.13)

See for instance [5, Theorem 4.2, p. 38]. The assertion (A.12) is not directly stated in this theorem but in its proof p. 39. Let $m\in {\mathbb {N}}$ be large enough to have,

$$\begin{aligned} \rho _m-\rho<(c^{\prime }(\rho )+2)(\rho _m-\rho )<\varepsilon _0. \end{aligned}$$

(A.14)

By (A.11), $\eta (\xi _m)\in \Gamma .$ Let $t\in [0,1]$.

• If $G_{\rho }\big (\xi _m(t)\big )\leqslant c(\rho )-(\rho _m-\rho )$ then by (A.12),

  $$\begin{aligned} G_{\rho }\big (\eta (\xi _m(t))\big )\leqslant c(\rho )-(\rho _m-\rho ). \end{aligned}$$

  (A.15)

• If $G_{\rho }\big (\xi _m(t)\big )>c(\rho )-(\rho _m-\rho )$ then by Lemma A.3 and (A.14), $\Vert \xi _m(t)\Vert \leqslant M$ and $G_{\rho }\big (\xi _m(t)\big )<c(\rho )+\varepsilon _0.$ It then follows from (A.13) and (A.14),

  $$\begin{aligned} G_{\rho }\big (\eta (\xi _m(t))\big )<c(\rho )-\varepsilon _0<c(\rho )-(\rho _m-\rho ). \end{aligned}$$

  (A.16)

  It follows from (A.15) and (A.16) that,

  $$\begin{aligned} c(\rho )=\inf _{\xi \in \Gamma }\max _{t\in [0,1]}G_{\rho }\big (\xi (t)\big )\leqslant \max _{t\in [0,1]}G_{\rho } \big (\eta (\xi _m(t))\big )\leqslant c(\rho )-(\rho _m-\rho ). \end{aligned}$$

  A contradiction, since $\rho _m-\rho >0$.

$\square$

Proof of Theorem A.1

The result follows by applying Lemma A.4 with any sequence $\varepsilon _n\searrow 0$.

$\square$

Proof of Proposition 5.1

Throughout this proof, we let $\kappa =1,$ if $N\geqslant 3$ and $\kappa =0,$ if $N\leqslant 2.$ We will denote by $C_1>1$ and $p_1$ the constants given by (1.8)–(1.9) for $\varepsilon =1.$ We proceed to the proof in 6 steps.

Step 1: $g:H^1({\mathbb {R}}^N)\longrightarrow H^{-1}({\mathbb {R}}^N)$ is well-defined, bounded on bounded sets and 2 holds.

By (1.8)–(1.9), $g(u)\in L^1_{\mathrm{loc}}({\mathbb {R}}^N).$ Let $\varphi \in {\mathscr {D}}({\mathbb {R}}^N).$ We have by (1.8)–(1.9), Hölder’s inequality and the Sobolev embeddings,

$$\begin{aligned}&\left| \langle g(u),\varphi \rangle _{{\mathscr {D}}^{\prime }({\mathbb {R}}^N);{\mathscr {D}}({\mathbb {R}}^N)}\right| =\left| \mathrm{Re}\displaystyle \int \limits _{{\mathbb {R}}^N}g(u)\overline{\varphi }\mathrm{d}x\right| \\&\quad \leqslant C_1\left( \Vert u\Vert _{L^2({\mathbb {R}}^N)}+\kappa \Vert u\Vert _{L^{{2^{\star }}} ({\mathbb {R}}^N)}^{{2^{\star }}-1} +\Vert u\Vert _{L^{p_1}({\mathbb {R}}^N)}^{p_1-1}\right) \Vert \varphi \Vert _{H^1({\mathbb {R}}^N)} \\&\quad \leqslant C\left( \Vert u\Vert _{H^1({\mathbb {R}}^N)}+\kappa \Vert u\Vert _{H^1({\mathbb {R}}^N)}^{{2^{\star }}-1} +\Vert u\Vert _{H^1({\mathbb {R}}^N)}^{p_1-1}\right) \Vert \varphi \Vert _{H^1({\mathbb {R}}^N)}. \end{aligned}$$

By density, it follows that $g:H^1({\mathbb {R}}^N)\longrightarrow H^{-1}({\mathbb {R}}^N)$ is well-defined, g is bounded on bounded sets and Property 2 holds.

Step 2: $\psi \in C(H^1({\mathbb {R}}^N);{\mathbb {R}})$, $\psi$ is bounded on bounded sets, Gâteaux-differentiable and its Gâteaux-differential is $\psi ^{\prime }_{\mathrm{g}}=g$.

Let $u\in H^1({\mathbb {R}}^N).$ By (1.8)–(1.9), Hölder’s inequality and the Sobolev embedding, $F(u)\in L^1({\mathbb {R}}^N;{\mathbb {R}})$ so that $\psi :H^1({\mathbb {R}}^N)\longrightarrow {\mathbb {R}}$ is well-defined and $\psi$ is bounded on bounded sets. Let $v\in H^1({\mathbb {R}}^N).$ Still by (1.8)–(1.9), Hölder’s inequality and the Sobolev embedding,

$$\begin{aligned}&|\psi (u+v)-\psi (u)|\leqslant \int \limits _{{\mathbb {R}}^N}\int \limits _{|u|}^{|u+v|}\big (t+\kappa t^{{2^{\star }}-1}+C_1t^{p_1-1}\big )\mathrm{d}t\mathrm{d}x \\&\quad \leqslant C\Big (\Vert u\Vert _{L^2}+\Vert v\Vert _{L^2}+\kappa (\Vert u\Vert _{L^{{2^{\star }}} }+\Vert v\Vert _{L^{{2^{\star }}} })^{{2^{\star }}-1}+(\Vert u\Vert _{L^{p_1}} +\Vert v\Vert _{L^{p_1}})^{p_1-1}\Big )\Vert v\Vert _{H^1({\mathbb {R}}^N)}. \end{aligned}$$

It follows that $\psi \in C(H^1({\mathbb {R}}^N);{\mathbb {R}}).$ Let $v\in H^1({\mathbb {R}}^N)$ and $0<|t|<1.$ Since $u,v\in L^2({\mathbb {R}}^N),$ the set

$$\begin{aligned} {\mathcal {N}}{\mathop {=}\limits ^{\mathrm{def}}}\big \{x\in {\mathbb {R}}^N; |u(x)|=\infty \text { or } |v(x)|=\infty \big \}, \end{aligned}$$

has Lebesgue measure 0. Let $x\in {\mathcal {N}}^{\mathrm{c}}.$ If $u(x)\ne 0$ then using that

$$\begin{aligned} |u(x)+tv(x)|=\sqrt{\big (u(x)+tv(x)\big )\overline{\big (u(x)+tv(x)\big )}}>0, \end{aligned}$$

for t small enough, we see that

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}F(x,|u(x)+tv(x)|)_{|t=0}=\mathrm{Re}\big (\varvec{f}(x,u(x))\overline{v(x)}\big ). \end{aligned}$$

If $u(x)=0$ then by (1.8)–(1.9),

$$\begin{aligned} \left| \frac{F(x,|tv(x)|)-F(x,0)}{t}\right| \leqslant C\big (|t||v(x)|^2 +\kappa |t|^{{2^{\star }}-1}|v(x)|^{{2^{\star }}} +C_1|t|^{p_1-1}|v(x)|^{p_1}\big )\xrightarrow {t\rightarrow 0}0. \end{aligned}$$

We then infer,

$$\begin{aligned} \frac{F(\,\cdot \,,|u+tv|)-F(\,\cdot \,,|u|)}{t}\xrightarrow [t\longrightarrow 0]{\text {a.e. in }{\mathbb {R}}^N}\mathrm{Re}\big (\varvec{f}(\,\cdot \,,u)\overline{v}\big ). \end{aligned}$$

By (1.8)–(1.9),

$$\begin{aligned}&\frac{F(\,\cdot \,,|u+tv|)-F(\,\cdot \,,|u|)-t\mathrm{Re}\big (\varvec{f}(x,u)\overline{v}\big )}{t} \\&\quad \leqslant \frac{1}{t}\int \limits _{|u|}^{|u+tv|}|f(\,\cdot \,,s)|\mathrm{d}s+|f(\,\cdot \,,|u|)||v| \\&\quad \leqslant C\big (|u|+|v|+\kappa (|u|+|v|)^{{2^{\star }}-1}+(|u|+|v|)^{p_1-1}\big )|v|\in L^1({\mathbb {R}}^N). \end{aligned}$$

It follows from the dominated convergence Theorem and Property 2 that,

$$\begin{aligned} \lim _{t\rightarrow 0}\frac{\psi (u+tv)-\psi (u)}{t}=\langle g(u),v\rangle _{H^{-1}({\mathbb {R}}^N),H^1({\mathbb {R}}^N)}. \end{aligned}$$

Hence Step 2.

Step 3: Let $u,v\in H^1({\mathbb {R}}^N)$ and $(u_n)_{n\in {\mathbb {N}}}\subset H^1({\mathbb {R}}^N)$ be bounded. Let $\varepsilon >0.$ Choose $\varepsilon ^{\prime }>0$ small enough to have,

$$\begin{aligned} 2\varepsilon ^{\prime }\left( \sup _{n\in {\mathbb {N}}}\Vert u_n\Vert _{L^2({\mathbb {R}}^N)}^2+\Vert u\Vert _{L^2({\mathbb {R}}^N)}^2 +\kappa \left( \sup _{n\in {\mathbb {N}}}\Vert u_n\Vert _{L^{{2^{\star }}} ({\mathbb {R}}^N)}^{{2^{\star }}-1} +\Vert u\Vert _{L^{{2^{\star }}} ({\mathbb {R}}^N)}^{{2^{\star }}-1}\right) \right) \leqslant \varepsilon . \end{aligned}$$

(A.17)

For such an $\varepsilon ^{\prime },$ let $p_{\varepsilon ^{\prime }}$ and $C_{\varepsilon ^{\prime }}$ be given by (1.8)–(1.9). For each $n\in {\mathbb {N}},$ let

$$\begin{aligned} A_n=\Big \{x\in {\mathbb {R}}^N;\varepsilon ^{\prime }\big (|u_n|+|u|+\kappa (|u_n|^{{2^{\star }}-1} +|u|^{{2^{\star }}-1})\big )\leqslant C_{\varepsilon ^{\prime }}(|u_n|^{p_{\varepsilon ^{\prime }}-1}+|u|^{p_{\varepsilon ^{\prime }}-1})\Big \}. \end{aligned}$$

It holds that,

$$\begin{aligned} \left| \langle g(u_n)-g(u),v\rangle _{H^{-1}({\mathbb {R}}^N),H^1({\mathbb {R}}^N)}\right| \leqslant \int \limits _{{\mathbb {R}}^N}\big |g(u_n)-g(u)\big ||v|{\mathbb {1}}_{A_n}\mathrm{d}x+\varepsilon \Vert v\Vert _{H^1({\mathbb {R}}^N)}. \end{aligned}$$

(A.18)

Indeed, by (1.8)–(1.9), Hölder’s inequality, the Sobolev embeddings and (A.17), we have,

$$\begin{aligned}&\left| \langle g(u_n)-g(u),v\rangle _{H^{-1}({\mathbb {R}}^N),H^1({\mathbb {R}}^N)}\right| \leqslant \int \limits _{{\mathbb {R}}^N}\big |g\big (u_n)-g(u)\big ||v|\mathrm{d}x \\&\quad = \int \limits _{{\mathbb {R}}^N}\big |g(u_n)-g(u)\big ||v|{\mathbb {1}}_{A_n}\mathrm{d}x+\int \limits _{{\mathbb {R}}^N}\big |g\big (u_n)-g(u)\big ||v|{\mathbb {1}}_{A_n^{\mathrm{c}}}\mathrm{d}x \\&\quad \leqslant \int \limits _{{\mathbb {R}}^N}\big |g(u_n)-g(u)\big ||v|{\mathbb {1}}_{A_n}\mathrm{d}x +2\varepsilon ^{\prime }\int \limits _{{\mathbb {R}}^N}\big (|u_n|+|u|+\kappa (|u_n|^{{2^{\star }}-1}+|u|^{{2^{\star }}-1})\big )|v|\mathrm{d}x \\&\quad \leqslant \int \limits _{{\mathbb {R}}^N}\big |g(u_n)-g(u)\big ||v|{\mathbb {1}}_{A_n}\mathrm{d}x+\varepsilon \Vert v\Vert _{H^1({\mathbb {R}}^N)}. \end{aligned}$$

Step 3 is proved.

Step 4: $\psi \in C^1(H^1({\mathbb {R}}^N);{\mathbb {R}})$ and $\psi ^{\prime }=g.$

By Step 2, it remains to show that $g\in C(H^1({\mathbb {R}}^N);H^{-1}({\mathbb {R}}^N))$ to have that $\psi$ is Fréchet-differentiable and $\psi ^{\prime }=\psi ^{\prime }_{\mathrm{g}}.$ Assume $u_n\xrightarrow [k\rightarrow \infty ]{H^1({\mathbb {R}}^N)} u.$ Let $\varepsilon >0.$ Let then $\varepsilon ^{\prime }$, $p_{\varepsilon ^{\prime }}$ and $C_{\varepsilon ^{\prime }}$ be given by Step 3. By Hölder’s inequality, we have for any $v\in H^1({\mathbb {R}}^N),$

$$\begin{aligned} \int \limits _{{\mathbb {R}}^N}\big |g(u_n)-g(u)\big ||v|{\mathbb {1}}_{A_n}\mathrm{d}x\leqslant \big \Vert \big (g(u_n)-g(u)\big ){\mathbb {1}}_{A_n}\big \Vert _{L^{p^{\prime }_{\varepsilon ^{\prime }}}({\mathbb {R}}^N)}\Vert v\Vert _{L^{p_{\varepsilon ^{\prime }}}({\mathbb {R}}^N)}. \end{aligned}$$

(A.19)

It follows from Sobolev’ embedding and (A.18)–(A.19) that,

$$\begin{aligned} \sup _{\Vert v\Vert _{H^1({\mathbb {R}}^N)}=1}\left| \langle g(u_n)-g(u),v\rangle _{H^{-1}({\mathbb {R}}^N),H^1({\mathbb {R}}^N)}\right| \leqslant C\big \Vert \big (g(u_n)-g(u)\big ){\mathbb {1}}_{A_n}\Vert _{L^{p^{\prime }_{\varepsilon ^{\prime }}}({\mathbb {R}}^N)}+\varepsilon \end{aligned}$$

(A.20)

We claim that,

$$\begin{aligned} \lim _{n\rightarrow \infty }\big \Vert \big (g(u_n)-g(u)\big ){\mathbb {1}}_{A_n}\Vert _{L^{p^{\prime }_{\varepsilon ^{\prime }}}({\mathbb {R}}^N)}=0. \end{aligned}$$

(A.21)

If not, for some $\varepsilon _0>0$ and a subsequence, that we will denote by $(u_n)_n,$ there would exist $h\in L^{p_{\varepsilon ^{\prime }}}({\mathbb {R}}^N;{\mathbb {R}})$ such that for any $n\in {\mathbb {N}}$, $\big \Vert \big (g(u_n)-g(u)\big ){\mathbb {1}}_{A_n}\Vert _{L^{p^{\prime }_{\varepsilon ^{\prime }}}({\mathbb {R}}^N)}\geqslant \varepsilon _0$, $|u_n|\overset{\text {a.e}}{\leqslant }h$ and $u_n\xrightarrow [n\rightarrow \infty ]{\text {a.e. in }{\mathbb {R}}^N}u.$ But then $\big (g(u_n)-g(u)\big ){\mathbb {1}}_{A_n}\xrightarrow [n\longrightarrow \infty ]{\text {a.e. in }{\mathbb {R}}^N}0$ and $\big |g(u_n)-g(u)\big | {\mathbb {1}}_{A_n}\leqslant Ch^{p_{\varepsilon ^{\prime }}-1}\in L^{p^{\prime }_{\varepsilon ^{\prime }}}({\mathbb {R}}^N).$ This would yield to a contradiction by the Lebesgue convergence Theorem. Hence (A.21). It then follows from (A.20)–(A.21) that,

$$\begin{aligned} \forall \varepsilon >0, \; \limsup _{n\rightarrow \infty }\Vert g(u_n)-g(u)\Vert _{H^{-1}({\mathbb {R}}^N)}\leqslant \varepsilon . \end{aligned}$$

Letting $\varepsilon \searrow 0,$ we get $g\in C(H^1({\mathbb {R}}^N);H^{-1}({\mathbb {R}}^N)).$

Step 5: Let $(u_n)_n,(v_n)_n \subset H^1({\mathbb {R}}^N)$ be bounded. If $\lim \limits \nolimits _{n\rightarrow \infty }\Vert u_n-v_n\Vert _{L^p({\mathbb {R}}^N)}=0,$ for some $p\in [1,\infty ],$ then $\lim \limits \nolimits _{n\rightarrow \infty }|\psi (u_n)-\psi (v_n)|=0.$

Let $\varepsilon >0.$ For such an $\varepsilon ,$ let $p_{\varepsilon }$ and $C_{\varepsilon }$ be given by (1.8)–(1.9). Let for any $t\in [0,1]$, $a(t)=\psi (v_n+t(u_n-v_n)).$ Then $a\in C^1([0,1];{\mathbb {R}})$ and by the mean value Theorem, there exists $t_n\in (0,1)$ such that $a(1)-a(0)=a^{\prime }(t_n)(1-0),$ that is

$$\begin{aligned} \psi (u_n)-\psi (v_n)=\langle g(w_n),u_n-v_n\rangle _{H^{-1}({\mathbb {R}}^N),H^1({\mathbb {R}}^N)}. \end{aligned}$$

where $w_n=v_n+t_n(u_n-v_n).$ Note that $(w_n)_{n\in {\mathbb {N}}}$ is bounded in $H^1({\mathbb {R}}^N).$ It follows from (1.8)–(1.9), Hölder’s inequality and Sobolev’s embedding that $\lim \limits \nolimits _{n\rightarrow \infty }\Vert u_n-v_n\Vert _{L^{p_{\varepsilon }}({\mathbb {R}}^N)}=0$ and

$$\begin{aligned}&|\psi (u_n)-\psi (v_n)| \\&\quad \leqslant \varepsilon \left( \Vert w_n\Vert _{L^2({\mathbb {R}}^N)}+\kappa \Vert w_n\Vert _{L^{{2^{\star }}} ({\mathbb {R}}^N)}^{{2^{\star }}-1}\right) \Vert u_n-v_n\Vert _{H^1({\mathbb {R}}^N)}\\&\qquad +C_{p_{\varepsilon }}\Vert w_n\Vert _{L^{p_{\varepsilon }}({\mathbb {R}}^N)}^{{p_{\varepsilon }}-1}\Vert u_n-v_n\Vert _{L^{p_{\varepsilon }}({\mathbb {R}}^N)} \\&\quad \leqslant C\varepsilon +C\Vert u_n-v_n\Vert _{L^{p_{\varepsilon }}({\mathbb {R}}^N)}. \end{aligned}$$

We infer,

$$\begin{aligned} \forall \varepsilon >0, \; \limsup \nolimits _{n\rightarrow \infty }|\psi (u_n)-\psi (v_n)|\leqslant C\varepsilon , \end{aligned}$$

from which the result follows.

Step 6: If $u_n \overset{H^1_{{ w}}}{-\!\!\!-\!\!\!-\!\!\!\rightharpoonup }u$ then $g(u_n) \overset{H^{-1}_{{ w}}}{-\!\!\!-\!\!\!-\!\!\!\rightharpoonup } g(u).$

Since $(g(u_n))_{n\in {\mathbb {N}}}$ is bounded in $H^{-1}({\mathbb {R}}^N)$ (Step 1), it is enough to show that $g(u_n)\xrightarrow [n\rightarrow \infty ]{{\mathscr {D}}^{\prime }({\mathbb {R}}^N)}g(u).$ Let $\varphi \in {\mathscr {D}}({\mathbb {R}}^N)$ with ${{\,\mathrm{supp}\,}}\varphi \subset B(0,R),$ for some $R>0.$ By compactness, $u_n\xrightarrow [n\rightarrow \infty ]{L^{p_{\varepsilon ^{\prime }}}(B(0,R))}u.$ Arguing by contradiction and using the dominated convergence Theorem, we show in the same way as in Step 4,

$$\begin{aligned} \lim _{n\rightarrow \infty }\int \limits _{{\mathbb {R}}^N}\big |g(u_n)-g(u)\big ||\varphi |{\mathbb {1}}_{A_n}\mathrm{d}x=0, \end{aligned}$$

from which we deduce, with help of (A.18),

$$\begin{aligned} \forall \varepsilon >0, \; \limsup _{n\rightarrow \infty }\left| \langle g(u_n)-g(u), \varphi \rangle _{{\mathscr {D}}^{\prime }({\mathbb {R}}^N);{\mathscr {D}}({\mathbb {R}}^N)}\right| \leqslant \varepsilon \Vert \varphi \Vert _{H^1({\mathbb {R}}^N)}. \end{aligned}$$

We conclude as in Step 4. $\square$

B Topological vector spaces over the field of complex numbers restricted to the field of real numbers

Throughout this paper, we consider Banach spaces (or, more generally, complete topological vector spaces) over ${\mathbb {R}}$ rather than ${\mathbb {C}}.$ The main motivations are the following. Firstly, the linear forms are real-valued and there is a relation of order over ${\mathbb {R}}.$ Secondly, if a function $\psi$ belongs to $C^1(X;{\mathbb {R}})$ (as in Proposition 5.1, for instance), where X is a real Banach space, then $\psi ^{\prime }\in C(X;X^{\star }),$ where $X^{\star }$ is the ${\mathbb {R}}$-vector space ${\mathscr {L}}(X;{\mathbb {R}}).$ If X is a complex Banach space then $X^{\star }$ is the ${\mathbb {C}}$-vector space ${\mathscr {L}}(X;{\mathbb {C}})$ and $\psi ^{\prime }\in C\big (X;{\mathscr {L}}(X;{\mathbb {R}})\big ).$ But then, when a Riesz representation theorem exists, we have two kinds of representation between the elements of ${\mathscr {L}}(X;{\mathbb {R}})$ and those of $X^{\star }={\mathscr {L}}(X;{\mathbb {C}}),$ since ${\mathscr {L}}(X;{\mathbb {R}})$ is not ${\mathbb {C}}$-linear. On the other hand, if X is a complex Banach space, it could be pleasant to consider $\lambda x,$ for $(\lambda ,x)\in {\mathbb {C}}\times X.$ So, if $X_{{\mathbb {C}}}$ is a complex topological vector space, throughout this paper we consider $X_{{\mathbb {R}}}$ as the elements of $X_{{\mathbb {C}}}$ over the field ${\mathbb {R}}.$ We then consider the real topological vector space $X^{\star }_{{\mathbb {R}}}$. For any $(\lambda ,x)\in {\mathbb {C}}\times X$, $\lambda x\in X_{{\mathbb {R}}},$ since $X_{{\mathbb {R}}}$ and $X_{{\mathbb {C}}}$ have the same elements. In the special case where $H_{{\mathbb {C}}}$ is a complex Hilbert space whose the inner product is $(\,\cdot \,,\,\cdot \,)_H$ then $H_{{\mathbb {R}}}$ is the real Hilbert space whose the scalar product is $\langle \,\cdot \,,\,\cdot \,\rangle _H{\mathop {=}\limits ^{\mathrm{def}}}\mathrm{Re}\,(\,\cdot \,,\,\cdot \,)_H.$ In particular, for any $(u,v)\in H_{{\mathbb {R}}}\times H_{{\mathbb {R}}}$, $\langle \mathrm{i}u,\mathrm{i}v\rangle _H=\langle u,v\rangle _H.$ Now, assume that $X_{{\mathbb {C}}}$ is a complex Banach space. Denote by $X^{\star }_{{\mathbb {C}}}$ and $X^{\star }_{{\mathbb {R}}}$ the topological dual spaces of $X_{{\mathbb {C}}}$ and $X_{{\mathbb {R}}},$ respectively. It follows that $X^{\star }_{{\mathbb {C}}}$ is a ${\mathbb {C}}$-linear space while $X^{\star }_{{\mathbb {R}}}$ is only a ${\mathbb {R}}$-linear space. Let us define the map,

$$\begin{aligned} I:X^{\star }_{{\mathbb {C}}}&\longrightarrow X^{\star }_{{\mathbb {R}}}, \nonumber \\ L&\longmapsto \mathrm{Re}\,L. \end{aligned}$$

(B.1)

Then I is a bijective isometry from $X^{\star }_{{\mathbb {C}}}$ onto $X^{\star }_{{\mathbb {R}}}$ (Brezis [3, Proposition 11.22, p. 361]). With help of this correspondance, we can identify some linear forms. For instance, let $X=L^p(\Omega ;{\mathbb {C}}),$ where $\Omega$ is an open subset of ${\mathbb {R}}^N$ and $1\leqslant p<\infty .$ If $p=2$ then the inner and scalar products are given by

$$\begin{aligned} (u,v)_X=\int \limits _{\Omega } u(x)\overline{v(x)}\mathrm{d}x \quad \text { and } \quad \langle u,v\rangle _X=\mathrm{Re}\int \limits _{\Omega } u(x)\overline{v(x)}\mathrm{d}x, \end{aligned}$$

respectively. Using the Riesz representation Theorem for the complex $L^p(\Omega ;{\mathbb {C}})_{{\mathbb {C}}}$ spaces (Yosida [15, Example 3, p. 115]) and the bijective isometric map (B.1), it follows that

$$\begin{aligned} L^p(\Omega ;{\mathbb {C}})^{\star }_{{\mathbb {R}}}=L^{p^{\prime }}(\Omega ;{\mathbb {C}})_{{\mathbb {R}}}, \end{aligned}$$

where $\dfrac{1}{p}+\dfrac{1}{p^{\prime }}=1.$ More precisely, for any $L\in L^p(\Omega ;{\mathbb {C}})^{\star }_{{\mathbb {R}}},$ there exists a unique $u\in L^{p^{\prime }}(\Omega ;{\mathbb {C}})_{{\mathbb {R}}}$ such that

$$\begin{aligned} \langle L,v\rangle _{L^p(\Omega )^{\star },L^p(\Omega )}=\mathrm{Re}\int \limits _{\Omega } u(x)\overline{v(x)}\mathrm{d}x, \end{aligned}$$

for any $v\in L^p(\Omega ;{\mathbb {C}})_{{\mathbb {R}}}.$ Furthermore, $\Vert u\Vert _{L^{p^{\prime }}(\Omega ;{\mathbb {C}})_{{\mathbb {R}}}}=\Vert L\Vert _{L^p(\Omega ;{\mathbb {C}})^{\star }_{{\mathbb {R}}}}.$ Finally, we end this appendix with the space of distributions ${\mathscr {D}}^{\prime }(\Omega ;{\mathbb {C}}).$ We consider the ${\mathbb {C}}$-complete topological vector space ${\mathscr {D}}(\Omega ;{\mathbb {C}})$ restricted to the field ${\mathbb {R}}$ as above. Then an element T belongs to the ${\mathbb {R}}$-complete topological vector space ${\mathscr {D}}^{\prime }(\Omega ;{\mathbb {C}})$ if T is a ${\mathbb {R}}$-linear continuous mapping from ${\mathscr {D}}(\Omega ;{\mathbb {C}})$ to ${\mathbb {R}}.$ In particular, a function $f\in L^1_{\mathrm{loc}}(\Omega ;{\mathbb {C}})$ (over the field ${\mathbb {R}})$ defines a distribution $T_f\in {\mathscr {D}}^{\prime }(\Omega ;{\mathbb {C}})$ by the formula,

$$\begin{aligned} \langle T_f,\varphi \rangle _{{\mathscr {D}}^{\prime }(\Omega ;{\mathbb {C}}),{\mathscr {D}}(\Omega ;{\mathbb {C}})} =\mathrm{Re}\int \limits _{\Omega } f(x)\overline{\varphi (x)}\mathrm{d}x, \end{aligned}$$

for any $\varphi \in {\mathscr {D}}(\Omega ;{\mathbb {C}}).$ Indeed, $T_f$ is clearly a ${\mathbb {R}}$-linear continuous mapping from ${\mathscr {D}}(\Omega ;{\mathbb {C}})$ to ${\mathbb {R}}.$ Furthermore, if $f\in L^1_{\mathrm{loc}}(\Omega ;{\mathbb {C}})$ satisfies,

$$\begin{aligned} \mathrm{Re}\int \limits _{\Omega } f(x)\overline{\varphi (x)}\mathrm{d}x=0, \end{aligned}$$

for any $\varphi \in {\mathscr {D}}(\Omega ;{\mathbb {C}}),$ then $f=0.$ To see this, we note that $\mathrm{Re}(f),\mathrm{Im}(f)\in L^1_{\mathrm{loc}}(\Omega ;{\mathbb {R}})$ and choosing $\varphi =\psi +\mathrm{i}0$ and then $\varphi =0+\mathrm{i}\psi$ in the above expression, we get

$$\begin{aligned} \int \limits _{\Omega }\mathrm{Re}\big (f(x)\big )\psi (x)\mathrm{d}x=\int \limits _{\Omega }\mathrm{Im}\big (f(x)\big )\psi (x)\mathrm{d}x=0, \end{aligned}$$

for any $\psi \in {\mathscr {D}}(\Omega ;{\mathbb {R}}).$ We infer that $\mathrm{Re}(f)=\mathrm{Im}(f)=0$ (Brezis [3, Corollary 4.24, p. 110]), from which the result follows. Obviously, if $f_n\xrightarrow [n\rightarrow \infty ]{L^1_{\mathrm{loc}}(\Omega ;{\mathbb {C}})}f$ then $T_{f_n}\xrightarrow [n\rightarrow \infty ]{{\mathscr {D}}^{\prime }(\Omega ;{\mathbb {C}})}T_f.$ We conclude that,

$$\begin{aligned} L^1_{\mathrm{loc}}(\Omega ;{\mathbb {C}})\hookrightarrow {\mathscr {D}}^{\prime }(\Omega ;{\mathbb {C}}), \end{aligned}$$

with embedding $T:f\in L^1_{\mathrm{loc}}(\Omega ;{\mathbb {C}})\longmapsto T_f\in {\mathscr {D}}^{\prime }(\Omega ;{\mathbb {C}})$.