Existence of solutions for a semirelativistic Hartree equation with unbounded potentials

TL;DR

This paper proves the existence of solutions for a semirelativistic Hartree equation with unbounded potentials, expanding the understanding of such equations under less restrictive growth conditions.

Contribution

It establishes existence results for solutions to a semirelativistic Hartree equation with unbounded potential functions, a case not previously well-addressed.

Findings

Existence of solutions under unbounded potential conditions

Applicable to potentials with specific growth assumptions

Extends previous results to more general potential functions

Abstract

We prove the existence of a solution to the semirelativistic Hartree equation $$\sqrt{-\Delta+m^2}u+ V(x) u = A(x)\left( W * |u|^p \right) |u|^{p-2}u $$ under suitable growth assumption on the potential functions $V$ and $A$. In particular, both can be unbounded from above.

Equations

$$\sqrt{-\Delta +m^2}u+V(x)u=A(x)\left(W\ast |u{|}^p\right)|u{|}^{p-2}u$$

$$i\frac{\partial \psi }{\partial t}=\left(\sqrt{-\Delta +m^2}-m\right)\psi -\left(\frac{1}{|x|}\ast |\psi {|}^2\right)\psi ,x\in {\mathbb{R}}^3$$

$$\left(\sqrt{-\Delta+m^{2}}-m\right)\phi-\left(\frac{1}{|x|}*|\phi|^{2}\right)\phi=\lambda\phi\quad\mbox{in $\mathbb{R}^{3}$}.$$

$$\sqrt{-\Delta +m^2}u+V(x)u=A(x)\left(W\ast |u{|}^p\right)|u{|}^{p-2}u$$

$$X=\left\{u\in L^2({\mathbb{R}}^N)\mid \Vert u{\Vert }_X<\infty \right\},$$

$$\Vert u{\Vert }_X^2={\int }_{{\mathbb{R}}^N}{\left|{\left(-\Delta +m^2\right)}^{1/4}u\right|}^{2}dx+{\int }_{{\mathbb{R}}^N}V(x)|u{|}^{2}dx.$$

$$\underset{{\mathbb{R}}^N}{\inf }V>-B$$

$${\lambda }_1=\underset{u\in X\setminus \{0\}}{\inf }\frac{{\int }_{{\mathbb{R}}^N}{\left|(-\Delta +m^2{)}^{1/4}u\right|}^2+{\int }_{{\mathbb{R}}^N}V|u{|}^2}{\Vert u{\Vert }_2^2}>0.$$

$$A(x{)}^{\frac{2r}{2r-1}}\le C_0\left(1+{\left(\max \{0,V(x)\}\right)}^{\frac{1}{\beta }}\right)$$

$$\underset{|x|\to +\infty }{\mathrm{limsup}}\frac{A(x{)}^{\frac{2r}{2r-1}}}{1+{\left(\max \left\{0,V(x)\right\}\right)}^{\frac{1}{\beta }}}<+\infty .$$

$$\nu \left(t,\Omega \right)$$

$$=+\infty \text{if Ω=∅}.$$

$$\underset{R\to +\infty }{\lim }\nu (t_0,{\mathbb{R}}^N\setminus \overline{B_R})=+\infty ,$$

$$\sqrt{-\Delta+m^{2}}\varphi_{1}+V(x)\varphi_{1}=\lambda_{1}\varphi_{1}\quad\mbox{in $\mathbb{R}^{N}$.}$$

$$\underset{n\to +\infty }{\lim }{\int }_{{\mathbb{R}}^N}{\left|{\left(-\Delta +m^2\right)}^{\frac{1}{4}}{u}_n\right|}^2+{\int }_{{\mathbb{R}}^N}V|u_n{|}^2$$

$${\int }_{{\mathbb{R}}^N}|u_n{|}^2$$

$${\int }_{{\mathbb{R}}^N}{\left|(-\Delta +m^2{)}^{\frac{1}{4}}{u}_n\right|}^2\le {\lambda }_1+1+B.$$

$$\begin{gathered} {\Vert u_n-u\Vert }_2^2\le {\Vert (1-\psi )(u_n-u)\Vert }_2^2+{\Vert \psi (u_n-u)\Vert }_2^2 \\ \le o_n(1)+\frac{1}{\nu (2,{\mathbb{R}}^N\setminus B_R)}\left({\int }_{{\mathbb{R}}^N}{\left|{\left(-\Delta +m^2\right)}^{\frac{1}{4}}(u_n-u)\right|}^2+{\int }_{{\mathbb{R}}^N}V|u_n-u{|}^2\right) \\ \le o_n(1)+o_R(1), \end{gathered}$$

$${\int }_{{\mathbb{R}}^N}{\left|(-\Delta +m^2{)}^{\frac{1}{4}}u\right|}^2\le \underset{n\to +\infty }{\mathrm{liminf}}{\left|(-\Delta +m^2{)}^{\frac{1}{4}}{u}_n\right|}^2.$$

$$\underset{n\to +\infty }{\lim }{\int }_{{\mathbb{R}}^N\setminus G}V|u_n{|}^2={\int }_{{\mathbb{R}}^N\setminus G}V|u{|}^2.$$

$$v\in L^2(G)\mapsto {\int }_{G}V|v{|}^2$$

$$\underset{n\to +\infty }{\mathrm{liminf}}{\int }_{{\mathbb{R}}^N}V|u_n{|}^2$$

$$\ge {\int }_{{\mathbb{R}}^N}V|u{|}^2.$$

$$\underset{|x|\to +\infty }{\lim }V(x)=+\infty$$

$$(-\Delta +m^2{)}^{s}u+V(x)u=\left(W\ast |u{|}^p\right)|u{|}^{p-2}u$$

$$L^{\alpha ,2}({\mathbb{R}}^N)=\left\{f:f=G_{\alpha }\star g\text{for some g∈L2(RN)}\right\},$$

$$G_{\alpha }(x)=\frac{1}{(4\pi {)}^{\alpha /2}\Gamma (\alpha /2)}{\int }_0^{\infty }\exp \left(-\frac{\pi }{t}|x{|}^2\right)\exp \left(-\frac{t}{4\pi }\right)t^{\frac{\alpha -N}{2}-1}dt$$

$$(I-\Delta {)}^{-\alpha }u=G_{2\alpha }\star u$$

$$G_{\alpha }={\mathcal{F}}^{-1}\circ \left({\left(1+|\xi {|}^2\right)}^{-\alpha /2}\circ \mathcal{F}\right),$$

$$\Vert f\Vert ={\Vert (I-\Delta {)}^{\alpha /2}f\Vert }_2.$$

$$X=\left\{u\in L^{1/2,2}({\mathbb{R}}^N)\mid \Vert u{\Vert }_X<\infty \right\}$$

$$\Vert u{\Vert }_X^2={\int }_{{\mathbb{R}}^N}{\left|{\left(-\Delta +m^2\right)}^{1/4}u\right|}^{2}dx+{\int }_{{\mathbb{R}}^N}V(x)|u{|}^{2}dx$$

Full text

[TABLE]

under suitable growth assumption on the potential functions $V$ and $A$. In particular, both can be unbounded from above.

To Francesca, always

1 Introduction

The mean field limit of a quantum system describing many self-gravitating, relativistic bosons with rest mass $m>0$ leads to the time-dependent pseudo-relativistic Hartree equation

[TABLE]

where $\psi:\mathbb{R}\times\mathbb{R}^{3}\to\mathbb{C}$ is the wave field. Such a physical system is often referred to as a boson star in astrophysics (see [16, 20, 21]). Solitary wave solutions $\psi(t,x)=e^{-it\lambda}\phi$, $\lambda\in\mathbb{R}$ to equation (1.1) satisfy the equation

[TABLE]

For the non-relativistic Hartree equation driven by the local differential operator $-\Delta+m^{2}$, existence and uniqueness (modulo translations) of a minimizer were proved by Lieb [22] by using symmetric decreasing rearrangement inequalities. Within the same setting, always for the negative Laplacian, P.-L. Lions [25] proved existence of infinitely many spherically symmetric solutions by application of abstract critical point theory both with and without constraints for a more general radially symmetric convolution potential. The non-relativistic Hartree equations is also known as the Choquard-Pekard or Schrödinger-Newton equation and recently a large amount of papers are devoted to the study of solitary states and its semiclassical limit: see [1, 4, 5, 7, 8, 9, 11, 12, 13, 14, 20, 23, 27, 26, 29, 30, 31, 38, 39] and references therein.

In this paper, which is somehow a continuation of the investigation we began in [33, 34], we consider the equation

[TABLE]

in the weighted space

[TABLE]

where

[TABLE]

In the case $A\equiv 1$, equation (1.2) has been recently investigated in [12], where least-energy solutions are constructed for a bounded potential $V$.

We collect our assumptions.

• (H1)

$N>1$, $\beta>1$, $2\leq p<\frac{2N\beta}{\beta(N-1)}$ and $r>\max\left\{1,\frac{1-N\beta}{2+\beta N(p-2)-\beta p}\right\}$.

• (H2)

$A\in L^{\infty}_{\mathrm{loc}}(\mathbb{R}^{N})$ satisfies $A(x)\geq 1$ for almost every $x\in\mathbb{R}^{N}$.

• (H3)

$V\colon\mathbb{R}^{N}\to\mathbb{R}$ is continuous and

[TABLE]

for some $B>0$; furthermore,

[TABLE]

• (H4)

There exist $C_{0}>0$ and $R_{0}>0$ such that

[TABLE]

for all $|x|>R_{0}$.

• (H5)

$W\colon\mathbb{R}^{N}\setminus\{0\}\to\mathbb{R}$ and $W=W_{1}+W_{2}$, where $W_{1}\in L^{r}$ and $W_{2}\in L^{\infty}$.

Equivalently, condition (H4) can be stated as

[TABLE]

Remark 1.1*.*

We highlight that both the potential $V$ and the potential $A$ in front of the nonlinear term in the right-hand side can be unbounded. To the best of our knowledge, this case is treated here for the first time. In the very recent paper [6], a similar equation has been studied under an assumption that, in our framework, reads as $V\equiv 1$ and $\lim_{|x|\to+\infty}A(x)=1$.

To state our last assumption, we define, for any open subset $\Omega\subset\mathbb{R}^{N}$ and any $t\geq 2$, the quantity (see Section 2 for the definition of the space $L^{1/2,2}(\Omega)$)

[TABLE]

We assume that

• ($\nu$)

there exists $t_{0}\in[2,2N(N-1))$ such that

[TABLE]

where $B_{R}=\left\{x\in\mathbb{R}^{N}\mid|x|<R\right\}$.

The number $\lambda_{1}$ in assumption (H3) can be actually seen as an eigenvalue, as the following result shows.

Proposition 1.2**.**

Assume that $V\colon\mathbb{R}^{N}\to\mathbb{R}$ is continuous and $\inf_{\mathbb{R}^{N}}V>-B$ for some $B>0$. Assume also that ($\nu$) holds with $t_{0}=2$. Then there exists $\varphi_{1}\in X$ such that

[TABLE]

Furthermore $\varphi_{1}$ can be assumed to be positive in $\mathbb{R}^{N}$.

Proof.

It follows immediately from the assumptions that $\lambda_{1}>-B>0$. Let $\{u_{n}\}_{n}$ be a minimizing sequence for $\lambda_{1}$, in the sense that

[TABLE]

As in [10, Lemma 2.1], we can assume that $u_{n}$ is non-negative. It follows from (1.3) that

[TABLE]

Let $u$ be the weak limit of $\{u_{n}\}_{n}$ in $L^{1/2,2}(\mathbb{R}^{N})$. Since $u_{n}\to u$ strongly in $L^{2}(B_{R})$ for any $R>0$, we may assume that $u_{n}$ converges to $u$ almost everywhere, and consequently $u\geq 0$. We fix a smooth cut-off function $\psi$ such that $\psi\equiv 0$ on $B_{R}$ and $\psi\equiv 1$ on $\mathbb{R}^{N}\setminus B_{R+1}$. Then

[TABLE]

where $\lim_{n\to+\infty}o_{n}(1)=0$ and $\lim_{R\to+\infty}o_{R}(1)=0$: this proves that $u_{n}\to u$ strongly in $L^{2}(\mathbb{R}^{N})$. In particular, $\int_{\mathbb{R}^{N}}|u|^{2}=1$ due to (1.4). Plainly,

[TABLE]

Set $G=\left\{x\in\mathbb{R}^{N}\mid V(x)>1\right\}$. Since $V\in L^{\infty}(\mathbb{R}^{N}\setminus G)$, we have

[TABLE]

On the other hand, by [19, Theorem 6.54] we deduce that

[TABLE]

is weakly lower semicontinuous. To summarize,

[TABLE]

Hence $u$ is a minimizer for $\lambda_{1}$. The strict positivity of $u$ follows from [17, Proposition 2]. ∎

Remark 1.3*.*

In the previous Proposition, we did not assume that $\lambda_{1}>0$. Under this additional assumption, the proof would be much easier.

It is important to remark that condition ($\nu$) is related to some other popular conditions. In Proposition 3.2 we prove that the coercivity assumption

[TABLE]

implies ($\nu$). Similarly, the condition

• ($\nu^{\prime}$)

For every $b>0$, the set $V^{b}$ has finite Lebesgue measure

also implies ($\nu$), see Proposition 3.4. Finally, Sirakov’s condition [35]

• ($\nu^{\prime\prime}$)

There exists $t_{1}\in[2,2N/(N-1))$ such that, for any $r>0$ and any sequence $\{x_{n}\}_{n}$ of points in $\mathbb{R}^{N}$ such that $\lim_{n\to+\infty}|x_{n}|=+\infty$, there results $\lim_{n\to+\infty}\nu(t_{1},B(x_{n},r))=+\infty$

is equivalent to ($\nu$): see Proposition 3.5.

Remark 1.4*.*

While dealing with the semirelativistic Hartree equation, it is customary to rewrite the operator $\sqrt{-\Delta+m^{2}}+V$ as $\sqrt{-\Delta+m^{2}}-m+(V+m)$ in order to exploit the fact that $\sqrt{-\Delta+m^{2}}-m>0$ in the sense of functional calculus. The natural assumption from a variational point of view is therefore $\inf_{\mathbb{R}^{N}}V>-m$. Our assumption (H3) is, in general, less restrictive as it only requires a suitable lower bound for $V$.

We can state the main result of this paper.

Theorem 1.5**.**

Suppose that (H1), (H2), (H3), (H4), (H5), and ($\nu$) are satisfied. Then equation (1.2) has infinitely many distinct solutions.

The proof is based on the ideas developed in [35] for a local equation and a pointwise nonlinearity.

Remark 1.6*.*

It should be noticed that our results continue to hold if we replace (1.2) with

[TABLE]

with $0<s<1$ and $N>2s$, for instance $N\geq 3$. Of course some numbers must be replaced: $2N/(N-1)$ should become $2N/(N-2s)$, and so on. We prefer to work out the details for $s=1/2$, which corresponds to the physical model of the Hartree equation.

In Section 2 we introduce the necessary preliminaries on function space. In Section 3 we compare several assumptions the ensure a compact embedding result. In Section 4 we prove a compactness theorem that is used in Section 5 to prove our main existence result.

Notation

The letters $c$ and $C$ will stand for a generic positive constant that may vary from line to line. 2. 2.

The operator $d$ will be reserved for the (Fréchet) derivative, so that $dI$ denotes the Fréchet derivative of a function $I$. 3. 3.

The symbol $\mathcal{L}^{N}$ will be reserved for the Lebesgue $N$-dimensional measure. 4. 4.

The Fourier transform of a function $u$ will be denoted by $\mathcal{F}u$.

2 Variational setting

Let us recall the definition of the Bessel function space defined for $\alpha>0$ by

[TABLE]

where the Bessel convolution kernel is defined by

[TABLE]

The norm of this Bessel space is $\|f\|=\|g\|_{2}$ if $f=G_{\alpha}\star g$. The operator

[TABLE]

is usually called Bessel operator of order $\alpha$. In Fourier variables the same operator reads

[TABLE]

so that

[TABLE]

For more detailed information, see [2, 36] and the references therein. The use of $(-\Delta+m^{2})^{\alpha/2}$ instead of $(-\Delta+I)^{\alpha/2}$ is clearly harmless. We summarize the embedding properties of Bessel spaces. For the proofs we refer to [18, Theorem 3.2], [36, Chapter V, Section 3] and [37, Section 4].

Theorem 2.1**.**

$L^{\alpha,2}(\mathbb{R}^{N})=W^{\alpha,2}(\mathbb{R}^{N})=H^{\alpha}(\mathbb{R}^{N})$. 2. 2.

If $\alpha\geq 0$ and $2\leq q\leq 2_{\alpha}^{*}=2N/(N-2\alpha)$, then $L^{\alpha,2}(\mathbb{R}^{N})$ is continuously embedded into $L^{q}(\mathbb{R}^{N})$; if $2\leq q<2_{\alpha}^{*}$ then the embedding is locally compact. 3. 3.

Assume that $0\leq\alpha\leq 2$ and $\alpha>N/2$. If $\alpha-N/2>1$ and $0<\mu\leq\alpha-N/2-1$, then $L^{\alpha,2}(\mathbb{R}^{N})$ is continuously embedded into $C^{1,\mu}(\mathbb{R}^{N})$. If $\alpha-N/2<1$ and $0<\mu\leq\alpha-N/2$, then $L^{\alpha,2}(\mathbb{R}^{N})$ is continuously embedded into $C^{0,\mu}(\mathbb{R}^{N})$.

Remark 2.2*.*

Although the Bessel space $L^{\alpha,2}(\mathbb{R}^{N})$ is topologically undistinguishable from the Sobolev fractional space $H^{\alpha}(\mathbb{R}^{N})$, we will not systematically confuse them, since our equation involves the Bessel norm.

For a general open subset $\Omega$ of $\mathbb{R}^{N}$, the Bessel space $L^{\alpha,2}(\Omega)$ is defined ad the space of the restrictions to $\Omega$ of functions in $L^{\alpha,2}(\mathbb{R}^{N})$. In this paper we will always take $\alpha=1/2$. As we said in the Introduction, we work in the weighted space

[TABLE]

where

[TABLE]

Lemma 2.3**.**

If assumption (H3) holds, then there exists a constant $\aleph>0$ such that

[TABLE]

for every $u\in X$.

Proof.

We proceed by contradiction. Assume that for some sequence $\{u_{n}\}_{n}$ in $X$ there results

[TABLE]

By definition of $\lambda_{1}>0$, the last inequality entails $\|u_{n}\|_{L^{2}}=o(1)$. But then the contradiction

[TABLE]

arises as $n\rightarrow+\infty$. ∎

Solutions to (1.2) correspond to critical points of the functional $I\colon X\rightarrow\mathbb{R}$ defined by

[TABLE]

We need to prove that $I$ is well-defined on $X$.

Proposition 2.4**.**

The space $X$ is continuously embedded into the weighted Lebesgue space

[TABLE]

for every $2\leq t\leq\frac{2(N\beta-1)}{\beta(N-1)}$.

Proof.

Let us decompose

[TABLE]

where $R_{0}>0$ is the number defined in assumption (H4). Now,

[TABLE]

where

[TABLE]

As a consequence, inserting this into (2.2),

[TABLE]

If we apply Hölder’s inequality to the last integral, we obtain

[TABLE]

But

[TABLE]

In conclusion,

[TABLE]

It is elementary to check that

[TABLE]

whenever

[TABLE]

so that we can invoke the continuous embedding of $X$ into $L^{s}(\mathbb{R}^{N})$ for $2\leq s\leq 2N/(N-1)$ and conclude from (2.3) that there exists a positive constant $C$ such that

[TABLE]

This completes the proof. ∎

Remark 2.5*.*

Observe that when $\inf_{\mathbb{R}^{N}}V>0$ and $A=1$ identically, we can let $\beta\to+\infty$ and recover the weaker assumption $2\leq t\leq 2N/(N-1)$.

To proceed further, we need the following inequality due to Hardy, Littlewood and Sobolev. We firstly recall that a function $h$ belongs to the weak $L^{q}$ space $L_{w}^{q}(\mathbb{R}^{N})$ if there exists a constant $C>0$ such that, for all $t>0$,

[TABLE]

Its norm is then

[TABLE]

Proposition 2.6** ([24]).**

Assume that $p$, $q$ and $t$ lie in $(1,+\infty)$ and $p^{-1}+q^{-1}+t^{-1}=2$. Then, for some constant $N_{p,q,t}>0$ and for any $f\in L^{p}(\mathbb{R}^{N})$, $g\in L^{t}(\mathbb{R}^{N})$ and $h\in L_{w}^{q}(\mathbb{R}^{N})$, we have the inequality

[TABLE]

Writing $W=W_{1}+W_{2}\in L^{r}(\mathbb{R}^{N})+L^{\infty}(\mathbb{R}^{N})$ and using $(\ref{eq:hls})$ we can estimate the convolution term as follows by means of Proposition 2.4:

[TABLE]

where we have used several times the fact that $A(x)\geq 1$ for almost every $x\in\mathbb{R}^{N}$. Since

[TABLE]

we can use Proposition 2.4 and from (2) we see that the convolution term in $I$ is finite. It is easy to check, by the same token and taking into account the assumption $p\geq 2$, that $I\in C^{1}(X)$.

Remark 2.7*.*

As before, if $\inf_{\mathbb{R}^{N}}V>0$ and $A=1$ identically, we can recover the assumption

[TABLE]

used in [12].

3 Comparison between conditions on $V$

In this section we prove that condition ($\nu$) is actually weaker than both the coercivity of $V$ and of condition ($\nu^{\prime}$). We start with a preliminary technical result.

Lemma 3.1**.**

Let $\Omega\subset\mathbb{R}^{N}$ be an open set. If $2\leq t<r<2N/(N-1)$, then there exist a constant $C=C(r,t,N,\lambda_{1})>0$ and a number $0<\theta_{1}<1$ such that

[TABLE]

If $2<r<t<2N/(N-1)$, then there exist a constant $C=C(r,t,N,\lambda_{1})>0$ and a number $0<\theta_{2}<1$ such that

[TABLE]

In particular,

[TABLE]

if

[TABLE]

Proof.

We write

[TABLE]

for some $0<\theta<1$. Then $\|u\|_{r}\leq\|u\|_{t}^{1-\theta}\|u\|_{\frac{2N}{N-1}}^{\theta}$ for every $u\in L^{1/2,2}(\Omega)$. The Gagliardo-Nirenberg inequality (see [28, Theorem 2.1] together with Theorem 2.1) yields $\|u\|_{r}\leq C\|u\|_{L^{1/2,2}}^{\theta}\|u\|_{t}^{1-\theta}$ for every $u\in L^{1/2,2}(\Omega)$. We invoke Lemma 2.3 to get

[TABLE]

As a consequence,

[TABLE]

The second inequality follows by the same token. ∎

Proposition 3.2**.**

If $\lim_{|x|\rightarrow+\infty}V(x)=+\infty$, then ($\nu$) holds true.

Proof.

By Lemma 3.1 it is sufficient to prove the validity of $(\nu$) with $t_{0}=2$. For any $u\in X$ and any $R>0$,

[TABLE]

Therefore $\nu(2,\mathbb{R}^{N}\setminus B_{R})\geq\inf_{\mathbb{R}^{N}\setminus B_{R}}V$, and we conclude by letting $R\to+\infty$. ∎

Lemma 3.3**.**

Let $\{\omega_{n}\}_{n}$ be a sequence of open subset of $\mathbb{R}^{N}$ such that

[TABLE]

Then, for every $\varrho>0$ and every $2\leq t<2N/(N-1)$, there results

[TABLE]

Proof.

Since

[TABLE]

by the Sobolev embedding we can find a constant $C_{1}>0$ such that

[TABLE]

whenever $\|u\|_{L^{1/2,2}}\leq\varrho$. The conclusion follows immediately. ∎

Proposition 3.4**.**

Condition ($\nu^{\prime}$) implies condition ($\nu$).

Proof.

For the sake of contradiction, let us suppose that there exist a sequence $R_{n}\to+\infty$ of positive numbers and a sequence $\{u_{n}\}_{n}$ of functions from $X$ such that

[TABLE]

By Lemma 2.3, the sequence $\{u_{n}\}$ is bounded in $L^{1/2,2}(\mathbb{R}^{N})$. For any $M>0$ we introduce the set

[TABLE]

so that $\lim_{n\to+\infty}\mathcal{L}^{N}(\Omega(M,n))=0$. For some $C>0$,

[TABLE]

By Lemma 3.3 the last term of (3.2) converges to zero as $n\to+\infty$. Since $M>0$ is arbitrary and (3.1) holds true, we derive a contradiction. ∎

Finally, we prove that our assumption ($\nu$) is logically equivalent to Sirakov’s condition ($\nu^{\prime\prime}$).

Proposition 3.5**.**

Condition ($\nu^{\prime\prime}$) is equivalent to ($\nu$).

Proof.

Step 1. Assume that ($\nu$) holds. Given any $r>0$, we know that

[TABLE]

where $R_{n}=\frac{1}{2}|x_{n}|-r$. Hence also ($\nu^{\prime\prime}$) holds.

Step 2. On the contrary, assume that ($\nu$) does not hold. Hence there are sequences $\{u_{n}\}_{n}$ in $X$ and $\{R_{n}\}_{n}$ in $(0,+\infty)$ such that $u_{n}\in L^{1/2,2}(\mathbb{R}^{N})$, $\operatorname{supp}u_{n}\subset\mathbb{R}^{N}\setminus\overline{B(0,R_{n})}$, $\lim_{n\to+\infty}R_{n}=+\infty$,

[TABLE]

As before, the sequence $\{u_{n}\}_{n}$ is bounded in $L^{1/2,2}(\mathbb{R}^{N})$ and by a Lemma of P.-L. Lions (see [32, Lemma 2.4]) there exist a sequence $\{x_{n}\}_{n}$ of points in $\mathbb{R}^{N}$, two constants $r_{0}>0$, $C_{0}>0$, such that, up to a subsequence,

[TABLE]

It follows from these properties that $|x_{n}|\to+\infty$. We fix a sequence of smooth cut-off functions $\varphi_{n}$ such that $\varphi_{n}\in C_{c}^{\infty}(\mathbb{R}^{N})$, $0\leq\varphi_{n}\leq 1$

[TABLE]

Let $v_{n}=\varphi_{n}u_{n}\in L^{1/2,2}(B(x_{n},2r_{0}))$, so that by (3.3)

[TABLE]

It is well known, see [15], that $\|v_{n}\|_{H^{1/2}}\leq C_{1}\|u_{n}\|_{H^{1/2}}$ for some constant $C_{1}>0$. By Theorem 2.1 and Lemma 2.3, $\|v_{n}\|_{X}\leq C_{2}\|u_{n}\|_{X}$. Thus for any $n\in\mathbb{N}$,

[TABLE]

We have proved that also ($\nu^{\prime\prime}$) does not hold. ∎

4 A compact embedding theorem

Theorem 1.5 will be proved by applying the Symmetric Mountain Pass Theorem of Ambrosetti and Rabinowitz [3] to the functional $I$. The required compactness is recovered by embedding the space $X$ into a weighted Lebesgue space, see Proposition 4.1.

We are ready to prove the main compactness result of this paper.

Proposition 4.1**.**

Assume that (H1), (H2), (H3), (H4) and ($\nu$) are satisfied. Then $X$ is compactly embedded into $L^{t}(\mathbb{R}^{N})$ for all $2\leq t\leq\frac{2(N\beta-1)}{\beta(N-1)}$, and compactly embedded into $L^{t}(\mathbb{R}^{N},A^{2r/(2r-1)}\,d\mathcal{L}^{N})$ if $2\leq t<\frac{2(N\beta-1)}{\beta(N-1)}$.

Proof.

Consider any bounded sequence $\{u_{n}\}_{n}$ in $X$. By reflexivity, we assume without loss of generality that $u_{n}\rightharpoonup u$ weakly in $X$ as $n\to+\infty$. Choose a smooth function $\varphi\colon\mathbb{R}^{N}\to[0,1]$ with the property that $\varphi(x)=0$ if $|x|\leq R$, while $\varphi(x)=1$ if $|x|>R+1$. Then

[TABLE]

The space $L^{1/2,2}(B_{R+1})$ is compactly embedded into $L^{t}(B_{R+1})$ because $\frac{2(N\beta-1)}{\beta(N-1)}<2N/(N-1)$, see Theorem 2.1. Hence, up to a subsequence,

[TABLE]

But by definition of $\nu$

[TABLE]

We deduce that, using the fact that $\|\varphi(u_{n}-u)\|_{H^{1/2}}\leq C\|u_{n}-u\|_{H^{1/2}}$ and that $L^{1/2,2}=H^{1/2}$,

[TABLE]

and we conclude by Lemma 3.1. The second part follows directly from Proposition 2.4 (see in particular (2.3)). ∎

5 Existence of critical points

In this section we apply the celebrated Mountain Pass Theorem and the embedding result proved earlier to find a critical point of the functional $I$ defined in (2.1).

Proposition 5.1**.**

The functional $I$ has the Mountain Pass geometry.

Proof.

Step 1: there exists a number $\rho>0$ such that $\|u\|_{X}=\rho$ implies $I(u)>0$.

Indeed, by (2) we find that for some constant $C>0$

[TABLE]

Hence $I(u)\leq 0$ implies $\|u\|_{X}\geq C$ for another constant $C>0$. The claim follows easily.

Step 2: there exists $e\in X$ such that $\|e\|_{X}>\rho$ and $I(e)<0$.

Simply, pick $u\in X$ such that $\int_{\mathbb{R}^{N}}(K*|u|^{2p})|u|^{2p}>0$ and compute

[TABLE]

∎

In order to apply the Mountain Pass Theorem, we need to ensure the validity of the Palais-Smale compactness condition. This follows from Proposition 4.1, as we now show.

Proposition 5.2**.**

If $\{u_{n}\}_{n}$ is any sequence from $X$ such that $dI(u_{n})=o(1)$ in $X^{*}$ and $I(u_{n})\leq C$ for every $n\in\mathbb{N}$, then $\{u_{n}\}_{n}$ converges strongly in $X$ up to a subsequence.

Proof.

First of all, the sequence $\{u_{n}\}_{n}$ is bounded in $X$. Indeed,

[TABLE]

By reflexivity, we assume without loss of generality that $u_{n}\rightharpoonup u$ weakly in $X$ as $n\to+\infty$. It is plain that $u$ is a critical point of $I$. Now Proposition 4.1 implies that — up to a subsequence — $u_{n}\to u$ strongly in $L^{\frac{2rp}{2r-1}}(\mathbb{R}^{N},|A|^{\frac{2r}{2r-1}}\,d\mathcal{L}^{N})$, so that (2) easily implies

[TABLE]

We conclude that $u_{n}\to u$ strongly in $X$ as $n\to+\infty$, and the proof is complete. ∎

Proof of Theorem 1.5.

Let us remark that the right-hand side of equation (1.2) is odd with respect to $u$. By Proposition 5.1 and Proposition 5.2, we can invoke the Symmetric Mountain Pass Theorem [3] and conclude that the functional $I$ possesses infinitely many critical points in $X$. ∎