A critical nonlinear elliptic equation with non local regional diffusion

TL;DR

This paper investigates the existence and behavior of solutions to a critical nonlinear nonlocal regional Schrödinger equation, focusing on the effects of a small parameter and the regional scope of the fractional Laplacian.

Contribution

It introduces a variational approach to analyze ground states and semi-classical solutions for a critical nonlocal regional Schrödinger equation with variable scope.

Findings

Existence of ground state solutions established.

Semi-classical solutions analyzed as epsilon approaches zero.

Behavior of solutions depends on the regional scope and parameters.

Abstract

In this article we are interested in the nonlocal regional Schr\"odinger equation with critical exponent \begin{eqnarray*} &\epsilon^{2\alpha} (-\Delta)_{\rho}^{\alpha}u + u = \lambda u^q + u^{2_{\alpha}^{*}-1} \mbox{ in } \mathbb{R}^{N}, \\ & u \in H^{\alpha}(\mathbb{R}^{N}), \end{eqnarray*} where $\epsilon$ is a small positive parameter, $\alpha \in (0,1)$, $q\in (1,2_{\alpha}^{*}-1)$, $2_{\alpha}^{*} = \frac{2N}{N-2\alpha}$ is the critical Sobolev exponent, $\lambda >0$ is a parameter and $(-\Delta)_{\rho}^{\alpha}$ is a variational version of the regional laplacian, whose range of scope is a ball with radius $\rho(x)>0$. We study the existence of a ground state and we analyze the behavior of semi-classical solutions as $\varepsilon\to 0$.

Equations

$${ϵ}^{2\alpha }(-\Delta {)}_{\rho }^{\alpha }u+u=\lambda u^q+u^{2_{\alpha }^{\ast }-1}\text{mbox}in{\mathbb{R}}^N,$$

$$u\in H^{\alpha }({\mathbb{R}}^N),$$

$${ϵ}^{2\alpha }(-\Delta {)}_{\rho }^{\alpha }u+u=\lambda u^q+u^{2_{\alpha }^{\ast }-1}\text{mbox}in{\mathbb{R}}^N,$$

$$u\in H^{\alpha }({\mathbb{R}}^N),$$

$${\int }_{{\mathbb{R}}^n}(-\Delta {)}_{\rho }^{\alpha }u(x)\phi (x)dx={\int }_{{\mathbb{R}}^n}{\int }_{B(0,\rho (x))}\frac{[u(x+z)-u(z)][\phi (x+z)-\phi (x)]}{|z{|}^{n+2\alpha }}dxdx$$

$${\rho }_0\le \rho (x)<{\rho }_{\infty },\forall x\in {\mathbb{R}}^n\text{mbox}and\underset{|x|\to \infty }{\lim }\rho (x)={\rho }_{\infty }.$$

$$|x|=\rho (x+x_0),x\in {\mathbb{R}}^n,$$

$$\underset{|x|\to \infty }{\mathrm{limsup}}\frac{\rho (x)}{|x|}\le a.$$

$$H^{\alpha }({\mathbb{R}}^N)=\left\{u\in L^2({\mathbb{R}}^N):\frac{|u(x)-u(z)|}{|x-z{|}^{\frac{N+2\alpha }{2}}}\in L^2({\mathbb{R}}^N\times {\mathbb{R}}^N)\right\}$$

$$\Vert u{\Vert }_{\alpha }={\left({\int }_{{\mathbb{R}}^N}|u(x){|}^{2}dx+{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x)-u(z){|}^2}{|x-z{|}^{N+2\alpha }}dzdx\right)}^{1/2}.$$

$$\Vert u{\Vert }_{L^{2_{\alpha }^{\ast }}}\le \mathfrak{C}\Vert u{\Vert }_{\alpha }$$

$$H_0^{\alpha }({\mathbb{R}}^N)$$

$$={\overline{C_0^{\infty }({\mathbb{R}}^N)}}^{\Vert .{\Vert }_0},$$

$$\Vert u{\Vert }_0^2={\int }_{{\mathbb{R}}^N}|\xi {|}^{2\alpha }|\hat{u}(\xi ){|}^{2}d\xi .$$

$$S=\underset{u\in H_0^{\alpha }({\mathbb{R}}^N)\setminus \{0\}}{\inf }\frac{{\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x)-u(z){|}^2}{|x-z{|}^{N+2\alpha }}dzdx}{{\left({\int }_{{\mathbb{R}}^N}|u(x){|}^{2_{\alpha }^{\ast }}dx\right)}^{2/2_{\alpha }^{\ast }}}.$$

$$u_0(x)=\frac{c}{({\theta }^2+|x-x_0{|}^2{)}^{\frac{N-2\alpha }{2}}},x\in {\mathbb{R}}^N,$$

$$U_{ϵ}(x)={ϵ}^{-\frac{N-2\alpha }{2}}\tilde{u}\left(\frac{x}{ϵS^{1/2\alpha }}\right),\tilde{u}(x)=\frac{u_0(x)}{\Vert u_0{\Vert }_{L^{2_{\alpha }^{\ast }}}}$$

$$(-\Delta {)}^{\alpha }u=|u{|}^{2_{\alpha }^{\ast }-2}u,x\in {\mathbb{R}}^N.$$

$$\Vert u{\Vert }_{\rho }^2={\int }_{{\mathbb{R}}^n}{\int }_{B(0,\rho (x))}\frac{|u(x)-u(z){|}^2}{|x-z{|}^{n+2\alpha }}dzdx+{\int }_{{\mathbb{R}}^n}u(x{)}^{2}dx$$

$$H_{\rho }^{\alpha }({\mathbb{R}}^n)=\{u\in L^2({\mathbb{R}}^n)/\Vert u{\Vert }_{\rho }^2<\infty \}.$$

$$\Vert u{\Vert }_{\alpha }\le C\Vert u{\Vert }_{\rho }.$$

$$\Vert u{\Vert }_{L^q}\le {\mathfrak{C}}_q\Vert u{\Vert }_{\rho },\text{mbox}foreveryu\in H_{\rho }^{\alpha }({\mathbb{R}}^n)\text{mbox}andq\in [2,2_{\alpha }^{\ast }].$$

$$\underset{n\to \infty }{\lim }\underset{y\in {\mathbb{R}}^N}{\sup }{\int }_{B(y,R)}|u_n(x){|}^{2}dx=0,$$

$$(-\Delta {)}^{\alpha }u+u=\lambda u^q+u^{2_{\alpha }^{\ast }-1}\text{mbox}in{\mathbb{R}}^N,$$

$$u\in H^{\alpha }({\mathbb{R}}^N).$$

$$I(u)=\frac{1}{2}\Vert u{\Vert }_{\alpha }^2--\frac{\lambda }{q+1}{\int }_{{\mathbb{R}}^n}{u}_{+}^{q+1}dx-\frac{1}{2_{\alpha }^{\ast }}{\int }_{{\mathbb{R}}^n}{u}_{+}^{2_{\alpha }^{\ast }}dx.$$

$$C^{\ast }=\underset{u\in \mathcal{N}}{\inf }I(u)$$

$$C^{\ast }=\underset{u\in H^{\alpha }({\mathbb{R}}^n)\setminus \{0\}}{\inf }\underset{t\ge 0}{\sup }I(tu)=C,$$

$$0<C<\frac{\alpha }{n}{S}^{n/2\alpha },$$

$$(-\Delta {)}_{\rho }^{\alpha }u+u=\lambda u^q+u^{2_{\alpha }^{\ast }-1}\text{mbox}in{\mathbb{R}}^N,$$

$$u\in H^{\alpha }({\mathbb{R}}^N).$$

$$⟨u,\phi {⟩}_{\rho }=\lambda {\int }_{{\mathbb{R}}^n}{u}_{+}^q\phi dx+{\int }_{{\mathbb{R}}^n}{u}_{+}^{2_{\alpha }^{\ast }-1}\phi dx,\forall \phi \in H_{\rho }^{\alpha }({\mathbb{R}}^n).$$

$$⟨u,\phi {⟩}_{\rho }={\int }_{{\mathbb{R}}^n}{\int }_{B(0,\rho (x))}\frac{[u(x+z)-u(x)][\phi (x+z)-\phi (x)]}{|z{|}^{n+2\alpha }}dzdx+{\int }_{{\mathbb{R}}^n}u\phi dx$$

Full text

A critical nonlinear elliptic equation with non local regional diffusion

César E. Torres Ledesma

Departamento de Matemáticas,

Universidad Nacional de Trujillo,

Av. Juan Pablo II s/n. Trujillo-Perú

ctl_576@yahoo.es

Abstract

In this article we are interested in the nonlocal regional Schrödinger equation with critical exponent

[TABLE]

where $\epsilon$ is a small positive parameter, $\alpha\in(0,1)$, $q\in(1,2_{\alpha}^{*}-1)$, $2_{\alpha}^{*}=\frac{2N}{N-2\alpha}$ is the critical Sobolev exponent, $\lambda>0$ is a parameter and $(-\Delta)_{\rho}^{\alpha}$ is a variational version of the regional laplacian, whose range of scope is a ball with radius $\rho(x)>0$. We study the existence of a ground state and we analyze the behavior of semi-classical solutions as $\varepsilon\to 0$.

1. Introduction

In the present paper, we consider the existence and concentration phenomena of solutions to the nonlinear Schrödinger equation with non local regional diffusion

[TABLE]

where $\epsilon$ is a small positive parameter, $\alpha\in(0,1)$, $q\in(1,2_{\alpha}^{*}-1)$, $2_{\alpha}^{*}=\frac{2N}{N-2\alpha}$ is the critical Sobolev exponent, $\lambda>0$ is a parameter and the operator $(-\Delta)_{\rho}^{\alpha}$ is a variational version of the non-local regional laplacian, with range of scope determined by the positive function $\rho\in C(\mathbb{R}^{n},\mathbb{R}^{+})$, which is defined as

[TABLE]

Now we make precise assumptions on the scope function $\rho$ we assume $\rho\in C(\mathbb{R}^{n},\mathbb{R}^{+})$ and it satisfies the following hypotheses:

• $(\rho_{1})$

There are numbers $0<\rho_{0}<\rho_{\infty}\leq\infty$ such that

[TABLE]

• $(\rho_{2})$

For any $x_{0}\in\mathbb{R}^{n}$, the equation

[TABLE]

defines an $(n-1)$-dimensional surface of class $C^{1}$ in $\mathbb{R}^{n}$.

• $(\rho_{3})$

In case $\rho_{\infty}=\infty$ we further assume that there exists $a\in(0,1)$ such that

[TABLE]

2. Preliminaries

For any $\alpha\in(0,1)$, the fractional Sobolev space $H^{\alpha}(\mathbb{R}^{N})$ is defined by

[TABLE]

endowed with the norm

[TABLE]

For the reader’s convenience, we review the main embedding result for $H^{\alpha}(\mathbb{R}^{N})$.

Lemma 2.1**.**

[2]** Let $\alpha\in(0,1)$ such that $2\alpha<N$. Then there exist a constant $\mathfrak{C}=\mathfrak{C}(N,\alpha)>0$, such that

[TABLE]

for every $u\in H^{\alpha}(\mathbb{R}^{N})$. Moreover, the embedding $H^{\alpha}(\mathbb{R}^{N})\subset L^{p}(\mathbb{R}^{N})$ is continuous for any $p\in[2,2_{\alpha}^{*}]$ and is locally compact whenever $p\in[2,2_{\alpha}^{*})$.

Furthermore, we introduce the homogeneous fractional Sobolev space

[TABLE]

where

[TABLE]

Now we consider the best Sobolev constant $S$ as follows:

[TABLE]

According to [1], $S$ is attained by the function $u_{0}(x)$ given by

[TABLE]

where $c\in\mathbb{R}\setminus\{0\}$, $\theta>0$ and $x_{0}\in\mathbb{R}^{N}$ are fixed constants. For any $\epsilon>0$ and $x\in\mathbb{R}^{N}$, let

[TABLE]

which is solution of the problem

[TABLE]

Given a function $\rho$ as above, we define

[TABLE]

and the space

[TABLE]

This space is very natural for the study of our problem. Furthermore, we have the following result

Proposition 2.1**.**

[3]** If $\rho$ satisfies ($H_{1}$) there exists a constant $C=C(n,\alpha,\rho_{0})>0$ such that

[TABLE]

Remark 2.1**.**

By Proposition 2.1 we have that $H_{\rho}^{\alpha}(\mathbb{R}^{n})\hookrightarrow H^{\alpha}(\mathbb{R}^{n})$ is continuous and then, by Theorem 2.1, we have that $H_{\rho}^{\alpha}(\mathbb{R}^{n})\hookrightarrow L^{q}(\mathbb{R}^{n})$ is continuous for any $q\in[2,2_{\alpha}^{*}]$, and there exists $\mathfrak{C}_{q}>0$ such that

[TABLE]

Furthermore $H_{\rho}^{\alpha}(\mathbb{R}^{n})\hookrightarrow L_{loc}^{q}(\mathbb{R}^{n})$ is compact for any $q\in[2,2_{\alpha}^{*})$.

Remark 2.2**.**

Since $\|u\|_{\rho}\leq\|u\|_{\alpha}$, under the condition $(\rho_{1})$ Proposition 2.1 implies $\|\cdot\|_{\rho}$ and $\|\cdot\|$ are equivalent norms in $H^{\alpha}(\mathbb{R}^{n})$.

Lemma 2.2**.**

[3]** Let $n\geq 2$. Assume that $\{u_{n}\}_{n\in\mathbb{N}}$ is bounded in $H_{\rho}^{\alpha}(\mathbb{R}^{N})$ and it satisfies

[TABLE]

where $R>0$. Then $u_{n}\to 0$ in $L^{q}(\mathbb{R}^{N})$ for $q\in(2,2_{\alpha}^{*})$.

Now, we consider the limit equations. namely

[TABLE]

This equation was study by Shang, Zhang and Yang in [5]. The solution of problem (2.4) are the critical point of the functional

[TABLE]

Furthermore, they studied the existence of ground state solutions to (2.4), namely, function in $\mathcal{N}=\{u\in H^{\alpha}(\mathbb{R}^{n})\setminus\{0\}:I^{\prime}(u)u=0\}$ such that

[TABLE]

is achieved and they got the following characterization:

[TABLE]

where $C=\inf_{\gamma\in\Gamma}\sup_{t\in[0,1]}I(\gamma(t))>0$ is the mountain pass critical value.

On the other hand, assuming that if $q\in(1,2_{\alpha}^{*}-1)$, they have proved that there exists $\lambda_{0}>0$ such that for all $\lambda\geq\lambda_{0}$, the critical value satisfies

[TABLE]

where $S$ is the best Sobolev constant given by (2.1) and problem (2.4) has a nontrivial ground state solution.

3. Ground state

Let $\epsilon=1$ and consider the following problem

[TABLE]

We recall that $u\in H_{\rho}^{\alpha}(\mathbb{R}^{n})\setminus\{0\}$ is a solution of (1.1) if $u(x)\geq 0$ and

[TABLE]

where

[TABLE]

and $u_{+}=\max\{u,0\}$.

In order to find solution for problem (3.1), we consider the functional $I_{\rho}:H_{\rho}^{\alpha}(\mathbb{R}^{n})\to\mathbb{R}$ defined as

[TABLE]

which is well defined and belongs to $C^{1}(H_{\rho}^{\alpha}(\mathbb{R}^{n}),\mathbb{R}^{n})$ with Fréchet derivative

[TABLE]

.

Now, we start recalling that the functional $I_{\rho}$ satisfies the mountain pass geometry conditions

Lemma 3.1**.**

The functional $I_{\rho_{\epsilon}}$ satisfies the following conditions:

1. (1)

There exist $\beta,\delta>0$, such that $I_{\rho}(u)\geq\beta$ if $\|u\|_{\rho}=\delta$. 2. (2)

There exists an $e\in H_{\rho}^{\alpha}(\mathbb{R}^{n})$ with $\|e\|_{\rho}$ such that $I_{\rho}(e)<0$.

From the previous Lemma, by using the mountain pass theorem without $(PS)_{c}$ condition ([6]) it follows that there exists a $(PS)_{C_{\rho}}$ sequence $\{u_{k}\}\subset H_{\rho}^{\alpha}(\mathbb{R}^{n})$ such that

[TABLE]

where

[TABLE]

and $\Gamma_{\rho}=\{\gamma\in C([0,1],H_{\rho}^{\alpha}(\mathbb{R}^{n})):\;\;\gamma(0)=0,I_{\rho}(\gamma(1))<0\}$.

Also, we define

[TABLE]

where

[TABLE]

Lemma 3.2**.**

$C^{*}>0$**

Proof.

Suppose by contradiction that $C^{*}=0$. Then there exists $u_{n}\in\mathcal{N}_{\rho}$. Then there exists $u_{k}\in\mathcal{N}_{\rho}$ such that

[TABLE]

But, since $u_{k}\in\mathcal{N}_{\rho}$ we have

[TABLE]

So

[TABLE]

Therefore

[TABLE]

On the other hand, since $0\neq u_{k}\in\mathcal{N}_{\rho}$, then by Remark 2.1 we have

[TABLE]

Combining this inequality with (3.4) we get a contradiction. This proves the Lemma. ∎

Lemma 3.3**.**

Let $C_{\rho}$ given by (3.2) and (3.3). Then

[TABLE]

.

Proof.

We note that our nonlinearity $f(t)=\lambda t^{q}+t^{2_{\alpha}^{*}-1}$, $t>0$, is a $C^{1}$ function and $\frac{f(t)}{t}$ is a strictly increasing function. Let $u\in\mathcal{N}_{\rho}$, then we can show that the function $h(t)=I_{\rho}(tu)$, $t\neq 0$ has a unique maximum point $t_{u}$ such that

[TABLE]

Furthermore, we can show that $t_{u}=1$. Now choose $t_{0}\in\mathbb{R}$ and $\tilde{u}=t_{0}u$ such that $I_{\rho}(\tilde{u})<0$. Then $\gamma(t)=t\tilde{u}\in\Gamma_{\rho}$ then $I_{\rho}(u)\geq C_{\rho}$, that is,

[TABLE]

On the other hand, let $\{u_{k}\}$ be the $(PS)_{C_{\rho}}$ sequence satisfying (3.2) and (3.3). Since $\{u_{k}\}$ is bounded, then $I^{\prime}_{\rho}(u_{k})u_{k}\to 0$ as $k\to\infty$, moreover, from (3.5) for each $k$, there is a unique $t_{k}\in\mathbb{R}^{+}$ such that

[TABLE]

Hence $t_{k}u_{k}\in\mathcal{N}_{\rho}$.

Now we note that by (3.7), we have

[TABLE]

So $t_{k}$ does not converge to [math]; otherwise, since $\{u_{k}\}$ is bounded in $H_{\rho}^{\alpha}(\mathbb{R}^{n})$, using (3.8) we obtain

[TABLE]

which is imposible since $C_{\rho}>0$. Also, $t_{k}$ does not go to infinity. In fact, by (3.8) we get

[TABLE]

So, assuming that $t_{k}\to\infty$, as $k\to\infty$, by (3.9) we get that

[TABLE]

Then using interpolation inequality it follows that

[TABLE]

Moreover, since $I^{\prime}_{\rho}(u_{k})u_{k}\to 0$, as $k\to\infty$, we obtain,

[TABLE]

So, by (3.10) and (3.11), we conclude that $\|u_{k}\|_{\rho}^{2}\to 0$, as $k\to\infty$, contradicting $C_{\rho}>0$. Hence, the sequence $\{t_{k}\}$ is bounded and there exists $t_{0}\in(0,\infty)$ such that (up to subsequence) $t_{k}\to t_{0}$ as $k\to\infty$.

Now, from (3.8) and (3.11) we obtain

[TABLE]

From where $t_{0}=1$, namely

[TABLE]

Therefore, by (3.13) and recalling that $t_{k}u_{k}\in\mathcal{N}_{\rho}$, we get

[TABLE]

Passing to the limit we obtain $C^{*}\leq C_{\rho}$.

On the other hand, By the previous comments, for any $u\in H_{\rho}^{\alpha}(\mathbb{R}^{n})\setminus\{0\}$ there is a unique $t_{u}=t(u)>0$ such that $t_{u}u\in\mathcal{N}_{\rho},$ then

[TABLE]

Moreover, for any $u\in\mathcal{N}_{\rho}$, we have

[TABLE]

so

[TABLE]

∎

Remark 3.1**.**

Suppose that $(\rho_{1})$ holds and without loss of generality take $\epsilon=1$, then

[TABLE]

In fact, let $u$ be a critical point of $I$ with critical value $C$ and for any $y\in\mathbb{R}^{n}$, define $u_{y}(x)=u(x+y)$. Then for any $t>0$ we have

[TABLE]

By Lemma 3.2, we can take $t^{*}>0$ such that $t^{*}u_{y}\in\mathcal{N}_{\rho}$ and

[TABLE]

consequently $C>I_{\rho}(t^{*}u_{y})\geq C_{\rho}$. In the same way we can show that

[TABLE]

Remark 3.2**.**

According to (2.5) and by Remark 3.1 we have

[TABLE]

Lemma 3.4**.**

Suppose that $C_{\rho}<C$. Then there are $\nu,R>0$ such that

[TABLE]

Proof.

By Lemma 3.3, for each $k\in\mathcal{N}$, there exist $t_{k}\subset\mathbb{R}$ such that $t_{k}\to 1$ and

[TABLE]

Furthermore

[TABLE]

Now we consider two cases, namely, when $\rho_{\infty}=\infty$ and $\rho_{\infty}<\infty$. In the first case, by $(\rho_{1})$, for every $\epsilon>0$ there exist $R>0$ such that

[TABLE]

Then

[TABLE]

If $\{\tau_{k}\}$ is the bounded real sequence seven by

[TABLE]

we obtain

[TABLE]

If

[TABLE]

from (3.14), (3.15) and (3.16) yield

[TABLE]

which is a contradiction with Remark 3.1.

Now we analyze the case $\rho_{\infty}<+\infty$. In this case we compare the functionals $I_{\rho}$ and $I_{\rho_{\infty}}$ writing

[TABLE]

By hypothesis ($\rho_{1}$), for any $\epsilon>0$ there is $R>0$ such that

[TABLE]

Then, we obtain

[TABLE]

where

[TABLE]

Proceeding as before, by (3.16) we get $C_{\rho}\geq C_{\rho_{\infty}}$, which is a contradiction with Remark 3.1. ∎

The next result shows the existence of positive solution to (1.1) with $\epsilon=1$.

Theorem 3.1**.**

Suppose that $\lambda>0$, $q>1$ and $(\rho_{1})$ hold. Then, problem (1.1) with $\epsilon=1$ possesses a positive ground state solution.

Proof.

Using (3.2), Lemma 3.4 and the Sobolev embedding we have

[TABLE]

which proves that $u\neq 0$. Furthermore, by standard arguments we have

[TABLE]

so choosing $\varphi=u_{-}(x)\max\{-u(x),0\}$ and noting that for $x,z\in\mathbb{R}^{n}$ we have

[TABLE]

we can conclude that $\|u_{-}\|_{\rho}=0$, thus $u(x)\geq 0$ a.e.$x\in\mathbb{R}^{n}$.

Moreover, from Lemma 3.3

[TABLE]

On the other hand, we have

[TABLE]

Applying Fatou’s Lemma to last inequality, we obtain

[TABLE]

From (3.18) and (3.19) we obtain

[TABLE]

and hence $u$ is a least energy solution and the proof is finished. ∎

Proof of Theorem LABEL:. In what follows, we denote by $\{u_{k}\}\subset H_{\rho_{\epsilon}}^{\alpha}(\mathbb{R}^{N})$ a sequence satisfying

[TABLE]

If $u_{k}\rightharpoonup 0$ in $H_{\rho_{\epsilon}}^{\alpha}(\mathbb{R}^{N})$, then

[TABLE]

By $(\rho_{1})$, we obtain

[TABLE]

where

[TABLE]

Now we know that there exists a bounded sequence $\{\tau_{k}\}$ such that

[TABLE]

where

[TABLE]

Thus,

[TABLE]

Taking the limit as $k\to\infty$, and after $\delta\to 0$, we find

[TABLE]

A standard argument shows that

[TABLE]

Therefore, if there is $\epsilon_{k}\to 0$ such that the $(PS)_{C_{\rho_{\epsilon_{k}}}}$ sequence has weak limit equal to zero, we must have

[TABLE]

leading to

[TABLE]

which contradicts Remark 3.1. This proves that the weak limit is non trivial for $\epsilon>0$ small enough and standard arguments show that its energy is equal to $C_{\rho_{\epsilon}}$, showing the desired result. $\Box$

4. Concentration Behaviour

In this section we make a preliminary analysis of the asymptotic behavior of the functional associated to equation (1.1) when $\epsilon\to 0$. As is point up in [3], the scope function $\rho$, that describes the size of the ball of the influential region of the non-local operator, plays a key role in deciding the concentration point of ground states of the equation. Even though, at a first sight, the minimum point of $\rho$ seems to be the concentration point, there is a non-local effect that needs to be taken in account. We define the concentration function

[TABLE]

where the sets $\mathcal{C}^{+}({x})$ and $\mathcal{C}^{-}({x})$ are defined as follows

[TABLE]

and

[TABLE]

We start with some basic properties of the function $\mathcal{H}$.

Lemma 4.1**.**

[3]** Assuming $\rho$ satisfies $(\rho_{1})-(\rho_{3})$, the function $\mathcal{H}$ is continuous and

[TABLE]

Moreover, there exists $x_{0}\in\mathbb{R}^{n}$ such that

[TABLE]

Along this section we will consider a sequence of functions $\{w_{m}\}\subset H^{\alpha}(\mathbb{R}^{n})$ such that $\|w_{m}-w\|_{L^{2}(\mathbb{R}^{2})}\to 0$, where $w\in H^{\alpha}(\mathbb{R}^{n})$. We will also consider sequences $\{z_{m}\}\subset\mathbb{R}^{n}$ and $\{\epsilon_{m}\}\subset\mathbb{R}$ and assume that $\epsilon_{m}\to 0$ as $m\to\infty$. We define $\overline{\rho}_{m}$ as

[TABLE]

and the functional $I_{\bar{\rho}_{m}}$ defined as

[TABLE]

We will be considering the functionals

[TABLE]

As in [3] we have the following key Theorem

Theorem 4.1**.**

Under hypotheses $(\rho_{1})-(\rho_{3})$, we assume as above that $w_{m},w\in H^{\alpha}(\mathbb{R}^{n})$ are such that $\|w_{m}-w\|_{L^{2}(\mathbb{R}^{2})}\to 0$ and $\epsilon_{m}\to 0$, as $m\to\infty$. Then we have:

i) If $\epsilon_{m}z_{m}\to\bar{x}$ then

[TABLE]

ii) If $|\epsilon_{m}|z_{{m}}\to\infty$ then

[TABLE]

Now, we rescaling equation (1.1), for this purpose we define $\rho_{\epsilon}(x)=\frac{1}{\epsilon}\rho(\epsilon x)$ and consider the rescaled equation

[TABLE]

and we see that $u$ is a weak solution of (1.1) if and only if $v_{\epsilon}(x)=u(\epsilon x)$ is a weak solution of (4.8).

In order to study equations (1.1) and (4.8), we consider the functional $I_{\rho_{\epsilon}}$ on the $\epsilon$-dependent Hilbert space $H_{\rho_{\epsilon}}^{\alpha}(\mathbb{R}^{n})$ with inner product $\langle\cdot,\cdot\rangle_{\rho_{\epsilon}}$. The functional $I_{\rho_{\epsilon}}$ is of class $C^{1}$ in $H_{\rho_{\epsilon}}^{\alpha}(\mathbb{R}^{n})$ and the critical points of $I_{\rho_{\epsilon}}$ are the weak solutions of (4.8). We further introduce

[TABLE]

[TABLE]

and the mountain pass minimax value

[TABLE]

From Lemma 3.3 we also have

[TABLE]

For comparison purposes we consider the functional $I$, whose critical points are the solutions of (2.4). We also consider the critical value $C$ that satisfies

[TABLE]

Now we start the proof of Theorem LABEL:Itm2 with some preliminary lemmas.

Lemma 4.2**.**

Suppose ($\rho_{1}$) holds. Then

[TABLE]

Proof.

Since we obviously have

[TABLE]

for all $u\in H_{\rho_{\epsilon}}^{\alpha}(\mathbb{R}^{n})$, then we have $I_{\rho_{\epsilon}}(u)\leq I(u)$ and therefore

[TABLE]

On the other hand, by ($\rho_{1}$) we have $\rho(\epsilon x)\geq\rho_{0}$ for all $x\in\mathbb{R}^{n}$ then

[TABLE]

By standard arguments we can show that

[TABLE]

Thus

[TABLE]

Therefore, by (4.11) and (4.12) we obtain (4.10). ∎

Lemma 4.3**.**

There are a $\epsilon_{0}>0$, a family $y_{\epsilon}\subset\mathbb{R}^{n}$, constants $\beta,R>0$ such that

[TABLE]

Proof.

By contradiction, there is a sequence $\epsilon_{m}\to 0$ such that for all $R>0$

[TABLE]

Using the following notation $v_{m}=v_{\epsilon_{m}}$ and $C_{\rho_{m}}=C_{\rho_{\epsilon_{m}}}$, by Lemma 2.2

[TABLE]

Furthermore, since $I^{\prime}_{\rho_{m}}(v_{m})v_{m}=0$ then

[TABLE]

Let $l\geq 0$ be such that

[TABLE]

Now, since $I_{\rho_{m}}(v_{m})=C_{\rho_{m}}$, we obtain

[TABLE]

then by Lemma 4.2

[TABLE]

hence $l>0$: Now, using the definition of the Sobolev constant $S$ and Remark 2.1, we have

[TABLE]

Therefore, by (4.14) and taking the limit in the above inequality as $m\to\infty$ we achieved that

[TABLE]

which is a contradiction with (2.5). ∎

Now let

[TABLE]

then by (4.13),

[TABLE]

To continue, we consider the rescaled scope function $\overline{\rho}_{\epsilon}$, as defined in (4.4),

[TABLE]

and then $w_{\epsilon}$ satisfies the equation

[TABLE]

Now we prove the convergence of $w_{\epsilon}$ as $\epsilon\to 0$.

Lemma 4.4**.**

For every sequence $\{\epsilon_{m}\}$ there is a subsequence, we keep calling the same, so that $w_{\epsilon_{m}}=w_{m}\to w$ in $H^{\alpha}(\mathbb{R}^{n})$, when $m\to\infty$, where $w$ is a solution of (2.4).

Proof.

Note that

[TABLE]

By Lemma 4.2 we obtain that

[TABLE]

On the other hand, by Fatou’s Lemma and the weak convergence of $\{w_{m}\}$, we get

[TABLE]

So, by (4.18) and (4.19), $\lim_{m\to\infty}\Lambda_{m}=C$, from where we get

[TABLE]

∎

We are now in a position to complete the proof of our second main theorem.

Proof of Theorem LABEL: We first obtain an upper bound for the critical values $C_{\rho_{\epsilon_{m}}}=C_{m}$, for the sequence $\{\epsilon_{m}\}$ given in Lemma 4.4. Next we consider the scope function

[TABLE]

where $x_{0}$ is a global minimum point of $\mathcal{H}$, see Lemma 4.1. To continue, we consider the function $w_{m}=w_{\epsilon_{m}}$ as given in (4.15) and let $t_{m}>0$ such that $t_{m}w_{m}\in\mathcal{N}_{\tilde{\rho}_{m}}$. According to Lemma 4.4, $\{w_{m}\}$ converges to $w\in\mathcal{N}$, then $t_{m}\to 1$ and $t_{m}w_{m}\to w$.

Now we apply Theorem 4.1 to obtain that

[TABLE]

We have used part (i) of Theorem 4.1 with $z_{m}=x_{0}/{\epsilon_{m}}$.

On the other hand, since $w_{m}\in H^{\alpha}(\mathbb{R}^{n})$ is a critical point of $I_{\bar{\rho}_{m}}$, we have that

[TABLE]

We write $y_{m}=y_{\epsilon_{m}}$. If $\epsilon_{m}|y_{m}|\to\infty$, then we may apply part (ii) of Theorem 4.1 with $z_{m}=y_{m}$ in (4.21) and obtain that

[TABLE]

which contradicts (4.20). We conclude then, that $\{\epsilon_{m}y_{m}\}$ is bounded and that, for a subsequence, $\epsilon_{m}y_{m}\to\bar{x}$, for some $\bar{x}\in\mathbb{R}^{n}$. Now we apply Theorem 4.1 again, but now part (i) with $z_{m}=y_{m}$ in (4.21), and we obtain that

[TABLE]

From (4.20) and (4.22) we finally get that

[TABLE]

and taking the limit as $m\to\infty$, we get

[TABLE]

completing the proof of the theorem. $\Box$