Competing power problem involving the half Laplacian

Abstract

We prove the existence and the limit profile of the least energy solution of a half Laplacian equation with competing powers.

Keywords

Fractional Laplacian existence blow-up analysis

1. Introduction

The motivation for the fractional Laplacian originates from probability and mathematical finance as the infinitesimal generators of stable Lévy processes [6–8] which play an important role in stochastic modelling in applied sciences and in financial mathematics. These operators also arise from mechanics in elasto–statics as Signorini obstacle problem in linear elasticity [10] and also from fluid mechanics as quasi-geostrophic fractional Navier–Stokes equation, see [11]. Apart from its significance in mathematics, the fractional Laplacian appears in theoretical physics in connection to the problem of stability of the matter.

Consider the problem $\begin{matrix} (1.1) & \{\begin{matrix} {(- Δ)}^{s} u = λ f (u) & in Ω, \\ u = 0 & in R^{N} ∖ Ω \end{matrix} \end{matrix}$ where Ω is a smooth bounded domain in $R^{N}$ , $N > 2 s$ , $s \in (0, 1)$ and λ is a parameter. It is well known by the Pohozaev identity [23], that if Ω is star-shaped, there exists no bounded positive non-trivial solution of (1.1) when $f (u) = u^{\frac{N + 2 s}{N - 2 s}}$ and $λ = 1$ . Bisci–Rădulescu [20] proved that if $f (u)$ is a sub-linear nonlinearity, then there exists an open interval $Λ \subset (0, \infty)$ and a $κ > 0$ such that for every $λ \in Λ$ , (1.1) admits at least two distinct, non-trivial weak solutions in $X_{0} = {u \in H^{s} (R^{N}) ∣ u = 0 in R^{N} ∖ Ω}$ whose $X_{0}$ -norms are less than κ. In [26], Servadei–Valdinoci studied existence of weak positive solutions of the Brezis–Nirenberg problem for the (1.1) with nonlinearity of the form $f (u) = u^{\frac{N + 2 s}{N - 2 s}} + α u$ , with $α > 0$ and $λ = 1$ .

Now we consider the case $N = 1$ . The critical exponent for ${(- Δ)}^{s}$ in one dimension is $\frac{2}{1 - 2 s}$ and when $s = 1 / 2$ the critical exponent is infinity. In one dimension, the quasi-relativistic approach to bounded states of the Schrödinger equation leads to the fractional operator corresponding to $s = \frac{1}{2}$ , $\sqrt{- \frac{d^{2}}{d x^{2}} + m}$ which is also known as the Klein–Gordon square root operator [2–5,17,28]. In the case of $m = 0$ , we obtain ${(- Δ)}^{1 / 2} = \sqrt{- \frac{d^{2}}{d x^{2}}}$ , which is called the ultra-relativistic operator. In this paper we consider, the following competing power problem $\begin{matrix} (1.2) & \{\begin{matrix} {(- Δ)}^{1 / 2} u = u^{p} - u^{p - 1} & in I, \\ u > 0 & in I, \\ u = 0 & in R ∖ I \end{matrix} \end{matrix}$ where $p > 2$ , $I = (- 1, 1)$ and $\begin{matrix} {(- Δ)}^{1 / 2} u (x) = \frac{1}{π} \int_{R} \frac{u (x) - u (y)}{| x - y |^{2}} d y for all x \in R . \end{matrix}$ The main goal of the paper is to existence of solution and the limiting profile of the least energy solution for (1.2) when the parameter $p \to \infty$ .

Define the Sobolev space $\begin{matrix} {\tilde{H}}^{1 / 2} (I) = {u \in H^{1 / 2} (R) ∣ u = 0 in R ∖ I} \end{matrix}$ and the norm $\begin{matrix} (1.3) & ‖ u ‖_{{\tilde{H}}^{1 / 2} (I)} = \int_{R} {| {(- Δ)}^{1 / 4} u |}^{2} d x = \frac{1}{2 π} \int_{R} \int_{R} \frac{| u (x) - u (y) |^{2}}{| x - y |^{2}} d x d y . \end{matrix}$

Definition 1.1 (Weak and classical solution).

We will call $u \in {\tilde{H}}^{1 / 2} (I)$ is a weak solution of (1.2) if $u > 0$ in I and for every $ϕ \in {\tilde{H}}^{1 / 2} (I)$ , we have $\begin{matrix} \frac{1}{2 π} \int_{R} \int_{R} \frac{(u (x) - u (y)) (ϕ (x) - ϕ (y))}{| x - y |^{2}} d x d y = \int_{I} u^{p} ϕ d x - \int_{I} u^{p - 1} ϕ d x . \end{matrix}$ Moreover, $u \in C^{1 / 2} (R) \cap C^{2} (I)$ is a classical solution of (1.2) if $u > 0$ in I, ${(- Δ)}^{1 / 2} u$ is defined pointwise for all $x \in I$ and (1.2) is satisfied pointwise.

Consider the following Liouville equation $\begin{matrix} (1.4) & {(- Δ)}^{1 / 2} W = e^{W}; \int_{R} e^{W} < + \infty . \end{matrix}$ Suppose $\begin{matrix} L_{1} (R) = {u \in L_{loc}^{1} (R) : \int_{R} \frac{| u (x) |}{(1 + | x |^{2})} d x < \infty} \end{matrix}$ and $W \in L_{1}$ and W is a solution to (1.4) in the sense that $\begin{matrix} (1.5) & \int_{R} W {(- Δ)}^{1 / 2} ϕ d x = \int_{R} e^{W} ϕ d x, \forall ϕ \in S \end{matrix}$ where $S$ denotes the Schwartz class function on $R$ . Then W is given by $\begin{matrix} (1.6) & W_{λ, a} (x) = log \frac{2 λ}{λ^{2} + | x - a |^{2}}, a \in R; λ \in R^{+} \end{matrix}$ see [12, Theorem 1.8].

In two dimensions, the Dirichlet problem $- Δ u = u^{p}$ was studied by Ren–Wei [22] in a star-shaped domain Ω for the least energy solution. It was extended to the general two-dimensional domain and a precise asymptotic behavior for the least energy solution was obtained in [1] as $p \to \infty$ . For the half Laplacian, the analogous problem was studied in an interval by Santra [25]. Ambrosio [5] studied the multiplicity and concentration of solutions for a fractional magnetic Schrödinger equation in $R$ with exponential critical growth. In [29], the authors studied the non-existence of positive solutions for the problem ${(- Δ)}^{s} u = u^{p}$ in Ω with $N ⩾ 1$ , $p > 0$ , $s \in (0, 1)$ and $Ω \subset R^{N - 1} \times [0, \infty)$ is an unbounded domain.

For the zero Dirichlet boundary condition, a similar type of problem $\begin{matrix} - ε^{2} Δ u = u^{p} - u^{q} in Ω \end{matrix}$ was studied in [13,14] when $ε \to 0$ and $1 < q < p < \frac{N + 2}{N - 2}$ . To our knowledge, there exists no result for the problem (1.2) when the two powers in the non-linearity are competing with each other even in the local case.

Theorem 1.1.
For each $p > 2$ , there exists a even least energy classical solution $u_{p}$ to ( 1.2 ) which is a decreasing on $(0, 1)$ .
Theorem 1.2.
Let $‖ u_{p} ‖_{\infty} = u_{p} (0)$ and define $W_{p}$ as $\begin{matrix} (1.7) & W_{p} (x) = \frac{p}{‖ u_{p} ‖_{\infty}} [u_{p} (\frac{x}{p ‖ u_{p} ‖_{\infty}^{p - 1}}) - u_{p} (0)] . \end{matrix}$ For any $δ \in (0, 1)$ , there exists a subsequence of $W_{p}$ which converges to W in $C_{loc}^{δ} (R)$ as $p \to \infty$ where W can be expressed as $\begin{matrix} (1.8) & W (x) = log \frac{4 {(1 - γ)}^{2}}{(4 {(1 - γ)}^{2} + x^{2})} \end{matrix}$ for some $0 < γ < 1$ .
Theorem 1.3.
For $p ≫ 1$ , there exists a $C > 0$ independent of p such that $\begin{matrix} 1 ⩽ ‖ u_{p} ‖_{\infty} ⩽ C . \end{matrix}$ Moreover, $\begin{matrix} 2 π (1 - γ) ⩽ \underset{p \to \infty}{lim inf} p \int_{I} u_{p}^{p} d x ⩽ \underset{p \to \infty}{lim sup} p \int_{I} u_{p}^{p} d x ⩽ C . \end{matrix}$

2. Existence of positive solutions

By [7], the Green function of ${(- Δ)}^{1 / 2}$ in $I = (- 1, 1)$ can be written as $\begin{matrix} (2.1) & G (x, y) = \frac{1}{π} log \frac{1}{| x - y |} + \frac{1}{π} log (\sqrt{(1 - | x |^{2}) (1 - | y |^{2})} + 1 - x y) \end{matrix}$ where $H (x, y) = \frac{1}{π} log (\sqrt{(1 - | x |^{2}) (1 - | y |^{2})} + 1 - x y)$ is the regular part of ${(- Δ)}^{1 / 2}$ of the Green function. Moreover, $H (x, y) ⩽ C$ for some $C > 0$ . Here $\frac{1}{π} log \frac{1}{| x - y |}$ is the fundamental solution of ${(- Δ)}^{1 / 2}$ .

Lemma 2.1.

Any weak solution to ( 1.2 ) is bounded in $R$ and hence belongs to $C^{1 / 2} (R)$ .

Proof.

By the Green function (2.1) we obtain $\begin{array}{rcl} u_{p} (x) & = & \int_{I} G (x, y) (u^{p} - u^{p - 1}) d y ⩽ \frac{1}{π} \int_{I} u^{p} log \frac{1}{| x - y |} d y + C \int_{I} u^{p} \\ ⩽ & \frac{1}{π} \int_{I \cap | x - y | < 1 / 2} u^{p} log (\frac{1}{| x - y |}) d y + \frac{1}{π} \int_{I \cap | x - y | > 1 / 2} u^{p} log (\frac{1}{| x - y |}) d y \\ (2.2) & + C \int_{I} u^{p} d y . \end{array}$ But $\int_{I} u^{p + 1} d y ⩽ C$ and $\int_{I} u^{p} d y ⩽ C$ which implies $\begin{array}{rcl} \int_{I \cap | x - y | < 1 / 2} u^{p} log (\frac{1}{| x - y |}) d y & ⩽ & {(\int_{I} u^{p + 1})}^{p / p + 1} {(\int_{I} {(log (\frac{1}{| x - y |}))}^{p + 1})}^{1 / p + 1} \\ ⩽ & C {(\int_{I} {(log (\frac{1}{| x - y |}))}^{p + 1})}^{1 / p + 1} \end{array}$ and $\begin{array}{rcl} \int_{I \cap | x - y | > 1 / 2} u^{p} log (\frac{1}{| x - y |}) d y & ⩽ & log 2 (\int_{I} u^{p}) . \end{array}$ Also we have $\begin{array}{rcl} \int_{I} {(log (\frac{1}{| x - y |}))}^{p + 1} d y & ⩽ & \int_{I} {| log | z | |}^{p + 1} d z = 2 \int_{0}^{\infty} t^{p + 1} e^{- t} d t \\ = & 2 Γ (p + 2) . \end{array}$ This implies that $u_{p} ⩽ C$ for some $C > 0$ which depends on p. Hence by Ros-Oton–Serra [24], $u \in C^{1 / 2} (R) \cap C^{2} (I)$ . □

Lemma 2.2.

There exist a classical mountain pass solution to ( 1.2 ).

Proof.

Define $u^{\pm} = max {\pm u, 0}$ and we modify the problem (1.2) to $\begin{matrix} (2.3) & \{\begin{matrix} {(- Δ)}^{1 / 2} u = {(u^{+})}^{p} - {(u^{+})}^{p - 1} & in I, \\ u = 0 & in R ∖ I . \end{matrix} \end{matrix}$ The associated functional to the problem (2.3); $Φ_{p} : {\tilde{H}}^{1 / 2} (I) \to R$ by $\begin{matrix} (2.4) & Φ_{p} (u) = \frac{1}{2} \int_{R} {| {(- Δ)}^{1 / 4} u |}^{2} d x - \frac{1}{p + 1} \int_{I} {(u^{+})}^{p + 1} d x + \frac{1}{p} \int_{I} {(u^{+})}^{p} d x . \end{matrix}$ Then $Φ_{p} \in C^{2} ({\tilde{H}}^{1 / 2} (I), R)$ and any critical point of (2.4) is a weak solution to (1.2). It is easy to show that any solution of (2.3) is non-negative, since if we multiply by $u^{-}$ and integrate by parts; we obtain $\int_{R} | {(- Δ)}^{1 / 4} u^{-} |^{2} = 0$ . Hence $u^{-} = 0$ and u is in fact a non-negative solution to (2.3). Since ${\tilde{H}}^{1 / 2} (I)$ is compactly embedded in $L^{q} (I)$ for any $q ⩾ 2$ , $Φ_{p}$ satisfies the Palais Smale condition and all the conditions of the mountain pass theorem and hence there exists a mountain pass solution $u_{p} > 0$ and a mountain pass critical value $\begin{matrix} 0 < c_{p} = inf_{γ \in Γ} max_{t \in [0, 1]} Φ_{p} (γ (t)) \end{matrix}$ where $\begin{matrix} Γ_{} = {γ \in C ([0, 1]; {\tilde{H}}^{1 / 2} (I)) : γ (0) = 0, γ (1) \neq 0, Φ_{p} (γ (1)) ⩽ 0} . \end{matrix}$ But since any weak solution of (1.2) is in $C^{1 / 2} (R)$ , $u_{p}$ is a classical solution to (1.2). □

We define the Nehari manifold $\begin{matrix} M_{p} = {u \in {\tilde{H}}^{1 / 2} (I) ∖ {0} : \int_{R} {| {(- Δ)}^{1 / 4} u |}^{2} = \int_{I} {(u^{+})}^{p + 1} - \int_{I} {(u^{+})}^{p}} . \end{matrix}$

Lemma 2.3.

For all $p > 1$ $\begin{matrix} c_{p} = inf_{γ \in Γ_{}} max_{t \in [0, 1]} Φ_{p} (γ (t)) = inf_{u \in M_{p}} Φ_{p} (u) = inf_{u \in {\tilde{H}}^{1 / 2} (I), u \neq 0} max_{t ⩾ 0} Φ_{p} (t u) . \end{matrix}$

Proof.

Let p be fixed. First note that $\begin{matrix} (2.5) & inf_{γ \in Γ} max_{t \in [0, 1]} Φ_{p} (γ (t)) ⩽ inf_{u \in {\tilde{H}}^{1 / 2} (I)} max_{t ⩾ 0} Φ_{p} (t u) . \end{matrix}$ We first claim that ${inf}_{u \in M_{p}} Φ_{p} (u) = {inf}_{u \in {\tilde{H}}^{1 / 2} (I)} {max}_{t ⩾ 0} Φ_{p} (t u)$ . Define $h (t) = Φ_{p} (t u)$ . We have $h (0) = 0$ , $h (t) > 0$ for small $t > 0$ and $h (t) < 0$ for $t > 0$ sufficiently large. Hence ${max}_{t \in [0, + \infty)} h (t)$ is achieved. Also note that $h^{'} (t) = 0$ implies $‖ u ‖_{{\tilde{H}}^{1 / 2} (I)}^{2} = g (t)$ where $\begin{matrix} g (t) = t^{p - 1} \int_{I} {(u^{+})}^{p + 1} - t^{p - 2} \int_{I} {(u^{+})}^{p} . \end{matrix}$ It is easy to see that g is an increasing function of t whenever $g (t) > 0$ . Thus there exists a unique t such that $‖ u ‖_{{\tilde{H}}^{1 / 2} (I)} = g (t)$ . Hence there exist a unique point $θ (u)$ such that $h^{'} (θ (u) u) = 0$ and $θ (u) u \in M_{p}$ . This implies that $M_{p}$ is radially homeomorphic to ${\tilde{H}}^{1 / 2} (I) ∖ {0}$ if we prove that $θ : {\tilde{H}}^{1 / 2} (I) ∖ {0} \to R^{+}$ is continuous. In order to do so let $u_{n} \to u$ in ${\tilde{H}}^{1 / 2} (I) ∖ {0}$ . Then $u_{n} \to u$ in ${\tilde{H}}^{1 / 2} (I)$ and $u_{n} \to u$ in $L^{r} (I)$ for all $r < \infty$ and $\begin{matrix} (2.6) & \int_{R} {| {(- Δ)}^{1 / 4} u_{n} |}^{2} = θ^{p - 1} (u_{n}) \int_{I} {(u_{n}^{+})}^{p + 1} - θ^{p - 2} (u_{n}) \int_{I} {(u_{n}^{+})}^{p} \end{matrix}$ which proves there exist constants $m > 0$ and $M > 0$ independent of n such that $m ⩽ θ (u_{n}) ⩽ M$ . By passing to the limit in (2.6) the whole sequence ${θ (u_{n})}$ converges as $u_{n}$ is convergent and hence $θ (u) = θ_{0}$ where $θ_{0} u \in M_{p}$ which proves our claim.

Next we claim that ${inf}_{γ \in Γ} {max}_{t \in [0, 1]} Φ_{p} (γ (t)) = {inf}_{u \in M_{p}} Φ_{p} (u)$ . It is easy to see that $\begin{matrix} inf_{γ \in Γ} max_{t \in [0, 1]} Φ_{p} (γ (t)) ⩾ inf_{u \in M_{p}} Φ_{p} (u) \end{matrix}$ by (2.5). It is enough to prove that any $γ \in Γ$ intersects $M_{p}$ . Note that $Φ_{p} (u) > 0$ for $‖ u ‖_{{\tilde{H}}^{1 / 2} (I)}$ sufficiently small and $Φ_{p} (γ (1)) < 0$ which implies the required result. □

Lemma 2.4.

The mountain pass solution of ( 1.2 ) is even and decreasing in $(0, 1)$ .

Proof.

Given a non-negative function $u \in {\tilde{H}}^{1 / 2} (I)$ , define $u^{*}$ as the symmetric rearrangement. Consequently, applying the symmetric rearrangement inequality for half Laplacian we have $\begin{matrix} \int_{R^{2}} \frac{| u^{*} (x) - u^{*} (y) |^{2}}{| x - y |^{2}} d x d y ⩽ \int_{R^{2}} \frac{| u (x) - u (y) |^{2}}{| x - y |^{2}} d x d y, \end{matrix}$ and $\int_{I} {(u_{p}^{*})}^{p} d x = \int_{I} u_{p}^{p} d x$ for all $p > 1$ . This implies $Φ_{p} (u^{*}) ⩽ Φ_{p} (u)$ , for all non-negative u of ${\tilde{H}}^{1 / 2} (I)$ . Therefore, if $u_{p}$ is a solution of (1.2), then $u_{p}$ is even in I and decreasing in $(0, 1)$ . □

Lemma 2.5.

There exists a $C > 0$ independent of p such that $p c_{p} ⩽ C$ for sufficiently large p.

Proof.

Define the Moser function $m_{p} : R \to R$ by $\begin{matrix} (2.7) & m_{p} (x) = \{\begin{matrix} {(log \frac{1}{2 ε})}^{\frac{1}{2}} & in 0 ⩽ | x | ⩽ ε, \\ \frac{(log \frac{1}{2 x})}{{(log \frac{1}{2 ε})}^{\frac{1}{2}}} & in ε < | x | ⩽ \frac{1}{2}, \\ 0 & in | x | > \frac{1}{2}, \end{matrix} \end{matrix}$ satisfying $\frac{1}{2 ε} = e^{\frac{p + 1}{2}}$ . Note that $p \to \infty$ implies $ε \to 0$ . We have $\begin{matrix} \int_{R} {| {(- Δ)}^{1 / 4} m_{p} |}^{2} d x = \frac{1}{2 π} \int_{R} \int_{R} \frac{{(m_{p} (x) - m_{p} (y))}^{2}}{| x - y |^{2}} d x d y . \end{matrix}$ Using the idea in Parini–Ruf [21] we obtain $\begin{matrix} \int_{R} \int_{R} \frac{{(m_{p} (x) - m_{p} (y))}^{2}}{| x - y |^{2}} d x d y = 4 \int_{0}^{\infty} \int_{0}^{\infty} {(m_{p} (x) - m_{p} (y))}^{2} \frac{x^{2} + y^{2}}{{(x^{2} - y^{2})}^{2}} d x d y = : I . \end{matrix}$ Next, we write $I = I_{1} + I_{2} + I_{3} + I_{4}$ where $\begin{array}{l} I_{1} = \frac{8}{| log \frac{1}{2 ε} |} \int_{ε}^{1 / 2} \int_{0}^{ε} {(log x - log ε)}^{2} \frac{x^{2} + y^{2}}{{(x^{2} - y^{2})}^{2}} d x d y, \\ I_{2} = \frac{4}{| log \frac{1}{2 ε} |} \int_{ε}^{1 / 2} \int_{ε}^{L} {(log x - log y)}^{2} \frac{x^{2} + y^{2}}{{(x^{2} - y^{2})}^{2}} d x d y, \\ I_{3} = 8 | log \frac{1}{2 ε} | \int_{1 / 2}^{\infty} \int_{0}^{ε} \frac{x^{2} + y^{2}}{{(x^{2} - y^{2})}^{2}} d x d y, \\ I_{4} = \frac{8}{| log \frac{1}{2 ε} |} \int_{ε}^{1 / 2} \int_{1 / 2}^{\infty} {(log \frac{1}{2 x})}^{2} \frac{x^{2} + y^{2}}{{(x^{2} - y^{2})}^{2}} d x d y . \end{array}$ By [21], it follows that $I_{j} = 0$ for $j = 1, 3, 4$ . Furthermore, by [18, Proposition 2.2], $\begin{matrix} I_{2} = 8 Γ (3) \sum_{k = 0}^{\infty} \frac{1}{{(1 + 2 k)}^{2}} = 2 π^{2} . \end{matrix}$ Hence (1.3), yields $\begin{matrix} \int_{R} {| {(- Δ)}^{1 / 4} m_{p} |}^{2} d x = π + o_{p} (1) \end{matrix}$ and $\begin{array}{rcl} \int_{I} | m_{p} |^{p} d x & = & 2 \int_{0}^{1 / 2} | m_{p} |^{p} d x = 2 \int_{0}^{ε} | m_{p} |^{p} d x + 2 \int_{ε}^{1 / 2} | m_{p} |^{p} d x \\ = & 2 ε {(log \frac{1}{2 ε})}^{\frac{p}{2}} + 2 {(log \frac{1}{2 ε})}^{- \frac{p}{2}} \int_{ε}^{1 / 2} {(log \frac{1}{2 x})}^{\frac{p}{2}} d x \\ = & 2 ε {(log \frac{1}{2 ε})}^{\frac{p}{2}} + {(log \frac{1}{2 ε})}^{- \frac{p}{2}} \int_{log \frac{1}{2 ε}}^{\infty} x^{p / 2} e^{- x} d x \\ = & {(\frac{p + 1}{2 e})}^{\frac{p}{2}} e^{- \frac{1}{2}} + {(\frac{p + 1}{2})}^{- \frac{p}{2}} o (1) \\ = & {(\frac{p + 1}{2 e})}^{\frac{p}{2}} e^{- \frac{1}{2}} [1 + o_{p} (1)] . \end{array}$ Furthermore, $\begin{array}{rcl} \int_{I} | m_{p} |^{p + 1} d x = {(\frac{p + 1}{2 e})}^{\frac{p + 1}{2}} [1 + o_{p} (1)] . \end{array}$ We have $\begin{matrix} c_{p} = inf_{u \in {\tilde{H}}^{1 / 2} (I)} max_{t ⩾ 0} Φ_{p} (t u) . \end{matrix}$ Then $\begin{matrix} (2.8) & c_{p} ⩽ max_{t ⩾ 0} Φ_{p} (t m_{p}) \end{matrix}$ and hence there exists a unique $t_{p} \in (0, + \infty)$ such that $\begin{matrix} max_{t ⩾ 0} Φ_{p} (t m_{p}) = Φ_{p} (t_{p} m_{p}) \end{matrix}$ and $\begin{array}{rcl} 0 & = & ⟨ Φ_{p}^{'} (t_{p} m_{p}), m_{p} ⟩ \\ = & t_{p}^{2} \int_{R} {| {(- Δ)}^{1 / 4} m_{p} |}^{2} - t_{p}^{p + 1} \int_{I} | m_{p} |^{p + 1} + t_{p}^{p} \int_{I} | m_{p} |^{p} \\ = & π t_{p}^{2} - t_{p}^{p + 1} {(\frac{p + 1}{2 e})}^{\frac{p + 1}{2}} [1 + o_{p} (1)] + t_{p}^{p} {(\frac{p + 1}{2 e})}^{\frac{p}{2}} e^{- \frac{1}{2}} [1 + o_{p} (1)] . \end{array}$ This implies that $\begin{matrix} (2.9) & \frac{p + 1}{2 e} t_{p}^{p - 1} - e^{- 1 / 2} t_{p}^{p - 2} = π {(\frac{2 e}{p + 1})}^{\frac{p + 1}{2}} [1 + o_{p} (1)] . \end{matrix}$ This implies that $(p + 1) t_{p} > 2 e^{1 / 2}$ and $(p + 1) t_{p}$ cannot converge to $2 e^{1 / 2}$ and hence we can choose p large such that $(p + 1) t_{p} > 3 e^{1 / 2}$ . As a result for $p ≫ 1$ , $\begin{matrix} t_{p}^{p - 1} ⩽ \frac{6 π e}{p + 1} {(\frac{2 e}{p + 1})}^{\frac{p + 1}{2}} ⩽ 3 π {(\frac{2 e}{p + 1})}^{\frac{p + 1}{2}} \end{matrix}$ which implies that $t_{p} ⩽ 2 {(\frac{2 e}{p + 1})}^{1 / 2}$ .

But by (2.8) we have $\begin{matrix} c_{p} ⩽ \frac{t_{p}^{2}}{2} \int_{R} {| {(- Δ)}^{1 / 4} m_{p} |}^{2} d x - \frac{t_{p}^{p + 1}}{p + 1} \int_{I} m_{p}^{p + 1} d x + \frac{t_{p}^{p}}{p} \int_{I} m_{p}^{p} d x \end{matrix}$ this implies $\begin{matrix} p c_{p} ⩽ \frac{p t_{p}^{2}}{2} \int_{R} {| {(- Δ)}^{1 / 4} m_{p} |}^{2} d x + \frac{p t_{p}^{p - 1} t_{p}^{2}}{p + 1} \int_{I} m_{p}^{p + 1} d x + \frac{p t_{p}^{p - 1} t_{p}}{p} \int_{I} m_{p}^{p} d x . \end{matrix}$ Using the estimates of $t_{p}^{p - 1}$ and $t_{p}$ we obtain $\begin{matrix} p c_{p} ⩽ C \end{matrix}$ where $C > 0$ independent of p. □

Lemma 2.6.

The least energy solution $u_{p}$ satisfies $\begin{matrix} (2.10) & p \int_{R} {| {(- Δ)}^{1 / 4} u_{p} |}^{2} ⩽ C; p \int_{I} u_{p}^{p + 1} ⩽ C; p \int_{I} u_{p}^{p} ⩽ C . \end{matrix}$

Proof.

By the Hölder’s inequality we obtain $\begin{array}{rcl} \int_{I} | u |^{p} d x & ⩽ & {(\int_{I} | u |^{p + 1} d x)}^{\frac{p}{p + 1}} {(\int_{I} 1)}^{1 - \frac{p}{p + 1}} \\ (2.11) & = & {(\int_{I} | u |^{p + 1} d x)}^{\frac{p}{p + 1}} | I |^{\frac{1}{p + 1}} . \end{array}$ Since $p c_{p} ⩽ C$ , we obtain $\begin{matrix} c_{p} = Φ_{p} (u_{p}) = (\frac{1}{2} - \frac{1}{p + 1}) \int_{R} {| {(- Δ)}^{1 / 4} u_{p} |}^{2} + (\frac{1}{p} - \frac{1}{p + 1}) \int_{I} u_{p}^{p + 1} . \end{matrix}$ This implies $p \int_{R} | {(- Δ)}^{1 / 4} u_{p} |^{2}$ and $p \int_{I} u_{p}^{p + 1}$ are uniformly bounded and hence by (2.11) we have $p \int_{I} u_{p}^{p} d x$ is uniformly bounded. □

Lemma 2.7.

Moreover, for sufficiently large p, $1 ⩽ ‖ u_{p} ‖_{\infty} ⩽ C$ where C is a constant independent of p.

Proof.

Let 0 be a point of maxima of $u_{p}$ in I. Then ${(- Δ)}^{1 / 2} u_{p} (0) ⩾ 0$ and hence from (1.2) we obtain $u_{p}^{p} (0) - u_{p}^{p - 1} (0) ⩾ 0$ and hence $u_{p} (0) ⩾ 1$ . Proceeding exactly as Lemma 2.1 and using the Stirling formula, $\begin{matrix} {(\int_{I} {(log (\frac{1}{| x - y |}))}^{p + 1} d x)}^{1 / p + 1} \sim p \end{matrix}$ we obtain for all $x \in I$ , $0 < u_{p} (x) ⩽ C$ holds for some constant $C > 0$ independent on p. This implies $‖ u_{p} ‖_{\infty} ⩽ C$ . □

By Lemma 2.7 $\begin{matrix} lim_{p \to \infty} p u_{p} (0) = + \infty . \end{matrix}$ Choose $ε_{p}$ such that $\begin{matrix} (2.12) & ε_{p} p u_{p}^{p - 1} (0) = 1 . \end{matrix}$ Then $ε_{p} \to 0$ as $p \to \infty$ . Let $I_{p} : = \frac{I}{ε_{p}}$ . Then $W_{p}$ defined in (1.7) is a classical solution of $\begin{matrix} (2.13) & \{\begin{matrix} {(- Δ)}^{1 / 2} W_{p} = [(1 + \frac{W_{p}}{p}) - \frac{1}{‖ u_{p} ‖_{\infty}}] {(1 + \frac{W_{p}}{p})}^{p - 1} & in I_{p}, \\ W_{p} ⩽ 0 & in I_{p}, \\ W_{p} = - p & in R ∖ I_{p} . \end{matrix} \end{matrix}$ We will show that there exists a subsequence of $W_{p}$ which converges to $C_{loc}^{δ} (R)$ for any $δ \in (0, 1)$ where W satisfies $\begin{matrix} (2.14) & \{\begin{matrix} {(- Δ)}^{1 / 2} W = (1 - γ) e^{W} in R, \\ W (0) = 0, \int_{R} e^{W} < \infty, \\ W \in L_{1} (R) \end{matrix} \end{matrix}$ for some $0 < γ < 1$ .

Lemma 2.8.

There exist a subsequence of $W_{p}$ which converges to W as $p \to \infty$ in $C_{loc}^{δ} (R)$ for any $δ \in (0, 1)$ . In particular, $\int_{R} e^{W} < + \infty$ .

Proof.

Let $R > 0$ be independent of p and let $A_{p}$ be a solution of $\begin{matrix} (2.15) & \{\begin{matrix} {(- Δ)}^{1 / 2} A_{p} = {(1 + \frac{W_{p}}{p})}^{p} & in (- R, R), \\ A_{p} = 0 & in R ∖ (- R, R) . \end{matrix} \end{matrix}$ Using the strong maximum principle, it follows that $A_{p} > 0$ . Moreover, as $0 ⩽ {(1 + \frac{W_{p}}{p})}^{p} ⩽ 1$ , we obtain ${(1 + \frac{W_{p}}{p})}^{p} \in L^{q} (- R, R)$ for any $1 < q < \infty$ . Therefore, applying [16, Proposition 1.2], it follows $A_{p} ⩽ C$ , for some $C > 0$ which is independent of p. For $x \in (- R, R)$ , we set $ψ_{p} (x) : = A_{p} (x) - W_{p} (x)$ . Therefore, $ψ_{p}$ is a sequence of functions satisfying $\begin{matrix} (2.16) & {(- Δ)}^{1 / 2} ψ_{p} = \frac{1}{‖ u_{p} ‖_{\infty}} {(1 + \frac{W_{p}}{p})}^{p - 1} \end{matrix}$ which is uniformly bounded below by 0. Since, by the definition of $ψ_{p}$ , $ψ_{p} (0) = A_{p} (0) ⩽ C$ . Hence by the Harnack inequality [27, Lemma 2.1], there exists a subsequence $ψ_{p}$ which is bounded in $L_{loc}^{\infty} (- R, R)$ . Hence there exists a subsequence of $W_{p}$ which is locally uniformly bounded. We will show that $‖ W_{p} ‖_{C_{loc}^{δ} ((- R^{'}, R^{'}))} ⩽ C$ for any $δ \in (0, 1)$ and $R^{'} < R$ where R is defined before.

Choose $0 < R_{1} < R$ and $η \in C_{0}^{\infty} (R)$ $\begin{matrix} (2.17) & η (x) = \{\begin{matrix} 1 & in (- R_{1}, R_{1}), \\ 0 & in R ∖ (- R, R) \end{matrix} \end{matrix}$ with $0 ⩽ η ⩽ 1$ in $(- R, R) ∖ (- R_{1}, R_{1})$ . Let $\begin{matrix} g_{p} (x) = [(1 + \frac{W_{p}}{p}) - \frac{1}{‖ u_{p} ‖_{\infty}}] {(1 + \frac{W_{p}}{p})}^{p - 1} . \end{matrix}$ Define $\begin{matrix} α_{p} (x) = \frac{1}{π} \int_{R} η (y) log \frac{1}{| x - y |} g_{p} (y) d y . \end{matrix}$ Then we obtain $\begin{matrix} {(- Δ)}^{1 / 2} α_{p} = η g_{p} in R . \end{matrix}$ Note that for any $δ \in (0, 1)$ there exists $C > 0$ such that $\begin{matrix} | log | x - y | - log | z - y | | ⩽ C | x - z |^{δ} max {| y - x |^{- δ}, | y - z |^{- δ}} . \end{matrix}$ Using the fact $g_{p}$ is uniformly bounded and for $x, z \in (- R_{1}, R_{1})$ , we obtain $\begin{array}{rcl} | α_{p} (x) - α_{p} (z) | & ⩽ & C | x - z |^{δ} \int_{R} η (y) max {| x - y |^{- δ}, | y - z |^{- δ}} d y \\ ⩽ & C | x - z |^{δ} \int_{R} η [| x - y |^{- δ} + | y - z |^{- δ}] d y \\ ⩽ & C | x - z |^{δ} [\int_{R} η [| x - y |^{- δ} d y + \int_{R} η | z - y |^{- δ} d y] \\ ⩽ & C | x - z |^{δ} . \end{array}$ As before, we define $ψ_{p} (x) : = W_{p} (x) - α_{p} (x)$ . Then $ψ_{p}$ is 1/2-harmonic in $(- R_{1}, R_{1})$ hence by [9] we have for any $x \in (- R^{'}, R^{'}) \subset (- R_{1}, R_{1})$ $\begin{array}{rcl} | ψ_{p}^{'} (x) | & ⩽ & C (‖ ψ_{p} ‖_{L^{\infty} (- R_{1}, R_{1})} - ψ_{p}) \\ ⩽ & C (‖ W_{p} ‖_{L^{\infty} (- R_{1}, R_{1})} + ‖ α_{p} ‖_{L^{\infty} (- R_{1}, R_{1})}) ⩽ C . \end{array}$ Hence we have $\begin{array}{rcl} \frac{| ψ_{p} (x) - ψ_{p} (y) |}{| x - y |^{δ}} = \frac{| ψ_{p} (x) - ψ_{p} (y) |}{| x - y |} | x - y |^{1 - δ} ⩽ C \frac{| ψ_{p} (x) - ψ_{p} (y) |}{| x - y |} . \end{array}$ This implies $ψ_{p}$ is uniformly δ-Hölder continuous in $(- R^{'}, R^{'})$ . Hence $‖ W_{p} ‖_{C_{loc}^{δ} ((- R^{'}, R^{'}))} ⩽ C$ . Using the Arzela–Ascoli theorem, we obtain $W_{p} \to W$ as $p \to \infty$ in $C_{loc}^{δ} (R)$ for every $0 < δ < 1$ . As $W_{p}$ is locally uniformly bounded, there exists a constant $C > 0$ independent of p such that $\begin{matrix} \int_{| x | ⩽ 1} \frac{| W_{p} (x) |}{1 + | x |^{2}} d x ⩽ C . \end{matrix}$ Now we claim that $\begin{matrix} \int_{| x | > 1} \frac{| W_{p} (x) |}{1 + | x |^{2}} d x ⩽ C_{1} \end{matrix}$ for some constant $C_{1} > 0$ independent of p. From (2.13) we have ${(- Δ)}^{1 / 2} W_{p} ⩽ 1$ in $I_{p}$ and $W_{p} ⩽ 0$ . Hence we obtain $\begin{array}{rcl} 1 & ⩾ & {(- Δ)}^{1 / 2} W_{p} (0) = \int_{R} \frac{(W_{p} (0) - W_{p} (y))}{| y |^{2}} d y \\ = & \int_{- 1}^{1} \frac{| W_{p} (y) |}{| y |^{2}} d y + \int_{| y | > 1} \frac{| W_{p} (y) |}{| y |^{2}} d y . \end{array}$ Hence $\begin{matrix} \int_{| y | > 1} \frac{| W_{p} (y) |}{| y |^{2}} d y ⩽ {(- Δ)}^{1 / 2} W_{p} (0) . \end{matrix}$ Note that when $| y | > 1$ and we obtain $\begin{array}{rcl} \int_{| y | > 1} \frac{| W_{p} (y) |}{1 + | y |^{2}} d y ⩽ {(- Δ)}^{1 / 2} W_{p} (0) ⩽ 1 . \end{array}$ This proves the claim. By the Fatou’s lemma $\begin{matrix} \int_{R} \frac{| W |}{1 + | x |^{2}} d x ⩽ \underset{p \to \infty}{lim inf} \int_{R} \frac{| W_{p} |}{1 + | x |^{2}} d x ⩽ C . \end{matrix}$ Now we claim that $\begin{matrix} (2.18) & lim_{p \to \infty} \int_{R} W_{p} {(- Δ)}^{1 / 2} ϕ d x = \int_{R} W {(- Δ)}^{1 / 2} ϕ d x \end{matrix}$ for any $ϕ \in C_{0}^{\infty} (R)$ . Note that $W_{p} {(- Δ)}^{1 / 2} ϕ \to W {(- Δ)}^{1 / 2} ϕ$ pointwise as $p \to \infty$ and $W_{p}$ is locally uniformly bounded. Any $ϕ \in C_{0}^{\infty} (R)$ satisfies $\begin{matrix} | {(- Δ)}^{1 / 2} ϕ | ⩽ \frac{‖ ϕ ‖_{\infty}}{1 + | x |^{2}} for all x \in R . \end{matrix}$ Hence $\begin{array}{rcl} \int_{R} | W_{p} | | {(- Δ)}^{1 / 2} ϕ | d x & ⩽ & ‖ ϕ ‖_{\infty} \int_{R} \frac{| W_{p} |}{1 + | x |^{2}} d x \\ ⩽ & C \int_{(- 1, 1)} \frac{| W_{p} |}{1 + | x |^{2}} d x + C \int_{R ∖ (- 1, 1)} \frac{| W_{p} |}{1 + | x |^{2}} d x \end{array}$ is uniformly bounded by an integrable function. By the dominated convergence theorem we obtain (2.18). Furthermore, as $W_{p}$ is a classical solution of (2.13), it is a distributional solution hence $\begin{matrix} \int_{R} W_{p} {(- Δ)}^{1 / 2} ϕ d x = \int_{R} {(1 + \frac{W_{p}}{p})}^{p} ϕ d x - \frac{1}{‖ u_{p} ‖_{\infty}} \int_{R} {(1 + \frac{W_{p}}{p})}^{p - 1} ϕ, \forall ϕ \in C_{0}^{\infty} (R) . \end{matrix}$ Using (2.18), we can pass through the limit in above expression to obtain $\begin{matrix} (2.19) & \int_{R} W {(- Δ)}^{1 / 2} ϕ d x = \int_{R} e^{W} ϕ d x - γ \int_{R} e^{W} ϕ d x \forall ϕ \in C_{0}^{\infty} (R) \end{matrix}$ where $\begin{matrix} \frac{1}{‖ u_{p} ‖_{\infty}} \to γ \end{matrix}$ along a subsequence. Since $W \in L_{1}$ , ${(- Δ)}^{1 / 2} W$ exists and the formulation is valid $\begin{matrix} ⟨ {(- Δ)}^{1 / 2} W, ϕ ⟩ = ⟨ W, {(- Δ)}^{1 / 2} ϕ ⟩, ϕ \in S . \end{matrix}$ In order to check the validity of (1.5), we need to show that (2.19) holds true even for every $ϕ \in S$ . Note that $C_{0}^{\infty} (R)$ is dense in $S$ , for any $ϕ \in S$ there exists a sequence $ϕ_{n} \in C_{0}^{\infty} (R)$ such that $ϕ_{n} \to ϕ$ . But by [19] (Lemma 20.8), there exists a uniform constant $C > 0$ such that $\begin{matrix} | {(- Δ)}^{1 / 2} ϕ_{n} (x) | ⩽ \frac{C}{1 + | x |^{2}} for all x \in R . \end{matrix}$ Again by the application of the dominated convergence theorem, we have $\begin{matrix} \int_{R} W {(- Δ)}^{1 / 2} ϕ d x = (1 - γ) \int_{R} e^{W} ϕ d x, \forall ϕ \in S . \end{matrix}$ Moreover, as ${(1 + \frac{W_{p}}{p})}^{p} \to e^{W}$ as $p \to \infty$ , we obtain by the Fatou’s Lemma and Lemma 2.6 $\begin{matrix} \int_{R} e^{W} ⩽ lim inf_{p \to \infty} {(1 + \frac{W_{p}}{p})}^{p} ⩽ C . \end{matrix}$ Hence we have $\begin{matrix} {(- Δ)}^{1 / 2} W = (1 - γ) e^{W}, W (0) = 0; W \in L_{1} (R) \end{matrix}$ for some $γ \in (0, 1]$ . □

Lemma 2.9.

We have $0 < γ < 1$ .

Proof.

We prove by contradiction. If possible, let $\begin{matrix} (2.20) & ‖ u_{p} ‖_{\infty} \to γ^{- 1} = 1 \end{matrix}$ along a subsequence. Then W satisfies $\begin{matrix} (2.21) & \{\begin{matrix} {(- Δ)}^{1 / 2} W = 0 in R, \\ W (0) = 0, W ⩽ 0, \\ W \in L_{1} (R) . \end{matrix} \end{matrix}$ The solution to (2.21) is given by $W (x) = a x + b$ for some $a, b \in R$ see [15]. But $W (0) = 0$ implies $b = 0$ . This implies $W (x) = a x$ but as $\int_{R} \frac{| W |}{1 + x^{2}} < \infty$ we have $a = 0$ which implies $W \equiv 0$ . Hence $W_{p} \to 0$ in $C_{loc}^{α} (R)$ .

Again by the Green function representation (2.1), we obtain, $\begin{matrix} (2.22) & u_{p} (x) = \int_{I} G (x, y) (u^{p} (y) - u^{p - 1} (y)) d y . \end{matrix}$ Thus we have $\begin{array}{rcl} u_{p} (0) & = & \int_{I} (- \frac{1}{π} log | y | + H (0, y)) (u^{p} (y) - u^{p - 1} (y)) d y \\ = & - \frac{1}{π} \int_{I} log | y | (u^{p} (y) - u^{p - 1} (y)) d y \\ + \int_{I} H (0, y) (u^{p} (y) - u^{p - 1} (y)) d y . \end{array}$ But by Lemma 2.6 we obtain $\begin{matrix} \int_{I} H (0, y) ({| u (y) |}^{p} + {| u (y) |}^{p - 1}) d y ⩽ \frac{C}{p} . \end{matrix}$ We study $\begin{array}{l} - \frac{1}{π} \int_{I} log | y | (u_{p}^{p} (y) - u_{p}^{p - 1} (y)) d y \\ = - \frac{ε_{p}}{π} \int_{I_{p}} (log ε_{p} | z |) (u_{p}^{p} (ε_{p} z) - u_{p}^{p - 1} (ε_{p} z)) d z \\ = - \frac{ε_{p}}{π} \int_{I_{p}} (log ε_{p} + log | z |) (‖ u_{p} ‖_{\infty}^{p} {(1 + \frac{W_{p}}{p})}^{p} - ‖ u_{p} ‖_{\infty}^{p - 1} {(1 + \frac{W_{p}}{p})}^{p - 1}) d z \\ = - \frac{ε_{p} ‖ u_{p} ‖_{\infty}^{p - 1} log ε_{p}}{π} \int_{I_{p}} (‖ u_{p} ‖_{\infty} {(1 + \frac{W_{p}}{p})}^{p} - {(1 + \frac{W_{p}}{p})}^{p - 1}) d z \\ - \frac{ε_{p} ‖ u_{p} ‖_{\infty}^{p - 1}}{π} \int_{I_{p}} log | z | (‖ u_{p} ‖_{\infty} {(1 + \frac{W_{p}}{p})}^{p} - {(1 + \frac{W_{p}}{p})}^{p - 1}) d z \\ = J_{1, p} + J_{2, p} . \end{array}$ Using the definition of $ε_{p}$ we get $\begin{array}{rcl} J_{2, p} = - \frac{1}{p π} \int_{I_{p}} log | z | (‖ u_{p} ‖_{\infty} {(1 + \frac{W_{p}}{p})}^{p} - {(1 + \frac{W_{p}}{p})}^{p - 1}) d z . \end{array}$ Since $W_{p} \to 0$ in $C_{loc}^{α} (R)$ , we have $\begin{array}{rcl} J_{2, p} = \frac{1}{p π} \int_{I_{p} \cap {| z | > 1}} log | z | (‖ u_{p} ‖_{\infty} {(1 + \frac{W_{p}}{p})}^{p} - {(1 + \frac{W_{p}}{p})}^{p - 1}) d z + O (\frac{1}{p}) . \end{array}$ On the other hand, since $W_{p} ⩽ 0$ in $I_{p}$ we obtain $\begin{array}{l} \int_{I_{p} \cap {| z | > 1}} log | z | (‖ u_{p} ‖_{\infty} {(1 + \frac{W_{p}}{p})}^{p} - {(1 + \frac{W_{p}}{p})}^{p - 1}) \\ = \int_{I_{p} \cap {| z | > 1}} log | z | {(1 + \frac{W_{p}}{p})}^{p - 1} (‖ u_{p} ‖_{\infty} - 1 + ‖ u_{p} ‖_{\infty} \frac{W_{p}}{p}) \\ ⩽ (‖ u_{p} ‖_{\infty} - 1) \int_{I_{p} \cap {| z | > 1}} log | z | {(1 + \frac{W_{p}}{p})}^{p - 1} d z \\ ⩽ (‖ u_{p} ‖_{\infty} - 1) (log \frac{1}{ε_{p}}) \int_{I_{p} \cap {| z | > 1}} {(1 + \frac{W_{p}}{p})}^{p - 1} d z \\ (2.23) & ⩽ C (‖ u_{p} ‖_{\infty} - 1) (log \frac{1}{ε_{p}}) . \end{array}$ Here we used the fact $z \in I_{p} = \frac{I}{ε_{p}}$ and $| z | ⩽ \frac{1}{ε_{p}}$ and $\int_{I_{p} \cap {| z | > 1}} {(1 + \frac{W_{p}}{p})}^{p - 1} d z ⩽ C$ . Note that by Lemma 2.6 we obtain $\begin{array}{rcl} \int_{I_{p}} {(1 + \frac{W_{p}}{p})}^{p - 1} d x & = & \int_{I_{p}} \frac{u_{p}^{p - 1} (ε_{p} x)}{u_{p}^{p - 1} (0)} d x \\ = & \frac{p}{p ε_{p} u_{p}^{p - 1} (0)} \int_{I} u_{p}^{p - 1} (x) d x \\ = & p {(\int_{I} u_{p}^{p} d x)}^{\frac{p - 1}{p}} | I |^{\frac{1}{p}} ⩽ C . \end{array}$ By definition of $ε_{p}$ we obtain $\begin{matrix} - log ε_{p} = log p + (p - 1) log ‖ u_{p} ‖_{\infty} \end{matrix}$ and since $‖ u_{p} ‖_{\infty} \to 1$ we obtain $\begin{matrix} J_{2, p} = o (1) . \end{matrix}$ Moreover, $\begin{array}{rcl} J_{1, p} & = & - \frac{ε_{p}^{} ‖ u_{p} ‖_{\infty}^{p - 1} log ε_{p}}{π} \int_{I_{p}} (‖ u_{p} ‖_{\infty} {(1 + \frac{W_{p}}{p})}^{p} - {(1 + \frac{W_{p}}{p})}^{p - 1}) \\ = & \frac{‖ u_{p} ‖_{\infty}}{π p} (log p + (p - 1) log ‖ u_{p} ‖_{\infty}) ({(1 + \frac{W_{p}}{p})}^{p} - \frac{1}{‖ u_{p} ‖_{\infty}} {(1 + \frac{W_{p}}{p})}^{p - 1}) . \end{array}$

Since $‖ u_{p} ‖_{\infty} \to 1$ , $\begin{matrix} \frac{‖ u_{p} ‖_{\infty}}{π p} (log p + (p - 1) log ‖ u_{p} ‖_{\infty}) \to 0 \end{matrix}$ and $\begin{array}{l} | \int_{I_{p}} ({(1 + \frac{W_{p}}{p})}^{p} - \frac{1}{‖ u_{p} ‖_{\infty}} {(1 + \frac{W_{p}}{p})}^{p - 1}) | \\ ⩽ \int_{I_{p}} {(1 + \frac{W_{p}}{p})}^{p} + \int_{I_{p}} {(1 + \frac{W_{p}}{p})}^{p - 1} ⩽ C . \end{array}$ This implies that $\begin{matrix} J_{1, p} \to 0 . \end{matrix}$ Hence $\begin{matrix} ‖ u_{p} ‖_{\infty} = J_{1, p} + J_{2, p} + o (1) = o (1) \end{matrix}$ and we obtain a contradiction to (2.20).

Using [12], it follows that $\begin{matrix} W (x) = log \frac{4 {(1 - γ)}^{2}}{(4 {(1 - γ)}^{2} + x^{2})} \end{matrix}$ and $\begin{matrix} (2.24) & lim_{R \to \infty} lim_{p \to \infty} \int_{- R}^{R} {(1 + \frac{W_{p}}{p})}^{p} d x = lim_{R \to \infty} \int_{- R}^{R} e^{W} = 2 π (1 - γ) \end{matrix}$ and hence $\int_{R} e^{W} d x = 2 (1 - γ) π$ . This completes the lemma. □

Proof of Theorem 1.2.

The proof follows from Lemma 2.8. □

Proof of Theorem 1.3.

The first part follows from Lemma 2.7. Using the Fatou’s lemma and Lemma 2.7, we have $\begin{array}{rcl} 2 (1 - γ) π & = & \int_{R} e^{W} ⩽ \underset{p \to \infty}{lim inf} \int_{I_{p}} {(1 + \frac{W_{p}}{p})}^{p} d x \\ = & \underset{p \to \infty}{lim inf} \frac{1}{ε_{p} ‖ u_{p} ‖_{\infty}^{p}} \int_{I} u_{p}^{p} = \underset{p \to \infty}{lim inf} \frac{p}{‖ u_{p} ‖_{\infty}} \int_{I} u_{p}^{p} d x \\ ⩽ & \underset{p \to \infty}{lim inf} p \int_{I} u_{p}^{p} d x . \end{array}$ The other side follows from Lemma 2.6. □

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