Existence and asymptotic properties of positive solutions for a general quasilinear Schrödinger equation

Abstract

By a change of variables with cut-off functions, we study the existence and the asymptotic behavior of positive solutions for a general quasilinear Schrödinger equation which arises from plasma physics. We extend the results of (Adv. Nonlinear Stud. 18(1):131-150, 2017) from $\alpha=1$ to $\alpha>\frac{1}{2}$. Especially, we can consider the exponent p in $(2,2^{*})$ for all $N\geq3$.

1 Introduction

In this paper, we study the existence and asymptotic behavior of positive solutions for the following general quasilinear elliptic equation:

$$ -\Delta u+V(x)u-\alpha\gamma\bigl(\Delta\bigl( \vert u \vert ^{2\alpha}\bigr)\bigr) \vert u \vert ^{2\alpha -2}u= \vert u \vert ^{p-2}u, \quad x\in\mathbb{R}^{N}, $$

(1)

where $\alpha>\frac{1}{2}$ is a positive constant, $\gamma>0$ is a parameter, $p>2$ and $N\geq3$.

Equation (1) is derived from a superfluid film equation in plasma physics [11]; see [7,8,9, 15] and the references therein for more physical backgrounds. When $\alpha=1$, the existence of solutions for Eq. (1) was extensively considered in recent years [2, 3, 9, 14,15,16, 19,20,21] since the change in [9, 14] was introduced. Furthermore, using the change of variables, for general $\alpha>\frac{1}{2}$, the existence of solutions of (1) have been studied; see [1, 4, 12] and the references therein. Comparing with the semilinear elliptic equations, it is much more challenging and interesting because of the existence of the term $(\Delta(|u|^{2\alpha}))|u|^{2\alpha-2}u$. It is worth mentioning that the authors in [20] considered problem (1) with $\alpha=1$. Using the change of variables introduced in [19] and the cut-off function technique in [5], the authors reduced Eq. (1) to a semilinear elliptic equation. Then the existence and boundedness of solution was obtained by the critical point theory when $p\in(2,2^{*})$ for $N\geq4$ or $p\in (2,4)$ for $N=3$. Moreover, they got the asymptotic properties of the solution of (1) by using the arguments in [1, 3]. But in [20], what will happen when $p\in[4,6)$ for $N=3$?

In this paper, we want to address the existence of Eq. (1) with $\alpha>\frac{1}{2}$ by using the technique of [5, 19, 20]. Furthermore, we can discuss the exponent p from 2 to 2^∗ for any $N\geq3$ by introducing different cut-off functions when $p<4\alpha$ and $p\geq4\alpha$. We also can get the asymptotic properties of the solution of (1) with the use of techniques in [1, 3, 20].

We assume that the potential function V satisfies $(V_{1})$ $0< V_{0}\leq V(x)\leq\lim_{|x|\to+\infty} V(x)=V_{\infty }<+\infty$.

Define the space $X=\{u\in H^{1}(\mathbb{R}^{N}): \int_{\mathbb {R}^{N}} |u|^{2(2\alpha-1)}|\nabla u|^{2}\,dx<\infty\}$. Then, for $u\in X$, the energy functional $I_{\gamma}(u)$ associated with (1) is

$$ I_{\gamma}(u)=\frac{1}{2} \int_{\mathbb{R}^{N}} \bigl( \vert \nabla u \vert ^{2}+V(x) \vert u \vert ^{2}\bigr)\,dx+\alpha^{2}\gamma \int_{\mathbb{R}^{N}} \vert u \vert ^{2(2\alpha-1)} \vert \nabla u \vert ^{2}\,dx-\frac{1}{p} \int_{\mathbb{R}^{N}} \vert u \vert ^{p} \,dx. $$

(2)

Theorem 1.1

Assume $V(x)=\mu>0$, then Eq. (1) has a positive solution $u_{\gamma}$ satisfying: (i) $u_{\gamma}$ is spherically symmetric and $u_{\gamma}$ decreases with respect to $|x|$; (ii) $u_{\gamma}\in C^{2}(\mathbb{R}^{N})$; (iii) $u_{\gamma}$ together with its derivatives up to order 2 have exponential decay at infinity $|D^{\alpha}u_{\gamma}|\le Ce^{-\delta|x|}$, $x\in\mathbb{R}^{N}$, for some $C,\delta>0$ and $|\alpha|\le2$. Passing to a subsequence if necessary, it follows that

$$ u_{\gamma}\to u_{0} \quad\textit{in } H^{2}\bigl(\mathbb {R}^{N}\bigr)\cap C^{2}\bigl(\mathbb{R}^{N}\bigr) \textit{ as }\gamma\to0^{+}, $$

where $u_{0}$ is the ground state of equation $-\Delta u+\mu u=|u|^{p-2}u$, $x\in\mathbb{R}^{N}$.

Theorem 1.2

Assume that $(V_{1})$ holds and $p\in(2,2^{*})$. Then there exists a $\gamma_{0}$ such that, for $\gamma\in(0,\gamma _{0})$, Eq. (1) has a positive solution $u_{\gamma}$ satisfying $\max_{x\in{\mathbb{R}^{N}}} |\gamma^{\mu}u_{\gamma}(x)|\to0\textit{ as }\gamma\to0^{+}\textit{ for any }\mu> \frac{1}{2(2\alpha-1)}$.

Remark 1.1

If $\alpha=1$, the above theorem is essentially Theorem 1.1 of [20]. When $N=3$, $p<4$ is necessary in [20]. But in here, we extend this result to $p<2^{*}$. Moreover, for general $\alpha>\frac {1}{2}$, [2, 15] obtain the existence of solutions of (1) for $p\geq4\alpha$. But we can obtain the existence of solutions for the case $p<4\alpha$.

In this paper, we use the following notations: C denotes constant, $\Vert u \Vert ^{2}=\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+u^{2})\,dx$ for $u\in H^{1}(\mathbb{R}^{N})$, $\Vert u \Vert _{p}$ denotes the norm of the space $L^{p}(\mathbb{R}^{N})$.

2 The cut-off technique and some lemmas

We introduce the cut-off function $\zeta(t):\mathbb{R}\rightarrow \mathbb{R}$ such that $\zeta(t)=0$ if $t\leq0$, $\zeta(t)=\frac {e^{-\frac{1}{t}}}{e^{-\frac{1}{t}}+e^{-\frac{1}{1-t}}}$ if $0< t<1$ and $\zeta(t)=1$ if $t\geq1$. The basic property of the function was already used in [17, 18, 20]. It is easy to see that $\zeta(t)\in C^{\infty}(\mathbb{R},[0,1])$, $0\le\zeta(t)\le1$ for all $t\in\mathbb{R}$. Moreover, $\zeta '(t)=\frac{(2t^{2}-2t+1)e^{\frac{1-2t}{t(1-t)}}}{t^{2} (1-t)^{2} [1+e^{\frac{1-2t}{t(1-t)}}]^{2}}$ if $0< t<1$ and $\zeta'(t)=0$ if $t<0$ or $t>1$. Let $\zeta'(0)=\zeta'(1)=0$, then $\zeta'(t)\ge0$ is uniformly bounded in $[0,1]$. This means there exists some $C_{0}>0$ such that $|\zeta'(t)|\le C_{0}$ for any $t\in \mathbb{R}$.

Case I: $4\alpha>p$. In this case, we assume that

$$ \rho(t)=\zeta^{2} \biggl[ \frac{2^{\frac{1}{2\alpha -1}}}{2^{\frac{1}{2\alpha-1}}-1} \biggl(1- \biggl( \frac{8\alpha ^{2}\gamma(4\alpha-p)}{p-2} \biggr)^{\frac{1}{2(2\alpha-1)}}t \biggr) \biggr]. $$

Then $\rho(t)\in C^{\infty}(\mathbb{R}^{+},[0,1])$ and

$$ \rho(t) \textstyle\begin{cases} =1 & \text{if }0\le t< (\frac{p-2}{32\alpha^{2} \gamma(4\alpha -p)} )^{\frac{1}{2(2\alpha-1)}}, \\ \in(0,1)&\text{if } (\frac{p-2}{32\alpha^{2} \gamma(4\alpha -p)} )^{\frac{1}{2(2\alpha-1)}}\le t < (\frac{p-2}{8\alpha ^{2} \gamma(4\alpha-p)} )^{\frac{1}{2(2\alpha-1)}},\\ =0 &\text{if } t\ge (\frac{p-2}{8\alpha^{2} \gamma(4\alpha -p)} )^{\frac{1}{2(2\alpha-1)}}. \end{cases} $$

Moreover, for any $t\in\mathbb{R}^{+}$, we have $0\ge\rho'(t)\ge- \frac{2^{\frac{2\alpha}{2\alpha-1}}}{2^{\frac {1}{2\alpha-1}}-1} (\frac{8\alpha^{2}\gamma(4\alpha -p)}{p-2} )^{\frac{1}{2(2\alpha-1)}}C_{0}\sqrt{\rho(t)}$. Nextly, we assume that $\eta(t)=\rho(-t)$ if $t\leq0$ and $\eta (t)=\rho(t)$ if $t\geq0$. It means that

$$ \eta(t) \textstyle\begin{cases} =\eta(-t)&\text{if }t\le0,\\ =1 & \text{if } 0\le t< (\frac{p-2}{32\alpha^{2} \gamma (4\alpha-p)} )^{\frac{1}{2(2\alpha-1)}}, \\ \in(0,1)& \text{if } (\frac{p-2}{32\alpha^{2} \gamma(4\alpha -p)} )^{\frac{1}{2(2\alpha-1)}}\le t < (\frac{p-2}{8\alpha ^{2} \gamma(4\alpha-p)} )^{\frac{1}{2(2\alpha-1)}},\\ =0 &\text{if } t\ge (\frac{p-2}{8\alpha^{2} \gamma(4\alpha -p)} )^{\frac{1}{2(2\alpha-1)}}, \end{cases} $$

(3)

$\eta(t)\in C_{0}^{\infty}(\mathbb{R},[0,1])$ and $\eta'(t)t\le0$ for $t\in\mathbb{R}^{+}$. Furthermore, for $t\in\mathbb{R}^{+}$, we have

$$ t\eta'(t)\ge \textstyle\begin{cases} -\frac{1}{2^{\frac{1}{2\alpha-1}}-1}C_{0}& \text{if }0\le t< (\frac{p-2}{32\alpha^{2} \gamma(4\alpha-p)} )^{\frac {1}{2(2\alpha-1)}}, \\ -\frac{2^{\frac{1}{2\alpha-1}}}{2^{\frac{1}{2\alpha-1}}-1}C_{0}\sqrt {\eta(t)}& \text{if } (\frac{p-2}{32\alpha^{2} \gamma(4\alpha -p)} )^{\frac{1}{2(2\alpha-1)}}\le t < (\frac{p-2}{8\alpha ^{2} \gamma(4\alpha-p)} )^{\frac{1}{2(2\alpha-1)}},\\ 0&\text{if } t\ge (\frac{p-2}{8\alpha^{2} \gamma(4\alpha -p)} )^{\frac{1}{2(2\alpha-1)}}. \end{cases} $$

Case II: $p\ge4\alpha$. In this case, we let

$$ \rho(t)=\zeta^{2} \biggl[\frac{2^{\frac{1}{2\alpha -1}}}{2^{\frac{1}{2\alpha-1}}-1} \biggl(1- \biggl( \frac{8\alpha ^{2}\gamma(6-p)}{p-2} \biggr)^{\frac{1}{2(2\alpha-1)}}t \biggr) \biggr]. $$

Similar to the case I, we assume that $\eta(t)=\rho(-t)$ if $t\leq0$ and $\eta(t)=\rho(t)$ if $t\geq0$. Then $0\ge\rho'(t)\ge-2 \frac {2^{\frac{1}{2\alpha-1}}}{2^{\frac{1}{2\alpha-1}}-1}(\frac{8\alpha ^{2}\gamma(6-p)}{p-2})^{\frac{1}{2(2\alpha-1)}}C_{0}\sqrt{\rho(t)}$ and

$$ \eta(t) \textstyle\begin{cases} =\eta(-t)&\text{if }t\le0,\\ =1 & \text{if } 0\le t< (\frac{p-2}{32\alpha^{2} \gamma (6-p)} )^{\frac{1}{2(2\alpha-1)}}, \\ \in(0,1)& \text{if } (\frac{p-2}{32\alpha^{2} \gamma (6-p)} )^{\frac{1}{2(2\alpha-1)}} \le t < (\frac {p-2}{8\alpha^{2} \gamma(6-p)} )^{\frac{1}{2(2\alpha-1)}},\\ =0 &\text{if } t\ge (\frac{p-2}{8\alpha^{2} \gamma(6-p)} )^{\frac{1}{2(2\alpha-1)}}. \end{cases} $$

(4)

For $p\in(2,2^{*})$, we construct an auxiliary function $g_{\gamma }(t)\text{: }\mathbb{R}\to\mathbb{R}^{+}$ just like:

$$ g_{\gamma}(t)= \sqrt{ \biggl(\frac{1}{2}+2\alpha ^{2}\gamma \vert t \vert ^{2(2\alpha-1)} \biggr)\eta(t)+\frac{1}{2}}, $$

where $\eta(t)$ take the form (3) if $p < 4\alpha$ and the form (4) if $p \geq4\alpha$. Then we know that $g_{\gamma }(0)=1$, $\frac{\sqrt{2}}{2}\leq g_{\gamma}(t)\leq\sqrt{\frac {14-3p}{4(4-p)}}$ if $p\leq4\alpha$, $\frac{\sqrt{2}}{2}\leq g_{\gamma}(t)\leq\sqrt{\frac{22-3p}{4(6-p)}}$ if $p\geq4\alpha$,

$$ g_{\gamma}^{\prime}(t)t=\frac{ (\frac{1}{2}+2\alpha^{2}\gamma \vert t \vert ^{2(2\alpha-1)} )\eta'(t)t+4(2\alpha-1)\gamma \vert t \vert ^{2(2\alpha-1)\eta(t) }}{2 [ (\frac{1}{2}+2\alpha ^{2}\gamma \vert t \vert ^{2(2\alpha-1)} )\eta(t)+\frac{1}{2} ]^{\frac{1}{2}}} $$

(5)

and $g_{\gamma}^{\prime}(t)t=-g_{\gamma}^{\prime}(-t)t$. Define $G_{\gamma}(t)=\int_{0}^{t} g_{\gamma}(s)\,ds$. Then the inverse function $G_{\gamma}^{-1}(t)$ exists and is an odd function. Furthermore, $G_{\gamma}, G_{\gamma}^{-1}\in C^{\infty}(\mathbb{R}, \mathbb{R})$.

Lemma 2.1

The following properties hold:

$$\begin{aligned} & \lim_{t\to0} \frac{G_{\gamma}^{-1}(t)}{t}=1;\qquad \lim _{t\to\infty } \frac{G_{\gamma}^{-1}(t)}{t}=\sqrt{2}; \end{aligned}$$

(6)

$$\begin{aligned} &\sqrt{\frac{4(4\alpha-p)}{16\alpha-2-3p}} \vert t \vert \le \bigl\vert G_{\gamma }^{-1}(t) \bigr\vert \le\sqrt{2} \vert t \vert ,\quad \textit{for all }t\in\mathbb{R}\textit{ and }p\leq4\alpha; \end{aligned}$$

(7)

$$\begin{aligned} &\sqrt{\frac{4(6-p)}{22-3p}} \vert t \vert \le \bigl\vert G_{\gamma}^{-1}(t) \bigr\vert \le\sqrt {2} \vert t \vert , \quad\textit{for all }t\in\mathbb{R}\textit{ and }p\geq 4\alpha; \end{aligned}$$

(8)

$$\begin{aligned} & {-}C\le\frac{g_{\gamma}'(t)t}{g_{\gamma}(t)}\le\frac{(8\alpha -2-p)(p-2)}{16\alpha-2-3p},\quad \textit{for all }t\in \mathbb{R}\textit{ and }p\leq4\alpha; \end{aligned}$$

(9)

$$\begin{aligned} & {-}C\le\frac{g_{\gamma}'(t)t}{g_{\gamma}(t)}\le\frac {(6-p)(p-2)}{14-3p},\quad \textit{for all }t\in \mathbb{R}\textit{ and }p\geq4\alpha. \end{aligned}$$

(10)

Proof

The proofs of (6)–(8) are similar to those of Lemma 2.1 in [20], so we omit them. For the case (9), By the definition of $g_{\gamma}$ and (3), we obtain

$$ \frac{g_{\gamma}'(t)t}{g_{\gamma}(t)} \ge\frac{-C(\frac {1}{2}+2\alpha^{2}\gamma t^{2(2\alpha-1)})\sqrt{\eta(t)}}{(1+4\alpha ^{2}\gamma t^{2(2\alpha-1)})\eta(t)+1} \ge \textstyle\begin{cases} -C & \text{if } 0\le t< (\frac{p-2}{8\alpha^{2} \gamma(4\alpha -p)})^{\frac{1}{2(2\alpha-1)}}, \\ 0 & \text{if } t\ge(\frac{p-2}{8\alpha^{2} \gamma(4\alpha -p)})^{\frac{1}{2(2\alpha-1)}}. \end{cases} $$

Moreover, for $0\le t<(\frac{p-2}{8\alpha^{2} \gamma(4\alpha -p)})^{\frac{1}{2(2\alpha-1)}}$, we know that $(p-2)+(4p-16\alpha)\alpha^{2}\gamma t^{2(2\alpha-1)}\ge\frac{p-2}{2}>0$. Hence

$$\begin{aligned} &\frac{p-2}{2}-\frac{g_{\gamma}'(t)t}{g_{\gamma}(t)}\\ &\quad= \frac {[(p-2)+(4p-16\alpha)\alpha^{2}\gamma t^{2(2\alpha-1)}]\eta(t)-\eta '(t)t(1+4\alpha^{2}\gamma t^{2(2\alpha-1)})+p-2}{4g_{\gamma}^{2}(t)} \\ &\quad\ge\frac{p-2}{2[(1+4\alpha^{2}\gamma t^{2(2\alpha-1)})\eta(t)+1]} \ge\frac{(p-2)(4\alpha-p)}{16\alpha-2-3p}, \end{aligned}$$

which yields the result.

For the case (10), since $p\geq4\alpha$, it is easy to see that $(p-2)+(4p-16\alpha)\alpha^{2}\gamma t^{2(2\alpha-1)}>0$. Then

$$ \begin{aligned} \frac{p-2}{2}-\frac{g_{\gamma}'(t)t}{g_{\gamma}(t)}\ge \frac {p-2}{2[(1+4\alpha^{2}\gamma t^{2(2\alpha-1)})\eta(t)+1]} \ge\frac {(p-2)(6-p)}{22-3p}. \end{aligned} $$

 □

According to the properties of $g_{\gamma}$, we will focus on the existence of positive solutions for the following general quasilinear Schrödinger equation:

$$ -\operatorname{div}\bigl(g_{\gamma}^{2}(u)\nabla u \bigr)+g_{\gamma}(u)g_{\gamma }'(u) \vert \nabla u \vert ^{2}+V(x)u= \vert u \vert ^{p-2}u ,\quad x\in \mathbb{R}^{N}. $$

(11)

The energy functional of (11) is

$$ E_{\gamma}(u)=\frac{1}{2} \int_{\mathbb{R}^{N}} g_{\gamma}^{2}(u) \vert \nabla u \vert ^{2} \,dx +\frac{1}{2} \int_{\mathbb{R}^{N}} V(x)u^{2} \,dx -\frac{1}{p} \int_{\mathbb{R}^{N}} \vert u \vert ^{p} \,dx. $$

Furthermore, we introduce $G_{\gamma}(t)=\int_{0}^{t} g_{\gamma }(s)\,ds$ and the change of variables $u=G_{\gamma}^{-1}(v)$. Then that functional $E_{\gamma}$ can be rewritten as

$$ J_{\gamma}(v)=\frac{1}{2} \int_{\mathbb{R}^{N}} \vert \nabla v \vert ^{2}\,dx+ \frac{1}{2} \int_{\mathbb{R}^{N}} V(x) \bigl\vert G_{\gamma }^{-1}(v) \bigr\vert ^{2}\,dx-\frac{1}{p} \int_{\mathbb{R}^{N}} \bigl\vert G_{\gamma }^{-1}(v) \bigr\vert ^{p} \,dx. $$

This means that the function v is the solution of the following equation:

$$ -\Delta v+V(x)\frac{G_{\gamma}^{-1}(v)}{g_{\gamma}(G_{\gamma }^{-1}(v))}-\frac{ \vert G_{\gamma}^{-1}(v) \vert ^{p-2}G_{\gamma }^{-1}(v)}{g_{\gamma}(G_{\gamma}^{-1}(v))}=0,\quad x\in \mathbb{R}^{N}. $$

(12)

From Lemma 2.1, $J_{\gamma}$ is well defined in $H^{1}(\mathbb {R}^{N})$ and of class $C^{1}$.

Lemma 2.2

Assume that $V(x)=\mu>0$ and $h(v)=\frac{|G_{\gamma }^{-1}(v)|^{p-2}G_{\gamma}^{-1}(v)}{g_{\gamma}(G_{\gamma }^{-1}(v))}-\mu\frac{G_{\gamma}^{-1}(v)}{g_{\gamma}(G_{\gamma }^{-1}(v))}$. Then

$$ \lim_{v\rightarrow0}\frac{h(v)}{v}=-\mu, \qquad \lim_{v\rightarrow\infty} \frac{h(v)}{v^{\frac{N+2}{N-2}}}=0 $$

and there is a $\xi>0$ such that $H(\xi)=\int_{0}^{\xi}h(s)\,ds>0$.

Proof

From Lemma 2.1, we have $G_{\gamma}^{-1}(v)\rightarrow0$ and $g_{\gamma}(G_{\gamma}^{-1}(v))\rightarrow1$ as $v\rightarrow0$. $G_{\gamma}^{-1}(v)\rightarrow\infty$ and $g_{\gamma}(G_{\gamma }^{-1}(v))\rightarrow\frac{1}{\sqrt{2}}$ as $v\rightarrow\infty$. Hence

$$\begin{aligned} &\lim_{v\rightarrow0}\frac{h(v)}{v}=\lim_{v\rightarrow0} \frac { \vert G_{\gamma}^{-1}(v) \vert ^{p-2}G_{\gamma}^{-1}(v)}{vg_{\gamma}(G_{\gamma }^{-1}(v))}- \mu\lim_{v\rightarrow0}\frac{G_{\gamma}^{-1}(v)}{vg_{\gamma }(G_{\gamma}^{-1}(v))}=-\mu, \\ &\lim_{v\rightarrow\infty}\frac{h(v)}{v^{\frac{N+2}{N-2}}}=\lim_{v\rightarrow\infty} \frac{ \vert G_{\gamma}^{-1}(v) \vert ^{p-2}G_{\gamma }^{-1}(v)}{G_{\gamma}^{-1}(v)^{\frac{N+2}{N-2}}}\frac{G_{\gamma }^{-1}(v)^{\frac{N+2}{N-2}}}{v^{\frac{N+2}{N-2}}g_{\gamma}(G_{\gamma }^{-1}(v))}-0=0. \end{aligned}$$

Moreover,

$$ \begin{aligned} \int_{0}^{G_{\gamma}(\xi)}h(s)\,ds&= \int_{0}^{G_{\gamma }(\xi)} \bigl\vert G_{\gamma}^{-1}(s) \bigr\vert ^{p-2}G_{\gamma}^{-1}(s) \,dG_{\gamma }^{-1}(s)-\mu \int_{0}^{G_{\gamma}(\xi)}G_{\gamma }^{-1}(s) \,dG_{\gamma}^{-1}(s) \\ &=\frac{\xi^{p}}{p}-\frac{\mu\xi}{2}. \end{aligned} $$

Hence, there is a $\xi>0$ such that $H(\xi)=\int_{0}^{\xi}h(s)\,ds>0$. □

Lemma 2.3

Assume that $(V_{1})$ holds. Then any $(PS)$ sequence $\{v_{n}\}$ of $J_{\gamma}$ is bounded.

Proof

Let $\{v_{n}\}$ be a $(PS)$ sequence, we have

$$\begin{aligned} & J_{\gamma}(v_{n})=\frac{1}{2} \int_{\mathbb{R}^{N}} \vert \nabla v_{n} \vert ^{2}\,dx+\frac{1}{2} \int_{\mathbb{R}^{N}} V(x) \bigl\vert G_{\gamma }^{-1}(v_{n}) \bigr\vert ^{2}\,dx-\frac{1}{p} \int_{\mathbb{R}^{N}} \bigl\vert G_{\gamma }^{-1}(v_{n}) \bigr\vert ^{p} \,dx \\ &\phantom{J_{\gamma}(v_{n})}=c_{\gamma}+o_{n}(1), \end{aligned}$$

(13)

$$\begin{aligned} & \bigl\langle J_{\gamma}'(v_{n}), \psi\bigr\rangle = \int_{\mathbb{R}^{N}} \nabla v_{n} \nabla\psi\,dx+ \int_{\mathbb{R}^{N}} V(x)\frac{G_{\gamma }^{-1}(v_{n})}{g_{\gamma}(G_{\gamma}^{-1}(v_{n}))}\psi\,dx \\ &\phantom{\bigl\langle J_{\gamma}'(v_{n}), \psi\bigr\rangle =}{}- \int_{\mathbb{R}^{N}} \frac{ \vert G_{\gamma}^{-1}(v_{n}) \vert ^{p-2}G_{\gamma }^{-1}(v_{n})}{g_{\gamma}(G_{\gamma}^{-1}(v_{n}))}\psi\,dx=o\bigl( \Vert \psi \Vert \bigr) \\ &\quad\text{for all }\psi\in H^{1}\bigl(\mathbb{R}^{N} \bigr). \end{aligned}$$

(14)

Taking $\psi_{n}=G_{\gamma}^{-1}(v_{n})g_{\gamma}(G_{\gamma }^{-1}(v_{n}))$. From Lemma 2.1, we can get

$$ \vert \nabla\psi_{n} \vert = \biggl\vert \biggl[1+ \frac{G_{\gamma }^{-1}(v_{n})g_{\gamma}'(G_{\gamma}^{-1}(v_{n}))}{g_{\gamma }(G_{\gamma}^{-1}(v_{n}))} \biggr]\nabla v_{n} \biggr\vert \le C_{0} \vert \nabla v_{n} \vert $$

and $|\psi_{n}|\le\sqrt{\frac{16\alpha-2-3p}{2(4\alpha-p)}}|v_{n}|$ if $p\leq4\alpha$, $|\psi_{n}|\le\sqrt{\frac {22-3p}{2(6-p)}}|v_{n}|$ if $p\geq4\alpha$.

If $p\leq4\alpha$, combining (13), (14) and (9) of Lemma 2.1, we get

$$ \begin{aligned} pc_{\gamma}+o(1)+o(1) \Vert v_{n} \Vert &\ge\frac{(p-2)(4\alpha-p)}{16\alpha -2-3p} \int_{\mathbb{R}^{N}} \vert \nabla v_{n} \vert ^{2}\,dx+\frac{p-2}{2} \int _{\mathbb{R}^{N}} V(x) \bigl\vert G_{\gamma}^{-1}(v_{n}) \bigr\vert ^{2}\,dx \\ &\ge\frac{(p-2)(4\alpha-p)}{16\alpha-2-3p} \Vert v \Vert ^{2}. \end{aligned} $$

If $p\geq4\alpha$, combining (13), (14) and (10) of Lemma 2.1, we get

$$ \begin{aligned} pc_{\gamma}+o(1)+o(1) \Vert v_{n} \Vert &\ge\frac{(p-2)(6-p)}{22-3p} \int _{\mathbb{R}^{N}} \vert \nabla v_{n} \vert ^{2}\,dx+\frac{p-2}{2} \int_{\mathbb {R}^{N}} V(x) \bigl\vert G_{\gamma}^{-1}(v_{n}) \bigr\vert ^{2}\,dx \\ &\ge\frac{(p-2)(6-p)}{22-3p} \Vert v \Vert ^{2}. \end{aligned} $$

This shows the boundedness of $\{v_{n}\}$ in $H^{1}(\mathbb{R}^{N})$. □

3 The proof of theorems

Proof of Theorem 1.1

If $V(x)=\mu>0$, from Lemma 2.2, a standard method similar to the proof of [6] indicates that there is a solution $v_{\gamma}$ of Eq. (12) satisfies: (i) $v_{\gamma}>0$ is spherically symmetric and $v_{\gamma}$ decrease with respect to $|x|$; (ii) $v_{\gamma}\in C^{2}(\mathbb {R}^{N})$; (iii) $v_{\gamma}$ together with its derivatives up to order 2 have exponential decay at infinity: $|D^{\alpha}v_{\gamma }|\le Ce^{-\delta|x|}\text{, }x\in\mathbb{R}^{N}$, for some $C,\delta>0$ and $|\alpha|\le2$. Then, according the techniques of [2, 10, 20], we can deduce that $u_{\gamma }=G^{-1}(v_{\gamma})$ is a solution of problem (1) and $\Vert \nabla u_{\gamma} \Vert _{\infty}\leq C$. Moreover, there is a $u_{0}$, such that $u_{\gamma}=G^{-1}(v_{\gamma})\rightarrow u_{0}$, where $u_{0}$ is a nonnegative solution of problem $-\triangle u+\mu u=|u|^{p-2}u$ in $\mathbb{R}^{N}$. Furthermore, similar to the proof of Lemma 4.5 in [20], we can deduce that $u_{\gamma}\to u_{0}$ in $H^{2}(\mathbb{R}^{N})$.

Similar to the proof of Lemma 5.5 in [3] or Lemma 4.6 in [20], we know that $|v_{\gamma}|\le\frac{C}{|x|}\|v_{\gamma}\|\le\frac{C}{|x|} \text{, }|x|\ge1$. Then, for any $\varepsilon>0$ and $q>2$, there exists $R>0$ independent of γ, such that

$$\begin{aligned} &\biggl\Vert -\mu\frac{G_{\gamma}^{-1}(v_{\gamma})}{g_{\gamma}(G_{\gamma }^{-1}(v_{\gamma}))}+\frac{ \vert G_{\gamma}^{-1}(v_{\gamma }) \vert ^{p-2}G_{\gamma}^{-1}(v_{\gamma})}{g_{\gamma}(G_{\gamma }^{-1}(v_{\gamma}))} \biggr\Vert _{L^{q}(\mathbb{R}^{N}\backslash B_{R}(0))}< \varepsilon, \\ &\Vert \mu u_{0} \Vert _{L^{q}(\mathbb{R}^{N}\backslash B_{R}(0))}+ \bigl\Vert \vert u_{0} \vert ^{p-2}u_{0} \bigr\Vert _{L^{q}(\mathbb{R}^{N}\backslash B_{R}(0))}< \varepsilon. \end{aligned}$$

From $\|u_{\gamma}\|_{\infty}=\|G_{\gamma}^{-1}(v_{\gamma})\| _{\infty}\le C$, we get $G_{\gamma}^{-1}(v_{\gamma})\to u_{0} \text{, a.e. in }\mathbb{R}^{N}$ and

$$ -\mu\frac{G_{\gamma}^{-1}(v_{\gamma})}{\sqrt {1+2\alpha^{2}\gamma \vert G_{\gamma}^{-1}(v_{\gamma}) \vert ^{2(2\alpha -1)}}}\to-\mu u_{0},\quad \text{a.e. in } \mathbb{R}^{N}. $$

Using the Lebesgue dominated convergence theorem, we have

$$ \biggl\Vert -\mu\frac{G_{\gamma}^{-1}(v_{\gamma})}{\sqrt {1+2\alpha^{2}\gamma \vert G_{\gamma}^{-1}(v_{\gamma}) \vert ^{2(2\alpha -1)}}}-\mu u_{0} \biggr\Vert _{L^{q}(B_{R}(0))}+ \bigl\Vert \vert u_{\gamma} \vert ^{p-2}u_{\gamma }- \vert u_{0} \vert ^{p-2}u_{0} \bigr\Vert _{L^{q}(B_{R}(0))}\to0. $$

Hence $\limsup_{\gamma\to0^{+}}\|\Delta(v_{\gamma}-u_{0})\| _{L^{q}}\le2\varepsilon$. From the arbitrariness of ε, we have $v_{\gamma}\to u_{0}$ in $W^{2,q}(\mathbb{R}^{N})$ for any $q>2$ as $\gamma\to0^{+}$. From the Sobolev embedding, we get $v_{\gamma}\to u_{0}$ in $C^{1,\alpha}(\mathbb{R}^{N})$. Moreover, the bootstrap arguments indicate that $v_{\gamma}\to u_{0}$ in $C^{2}(\mathbb{R}^{N})$.

From the definition of $v_{\gamma}$, we have

$$ \begin{aligned} \vert v_{\gamma}-u_{\gamma} \vert = \biggl\vert \int_{0}^{u_{\gamma}} \bigl(\sqrt {1+2\alpha^{2} \gamma \vert t \vert ^{2(2\alpha-1)}}-1\bigr)\,dt \biggr\vert \le \frac {\alpha^{2}\gamma u_{\gamma}^{4\alpha-1}}{4\alpha-1}. \end{aligned} $$

Hence $\sup_{x\in\mathbb{R}^{N}} |v_{\gamma}(x)-u_{\gamma}(x)|\le C\gamma \Vert u_{\gamma} \Vert _{\infty}^{3}\to0$ as $\gamma\rightarrow0$.

Furthermore, from the definition of $v_{\gamma}$, we know that $\nabla v_{\gamma}=g_{\gamma}(u_{\gamma})\nabla u_{\gamma}$ and

$$\begin{aligned} &\sup_{x\in\mathbb{R}^{N}} \bigl\vert \nabla v_{\gamma}(x)-\nabla u_{\gamma }(x) \bigr\vert =\sup _{x\in\mathbb{R}^{N}} \bigl\vert \bigl(g_{\gamma}(u_{\gamma})-1 \bigr)\nabla u_{\gamma} \bigr\vert =\sup_{x\in\mathbb{R}^{N}} \biggl\vert \frac{2\alpha ^{2}\gamma u_{\gamma}^{2(2\alpha-1)}\nabla u_{\gamma}}{\sqrt {1+2\alpha^{2}\gamma u_{\gamma}^{2(2\alpha-1)}}+1} \biggr\vert \\ &\phantom{\sup_{x\in\mathbb{R}^{N}} \bigl\vert \nabla v_{\gamma}(x)-\nabla u_{\gamma }(x) \bigr\vert }\le\sup_{x\in\mathbb{R}^{N}} \bigl\vert \alpha^{2}\gamma u_{\gamma }^{2(2\alpha-1)}\nabla u_{\gamma} \bigr\vert \le \alpha^{2}\gamma \bigl\vert \vert u_{\gamma } \vert \bigl\vert _{\infty}^{2(2\alpha-1)} \bigr\vert \vert \nabla u_{\gamma} \vert \bigr\vert _{\infty}\to0 , \\ &\sup_{x\in\mathbb{R}^{N}}\biggl|{-}\mu\frac{G_{\gamma}^{-1}(v_{\gamma })}{g_{\gamma}(G_{\gamma}^{-1}(v_{\gamma}))}+\frac{ \vert G_{\gamma }^{-1}(v_{\gamma}) \vert ^{p-2}G_{\gamma}^{-1}(v_{\gamma})}{g_{\gamma }(G_{\gamma}^{-1}(v_{\gamma}))}-\mu u_{\gamma}- \vert u_{\gamma } \vert ^{p-2}u_{\gamma}\biggr| \to0 \end{aligned}$$

as $\gamma\rightarrow0$. On the other hand,

$$ \vert \Delta u_{\gamma} \vert = \biggl\vert \frac{1}{1+2\alpha ^{2}\gamma \vert u_{\gamma} \vert ^{2(2\alpha-1)}} \bigl[2(2\alpha-1)\alpha ^{2}\gamma \vert u_{\gamma} \vert ^{4\alpha-4}u_{\gamma} \vert \nabla u_{\gamma } \vert ^{2}-\mu u_{\gamma}+ \vert u_{\gamma} \vert ^{p-2}u_{\gamma}\bigr] \biggr\vert \le C. $$

It indicates that

$$\begin{aligned} &\sup_{x\in\mathbb{R}^{N}}\bigl\vert \Delta(v_{\gamma}-u_{\gamma}) \bigr\vert \\ &\quad \le\sup _{x\in\mathbb{R}^{N}} \bigl\vert 2\alpha^{2}\gamma u_{\gamma}^{2(2\alpha -1)}\Delta u_{\gamma} \bigr\vert +\sup _{x\in\mathbb{R}^{N}} \bigl\vert 2(2\alpha-1)\alpha^{2}\gamma u_{\gamma}^{4\alpha-3} \vert \nabla u_{\gamma} \vert ^{2} \bigr\vert \\ &\qquad{} +\sup_{x\in\mathbb{R}^{N}}\biggl|{-}\mu\frac{G_{\gamma }^{-1}(v_{\gamma})}{g_{\gamma}(G_{\gamma}^{-1}(v_{\gamma}))}+\frac { \vert G_{\gamma}^{-1}(v_{\gamma}) \vert ^{p-2}G_{\gamma}^{-1}(v_{\gamma })}{g_{\gamma}(G_{\gamma}^{-1}(v_{\gamma}))}-\mu u_{\gamma }- \vert u_{\gamma} \vert ^{p-2}u_{\gamma}\biggr| \to0. \end{aligned}$$

(15)

As in [3] Lemma 5.5, or [20] Lemma 4.6, (15) together with the Sobolev interpolation inequality yields

$$ \sup_{x\in\mathbb{R}^{N}} \bigl\vert D^{j}(v_{\gamma }-u_{\gamma}) \bigr\vert \to0, \quad \vert j \vert \le2. $$

Multiplying $u_{\gamma}$ by (1), we have

$$ \int_{x\in\mathbb{R}^{N}}\bigl(1+4\alpha^{3}\gamma u_{\gamma }^{2}\bigr) \vert \nabla u_{\gamma} \vert ^{2}+\mu u_{\gamma}^{2}-u_{\gamma}^{p} \,dx=0. $$

This implies that

$$ \int_{x\in\mathbb{R}^{N}}\mu u_{\gamma }^{2}-u_{\gamma}^{p} < 0. $$

If $u_{\gamma}(0)=\|u_{\gamma}\|_{L^{\infty}} \leq\mu^{\frac {1}{p-2}}$, one has $\mu u_{\gamma}^{2}-u_{\gamma}^{p} \geq0$, from which we arrive at a contradiction. Then we get $u_{\gamma}(0)>\mu ^{\frac{1}{p-2}}$. Since $u_{\gamma}\to u_{0}$ in $C^{2}$, we can obtain $u_{0}(0) \geq\mu^{\frac{1}{p-2}}$. By the maximum principle, we finally get $u_{0}>0$. □

The proof of Theorem 1.2

From Lemma 2.1, a standard discussion shows that $J_{\gamma}$ satisfies the mountain pass geometric hypothesis. Hence, there exists a $(PS)$ sequence $\{v_{n}\} \subset H^{1}(\mathbb{R}^{N})$, such that $J_{\gamma}(v_{n})\to c_{\gamma}$ and $J_{\gamma}'(v_{n})\to0$, where $c_{\gamma}=\inf_{\xi\in\varGamma_{\gamma}}\sup_{t\in [0,1]}J_{\gamma}(\xi(t))$, $\varGamma_{\gamma}=\{\xi(t)\in C([0,1],H^{1}(\mathbb{R}^{N})):\xi(0)=0,\xi(1)\neq0,J_{\gamma}(\xi (1))<0\}$. Then, from Lemma 2.3, we see that the sequence $\{v_{n}\}$ is bounded. This indicates that there is a subsequence of $\{v_{n}\}$, denoted still by $\{v_{n}\}$, there is $v_{\gamma}\in H^{1}(\mathbb {R}^{N})$ such that $v_{n}\rightharpoonup v_{\gamma}$ in $H^{1}(\mathbb {R}^{N})$, $v_{n}\to v_{\gamma}$ in $L_{\mathrm{loc}}^{q}(\mathbb{R}^{N}), q\in[2,2^{*})$. Hence, using Lebesgue dominated convergence theorem, it is easy to see that $J_{\gamma}'(v_{\gamma})=0$. Furthermore, we can replace $v_{n}$ by $|v_{n}|$. Hence, we can assume that $v_{n}\geq 0$ in $\mathbb{R}^{N}$ and $v_{\gamma}\geq0$. If $v_{\gamma}\neq0$, then $v_{\gamma}$ is a positive solution of Eq. (12). By contradiction, we assume that $v_{\gamma}= 0$. In this time, consider the functional $J_{\gamma}^{\infty}:H^{1}(\mathbb{R}^{N})\to\mathbb {R}$ by

$$ J_{\gamma}^{\infty}=\frac{1}{2} \int_{\mathbb {R}^{N}}\bigl( \vert \nabla v_{n} \vert ^{2}+V_{\infty} \bigl\vert G_{\gamma}^{-1}(v_{n}) \bigr\vert ^{2}\bigr)\,dx -\frac{1}{p} \int_{\mathbb{R}^{N}} \bigl\vert G_{\gamma}^{-1}(v_{n}) \bigr\vert ^{p} \,dx. $$

Then we get a contradiction as in a similar proof to [9, 19, 20] by using the compactness lemma [13]. Hence, $v_{\gamma}$ is a nontrivial solution of Eq. (12). By using the fact that $G_{\gamma}^{-1}(t)\in C^{2}$ together with Lemma 2.1, a direct computation shows that $u=G_{\gamma}^{-1}(v)\in C^{2}(\mathbb{R}^{N})\cap H^{1}(\mathbb{R}^{N})$. If $v_{\gamma}$ is a critical point for $J_{\gamma}$, we know that

$$\begin{aligned} &\int_{\mathbb{R}^{N}} \biggl[\nabla v\nabla\psi+V(x)\frac{G_{\gamma }^{-1}(v)}{g_{\gamma}(G_{\gamma}^{-1}(v))}\psi- \frac{ \vert G_{\gamma }^{-1}(v) \vert ^{p-2}G_{\gamma}^{-1}(v)}{g_{\gamma}(G_{\gamma }^{-1}(v))}\psi\biggr]\,dx=0 \\ &\quad \text{for all }\psi\in H^{1}\bigl( \mathbb{R}^{N}\bigr). \end{aligned}$$

(16)

Taking $\psi=g_{\gamma}(u)\varphi\in C_{0}^{2}(\mathbb {R}^{N})\subset H^{1}(\mathbb{R}^{N})$ in (16), we have

$$ \int_{\mathbb{R}^{N}} \bigl[g_{\gamma}^{2}(u)\nabla u\nabla \varphi+g_{\gamma}(u)g_{\gamma}'(u) \vert \nabla u \vert ^{2} \varphi +V(x)u\varphi+ \vert u \vert ^{p-2}u \varphi\bigr]\,dx=0. $$

It means that u is a classical solution of (11). In the next part of this section, we will prove that $u=G^{-1}(v_{\gamma})$ is the solution of Eq. (1).

If $p\leq4\alpha$, we define the functional $P:H^{1}(\mathbb {R}^{N})\to\mathbb{R}$ by

$$ P(v)=\frac{1}{2} \int_{\mathbb{R}^{N}} \vert \nabla v \vert ^{2} \,dx+2V_{\infty} \int_{\mathbb{R}^{N}} \vert v \vert ^{2}\,dx- \frac{1}{p} \biggl[\frac{4(4\alpha-p)}{16-2\alpha-3p} \biggr]^{\frac{p}{2}} \int _{\mathbb{R}^{N}} \vert v \vert ^{p} \,dx. $$

Then the function v satisfies the equation

$$ -\Delta v+4V_{\infty}v= \biggl[\frac{4(4\alpha-p)}{16-2\alpha -3p} \biggr]^{\frac{p}{2}} \vert v \vert ^{p-2}v,\quad x\in\mathbb{R}^{N}. $$

(17)

From Jeanjean and Tanaka [10], if we consider the set $\varGamma=\{\xi\in C([0,1],H^{1}(\mathbb{R}^{N})):\xi(0)=0,\xi(1)\neq 0, P(\xi(1))<0 \}$. Then $m=\inf_{\xi\in\varGamma} \sup_{t\in[0,1]} P(\xi(t))$ is the least energy level of the functional $P(v)$.

Since $v_{\gamma}$ is a critical point of $J_{\gamma}$, one has

$$ pc_{\gamma}=pJ_{\gamma}(v_{\gamma})-\bigl\langle J_{\gamma}'(v_{\gamma}),G_{\gamma}^{-1}(v_{\gamma})g_{\gamma } \bigl(G_{\gamma}^{-1}(v_{\gamma})\bigr)\bigr\rangle \ge \frac{(p-2)(4\alpha -p)}{16-2\alpha-3p} \int_{\mathbb{R}^{N}} \vert \nabla v_{\gamma} \vert ^{2}\,dx. $$

This indicates that

$$ \Vert \nabla v_{\gamma} \Vert _{2}^{2}\le \frac{p(16-2\alpha -3p)}{(p-2)(4\alpha-p)}c_{\gamma}. $$

Furthermore, from the property (7) of Lemma 2.1, we can deduce that $J_{\gamma}(v)\le P(v)$ and $\varGamma\subset\varGamma_{\gamma}$. Hence

$$ c_{\gamma}=\inf_{\xi\in\varGamma_{\gamma}} \sup_{t\in[0,1]} J_{\gamma}\bigl(\xi(t)\bigr)\le\inf_{\xi\in\varGamma} \sup _{t\in[0,1]} J_{\gamma}\bigl(\xi(t)\bigr)\le\inf _{\xi\in\varGamma} \sup_{t\in[0,1]} P\bigl(\xi(t)\bigr):=m $$

and

$$ \Vert \nabla v_{\gamma} \Vert _{2}^{2} \le\frac{p(16-2\alpha -3p)}{(p-2)(4\alpha-p)}m. $$

(18)

Using the Sobolev inequality, we can get

$$ \Vert v_{\gamma} \Vert _{2^{*}}\le\sqrt{\frac{p m(16-2\alpha -3p)}{S(p-2)(4\alpha-p)}}, $$

(19)

where S is the best Sobolev constant.

If $p\geq4\alpha$, we define the function $P:H^{1}(\mathbb{R}^{N})\to \mathbb{R}$ by

$$ P(v)=\frac{1}{2} \int_{\mathbb{R}^{N}} \vert \nabla v \vert ^{2} \,dx+2V_{\infty} \int_{\mathbb{R}^{N}} \vert v \vert ^{2}\,dx- \frac{1}{p} \biggl[\frac{4(6-p)}{22-3p} \biggr]^{\frac{p}{2}} \int_{\mathbb{R}^{N}} \vert v \vert ^{p} \,dx, $$

the set Γ and m are defined like $p\leq4\alpha$. In this time, if $v_{\gamma}$ is a critical point of $J_{\gamma}$,

$$ pc_{\gamma}=pJ_{\gamma}(v_{\gamma})-\bigl\langle J_{\gamma}'(v_{\gamma}),G_{\gamma}^{-1}(v_{\gamma})g_{\gamma } \bigl(G_{\gamma}^{-1}(v_{\gamma})\bigr)\bigr\rangle \ge \frac {(p-2)(6-p)}{22-3p} \int_{\mathbb{R}^{N}} \vert \nabla v_{\gamma} \vert ^{2}\,dx. $$

Hence, we can deduce that

$$ \Vert \nabla v_{\gamma} \Vert _{2}^{2} \le\frac{p(22-3p)}{(p-2)(6-p)}m $$

(20)

and

$$ \Vert v_{\gamma} \Vert _{2^{*}}\le S^{-\frac{1}{2}} \Vert \nabla v_{\gamma} \Vert _{2}\le\sqrt{\frac{pm(22-3p)}{S(p-2)(6-p)}}. $$

(21)

Then, by the same proof as Proposition 3.6 of [20], we can deduce that there exists a constant $K>0$ independent of γ such that $\|v_{\gamma}\|_{\infty}\le K$. If $p\leq4\alpha$, let $\gamma_{0}:=\frac{p-2}{32\alpha^{2}(4\alpha -p)(2K)^{2(2\alpha-1)}}$, we have

$$ \Vert u_{\gamma} \Vert _{\infty}= \bigl\Vert G_{\gamma}^{-1}(v_{\gamma }) \bigr\Vert \le2 \Vert v_{\gamma} \Vert _{\infty}\le2K\le \biggl(\frac{p-2}{32\alpha ^{2} \gamma(4\alpha-p)} \biggr)^{\frac{1}{2(2\alpha-1)}}\quad\text{for all }\gamma\in(0,\gamma_{0}). $$

If $p\geq4\alpha$, let $\gamma_{0}:=\frac{p-2}{32\alpha ^{2}(6-p)(2K)^{2(2\alpha-1)}}$, we get

$$ \Vert u_{\gamma} \Vert _{\infty}= \bigl\Vert G_{\gamma}^{-1}(v_{\gamma }) \bigr\Vert \le2 \Vert v_{\gamma} \Vert _{\infty}\le2K\le \biggl(\frac{p-2}{32\alpha ^{2} \gamma(6-p)} \biggr)^{\frac{1}{2(2\alpha-1)}}\quad\text{for all }\gamma\in(0,\gamma_{0}). $$

Hence, we can deduce that $g_{\gamma}(u_{\gamma})=\sqrt{1+2\alpha ^{2}\gamma|u_{\gamma}|^{2(2\alpha-1)}}$ if $\gamma\in(0,\gamma _{0})$ and so $u_{\gamma}=G_{\gamma}^{-1}(v_{\gamma})$ is a positive solution of (1). □