2.1 The subcritical case

For the subcritical exponential growth case, we make the following assumptions on the nonlinearity $f$.

1. (f ₁) For every $\theta >0$, there exists $C_\theta >0$ such that $|f(s)|\leq C_\theta \min \{1,|s|\}\,{\rm e}^{\theta |s|^2}, \forall s>0$.

2. (f ₂) $f\in C(\mathbb {R},\mathbb {R})$ and $f(s)=o(s)$ as $s\rightarrow 0$, $f(s)\equiv 0$ for $s\leq 0$ and $f(s)>0$ for $s>0$.

3. (f ₃) The function $\frac {f(t)t-F(t)}{t^3}: t\mapsto \mathbb {R}$ is nondecreasing in $(0,+\infty )$.

4. (f ₄) There exist $M_0>0$ and $t_0>0$ such that

   \[ F(t)\leq M_0|f(t)|,\quad \forall |t|\geq t_0. \]

Now we state our result about the existence of positive ground state solutions to problem (1.3).

Theorem 2.2 Assume (V1)–(V3) and $(f_1)$–$(f_4)$ hold. Then, for any $\nu >0$, problem (1.3) has at least a positive ground state solution $u\in X$ satisfying

(2.2)\begin{equation} \left|\int_{\mathbb{R}^2}\int_{\mathbb{R}^2}\ln|x-y|u^2(x)u^2(y)\,{\rm d}\kern 0.06em x\,{\rm d}y\right|<{+}\infty. \end{equation}

Remark 2.3 In the present paper, we can also endow $X$ with the norm (see [Reference Chen and Tang19, Reference Chen and Tang20])

\[ \|u\|_{X}^{2}=\int_{\mathbb{R}^{2}}\left(|\nabla u|^{2}+V(x)|u|^{2}\right) {\rm d}\kern 0.06em x+\int_{\mathbb{R}^{2}} \ln (2+|x|)|u(x)|^{2}\,{\rm d}\kern 0.06em x, \]

from which we can easily see that

\[ \int_{\mathbb{R}^{2}} \ln (2+|x|)|u(x)|^{2}\,{\rm d}\kern 0.06em x>\int_{\mathbb{R}^{2}} \ln 2|u(x)|^{2}\,{\rm d}\kern 0.06em x. \]

As a clear estimate on bound from below for the term $\int _{\mathbb {R}^{2}} \ln (2+|x|)|u(x)|^{2}\,{\rm d}\kern 0.06em x$, it is good for us to develop an energy estimate inequality in analysis. Thus, it seems possible to improve condition (V3) by relaxing the lower bound of $V$.

Remark 2.4 In order to obtain the compactness directly, the authors in [Reference Chen and Tang19–Reference Chen and Tang22, Reference Wen, Chen and Rădulescu55] studied system (1.3) in an axially symmetric space $E:=X\cap H_{as}^1$ with

\[ H_{as}^1=\left\{u\in H^1(\mathbb{R}^2):\, u(x):=u(x_1,x_2)=u(|x_1|,|x_2|),\ \forall x\in\mathbb{R}^2\right\}. \]

This is a natural constraint set, since critical points of the functional $I$ restricted to $E$ are also critical points of the functional $I$ in $X$. More importantly, for any axially symmetric function $u\in E$, by decomposing the convolution term in functional $I$, they can obtain an estimate

\[ \int_{\mathbb{R}^4}\ln(1+|x-y|)u^2(x)u^2(y)\,{\rm d}\kern 0.06em x\,{\rm d}y\geq \frac{1}{4}\|u\|_2^2\int_{\mathbb{R}^2}\ln(1+|x|)u^2\,{\rm d}\kern 0.06em x,\quad u\in E, \]

which is crucial in proving that $I$ satisfies the Cerami condition at arbitrary energy level in $E$. Moreover, compared with [Reference Chen and Tang21] where the authors studied the case of subcritical polynomial growth, and used a monotonicity condition on $V$:

\begin{align*} & V\in C^1(\mathbb{R}^2,\mathbb{R}),\, t\mapsto t^2[2V(tx)\\ & \quad - \nabla V(tx)\cdot(tx)]\ \text{is nondecreasing in}\ (0,\infty)\ \text{for all}\ x\in\mathbb{R}^2 \end{align*}

to prove the existence of ground state solutions of (1.3), this condition is removed in theorem 2.2. However, we do not provide a minimax characterization of this ground state energy.

In addition to the difficulties that the quadratic part of $I$ is not coercive on $X$ and the norm of $X$ is not translation invariant, the boundedness and compactness of any Cerami sequence $\{u_n\}$ associated with functional $I$ are also the main obstacles that we need to overcome in our arguments. In the autonomous or periodic case, see [Reference Alves and Figueiredo5], some strong compactness lemma can be established with the help of the translation invariance of $I$. However, their approaches do not work any more for the non-autonomous equation (1.3), since it seems difficult to construct a Cerami sequence satisfying asymptotically some Pohozaev identity due to the lack of the translation invariance of $I$.

In contrast to the symmetry or axial symmetry case (see [Reference Chen and Tang21]), where one can establish a new inequality on $V_1(u)$ to prove the boundedness of any Cerami sequence $\{u_n\}$ for functional $I$, as mentioned in remark 2.4, it becomes tougher to prove the boundedness of $\{u_n\}$ in $X$ for our case without any symmetry assumption on $V$. To bypass this obstacle, a new nonlocal perturbation approach is introduced to prove firstly that any Cerami sequence $\{u_n\}$ is bounded in $H^1(\mathbb {R}^2)$. And then by virtue of Lions’ vanishing lemma combined with estimates on the convolution term, we can rule out the vanishing case for sequence $\{u_n\}$. Finally, a delicate analysis on the mountain-pass values corresponding to the functional $I$ and the associated limit functional $I_\infty$ is given to exclude energy bubbling of sequence $\{u_n\}$ at infinity.

2.2 The critical case

We now turn our attention to the critical case. In the literature, there have been a few results on planar Schrödinger–Poisson systems with critical growth of Trudinger–Moser type. Recently, Alves and Figueiredo [Reference Alves and Figueiredo5] investigated the existence of positive ground state solutions for (1.3) when $V(x)\equiv 1$ and $f$ satisfies ($f_0$),($f_2$) and the following conditions:

1. (f'₃) $\frac {f(t)}{t^3}$ is increasing in $(0,\infty )$;

2. (f ₅) there exists $\theta >2$ such that $0<\theta F(t)\leq f(t)t$ for all $t>0$;

3. (f ₆) there exist constants $p>4$ and

   \[ \lambda_0>\max\left\{1,\left[\frac{2(p-2)\alpha_0c_p}{\pi(p-4)}\right]^{(p-2)/2}\right\} \]
   such that $f(t)\geq \lambda _0t^{p-1}$ for $t\geq 0$, where $c_p=\inf _{\mathcal {N}_p} I_p$ with
   \[ \mathcal{N}_p:=\{u\in X{\setminus}\{0\}:\,I'_p(u)u=0\} \]
   and
   \begin{align*} I_p(u)& =\frac{1}{2}\int_{\mathbb{R}^2}(|\nabla u|^2+u^2)\,{\rm d}\kern 0.06em x\\ & \quad +\frac{\nu}{8\pi}\int_{\mathbb{R}^4}\ln(|x-y|)u^2(x)u^2(y)\,{\rm d}\kern 0.06em x{\rm d}y-\frac{1}{p}\int_{\mathbb{R}^2}|u|^p\,{\rm d}\kern 0.06em x. \end{align*}

Observe from [Reference Alves and Figueiredo5] that the monotonicity condition ($f'_3$) is often used to guarantee the boundedness of the Palais–Smale sequence $\{u_n\}$ associated with $I$. With the aid of ($f_6$), the authors established directly an estimate on the norm of sequence $\{u_n\}$ in $H^1(\mathbb {R}^2)$. Then thanks to the Trudinger–Moser inequality, the compactness was obtained. However, as a global condition, ($f_6$) requires $f(t)$ to be super-cubic for all $t\geq 0$, which seems a little bit strict especially for $t>0$ small.

Later, under weaker assumptions than ($f'_3$) and ($f_6$), Chen and Tang [Reference Chen and Tang22] studied the existence of nontrivial solutions to (1.3) when $f(u)$ is replaced by $f(x,u)$. Motivated by [Reference de Figueiredo, Miyagaki and Ruf24], $f(x,u)$ is required to satisfy the following conditions:

1. (F1) $f(x,t)t>0$ for all $(x,t)\in \mathbb {R}^2\times (\mathbb {R}{\setminus} \{0\})$ and there exist $M_0>0$ and $t_0>0$ such that

   \[ F(x,t)\leq M_0|f(x,t)|,\quad \forall x\in\mathbb{R}^2,\ |t|\geq t_0; \]

2. (F2) $\liminf _{t\rightarrow \infty }\frac {t^2F(x,t)}{{\rm e}^{\alpha _0 t^2}}\geq \kappa >\frac {2}{\alpha _0^2\rho ^2}$ where $\rho \in (0,1/2)$ such that $\rho ^2\max _{|x|\leq \rho } V(x) \leq 1$.

Conditions (F1) and (F2) are devoted to giving a sharp estimate on the minimax level to guarantee that any Cerami sequence or any minimizing sequence $\{u_n\}$ of the associated functional does not vanish. Moreover, (F1) together with a weaker monotonicity condition

1. (f ₇) for all $x\in \mathbb {R}^2$, the mapping $(0,\infty )\ni t\mapsto \displaystyle \frac {f(t)-V(x)t}{t^3}$ is non-decreasing,

than ($f'_3$) can be used to verify that the weak limit of any Cerami sequence $\{u_n\}$ is a nontrivial solution of (1.3) in [Reference Chen and Tang22]. It is worth pointing out that the authors in [Reference Chen and Tang22] need to introduce some sort of axially symmetric assumptions on $V$ and $f$.

Another feature of the present paper is that we study the existence of nontrivial solutions to system (1.3) without hypotheses (F1) and (F2). Let $\mathcal {S}_2$ be the best constant of Sobolev embedding $H_0^1(B_{1/4}(0))\hookrightarrow L^2(B_{1/4}(0))$, that is,

(2.3)\begin{equation} \mathcal{S}_2\left(\int_{B_{1/4}(0)}u^2\,{\rm d}\kern 0.06em x\right)^{1/2}\leq\left(\int_{B_{1/4}(0)}|\nabla u|^2+V(x)u^2\,{\rm d}\kern 0.06em x\right)^{1/2}, \end{equation}

which has been one well-known fact. Moreover, the compactness of the embedding guarantees the achievement of $\mathcal {S}_2$, since $B_{1/4}(0)$ is a specific bounded domain. Recently, one fine bound of constant $\mathcal {S}_2$ has been also obtained in Du [Reference Du25], which makes possible to give one specific estimate of $\nu$ from below in the following result.

Theorem 2.5 Assume that conditions (V2)–(V3) and (f$_0$), (f$_2$), (f$_7$) hold. Then, for

\[ \nu>\frac{\alpha_0 \mathcal{S}_2^4}{4\pi\ln 2}, \]

equation (1.3) has at least a positive ground state solution $u\in X$ satisfying

(2.4)\begin{equation} \left|\int_{\mathbb{R}^2}\int_{\mathbb{R}^2}\ln|x-y|u^2(x)u^2(y)\,{\rm d}\kern 0.06em x\,{\rm d}y\right|<{+}\infty. \end{equation}

Essentially, we adopt some ideas used in the subcritical exponential growth case to prove theorem 2.5. However, instead of the nonlocal perturbation approach used in theorem 2.2, the boundedness of any Cerami sequence $\{u_n\}$ in $H^1(\mathbb {R}^2)$ is obtained by a contradiction argument combining with Lions’ vanishing lemma.

In order to obtain the existence of weak solutions to the limiting problem and further to exclude energy bubbling of sequence $\{u_n\}$ at infinity, a refined estimate on the mountain pass value is given under an additional restriction on $\nu$.

Throughout the paper, we need the following notations. Denote by $L^s(\mathbb {R}^2)$, $s\in [1,\infty ]$ the usual Lebesgue space with the norm $\|\cdot \|_s$. For any $r>0$ and any $z\in \mathbb {R}^2$, $B_r(z)$ stands for the ball of radius $r$ centred at $z$. $X^*$ denotes the dual space of $X$. At last, $C,C_1,C_2,\ldots$ denote various positive generic constants.

4.1 Proof of theorem 2.2

In view of lemma 4.5, for fixed $\lambda \in (0,1]$, we have $I'_{\lambda }(u_\lambda )=0$ for some $u_\lambda \in X{\setminus} \{0\}$. Choosing a sequence $\{\lambda _n\}\subset (0,1]$ satisfying $\lambda _n\rightarrow 0^+$, there exists a sequence of nontrivial critical points $\{u_{\lambda _n}\}$(still denoted by $\{u_n\}$) of $I_{\lambda _n}$ with $I_{\lambda _n}(u_n)=c_{\lambda _n}$. We now show that $\{u_n\}$ is bounded in $H^1(\mathbb {R}^2)$. In fact, according to the definition of $I'_{\lambda _n}(u_n)u_n=0$, we have

(4.20)\begin{equation} V_0(u_n)={-}(\|\nabla u_n\|_2^2+\int_{\mathbb{R}^2}V(x)u_n^2\,{\rm d}\kern 0.06em x) -\lambda_n\|u_n\|_2^{5/2} +\lambda_n\|u_n\|_r^r+\int_{\mathbb{R}^2}f(u_n)u_n\,{\rm d}\kern 0.06em x. \end{equation}

Substituting $V_0(u_n)$ into $P_\lambda (u_n)=0$ gives

(4.21)\begin{align} & -\|\nabla u_n\|_2^2 +\frac{1}{2}\int_{\mathbb{R}^2}(\nabla V(x),x)u_n^2+\frac{1}{4}\|u_n\|_2^4\nonumber\\ & \quad +\int_{\mathbb{R}^2}f(u_n)u_n-2F(u_n)\,{\rm d}\kern 0.06em x +\frac{r-2}{r}\lambda_n\|u_n\|_r^r=0. \end{align}

We use the same fashion to get

(4.22)\begin{align} I_{\lambda_n}(u_n)& =\frac{1}{4}\|\nabla u_n\|_2^2+\frac{1}{4}\int_{\mathbb{R}^2}V(x)u_n^2\,{\rm d}\kern 0.06em x\nonumber\\ & \quad+\int_{\mathbb{R}^2}\frac{1}{4}f(u_n)u_n-F(u_n)\,{\rm d}\kern 0.06em x+\frac{3\lambda_n}{20}\|u_n\|_2^{5/2} +\frac{\lambda_n(r-4)}{4r}\|u_n\|_r^r. \end{align}

Putting (4.22) into (4.21), we get

(4.23)\begin{equation} \begin{aligned} & -[4I_{\lambda_n}(u_n)-\int_{\mathbb{R}^2}V(x)u_n^2\,{\rm d}\kern 0.06em x+\left(\frac{4}{r}-1\right)\lambda_n\|u_n\|_r^r\\ & \quad +\int_{\mathbb{R}^2}4F(u_n)-f(u_n)u_n\,{\rm d}\kern 0.06em x-\frac{3\lambda_n}{5}\|u_n\|_2^{5/2}] \\ & +\frac{1}{2}\int_{\mathbb{R}^2}(\nabla V(x),x)u_n^2 +\frac{1}{4}\|u_n\|_2^4 +\int_{\mathbb{R}^2}f(u_n)u_n-2F(u_n)\,{\rm d}\kern 0.06em x +\frac{r-2}{r}\lambda_n\|u_n\|_r^r=0, \end{aligned} \end{equation}

which can be rewritten as

(4.24)\begin{align} & \frac{1}{4}\|u_n\|_2^4+\int_{\mathbb{R}^2}V(x)u_n^2\,{\rm d}\kern 0.06em x+\left(2-\frac{6}{r}\right)\lambda_n\|u_n\|_r^r \nonumber\\ & \quad + \frac{1}{2}\int_{\mathbb{R}^2}(\nabla V(x),x)u_n^2\,{\rm d}\kern 0.06em x+\frac{3\lambda_n}{5}\|u_n\|_2^{5/2}\nonumber\\ & \quad +\int_{\mathbb{R}^2}2f(u_n)u_n-6F(u_n){\rm d}\kern 0.06em x=4I_{\lambda_n}(u_n). \end{align}

From condition ($f_3$), we can conclude that $f(u_n)u_n\geq 3F(u_n)$, see lemma 4.2 in [Reference Chen and Tang21]. Thus, (4.24) implies that $\{u_n\}$ is bounded in $L^2(\mathbb {R}^2)$ uniformly for $n$. Moreover, observe from (4.24) that

(4.25)\begin{equation} \begin{aligned} & \int_{\mathbb{R}^2}V(x)u_n^2\,{\rm d}\kern 0.06em x+\frac{1}{2}\int_{\mathbb{R}^2}\nabla V(x)\cdot xu_n^2\,{\rm d}\kern 0.06em x\leq C,\\ & \left(2-\frac{6}{r}\right)\lambda_n\|u_n\|_r^r\leq C,\quad \frac{3\lambda_n}{5}\|u_n\|_2^{5/2}\leq C,\\ & \int_{\mathbb{R}^2}2f(u_n)u_n-6F(u_n)\,{\rm d}\kern 0.06em x\leq C. \end{aligned} \end{equation}

In view of lemma 4.1, we have

(4.26)\begin{align} & 2I'_{\lambda_n}(u_n)u_n-P_{\lambda_n}(u_n)= 2\|\nabla u_n\|_2^2\nonumber\\ & \qquad +\int_{\mathbb{R}^2} (V(x)-\frac{1}{2}\nabla V(x)\cdot x) u_n^2\,{\rm d}\kern 0.06em x+\lambda_n\|u_n\|_2^{5/2} +V_0(u)\nonumber\\ & \qquad-\frac{1}{4}\|u_n\|_2^4-2\int_{\mathbb{R}^2}(f(u_n)u_n-F(u_n))\,{\rm d}\kern 0.06em x-\frac{2(r-1)\lambda_n}{r}\|u_n\|_r^r=0. \end{align}

For $t>0$, from (4.26) we deduce that

(4.27)\begin{align} & I_{\lambda_n}(u_n)-I_{\lambda_n}(u_{nt})= \frac{1-t^4}{2}\|\nabla u_n\|_2^2 +\frac{1}{2}\int_{\mathbb{R}^2} [V(x)-t^2V(t^{{-}1}x)] u_n^2\,{\rm d}\kern 0.06em x+\lambda_n\frac{2(1-t^{5/2})}{5}\|u_n\|_2^{5/2}\nonumber\\ & \quad+\frac{1-t^4}{4}V_0(u_n) +\frac{t^4\ln t}{4}\|u_n\|_2^4+\int_{\mathbb{R}^2}\left[\frac{F(t^2u_n)}{t^2}-F(u_n)\right]{\rm d}\kern 0.06em x-\frac{(1-t^{2(r-1)})\lambda_n}{r}\|u_n\|_r^r\nonumber\\ & \quad= \frac{1-t^4}{4}[2I'_{\lambda_n}(u_n)u_n-P_{\lambda_n}(u_n)]+\frac{1}{t^2}F(t^2u_n)\bigg]{\rm d}\kern 0.06em x\nonumber\\ & \quad+\frac{1}{4}\int_{\mathbb{R}^2}\left[(1+t^4)V(x)-2t^2V(t^{{-}1}x) +\frac{1-t^4}{2}\nabla V(x)\cdot x\right]u_n^2\,{\rm d}\kern 0.06em x\nonumber\\ & \quad+\frac{1-t^4+4t^4\ln t}{16}\|u_n\|_2^4+\int_{\mathbb{R}^2}\left[\frac{1-t^4}{2}f(u_n)u_n +\frac{t^4-3}{2}F(u_n)\right.\nonumber\\ & \quad+\left(\frac{3}{20}+\frac{t^4}{4}-\frac{2t^{5/2}}{5}\right)\lambda_n\|u_n\|_2^{5/2}+\left[\frac{(r-1)(1-t^4)}{2r}-\frac{(1-t^{2(r-1)})} {r}\right]\lambda_n\|u_n\|_r^r. \end{align}

where $u_{nt}(x)=t^2u_{n}(tx)$. Now we show that $\{\|\nabla u_n\|_2\}$ is bounded uniformly for $n$. Suppose by contradiction that $\|\nabla u_n\|_2\rightarrow \infty$. Take $t_n=(\sqrt {M}/\|\nabla u_n\|_2)^{1/2}$ for some $M>0$ large, then $t_n\rightarrow 0$. Obviously, $t_n^4\ln t_n\rightarrow 0$. Letting $t=t_n$ in (4.27), since $\{u_n\}$ is bounded in $L^2(\mathbb {R}^2)$ uniformly for $n$, by (V2) and ($f_1$) and (4.4), we have for large $t_n$

(4.28)\begin{align} & I_{\lambda_n}(u_n)-I_{\lambda_n}(t_n^2u_{nt_{n}})\nonumber\\ & = \frac{1}{4}\int_{\mathbb{R}^2}\left[V(x) +\frac{1}{2}\nabla V(x)\cdot x\right]u_n^2\,{\rm d}\kern 0.06em x\nonumber\\ & \quad+\frac{1}{16}\|u_n\|_2^4+\int_{\mathbb{R}^2}\left[\frac{1}{2}f(u_n)u_n-\frac{3}{2}F(u_n)\right]{\rm d}\kern 0.06em x,\nonumber\\ & \quad+ \frac{3}{20}\lambda_n\|u_n\|_2^{5/2}+\frac{r-3}{2r}\lambda_n\|u_n\|_r^r+o(1). \end{align}

Therefore, it follows from (3.3), (4.4), (4.25), (4.28) and (V1), ($f_1$), the Gagliardo–Nirenberg inequality that

(4.29)\begin{align} c_{\lambda_n}& =I_{\lambda_n}(u_n)\geq I_{\lambda_n}(t_n^2u_{nt_{n}})+o(1)\nonumber\\ & = \frac{t_n^4}{2}\|\nabla u_n\|_2^2+\frac{t_n^4}{4}V_0(u_n)-\frac{t_n^4\ln t_n}{4}\|u_n\|_2^4 -\frac{1}{t_n^2}\int_{\mathbb{R}^2}F(t_n^2u_n)\,{\rm d}\kern 0.06em x\nonumber\\ & \quad+\frac{1}{2}\int_{\mathbb{R}^2} t_n^2V(t_n^{{-}1}x) u_n^2\,{\rm d}\kern 0.06em x+\lambda_n\frac{2t_n^{5/2}}{5}\|u_n\|_2^{5/2}-\frac{t_n^{2r-2}}{r}\lambda_n\|u_n\|_r^{r}+o(1)\nonumber\\ & \geq\frac{t_n^4}{2}\|\nabla u_n\|_2^2-\frac{t_n^4}{4}V_2(u_n)-t_n^{2(p-1)}C\|u_n\|_p^p+o(1)\nonumber\\ & \geq\frac{M}{2}-\frac{t_n^4C}{4}\|u_n\|_2^3\|\nabla u_n\|_2-t_n^{2(p-1)}C\|u_n\|_2^2\|\nabla u_n\|_2^{p-2}+o(1)\nonumber\\ & \geq\frac{M}{2}-\frac{CM}{4\|\nabla u_n\|_2}\|u_n\|_2^3-\frac{CM^{(p-1)/2}}{\|\nabla u_n\|_2}\|u_n\|_2^2+o(1), \end{align}

which, together with the fact that $\|\nabla u_n\|_2\to +\infty$, implies a contradiction by letting $M>0$ large enough. Hence, $\{u_n\}$ is bounded in $H^1(\mathbb {R}^2)$. Arguing similarly as in lemma 4.5, we obtain that $u_n\rightarrow u_0$ in $X$. Moreover, by letting $n\rightarrow +\infty$, one has $c_{\lambda _n}\rightarrow c^*>a>0$ and $I_{\lambda _n}(u_{n})=c^*+o(1)$. Here, $a$ is positive number given in (4.2) below. For any $\varphi \in C_0^\infty (\mathbb {R}^2)$, we have

\[ I'_{\lambda_n}(u_{n})\varphi= I'(u_{n})\varphi+\lambda_n\|u_n\|_2^\frac{1}{2}\int_{\mathbb{R}^2}u_{n}\varphi\,{\rm d}\kern 0.06em x +\lambda_n\int_{\mathbb{R}^2}|u_n|^{r-2}u_n\varphi\,{\rm d}\kern 0.06em x=o(1)\|\varphi\|_X. \]

Thus, $\{u_n\}$ is a Palais–Smale sequence of $I$ with level $c^*$. Therefore, arguing as that in lemma 4.5, there exists a nontrivial $u_0\in X$ such that $I'(u_0)=0$ and $I(u_0)=c^*$. Now let us define the set of solutions

\[ \mathcal{S}:=\{u\in X{\setminus}\{0\}:\,I'(u)=0\}. \]

It is clear that $\mathcal {S}\not =\emptyset$ and $\mathcal {S}$ is bounded away from zero. To be precise, a short estimate yields that for any $u\in \mathcal {S}$, the following holds:

(4.30)\begin{equation} \|u\|\geq C\quad\text{for}\; \text{some}\ C>0. \end{equation}

We claim that

\[ c_*:=\inf\limits_{u\in\mathcal{S}}I(u)>0. \]

For a contradiction, we assume $c_*=0$. Then there exists $\{u_n\}\subset \mathcal {S}$ such that $I(u_n)\rightarrow 0$ as $n\rightarrow \infty$. In view of (4.24), $I(u)\geq \frac {1}{16}\|u\|_2^4$ for all $u\in \mathcal {S}$. So, $\|u_n\|_2^4\rightarrow 0$. Arguing as above, we have $u_n$ is bounded in $H^1(\mathbb {R}^2)$ uniformly for $n$. Using the Gagliardo–Nirenberg inequality:

\[ \|u_n\|_p^p\leq C_p\|u_n\|_2^2\|\nabla u_n\|_2^{p-2} \]

we infer that $u_n\rightarrow 0$ in $L^p(\mathbb {R}^2)$ for $p\in [2,+\infty )$. By (4.13), we have ${\int _{\mathbb {R}^2}f(u_n)u_n{\rm d}\kern 0.06em x=o(1)}$ for large $n$, and furthermore $u_n\rightarrow 0$ in $H^1(\mathbb {R}^2)$ which contradicts (4.30). The claim is true. Take finally a minimizing sequence $\{u_n\}\subset \mathcal {S}$ so that $I(u_n)\rightarrow c_*$. Similarly to lemma 4.5, there exists $u_{*}\in X$ so that $u_n\rightarrow u_{*}$ in $X$ and $I'(u_{*})=0$. It follows that $u_{*}$ is a positive ground state solution of problem (1.3).

5.1 Proof of theorem 2.5

Similarly to theorem 2.2, let us define the set of solutions

\[ \mathcal{\tilde{S}}:=\{u\in X{\setminus}\{0\}:\,I'(u)=0\}. \]

By lemma 5.3, $\mathcal {\tilde {S}}\not =\emptyset$. For $u\in \mathcal {\tilde {S}}$ by ($f_0$), for any $\varepsilon >0$ there exists $C_\varepsilon >0$ such that

\begin{align*} \|u\|^2& \leq \|u\|^2+\nu V_1(u)\\ & \leq\varepsilon\int_{\mathbb{R}^2}u^2\,{\rm d}\kern 0.06em x+C_\varepsilon\int_{\mathbb{R}^2}({\rm e}^{\alpha_0 u^2}-1)u^3\,{\rm d}\kern 0.06em x+\nu V_2(u)\\ & \leq \varepsilon\|u\|^2+C_\varepsilon\left(\int_{\mathbb{R}^2}\,{\rm e}^{2\alpha_0 u^2}-1)\right)^{\frac{1}{2}}\|u\|_6^3+\nu C\|u\|^4\\ & \leq\varepsilon\|u\|^2+C_\varepsilon C\|u\|^3+\nu C\|u\|^4, \end{align*}

which implies that there exists $C>0$ such that $\|u\|\geq C$ for any $u\in \mathcal {\tilde {S}}$.

We claim that

(5.26)\begin{equation} \tilde{c}_*:=\inf\limits_{u\in\mathcal{\tilde{S}}}I(u)>0. \end{equation}

Assume by contradiction that $\tilde {c}_*=0$ and $\{u_n\}\subset \mathcal {\tilde {S}}$ satisfies $I(u_n)\rightarrow 0$ as $n\rightarrow \infty$. Recalling lemma 5.2, we deduce that $\{u_n\}$ is bounded in $H^1(\mathbb {R}^2)$ uniformly for $n$, and then by (5.5), $I(u_n)\geq \frac {1}{4}\|\nabla u_n\|_2^2$. Obviously, $\|\nabla u_n\|_2^2\rightarrow 0$. By the Gagliardo–Nirenberg inequality, we have $u_n\rightarrow 0$ in $L^p(\mathbb {R}^2)$ for $p>2$. The Trudinger–Moser inequality implies $\int _{\mathbb {R}^2}f(u_n)u_n{\rm d}\kern 0.06em x=o(1)$ as $n\rightarrow \infty$. And so using $I'(u_n)=0$ and (3.3), we have $V_1(u_n)\rightarrow 0$ and $u_n\rightarrow 0$ in $H^1(\mathbb {R}^2)$ which is impossible. (5.26) holds true. It is easy to obtain from lemma 5.1 that $\tilde {c}_*<\frac {\pi }{\alpha _0}$.

Finally, let $\{u_n\}\subset \mathcal {\tilde {S}}$ be a minimizing sequence, hence $I(u_n)\rightarrow \tilde {c}_*\in \left (0,\frac {\pi }{\alpha _0}\right )$. By lemma 5.2, we know that $\{u_n\}$ is bounded in $H^1(\mathbb {R}^2)$. Similar to the proof of lemma 5.3, there exists $\tilde {u}_{*}\in X$ such that $u_n\rightarrow \tilde {u}_{*}$ in $X$ and $I'(\tilde {u}_{*})=0$ and $I(\tilde {u}_{*})=\tilde {c}_*$. Thus, $\tilde {u}_{*}$ is a positive ground state solution of problem (1.3). The proof is now complete.