Existence of solution for a class of activator–inhibitor systems

Abstract

We prove the existence of a solution for a class of activator–inhibitor system of type $- \Delta u +u = f(u) -v$ , $-\Delta v+ v=u$ in $\mathbb{R}^{N}$ . The function f is a general nonlinearity which can grow polynomially in dimension $N\geq 3$ or exponentiallly if $N=2$ . We are able to treat f when it has critical growth corresponding to the Sobolev space we work with. We transform the system into an equation with a nonlocal term. We find a critical point of the corresponding energy functional defined in the space of functions with norm endowed by a scalar product that takes into account such nonlocal term. For that matter, and due to the lack of compactness, we deal with weak convergent minimizing sequences and sequences of Lagrange multipliers of an action minima problem.

1. Introduction

In this paper, we prove the existence of a solution of the following system

(1.1) \begin{equation} \left\{\!\begin{array}{lll}-\Delta u + u & = f(u) -v & \ \ \mbox{in} \ \ \mathbb{R}^{N} \\ -\Delta v + v & = u & \ \ \mbox{in} \ \ \mathbb{R}^{N},\end{array}\right.\end{equation}

where $f\;:\;\mathbb{R} \to \mathbb{R}$ is a continuous function which is allowed to have critical growth: polynomial in case $N\geq 3$ and exponential if $N=2$ , quoted in (1.2)–(1.6) below.

System (1.1) fits in the class of activator–inhibitor models, where u is the concentration of the activator and v is the concentration of some inhibitor. The function $f(u)-v$ is the kinetics and it is responsible for the balance between u and v in the medium given by $\mathbb{R}^{N}$ . The general theory of such systems can be found in [37] and at Remark 3 below.

Let $H_{rad}^1(\mathbb{R}^N)$ be the space of $H^1$ functions defined in $\mathbb{R}^N$ which are radially symmetric. A weak solution of (1.1) is a pair $(u,v) \in H^{1}_{rad}(\mathbb{R}^{N})\times H^{1}_{rad}(\mathbb{R}^{N})$ with radially symmetric components that satisfies

\begin{equation*}\displaystyle\int_{\mathbb{R}^{N}}\nabla u\nabla \phi \ dx +\displaystyle\int_{\mathbb{R}^{N}} u \phi \ dx-\displaystyle\int_{\mathbb{R}^{N}}f(u)\phi \ dx +\displaystyle\int_{\mathbb{R}^{N}}v\phi \ dx =0, \ \ \forall \ \phi\in H^{1}_{rad}(\mathbb{R}^{N})\end{equation*}

and

\begin{equation*}\displaystyle\int_{\mathbb{R}^{N}}\nabla v\nabla \psi \ dx +\displaystyle\int_{\mathbb{R}^{N}}v\psi \ dx -\displaystyle\int_{\mathbb{R}^{N}}u\psi \ dx =0, \ \ \forall \ \psi \in H^{1}_{rad}(\mathbb{R}^{N}).\end{equation*}

Since we intend to find a nonnegative solution, we will assume that $f(s) =0$ for every $s \leq 0$ .

The continuous function $f\;:\;\mathbb{R} \to \mathbb{R}$ is subject to the following conditions:

(1.2) \begin{equation} \mbox{ $\displaystyle\lim_{s \rightarrow 0^+} \frac{f(s)}{s} =0$; }\end{equation}

(1.3) \begin{equation} \mbox{ $\displaystyle \limsup_{s \rightarrow + \infty} \frac{f(s)}{s^{2^{*} -1}} \leq 1$, where we denote $\displaystyle 2^{*}= \frac{2N}{N-2}$; }\end{equation}

(1.4) \begin{equation} \mbox{ $ f(s) s - 2F(s) \geq 0$ for $s > 0$, where $\displaystyle F(s)=\int_0^s f(t) dt$; }\end{equation}

There are $q \in (2, 2^{*})$ in the case $N\geq3$ , $q >2$ in the case $N=2$ and $\tau > \tau^{*}$ such that

(1.5) \begin{equation} \mbox{$f(s) \geq \tau s^{q-1}$ for $s \geq 0$, where $\tau^{*}>0$ will be fixed later;}\end{equation}

(1.6) \begin{equation} \text{ if $N=2$, there exists $\beta_0 >0$ such that $ \forall \beta > \beta_0, \ \lim_{s\rightarrow +\infty}\frac{f(s)}{e^{\beta s^2}}=0 \ \ \mbox{and} \ \ \forall \beta < \beta_0, \lim_{s \rightarrow +\infty}\frac{f(s)}{e^{\beta s^2}}=+\infty. $ }\end{equation}

As we can see, the critical growth for the function f is allowed by (1.3) and (1.6). The constant $\beta_0=0$ in (1.6) represents the subcritical behavior of f, and in this case one has to suppress the $+\infty$ limit. This situation is also covered in this paper. Notice that we are imposing a weaker condition, namely (1.4), than the classical Ambrosetti–Rabinowitz hypothesis on f.

We stress two estimates which will be frequently used henceforth. From (1.2) and (1.3), for every $\varepsilon > 0$ there exists $C_{\varepsilon} > 0$ such that

(1.7) \begin{equation} |\,f(s) | \leq \varepsilon |s|+ C_{\varepsilon} |s|^{2^{*} -1} \quad \hbox{for all $s >0$}\end{equation}

and, then by integration,

(1.8) \begin{equation} |F(s) | \leq \frac{\varepsilon}{2} s^{2} + \frac{C_{\varepsilon}}{2^{*}} |s|^{2^{*} } \quad \hbox{for all $s >0$}.\end{equation}

We intend to use variational methods to get a weak solution of (1.1). We recall that for each $u \in H^{1}(\mathbb{R}^{N})$ , the linear problem

(1.9) \begin{equation} \left. \begin{array}{rll} \Delta v + v= u \ \ \mbox{in} \ \ \mathbb{R}^{N}, \end{array} \right.\end{equation}

has a unique solution $v\in H^{1}(\mathbb{R}^{N})$ , that is,

\begin{equation*}\text{ $v=Bu$, where $B=(-\!\Delta +1)^{-1}$}.\end{equation*}

Notice that B denotes the solution operator associated to (1.9). It is well known that $B\;:\;H^{1}(\mathbb{R}^{N})\rightarrow L^{2}(\mathbb{R}^{N})$ is linear, continuous, symmetric, and positive, that is,

\begin{equation*}\displaystyle\int_{\mathbb{R}^{N}}uBu \ dx\geq 0, \quad \forall u \in H^{1}(\mathbb{R}^{N}).\end{equation*}

Notice that (1.1) is equivalent to the nonlocal problem

(1.10) \begin{equation} \left. \begin{array}{rll} -\Delta u = f(u)-\!u-\!Bu \ \ \mbox{in} \ \ \mathbb{R}^{N}. \end{array} \right.\end{equation}

Using the fact that B is linear, continuous, and positive, we conclude that $X=(H^{1}(\mathbb{R}^{N}), \langle.,.\rangle)$ is a Hilbert space with scalar product

\begin{equation*}\left\langle u,w \right\rangle =\displaystyle\int_{\mathbb{R}^{N}}\nabla u\nabla w \ dx +\displaystyle\int_{\mathbb{R}^{N}}uw \ dx+\displaystyle\int_{\mathbb{R}^{N}}wBu \ dx\end{equation*}

and corresponding norm

\begin{equation*}\|u\|=\left(\displaystyle\int_{\mathbb{R}^{N}}|\nabla u|^{2} \ dx+ \displaystyle\int_{\mathbb{R}^{N}} u^{2} \ dx +\displaystyle\int_{\mathbb{R}^{N}}uBu \ dx \right)^{\frac{1}{2}}.\end{equation*}

Moreover, since B is linear, this norm is bounded from above by the standard norm of $H^{1}(\mathbb{R}^{N})$ and X is continuously embedded in $L^{p}(\mathbb{R})$ for $2\leq p\leq 2^{*}$ .

Let

\begin{equation*}g(u) = f(u) - u - Bu\end{equation*}

and

\begin{equation*}G(u) = F(u) - \frac{1}{2} u^2 - \frac{1}{2} u \; Bu.\end{equation*}

We define the energy functional $I\;:\;X \to \mathbb{R}$ corresponding to (1.10) by

(1.11) \begin{equation} I(u) = \displaystyle\frac{1}{2}\displaystyle\int_{\mathbb{R}^{N}}|\nabla u|^{2} \ dx -\displaystyle\int_{\mathbb{R}^{N}} G(u) \ dx,\end{equation}

which is well defined in view of (1.2)–(1.3). From the Sobolev embeddings, we conclude that $I \in C^1(X ,\mathbb{R})$ and

\begin{equation*}I'(u)w=\displaystyle\frac{1}{2}\displaystyle\int_{\mathbb{R}^{N}}\nabla u \nabla w\ dx -\displaystyle\int_{\mathbb{R}^{N}}g(u)w \ dx \quad \forall u,w \in H^{1}(\mathbb{R}^{N}).\end{equation*}

In synthesis, we see that if u is a critical point of I, then the pair (u, v), with $v=Bu$ , is a weak solution of (1.1).

We consider the set of nontrivial critical points of I, corresponding to nontrivial solutions of (1.10),

\begin{equation*}\Sigma = \{ u \in X\;:\; I'(u) =0, u \neq 0 \}.\end{equation*}

We define the ground state level (or least energy level) associated to (1.10) by

(1.12) \begin{equation} m = \inf_{u \in \Sigma} I(u)\end{equation}

We introduce the constrained minimum

(1.13) \begin{equation} D = \inf_{u \in \mathcal{M}} \displaystyle \frac{1}{2} \int_{\mathbb{R}^{N}}|\nabla u|^{2} \ dx\end{equation}

in the set

(1.14) \begin{equation} \mathcal{M}= \left\{ u \in X\;:\; \int_{\mathbb R^N} G(u) \, dx =1, u \neq 0 \right \} \quad if \quad N \geq 3\end{equation}

or

(1.15) \begin{equation} \mathcal{M}= \left\{ u \in X\;:\;\int_{\mathbb R^{2}} G(u) \, dx =0, u \neq 0 \right \} \quad if \quad N=2.\end{equation}

The underlying idea in the proof of our results is to show that D is attained.

We state the main theorems.

Theorem 1.1. Suppose that $N \geq 3$ and f satisfies (1.2)–(1.5). Then system (1.1) admits a solution (u, v) where each component is nonnegative, radially symmetric, decreasing and such that $u(x) \to 0$ and $v(x) \to 0$ as $|x| \to \infty$ .

Theorem 1.2. Suppose that $N = 2$ and f satisfies (1.2), (1.4), (1.5) and (1.6). Then system (1.1) admits a solution (u, v) where each component is nonnegative, radially symmetric, decreasing and such that $u(x) \to 0$ and $v(x) \to 0$ as $|x| \to \infty$ .

In addition, we define the minimax level associated to the functional I,

(1.16) \begin{equation} b = \inf_{\gamma \in \Gamma} \max_{t \in [0, 1]} I(\gamma(t)),\end{equation}

where

\begin{equation*} \Gamma = \{\gamma \in C\left([0, 1], X \right):\; \gamma(0) =0 \ \hbox{ and } \ I(\gamma(1)) < 0\},\end{equation*}

is nonempty since I satisfies the conditions of the Mountain Pass Theorem.

Remark 1. Focusing only on equation (1.10), the above defined quantities satisfy,

\begin{equation*} m = D = b, \quad if \quad N = 2 \end{equation*}

and

\begin{equation*} m = \frac{1}{N} \left(\frac{N-2}{2N} \right)^{\frac{N-2}{2}} (2D)^{\frac{N}{2}} = b, \quad if \quad N \geq 3. \end{equation*}

The proof follows after accomplishing the arguments of Sections 2–5. The conclusion follows from the ideas of [10, Theorem 3] in dimension $N \geq 3$ and [3, Corollary 1.5] joint with [25] in the case $N=2$ . We will not dwell on this since these equalities are not fundamental in the present paper.

Remark 2. A simple application of the strong maximum principle in the scalar case for a solution pair (u, v) with $u \geq 0$ and $v \geq 0$ , both nontrivial, gives $u \geq 0$ , $u \not\equiv 0$ and $v>0$ . We do not know if $u>0$ , due to the fact that we are not in position to use the maximum principle for systems as in [16, 19, 33], where they adjust the parameters d and $\gamma$ to do so. The system studied in [16] reads as

\begin{equation*} \left\{ \!\!\begin{array}{rll} & - \Delta u = f(u) - v & \ \ \mbox{in} \ \ \mathbb{R}^N \\ & - d \Delta v + \gamma v = u & \ \ \mbox{in} \ \ \mathbb{R}^N, \end{array} \right. \end{equation*}

with $d>0$ and $\gamma > 0$ . In [16], the authors conclude that $u>0$ and $v>0$ , assuming that there exist $\zeta_0 > 0$ and $\gamma_0 >0$ such that $F(\zeta_0) \geq \zeta_0^2/ 2 \gamma_0$ , $\gamma > \max\{\gamma_0, 2 \sqrt{d}\}$ and either

if $f(s) > 0 \ \forall s \geq \zeta_0$ , then $f(s) +\displaystyle \frac{1}{d}\left(\gamma_0-2\sqrt{d}\right)s \geq 0, \ \forall s \in [0, \zeta_0]$

or

if there is $\zeta_1>\zeta_0$ such that $f(\zeta_1) = 0$ , then $f(s)+\displaystyle \frac{1}{d}\left(\gamma_0-2\sqrt{d}\right)s \geq 0, \ \forall s \in [0, \zeta_1].$

Our approach in the present paper contrasts with the procedure adopted in [16, 33], and actually Chen–Tanaka leave the question on page 113 if the constrained minimization (1.13) is a good strategy to treat their problem. We answer affirmatively such question. The system (1.1) was studied in [16] with a subcritical nonlinearity f, where the authors showed that the mountain pass level of the energy functional $\tilde{I}_\gamma$ associated to (1.1) is attained by some pair (u, v), corresponding to a solution for a larger interval of parameters $\gamma,d$ when compared to [33]. The assumptions (1.2)–(1.6) are considerable improvements compared to [16], since we are capable of considering a general nonlinearity f not only with subcritical growth, but also with critical polynomial growth in case $N\geq 3$ or critical exponential behavior if $N=2$ , and we do not need further assumptions on $\gamma$ and d to show existence of solution.

In [16], the authors employ a truncation of f and apply minimax arguments to the corresponding augmented functional $\tilde{I}_\gamma \;:\; \mathbb{R} \times H_{rad}^1 (\mathbb{R}^N) \times H_{rad}^1 (\mathbb{R}^N) \to \mathbb{R}$ defined by

(1.17) \begin{equation} \tilde{I}_\gamma (\theta, u, v) = \frac{e^{(N-2) \theta}}{2} \int |\nabla u|^2- d |\nabla v|^2 - e^{N \theta} \int F(u) + \frac{\gamma}{2} v^2 - uv,\end{equation}

instead of working directly with the strongly indefinite functional $\tilde{I}_\gamma (0, u, v)$ defined on $H_{rad}^1 (\mathbb{R}^N) \times H_{rad}^1 (\mathbb{R}^N)$ , see also [22]. The functional $\tilde{I}_\gamma (\theta, u, v)$ is more adequate to show the Palais–Smale condition. This strategy is based on the ideas in [4, 6, 7, 24, 30]. The positivity of the solution follows from the maximum principle for systems due to the fact that the parametres d and $\gamma$ verify the assumptions of [19].

We do not work with $\tilde{I}_\gamma (\theta, u, v)$ nor with $\tilde{I}_\gamma (0, u, v)$ . Instead, we invert the second equation of (1.1) for v and replace the solution in the first one, dealing with the single equation (1.10) with a nonlocal term Bu. The resulting functional I defined in (1.11) on the space X with norm endowed by a scalar product that takes into account the aforesaid nonlocal term, is the one that prevails in our work.

The critical nonlinearity f brings some difficulties, e.g., choosing the adequate space of functions and dealing with the lack of compactness that spoils the Palais–Smale condition. We show that the solution exists by solving equivalently, say, the action minima problem (1.13), where the corresponding minimizing sequence is weak convergent and the sequence of Lagrange multipliers is also convergent. This strategy has been developed by Coleman–Glaser–Martin [17] and successfully adopted in [8, 10], and subsequently undertaken in [3, 11, 25, 38], respectively, for subcritical and critical scalar equations. We substantially improve the results of [16, 33] using different techniques to find a solution of (1.1).

Remark 3. We do not study system (1.1) when $f(u)=u(1-u)(u-\alpha)$ with $0<\alpha<1/2$ . This is the case of the so-called FitzHugh–Nagumo (FHN) model, which has been introduced by the authors of the papers FitzHugh [21] and Nagumo–Arimoto–Yoshizawa [31]. They proposed a way of describing an excitable medium with a simplified counterpart of the Hodgkin–Huxley pattern [23]. In fact, they studied the activator–inhibitor dynamics of a neuron electric pulse along the axon.

Let

(1.18) \begin{equation} \left\{ \begin{array}{rll} u_t - d_1 \Delta u & = f(u) - v & \ \ \mbox{in} \ \ \mathbb{R}^N \times (0,\infty) \\ \tau v_t - d_2 \Delta v & = u + a - b v & \ \ \mbox{in} \ \ \mathbb{R}^N \times (0,\infty) \end{array} \right.,\end{equation}

where $\tau$ , $d_1 \geq 0$ and $d_2 \geq 0$ are constants. The original FHN model is an ODE system where $f(u)=u(1-u)(u-\alpha)$ with $0<\alpha<1/2$ and with $d_1=d_2=0$ in (1.18).

Periodic solutions and stability involving the equilibrium solutions have been studied in [27, 28] for $d_1 > 0$ , $d_2>0$ and $N=1$ , and in [32] on bounded domains and more dimensions, see also [34].

More recently, other stability issues, pattern formation, standing, and traveling waves were addressed in [12–15].

The stationary system on bounded domains $\Omega \subset \mathbb{R}^{N}$ ,

(1.19) \begin{equation} \left\{ \!\begin{array}{lll} - d_1 \Delta u & = f(u) - v & \ \ \mbox{in} \ \ \Omega \\ - d_2 \Delta v & = u + a - b v & \ \ \mbox{in} \ \ \Omega \end{array} \right.\end{equation}

has been treated in [18, 35, 36] with suitable boundary conditions.

System (1.19) with $\Omega = \mathbb{R}^{N}$ , $a=0$ , $d_1=d_2=1$ has been considered in [33] by transforming it into a quasimonotone system. Restricted to a ball, the authors found a subsolution and a supersolution, which are ordered and give rise to a solution by the method of [2]. Such a solution defined on a ball tends to a genuine solution of (1.1) as the radius of the ball tends to infinity, it is an idea already used in [9]. There the positivity of the solution follows from the results of [19]. An extension of the result on existence of solution of [33] by means of variational methods is in [16].

The plan of the paper is as follows. In Section 2, we show that the set $\mathcal{M}$ is nonempty and we introduce the Pohozaev manifold to show a lower bound for b. Section 3 is devoted to the extensive task of recovering compactness. For that matter we deal with minimizing sequences and Lagrange multipliers sequences. As a consequence, we find a weak limit function which is the candidate to be a solution of the nonlocal equation. Such fact is confirmed in Section 4, thus proving Theorem 1.1. Theorem 1.2 is proved in Section 5. The constant $\tau^*$ is determined in Lemmas 3.6 and 5.4.

2. The set $\mathcal{M}$ and the Pohozaev manifold

We will use the set $\mathcal{M}$ in the minimizing arguments, next we show that it is a nontrivial manifold.

Lemma 2.1. The set $\mathcal{M}$ defined in (1.14) is nonempty and it is a $C^1$ manifold.

Proof. Observe that for a fixed $\varphi\in C^{\infty}_{0}(\mathbb R^{N})$ , $\varphi\geq0$ , $\varphi \not\equiv 0$ , using (1.2), the function $h(t)=\int_{\mathbb R^{N}}G(t\varphi)dx $ is strictly negative for small t and by (1.5), we have that $h(t)>0$ for t large; this implies that there exists some $\bar t>0$ such that $\bar t\varphi\in \mathcal M.$ Moreover, $\mathcal M$ is a $C^{1}$ manifold. In fact, define the $C^{1}$ functional $J(u) = \int_{\mathbb R^{N}} G(u)dx -1$ , it holds from (1.4) that

(2.1) \begin{equation} J'(u)[u]=\int_{\mathbb R^{N}}(f(u)u - u^{2}-uBu) dx = \int_{\mathbb R^{N}} (f(u) u - 2F(u))dx +2\int_{\mathbb R^{N}}G(u)\geq 2, \ \forall u\in \mathcal M.\end{equation}

Lemma 2.2. The set $\mathcal{M}$ defined in (1.15) is nonempty and it is a $C^1$ manifold.

Proof. Consider $\zeta \in C^{\infty}_{0}(\mathbb R)$ with $\zeta(x)>0$ and define the function

\begin{equation*}h(t)=\displaystyle\int_{\mathbb R^{2}}G(t\zeta)dx=\displaystyle\int_{\mathbb R^{2}}F(t\zeta) dx-\frac {t^{2}}{2} \displaystyle\int_{\mathbb R^{2}}\zeta B(\zeta)dx-\frac {t^{2}}{2} \displaystyle\int_{\mathbb R^{2}}\zeta^{2}dx.\end{equation*}

From (1.2) for $t>0$ small we have

\begin{equation*}h(t)\leq \frac{\varepsilon -1}{2}t^{2}\displaystyle\int_{\mathbb R^{2}}\zeta^{2}dx.\end{equation*}

For $\varepsilon <1$ , we get $h(t)<0$ for $t>0$ small.

By virtue of (1.5) we obtain

\begin{equation*}h(t) \geq \tau \frac{t^{q}}{q}\displaystyle\int_{\mathbb R^{2}}\zeta^{q}dx-\frac {t^{2}}{2} \displaystyle\int_{\mathbb R^{2}}\zeta^{2}dx -\frac {t^{2}}{2} \displaystyle\int_{\mathbb R^{2}}\zeta B(\zeta)dx\end{equation*}

and

\begin{equation*}h'(t) \geq \lambda t^{q-1}\displaystyle\int_{\mathbb R^{2}}\zeta^{q}dx- t \displaystyle\int_{\mathbb R^{2}}\zeta^{2}dx - t \displaystyle\int_{\mathbb R^{2}}\zeta B(\zeta)dx.\end{equation*}

Then, $h(t)>0$ for $t>0$ large and $h'(t)>0$ for $t>0$ large. Then there is a $\overline{t}>0$ such that

\begin{equation*}\displaystyle\int_{\mathbb R^{2}}G(\overline{t}\zeta)\,dx=h(\overline{t})=0.\end{equation*}

Now we prove that $\mathcal{M}$ is a manifold. Indeed, if $\zeta\in \mathcal{M}$ , then $\zeta \not=0$ . Then, from (1.2) and the fact that $\lim_{|x|\to \infty}\zeta(x)=0 $ and $B\geq 0$ , there exists $ x_0 \in \mathbb R^{2}$ such that $g(\zeta(x_0))<0$ . Thereby, the continuity implies that there is an open interval $B_\delta(x_0)$ such that

\begin{equation*}g(\zeta(x))<0, \quad \forall x \in B_\delta(x_0).\end{equation*}

As a consequence we can always find $\phi \in C_0^{\infty}(\mathbb R^{2}) \subset X$ such that $J'(\zeta)[\phi]=\displaystyle\int_{\mathbb R^{2}}g(\zeta)\phi \,dx < 0$ , where $J(u) = \int_{\mathbb R} G(u)dx$ , showing that $J'(\zeta) \not=0$ .

We will need an estimate for the minimax value b defined by (1.16). For that matter we use the Pohozaev manifold associated to the first equation in (1.1) for $N \geq 3$ , given by

\begin{equation*}\mathcal{P}= \left\{u \in H^{1}(\mathbb{R}^{N})\;:\;\frac{N-2}{2}\displaystyle\int_{\mathbb{R}^{N}}|\nabla u|^{2} \ dx = N \displaystyle\int_{\mathbb{R}^{N}}[F(u) -u^{2}- u v] \ dx, u \neq 0 \right\},\end{equation*}

where $v= Bu$ . Then we can rewrite it as

\begin{equation*}\mathcal{P} = \left\{u \in X\;:\;\frac{N-2}{2} \int_{\mathbb R^N} | \nabla u|^2 \, dx = N \int_{\mathbb R^N} G(u) \, dx, u \neq 0 \right\},\end{equation*}

which, by a direct computation, contains every weak solution of (1.10).

We set the notation

\begin{equation*}p_0 = \inf_{u \in \mathcal{P}} I(u).\end{equation*}

Since B is linear and arguing as in [25, proof of Lemma 3.1], we can see that

(2.2) \begin{equation}p_0 = \frac {1}{N} \left(\frac{N-2}{2N} \right)^{(N-2)/2} (2D)^{N/2},\end{equation}

where D is as in (1.13).

We prove next some results involving the Pohozaev manifold.

Lemma 2.3. It holds

\begin{equation*}\frac {1}{N} \left(\frac{N-2}{2N} \right)^{(N-2)/2} (2D)^{N/2} \leq b,\end{equation*}

where b is the minimax level of I defined in (1.16).

Proof. Indeed, since B is linear and arguing as [25, step 3 of Lemma 1.4], for each $\overline \gamma \in \Gamma$ , it results $\overline{\gamma}([0, 1]) \cap \mathcal{P} \neq \emptyset$ . Then, there exists $t_0 \in [0, 1]$ such that $\overline{\gamma}(t_0) \in \mathcal{P}$ . The property $\int_{\mathbb{R}^{N}}u B(u) dx \geq 0$ , implies

\begin{equation*}p_0 \leq I(\overline{\gamma}(t_0))\leq \max_{t \in [0, 1]} I(\overline{\gamma}(t)).\end{equation*}

Hence, $p_0 \leq b$ and the result follows from (2.2).

3. Minimizing sequences and Lagrange multipliers

We establish some preliminary results which will be useful in order to prove Theorem 1.1. Along the paper, we will take advantage of the following rephrased form of (1.13)–(1.14), namely

(3.1) \begin{equation} D = \inf_{u\in \mathcal M} T(u), \qquad T(u) =\displaystyle\frac{1}{2}\displaystyle\int_{\mathbb{R}^{N}}|\nabla u|^{2} \ dx .\end{equation}

The next steps consist in proving the boundedness of the minimizing sequences in $H^{1}(\mathbb{R}^N)$ for D.

Lemma 3.1. Any minimizing sequence $\{ u_n \}\subset \mathcal M$ for T is bounded in X.

Proof. Let $\{ u_n \}\subset \mathcal M $ be a minimizing sequence for T, then

\begin{equation*}T(u_{n}) =\frac 12 \int_{\mathbb R^N} |\nabla u_n|^2 dx \longrightarrow D \quad \hbox{as $n \rightarrow + \infty$} \end{equation*}

and

\begin{equation*} \int_{\mathbb R^N} G(u_n) dx = 1, \quad \hbox{that is}, \quad \int_{\mathbb R^N} \left( F(u_n) -\frac{1}{2}u_n^{2} - \frac{1}{2} u_nB(u_n) \right) dx =1. \end{equation*}

Then

(3.2) \begin{equation} \frac 12\int_{\mathbb R^N} |\nabla u_{n}|^2 \, dx \leq C \quad \hbox{for all $n \in \mathbb N$ and for some constant $C > 0$ } \end{equation}

and

\begin{equation*}\int_{\mathbb R^N} F(u_n) \,dx = 1 +\frac{1}{2} \int_{\mathbb R^N} u_n^{2} \, dx+\frac{1}{2} \int_{\mathbb R^N} u_n B(u_n) \, dx.\end{equation*}

By using (1.7) with $\varepsilon = 1/2$ , we get

\begin{equation*} 1 + \frac{1}{2} \int_{\mathbb R^N} u_n^{2} \, dx+\frac{1}{2} \int_{\mathbb R^N} u_nB(u_n) \, dx \leq \frac 14 \int_{\mathbb R^N} u_n ^2 \, dx + C_{1/2} \int_{\mathbb R^N} |u_n|^{2^{*}} \, dx. \end{equation*}

Then, for every $n \in \mathbb N$ , by using (3.2), it follows

\begin{equation*}\frac 12 \int_{\mathbb R^N} u_n^2 dx \leq C_{1/4} \, \int_{\mathbb R^N} |u_n|^{2^{*}} dx \leq C_{1/4} C \int_{\mathbb R^N} |\nabla u_{n}|^2 \, dx \leq \overline C. \end{equation*}

Consequently $\{ u_n \} $ is bounded also in $L^2(\mathbb R^N)$ . Moreover, $ \displaystyle\int_{\mathbb R^N} u_n B(u_n) \, dx \leq \|B\|_{X^{-1}} \int_{\mathbb R^N} u_n^2 dx$ and this ensures its boundedness in X (here $X^{-1}$ is the dual space of X).

By the Ekeland Variational Principle (see [20]), we can assume that the minimizing sequence $\{u_{n}\}$ is also a Palais–Smale sequence, that is, using (2.1) there exists a sequence of Lagrange multipliers $\{ \lambda_n \} \subset \mathbb R$ such that

(3.3) \begin{equation}\frac 12 \int_{\mathbb R^N} |\nabla u_n|^2 dx \longrightarrow D \qquad \hbox{as $n \rightarrow + \infty$}\end{equation}

and

(3.4) \begin{equation}T'(u_n) - \lambda_n J'(u_n) \longrightarrow 0 \ \hbox{in } X^{-1}\qquad \text{ as } n \rightarrow + \infty.\end{equation}

In the remaining part of this section, $\{\lambda_{n}\}$ will be the associated sequence of Lagrange multipliers. At this point, it is useful to establish some properties of the levels D and b.

Lemma 3.2. The number D given by (3.1) is positive.

Proof. Clearly by definition $D \geq 0$ . Suppose, by contradiction, that $D =0$ . If $\{ u_n \} $ is a minimizing sequence for $D =0$ , then

\begin{equation*}\frac 12 \int_{\mathbb R^N} |\nabla u_n|^2 dx \to 0 \quad \hbox{as} \quad n \to + \infty\end{equation*}

and

\begin{equation*} 1 = \int_{\mathbb R^N} G(u_n) \, dx = \int_{\mathbb R^N} \left( F(u_n) - \frac 12 u_n^2 - \frac 12 u_nB(u_n) \right) dx. \end{equation*}

Then, for any $\varepsilon>0$ , see (1.8),

\begin{equation*}1 + \frac 12\int_{\mathbb R^N} u_n^2 \, dx + \frac 12\int_{\mathbb R^N} u_nB(u_n) \, dx = \int_{\mathbb R^N} F(u_n) \, dx \leq \frac{\varepsilon}{2} \int_{\mathbb R^N} u_n^2 \, dx+ {C_{\varepsilon}} \int_{\mathbb R^N} |u_n|^{2^{*}} \, dx \end{equation*}

so that

\begin{equation*} 1+ \frac 12(1 - \varepsilon) \int_{\mathbb R^N} u_n^2 \, dx \leq C_{\varepsilon} \int_{\mathbb R^N} |u_n|^{2^{*}} \, dx \leq C_{\varepsilon} C \int_{\mathbb R^N} |\nabla u_n|^2 dx.\end{equation*}

By choosing $\varepsilon = 1/2$ , we obtain

\begin{equation*}1 \leq C_{ 1/2} C \int_{\mathbb R^N} |\nabla u_n|^2 dx \longrightarrow 0 \qquad \hbox{ as $n \longrightarrow + \infty$}.\end{equation*}

This contradiction concludes the proof that $D > 0$ .

Lemma 3.3. The sequence of Lagrange multipliers $\{ \lambda_n \}$ associated to the minimizing sequence $\{u_{n}\}$ is bounded. More precisely, we have that

\begin{equation*}0<\liminf_{n \to +\infty}\lambda_n \leq \limsup_{n \to + \infty} \lambda_n \leq D.\end{equation*}

Hence, for some subsequence, still denoted by $\{\lambda_n\}$ , we can assume that $\lambda_n \to \lambda^{*}$ , for some $\lambda^{*} \in (0, D]$ .

Proof. By (3.4),

(3.5) \begin{equation}2T(u_{n}) =T'(u_n)[u_n] = \lambda_n J'(u_n)[u_n] +o_{n}(1).\end{equation}

Then, from (2.1)

\begin{equation*}2T(u_n) \geq 2\lambda_{n} +o_{n}(1)\end{equation*}

which implies, taking into account (3.3),

\begin{equation*}\limsup_{n \rightarrow + \infty} \lambda_n \leq \frac 12 \limsup_{n \rightarrow + \infty}\|u_{n}\|^2 =D.\end{equation*}

Since $\{u_{n}\}$ is a bounded minimizing sequence, it is easy to see that $|J'(u_{n})[u_{n}] |= |\int_{\mathbb R^{N}} g(u_{n}) u_{n}| \leq C$ , and then by (3.5) and the fact that $2T(u_{n})\to 2D>0$ , we infer that

\begin{equation*}\liminf_{n \to +\infty}\lambda_n >0.\end{equation*}

The proof is thereby completed.

In the sequel, we will show that a minimizing sequence for (3.1) can be chosen to be nonnegative and radially symmetric around the origin. Note that for our proof we do not need to consider the “odd extension” of the nonlinearity, as it is usually done in the literature to show that the minimizing sequence can be replaced by the sequence of the absolute values. In fact we will prove that the minimizing sequence can be replaced, roughly speaking, by the sequence of the positive parts.

Lemma 3.4. Any minimizing sequence $\{u_n\}$ for (3.1) can be assumed to be radially symmetric around the origin and nonnegative.

Proof. To begin with, we recall that $F(s)=0$ for all $s \leq 0$ . Thus, $F(u_n)=F(u_n^{+})$ for all $n \in \mathbb{N}$ with $u_n^{+}=\max\{0,u_n\}$ . From this, the equality

\begin{equation*}\int_{\mathbb R^N}G(u_n)\,dx=1, \quad \forall n \in \mathbb{N}\end{equation*}

leads to

\begin{equation*}\int_{\mathbb R^N}G(u_n^+)\,dx \geq 1, \quad \forall n \in \mathbb{N}.\end{equation*}

Defining the function $h_n\;:\;[0,1] \to \mathbb R$ by

\begin{equation*}h_n(t)=\int_{\mathbb R^N}G(t u_n^+)\,dx\end{equation*}

the conditions on f yield that h is continuous with $h_n(1) \geq 1$ . Once $u_n^+ \not=0$ for all $n \in \mathbb{N}$ , the condition (1.2) ensures that $h_n(t)<0$ for t close to 0. Thus, there is $t_n \in (0,1]$ such that $h_n(t_n)=1$ , that is,

\begin{equation*}\int_{\mathbb R^N}G(t_n u_n^+)\,dx = 1, \quad \forall n \in \mathbb{N},\end{equation*}

implying that $t_n u_n^+ \in \mathcal{M}$ . We also know that

\begin{equation*}\|u_n^+\|^{2} \leq \|u_n\|^{2}.\end{equation*}

Since $t_n \in (0,1]$ , the last inequality gives

\begin{equation*}D \leq T(t_n u_n^+)\leq T(u_n)=D+o_n(1)\end{equation*}

that is,

\begin{equation*}t_n u_n^{+} \in \mathcal{M} \quad \mbox{and} \quad T(t_n u_n^+) \to D,\end{equation*}

showing that $\{t_n u_n^+\}$ is a minimizing sequence for T. Hence, without lost of generality, we can assume that $\{u_n\}$ is a nonnegative sequence.

Moreover,

\begin{equation*} \int_{\mathbb R^N} |\nabla u_n^{*}|^2 \, dx \leq \int_{\mathbb R^N} |\nabla u_n|^2 \, dx, \quad \forall n \in \mathbb{N} \end{equation*}

(see [5, Theorem 3]) and

\begin{equation*} \int_{\mathbb{R}^N}u_{n}^{*}B(u_{n}^{*})\, dx=\int_{\mathbb{R}^N}u_{n}B(u_{n})\, dx, \quad \forall n \in \mathbb{N}, \end{equation*}

\begin{equation*} \int_{\mathbb{R}^N}|u_{n}^{*}|^{2}\, dx=\int_{\mathbb{R}^N}|u_{n}|^{2}\, dx, \quad \forall n \in \mathbb{N}, \end{equation*}

which implies

\begin{equation*}\int_{\mathbb{R}^N}G(u_{n}^{*})\, dx= \int_{\mathbb{R}^N}G(u_{n})\, dx, \quad \forall n \in \mathbb{N},\end{equation*}

where $u_n^{*}$ is the Schwartz symmetrization of $u_n$ . Then, every minimizing sequence can be assumed radially symmetric, nonnegative and decreasing in $r=|x|$ .

Let $X_{rad}$ be the space of radially symmetric functions of X. In what follows, we will use that the embedding

(3.6) \begin{equation} X_{rad} \hookrightarrow L^{p}(\mathbb R^N)\end{equation}

is compact for all $p \in (2,2^{*})$ , recall the similar embedding of $H^1_{rad}(\mathbb R^N)$ and see Lions [29] for more details.

It turns out that the key point in the proof of Theorem 1.1 is to verify that the weak limit of $\{u_{n}\}$ in $X_{rad}$ , denoted hereafter by u, is a solution of the minimizing problem (3.1). Before verifying this in Section 4, some preliminary lemmas are required in order to recover compactness.

Remark 4. Due to the boundedness in $X_{rad}$ of the nonnegative and radially symmetric minimizing sequence $\{u_{n}\}$ , assume for the moment that its weak limit u in $X_{rad}$ is a solution of (1.10). Observe also that, by the boundedness in $L^{2}(\mathbb R^{N})$ we have the uniform decay $|u_{n}(x)| \leq C |x|^{-(N-1)/2}$ , see [26, Lemma 1.1. p. 250]. Therefore, arguing with a subsequence, if necessary, we deduce that the weak limit u is nonnegative, radially symmetric, and decreasing. These properties do also hold for $v=B(u)$ . Therefore, one has the decay of $u(x) \to 0$ and $v(x) \to 0$ as $|x| \to \infty$ , where (u, v) is a solution of (1.1). This observation is valid for $N \geq 2$ .

In the next result, we use the following constant

\begin{equation*}S= \displaystyle\inf_{u\in \mathcal{D}^{1,2}(\mathbb{R}^{N})\setminus \{0\}}\frac{\displaystyle\int_{\mathbb{R}^{N}}|\nabla u|^{2} dx}{\left(\displaystyle\int_{\mathbb{R}^{N}}|u|^{2^{*}} dx\right)^{2/2^{*}}}.\end{equation*}

Lemma 3.5. Assume that $w_n=u_n-u\rightharpoonup 0$ in $X_{rad}$ and $\| w_n\|^{2} \to L>0$ . Then

\begin{equation*}D \geq 2^{-2 /N}S.\end{equation*}

Proof. First of all, we recall (2.1) and that the limit $T'(u_n) - \lambda_n J'(u_n) \rightarrow 0$ as $n \rightarrow + \infty$ gives

\begin{equation*}T'(u_n)[u_n]-\lambda_{n} J'(u_n)[u_n]=o_n(1).\end{equation*}

Using standard arguments, it is possible to prove that

\begin{equation*}T'(u_n)[u_n]-\lambda_{n} J'(u_n)[u_n]=T'(w_n)[w_n]-\lambda_{n} J'(w_n)[w_n]+T'(u)[u]-\lambda^{*} J'(u)[u] + o_n(1)\end{equation*}

and

\begin{equation*}T'(u)-\lambda^{*} J'(u)=0 \quad \mbox{in} \quad X^{-1},\end{equation*}

where $\lambda^{*}$ is the same which appear in Lemma 3.3. Then $T'(w_n)[w_n]-\lambda_{n} J'(w_n)[w_n]=o_n(1)$ , or equivalently,

\begin{equation*} \int_{\mathbb R^N} |\nabla w_n|^2 \, dx =\lambda_n \, \int_{\mathbb R^N} f(w_n) w_n \, dx - \lambda_n \int_{\mathbb R^N} w_n^2 \, dx - \lambda_n \int_{\mathbb R^N} w_nB(w_n) \, dx + o_n(1).\end{equation*}

Using the growth conditions on f, fixed $q \in (2,2^{*})$ and given $\varepsilon >0$ , there exists $C=C(\varepsilon, q)>0$ such that

\begin{equation*}f(t)t \leq \varepsilon t^{2}+C|t|^{q}+(1+\varepsilon)|t|^{2^{*}}, \quad \forall t \in \mathbb{R}.\end{equation*}

From this,

\begin{equation*} \int_{\mathbb R^N} |\nabla w_n|^2 \, dx \leq\lambda_n \left(\varepsilon \int_{\mathbb R^N} w_n^2\, dx + C \int_{\mathbb R^N} |w_n|^q dx +(1+\varepsilon) \int_{\mathbb R^N} |w_n|^{2^{*}} \, dx\right) + o_n(1).\end{equation*}

Now, using the definition of S, we get

(3.7) \begin{eqnarray} \int_{\mathbb R^N} |\nabla w_n|^2 \, dx &\leq &\lambda_n \left(\varepsilon \int_{\mathbb R^N} w_n^2 dx + C \int_{\mathbb R^N} |w_n|^q dx +(1+\varepsilon) \left( \frac{1}{S} \int_{\mathbb R^N} |\nabla w_n|^{2} \, dx\right)^{2^{*}/ 2} \right)+ o_n(1)\nonumber\\&\leq & \lambda_n \left(\varepsilon \int_{\mathbb R^N} w_n^2 dx + C \int_{\mathbb R^N} |w_n|^q dx +(1+\varepsilon) \left( \frac{1}{S} \int_{\mathbb R^N} |\nabla w_n|^2 \, dx \right)^{2^{*}/ 2} \right)+ o_n(1).\end{eqnarray}

Taking to the limit in (3.7), recalling that $\{w_n\}$ is bounded,

(3.8) \begin{equation} \int_{\mathbb R^N} |\nabla w_n|^2 \, dx \longrightarrow L\end{equation}

and that $w_n \to 0$ in $L^{q}(\mathbb R^N)$ (see (3.6)), we obtain

\begin{equation*}L\leq \lambda^*\left( \varepsilon C_1 + (1+\varepsilon)\left(\frac{L}{S}\right)^{2^{*}/2} \right).\end{equation*}

Since $\varepsilon$ arbitrary and the fact that $\lambda^{*}\leq D$ (recall Lemma 3.3), we derive $L\leq D \left(L/S \right)^{2^{*}/2}$ , or equivalently,

(3.9) \begin{equation}S^{2^{*}/2}\leq D L^{2 /(N-2)}.\end{equation}

On the other, hand (3.8) implies that $L = 2D -\|u\|^2 \leq 2D$ . Hence (3.9) becomes

\begin{equation*}S^{2^{*}/2}\leq 2^{2 /(N-2 )} D^{2^{*}/2},\quad \text{ i.e. } \quad D \geq 2^{-2/N}S\end{equation*}

and the proof is finished.

Now we consider the equation

(3.10) \begin{equation}\left\{\begin{array}{l}-\Delta \omega + \omega + B(\omega)= |\omega|^{q-2} \omega\ \ \mbox{in $\Omega$}\\\omega \in H^{1}_{0}(\Omega)\end{array}\right.,\end{equation}

where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ and q is the constant which appears in the hypothesis (1.5). The functional $I_q\;:\; H^{1,}_0(\Omega)\rightarrow \mathbb{R}$ associated to (3.10) is given by

\begin{equation*}I_q(u)=\dfrac{1}{2}\displaystyle\int\limits_{\Omega}\left[|\nabla u|^2+|u|^2 + B(u)u \right]dx-\dfrac{1}{q}\displaystyle\int\limits_{\Omega}|u|^q dx .\end{equation*}

Since $\Omega$ is bounded, the expression

\begin{equation*}\| u \|_{\Omega} = \left( \int\limits_{\Omega}\left[|\nabla u|^2+|u|^2 + B(u)u \right]dx \right)^\frac{1}{2}\end{equation*}

is equivalent the usual norm of $H^{1}_{0}(\Omega)$ , by the Mountain Pass Theorem [1], there exists $\omega_q\in H^{1}_{0}(\Omega)$ such that

\begin{eqnarray*}I_q(\omega_q)=c_q, \ \ \|\omega_q\|^{2}_{\Omega}=\displaystyle \int\limits_{\Omega}|\omega_q|^qdx &\text{and}&I'_q(\omega_q)=0.\end{eqnarray*}

Moreover,

(3.11) \begin{equation}c_q= \left(\dfrac{q-2}{2q}\right)\displaystyle\int\limits_{\Omega}|\omega_q|^qdx,\end{equation}

In the next result, the condition $\tau>\tau^{*}$ given in (1.5) plays a crucial role. We fix

\begin{equation*}\tau^{*}= \left[\frac{c_q N(2N/N-2)^{(N-2)/2}}{2^{(N-2)/2}S^{N/2}}\right]^{(q-2)/2}.\end{equation*}

Lemma 3.6. It holds

\begin{equation*}b < \frac {1}{ N} \left(\frac{N-2}{2N} \right)^{(N-2)/2} 2^{(N-2)/2} S^{N/2}, \quad \forall \tau > \tau^{*}.\end{equation*}

Proof. Take $\omega_q \in H^{1}_{0}(\Omega)$ a solution of problem (3.10). By (1.5) and the definition of b at (1.16), we get

\begin{eqnarray*}b \leq \max_{ t \geq 0} I(t \omega_q) & \leq &\max_{t \geq 0} \left\{ \frac{t^2}{2}\|\omega_q\|^2_{\Omega} - \tau \frac{t^q}{q} \int_{\mathbb R^N} |\omega_q|^q \, dx \right\} \leq \max_{t \geq 0} \left\{ \frac{t^2}{2} - \tau \frac{t^q}{q} \right\} \int_{\mathbb R^N} |\omega_q|^q \, dx\\& = & \max_{t \geq 0} \left\{ \frac{t^2}{2} - \tau \frac{t^q}{q} \right\} c_q \frac{2q}{q-2} = \frac{c_q}{\tau^{2/(q-2)}}.\end{eqnarray*}

The conclusion of the lemma follows.

Lemma 3.7. If $u_n \rightharpoonup u$ in $X_{rad}$ , then $u_n \to u$ in $D^{1,2}(\mathbb{R}^N)$ . In particular, $u_n \to u$ in $L^{2^{*}}(\mathbb{R}^N)$ .

Proof. Suppose by contradiction that $u_n \not\to u $ in $D^{1,2}(\mathbb{R}^N)$ and let $w_n = u_n - u$ . Thus $\int_{\mathbb R^N} |\nabla w_n|^{2} \, dx \to L>0$ for some subsequence. Then, by Lemma 3.5,

(3.12) \begin{equation}D \geq 2^{- 2/N} S.\end{equation}

From Lemma 2.3

\begin{equation*}\frac 1{N} \left(\frac{N-2}{2N} \right)^{(N-2)/2} (2D)^{N/2} \leq b,\end{equation*}

from which, using (3.12), it follows that

\begin{equation*}\frac {1}{N} \left(\frac{N-2}{2N} \right)^{(N-2)/2} 2^{(N-2 )/2} S^{N/2} \leq b.\end{equation*}

This contradicts Lemma 3.6 and accomplishes the proof.

4. Proof of Theorem 1.1

The results of the last section will be used next.

Proof of Theorem 1.1. We wish to show that D is attained by u, where u is the weak limit of $\{ u_n \}$ . First of all, we know that

(4.1) \begin{equation}T(u) = \frac 12\int_{\mathbb R^N} |\nabla u|^2 \, dx \leq \liminf_{n \rightarrow + \infty} \frac 12 \int_{\mathbb R^N}|\nabla u_n|^2 \, dx =D\end{equation}

so we just need to prove that $u\in \mathcal M$ .

Since

\begin{equation*}u(R)\leq \biggl(\frac{N}{w_{N-1}}\biggl)^{1/t}R^{-N/t}|u|_{L^{t}(\mathbb{R}^{N})},\end{equation*}

for all $t\geq 1$ , for $R>0$ large, we have

\begin{equation*}\frac{1}{2}u_n^{2}-F(u_n) \geq 0 \quad \forall n \in \mathbb{N} \quad \mbox{in } \mathbb R^{N} \setminus B_{R},\end{equation*}

$B_{R}$ being the ball of radius R centered in $0.$ Since $u_n \to u $ in $L^{2^{*}}(B_R)$ , reasoning with a subsequence if needed, we have $u_{n} \to u $ a.e. in $B_{R}$ and there exists $h\in L^{2^{*}}(B_{R})$ such that $|u_{n}(x)|\leq h(x)$ . Moreover, we have $F(u_{n}(x)) \to F(u(x))$ a.e. and $|F(u_{n})|\leq 2^{-1}\varepsilon u_{n}^{2} + C_{\varepsilon} |u_{n}|^{2^{*}}\leq2^{-1}\varepsilon u_{n}^{2} + C_{\varepsilon} |v|^{2^{*}} $ which, joint with the fact that $\varepsilon$ is arbitrary and the Dominated Convergence Theorem, give $\int_{B_{R}} F(u_{n})dx \to \int_{B_{R}} F(u)dx$ . Then by

\begin{equation*}\int_{B_R}F(u_n)\,dx=\frac{1}{2}\int_{\mathbb{R}^{N}}u_nB(u_n)\,dx+\frac{1}{2}\int_{B_R}u_n^{2}\,dx+\int_{\mathbb{R}^N \setminus B_R}\left(\frac{1}{2}u_n^{2}-F(u_n)\right)\,dx +1,\end{equation*}

taking into account the above considerations and the Fatou Lemma we infer

\begin{equation*}\displaystyle\int_{B_R}F(u)\, dx \geq \frac{1}{2}\int_{\mathbb{R}^{N}}uB(u)\,dx+\frac{1}{2}\int_{B_R}u^{2}dx+\int_{\mathbb{R}^N \setminus B_R}\left(\frac{1}{2}u^{2}-F(u)\right)\,dx +1\end{equation*}

which leads to

\begin{equation*}\displaystyle\int_{\mathbb R^N} G(u) \, dx \geq 1.\end{equation*}

Suppose by contradiction that

\begin{equation*}\int_{\mathbb R^N} G(u) \, dx > 1\end{equation*}

and define $h\;:\;[0, 1] \rightarrow \mathbb R$ by $h(t) = \int_{\mathbb R^N} G(tu) \, dx$ . The growth conditions on f ensure that $h(t) < 0$ for t close to 0 and $h(1) = \int_{\mathbb R^N} G(u) \, dx > 1$ . Then, by the continuity of h, there exists $t_0 \in (0, 1)$ such that $h(t_0)=1$ . Then,

\begin{equation*}\int_{\mathbb R^N} G(t_0 u) \, dx = 1 \text{ if and only if } t_0 u \in \mathcal{M}.\end{equation*}

Consequently, by (4.1)

\begin{equation*}D \leq T(t_0 u) = \frac{t_0^2}{2} \int_{\mathbb R^N} |\nabla u|^2 \, dx = t_0^2 \, T(u) \leq t_0^2 \, D < D\end{equation*}

which is an absurd. Thus $ \int_{\mathbb R^N} G(u) \, dx =1 $ , i.e. $ u\in \mathcal M$ . The fact that the solution u of the minimizing problem gives rise to a solution of (1.10), follows by standard arguments; indeed, since u is a solution of the minimizing problem (3.1), i.e. $D =T(u)= \inf_{w\in \mathcal M}T(w)$ , then there exists an associated Lagrange multiplier $\lambda$ such that, in the weak sense,

\begin{equation*}-\Delta u = \lambda g( u).\end{equation*}

Testing the previous equation on the same minimizer u, we deduce

\begin{equation*}2T(u)=\lambda\int_{\mathbb R^{N}}g(u)u dx = \lambda J'(u)[u] \geq 2\lambda .\end{equation*}

so that, by Lemma 3.2, $T(u)\geq\lambda>0$ . Setting $u_{\sigma}(x)=u(\sigma x)$ for $\sigma>0$ , we easily see that

\begin{equation*}-\Delta u_{\sigma} = \lambda \sigma^{2}g(u_{\sigma}).\end{equation*}

Choosing $\sigma=\lambda^{-1/2}$ we obtain a solution of (1.10). Arguing as in [10, Theorem 3], $u_\sigma$ is a solution of (1.10). Rewriting $u_{\sigma}$ as u, and since $v=B(u)$ , we conclude that (u, v) is a solution of (1.1).

5. Proof of Theorem 1.2

We establish preliminary results in the entire plan which will be useful in order to prove Theorem 1.2. Here we point out that, we will deal with minimizing sequences $\{u_{n}\}$ for the minimization problem (1.13)–(1.15), rewritten in the form

(5.1) \begin{equation} D = \inf_{u\in \mathcal M} T(u), \qquad T(u) =\displaystyle\frac{1}{2}\displaystyle\int_{\mathbb{R}^{N}}|\nabla u|^{2} \ dx .\end{equation}

We suppose that $u_{n}$ is nonnegative and radially symmetric.

It is useful to state the next result which is a consequence of [3, Lemma 5.1]. The proof is based on Trudinger–Moser inequality and the Compactness Lemma of Strauss [10, Theorem A.I].

Lemma 5.1. Assume that f satisfies (1.2), (1.5), and (1.6). Let $\{w_n\}\subset X_{rad}$ be a sequence of radially symmetric functions such that

\begin{equation*}w_n \rightharpoonup w \ \ \mbox{in} \ \ X\end{equation*}

and

\begin{equation*}\displaystyle\sup_{n}|\nabla w_n|^{2}_{2}=\rho \leq \frac 12 \ \ \mbox{and} \ \ \displaystyle\sup_{n}|w_n|^{2}_{2}=M <\infty.\end{equation*}

Then,

\begin{equation*}\displaystyle\int_{\mathbb R^{2}}F(w_n)dx\longrightarrow \displaystyle\int_{\mathbb R^{2}}F(w)dx.\end{equation*}

The relation between the minimum D and the minimax level b defined in (1.16) is stated in the following lemma.

Lemma 5.2. The numbers D and b satisfy the inequality $D\leq b$ .

Proof. Arguing as in Lemma 2.2, given $\eta \in X$ with $\eta^{+}=\max\{\eta,0\}\neq 0$ , there is $t_{0}>0$ such that $t_0 \eta^{+} \in \mathcal{M}$ . Then,

\begin{equation*}D\leq \frac{t_{0}^{2}}{ 2} \int_{\mathbb R^{2}} |\nabla \eta^{+}|^2 \, dx = I(t_{0}\eta^{+})\leq \displaystyle\max_{t\geq 0}I(t\eta^{+}).\end{equation*}

Due to the fact that $f(s)=0$ for $s\leq 0$ , for $\eta\in X$ , $\eta\neq 0$ with $\eta^{+}=0$ , we obtain $\max_{t\geq 0}I(tv)=\infty$ . Hence in any case $D\leq b$ .

Lemma 5.3. The number D given by (5.1) is positive.

Proof. By definition $D\geq 0$ . Assume by contradiction that $D=0$ an let $\{u_n\}$ be a (nonnegative and radially symmetric) minimizing sequence in $X_{rad}$ for T, that is,

\begin{equation*}\displaystyle\int_{\mathbb R^{2}} |\nabla u_n|^2 dx \to 0 \ \ \mbox{and} \ \ \displaystyle\int_{\mathbb R^{2}} G(u_n) \, dx =0.\end{equation*}

For each $\mu_n>0$ , the function $\xi_n(x)=u_n( x / \mu_n)$ satisfies

\begin{equation*}\displaystyle\int_{\mathbb R^{2}} |\nabla \xi_n|^2dx=\displaystyle\int_{\mathbb R^{2}} |\nabla u_n|^2dx \ \ \mbox{and} \ \ \displaystyle\int_{\mathbb R^{2}} G(\xi_n) \, dx =0.\end{equation*}

Since

\begin{equation*}\displaystyle\int_{\mathbb R^{2}} \xi_n^{2}dx+ \displaystyle\int_{\mathbb R^{2}} \xi_nB(\xi_n)dx = \mu_{n}^{2}\biggl[\displaystyle\int_{\mathbb R^{2}} u_n^{2}dx+ \displaystyle\int_{\mathbb R^{2}} u_nB(u_n)dx\biggl],\end{equation*}

we choose $\mu_{n}^{2}=\biggl[\displaystyle\int_{\mathbb R^{2}} u_n^{2}dx+ \displaystyle\int_{\mathbb R^{2}} u_nB(u_n)dx\biggl]^{-1}$ to obtain

\begin{equation*}\displaystyle\int_{\mathbb R^{2}} |\nabla \xi_n|^2 dx\to 0, \ \ \biggl[\displaystyle\int_{\mathbb R^{2}} u_n^{2}dx+ \displaystyle\int_{\mathbb R^{2}} u_nB(u_n)dx\biggl]=1 \ \ \mbox{and} \ \ \displaystyle\int_{\mathbb R^{2}} G(\xi_n) \, dx =0.\end{equation*}

We can assume that there exists $ v \in X_{rad}$ , radially symmetric, such that $\xi_n \rightharpoonup \xi$ in $X_{rad}$ . From Lemma 5.1, we get

\begin{equation*}\displaystyle\int_{\mathbb R^{2}}F(\xi_n)dx\longrightarrow \displaystyle\int_{\mathbb R^{2}}F(\xi)dx.\end{equation*}

Notice that $\int_{\mathbb R^{2}} G(\xi_n) \, dx =0$ implies $\int_{\mathbb R^{2}} F(\xi_n) \, dx =1/2$ and $\int_{\mathbb R^{2}} F(\xi) \, dx = 1/2$ . Then $v \neq 0$ . But

\begin{equation*}\displaystyle\int_{\mathbb R^{2}} |\nabla \xi|^2dx \leq \liminf_{n\to+\infty} \displaystyle\int_{\mathbb R^{2}} |\nabla \xi_n|^2dx\longrightarrow 0,\end{equation*}

implies $\xi=0$ , which is an absurd.

Lemma 5.4. We have $ b < 1/2.$

Proof. It is enough to repeat the same argument of the proof of Lemma 3.6, recalling now that $N=2$ and by (1.5) we choose

\begin{equation*}\tau* = \left(\frac{c_q}{4}\right)^{(q-2)/2}.\end{equation*}

Proof of Theorem 1.2. We will show that D is attained by u, where u is the weak limit of $\{ u_n \}$ . Indeed, since $u_n \rightharpoonup u$ in $X_{rad}$ we have

(5.2) \begin{equation}T(u) = \frac 12\int_{\mathbb R^{2}} |\nabla u|^2 \, dx \leq \liminf_{n \rightarrow + \infty} \frac 12 \int_{\mathbb R^{2}}|\nabla u_n|^2 \, dx =D.\end{equation}

Moreover, by Lemma 5.1 we have

\begin{equation*}\displaystyle\int_{\mathbb{R}^{2}}F(u) dx= \displaystyle\liminf_{n\to\infty}\displaystyle\int_{\mathbb{R}^{2}}F(u_n) dx\geq \frac 12 \displaystyle\int_{\mathbb{R}^{2}}u^{2} dx + \frac 12 \displaystyle\int_{\mathbb{R}^{2}}uB(u)dx\end{equation*}

leading to

\begin{equation*}\displaystyle\int_{\mathbb R^{2}} G(u) \, dx \geq 0.\end{equation*}

As in the previous case $N\geq3$ , we just need to prove that $u\in \mathcal M$ , i.e. $\int_{\mathbb R^N} G(u) \, dx = 0$ .

We again argue by contradiction. Suppose that

\begin{equation*}\int_{\mathbb R^{2}} G(u) \, dx > 0.\end{equation*}

As in the proof of Lemma 2.2, we set $h\;:\;[0, 1] \rightarrow \mathbb R$ by $h(t) = \int_{\mathbb R} G(tu) \, dx$ . Using the growth condition of f, we have $h(t) < 0$ for t close to 0 and $h(1) = \int_{\mathbb R} G(u) \, dx > 0$ . Hence, by the continuity of h, there exists $t_0 \in (0, 1)$ such that $h(t_0)=0$ , that is, $t_{0}u\in \mathcal M$ . Consequently, by (5.2)

\begin{equation*}D \leq T(t_0 u) = \frac{t_0^2}{2} \int_{\mathbb R^2} |\nabla u|^2 \, dx = t_0^2 \, T(u) \leq t_0^2 \, D < D\end{equation*}

which is an absurd.

As for the case $N\geq3$ , one shows that the minimizer u of (5.1) gives rise to a solution of (1.10), and then (u, v) with $v=B(u)$ solves (1.1).