---
abstract: 'In this paper we prove the existence of a positive solution to the equation $-\Delta u + V(x)u=g(u)$ in ${{\mathbb{R}^N}},$ assuming the general hypotheses on the nonlinearity introduced by Berestycki & Lions. Moreover we show that a minimizing problem, related to the existence of a ground state, has no solution.'
author:
- 'A. Azzollini [^1] & A. Pomponio[^2]'
title: |
    [**On the Schrödinger equation in ${{\mathbb{R}^N}}$\
    under the effect of a general\
    nonlinear term**]{}
---

Introduction
============

This paper deals with the following equation: $$\label{eq:V}
\left\{
\begin{array}{ll}
-\Delta u + V(x)u=g(u), & x\in{{\mathbb{R}^N}}, \quad N{\geqslant}3;
\\
u>0.
\end{array}
\right.$$ An existence result of nontrivial solutions for this kind of problem has been obtained by Rabinowitz [@R] assuming that $g$ is superlinear and subcritical at infinity and satisfies the global growth Ambrosetti-Rabinowitz condition $$\label{eq:ambr}
        \exists \mu>2 \hbox{ s.t. } 0<\mu \int_0^t g(s) \,d s{\leqslant}g(t)t,
        \hbox{ for all }t\in {\mathbb{R}}.$$ This condition is used to get the boundedness of Palais-Smale sequences. In [@JT2], L. Jeanjean and K. Tanaka have been able to remove the hypothesis by means of an abstract tool, consisting in a suitable approximating method (see [@J Theorem 1.1]). However they preserved the condition on the superlinear growth at infinity.

On the other hand, in the fundamental paper [@BL1], Berestycki & Lions proved the existence of a ground state, namely a solution which minimizes the action among all the solutions, for the problem $$\label{eq}
\left\{
\begin{array}{l}
-\Delta u =g(u), \hbox{ in } {{\mathbb{R}^N}};
\\
u \in H^1({{\mathbb{R}^N}}), u\neq 0,
\end{array}
\right.$$ under the following assumptions on the nonlinearity $g$:

-   $g\in C({{\mathbb{R}^N}},{\mathbb{R}})$, $g$ odd;

-   $-\infty <\liminf_{s\to 0^+} g(s)/s{\leqslant}\limsup_{s\to 0^+}
    g(s)/s=-m<0$;

-   $-\infty <\limsup_{s\to +\infty}
    g(s)/s^{2^*-1}{\leqslant}0$;

-   there exists $\zeta>0$ such that $G(\zeta):=\int_0^\zeta g(s)\,d s>0$;

here $2^*=2N/(N-2)$.

So it seems that the “natural" assumptions on the nonlinearity do not require the superlinearity at infinity.

In view of this result, the aim of this paper is to study the problem preserving the same general assumptions of [@BL1] on $g$. Moreover we assume the following hypotheses on $V:$

-   $V\in C^1({{\mathbb{R}^N}},{\mathbb{R}})$ and $V(x){\geqslant}0$, for all $x\in {{\mathbb{R}^N}}$, and the inequality is strict somewhere;

-   $\|({\nabla }V(\cdot)\mid \cdot)^+\|_{N/2}<2S$;

-   $\lim_{|x|\to \infty}V(x)=0$;

-   $V$ is radially symmetric;

here $({\nabla }V(x)\mid x)^+=\max \big\{({\nabla }V(x)\mid x),0\big\}$ and $S$ is the best Sobolev constant of the embedding $\D\hookrightarrow L^{2^*}({{\mathbb{R}^N}}),$ namely $$S  = \inf_{u\in\mathcal{D}^{1,2}\setminus\{0\}}\frac{\|\n
u\|_2^2}{\|u\|_{2^*}^2}.$$

Our first main result is the following

\[main\] Assume that ([**g1-4**]{}) and ([**V1-4**]{}) hold, then the problem possesses at least a radially symmetric solution.

Up to our knowledge, this is the first result on a problem as , where exactly the same general hypotheses of Berestycki & Lions [@BL1] are assumed on the nonlinearity $g$.

As a consequence of Theorem \[main\], we can prove the following

\[th:radial\] Assuming that ([**g1-4**]{}) and ([**V1-4**]{}) hold, then the problem possesses a radial ground state solution, namely a solution minimizing the action among all the nontrivial radial solutions.

We can generalize the hypotheses ([**V1**]{}) and ([**V3**]{}), requiring:

-   $V\in C^1({{\mathbb{R}^N}},{\mathbb{R}})$ and $V(x){\geqslant}V_0>-m$, for all $x\in {{\mathbb{R}^N}}$, and the inequality is strict somewhere;

-   $\lim_{|x|\to \infty}V(x)=V_0$;

and supposing that the function $\tilde g(s)=g(s)-V_0s$ satisfies ([**g1-4**]{}).

The geometrical hypotheses on the potential $V$ do not allow us to use concentration-compactness arguments as in [@JT2]. As a consequence, we have to require a symmetry property on $V$ to prevent any possible loss of mass at infinity.

In the second part of the paper, we are interested in solving a minimization problem strictly related to the existence of a ground state solution for . Let $$I(u)=\frac 12 {\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 + V(x) u^2 -{\int_{{{\mathbb{R}^N}}}}G(u),\quad u\in{H^1({{\mathbb{R}^N}})}.$$ When the nonlinearity $g$ satisfies , a standard method to look for the existence of a ground state solution for an equation as is to study the minimizing problem $$\label{eq:minimizing}
        I(\bar u)=\inf_{u\in{\mathcal{N}}} I(u),\quad \bar u \in {\mathcal{N}},$$ where ${\mathcal{N}}$ is the Nehari manifold related to $I$. In [@R] the geometrical assumption on $V$ $$\label{eq:geo}
        V(y){\leqslant}\lim_{x\to\infty} V(x),\quad\hbox{for all }
        y\in{\mathbb{R}}^N$$ is used to solve such a minimizing problem. Moreover, in [@JT2] it has been proved that, assuming , there exists a ground state solution for also without the condition . On the other hand, it is well known (see for example [@BGM]) that if $g$ satisfies , and holds with the reverse inequality, the minimizing problem cannot be solved. In fact, a contradiction argument deriving from the comparison of the level $\eta:=\inf_{u\in{\mathcal{N}}} I(u)$ with $\eta_0
:= \inf_{u\in{\mathcal{N}}_{0}}I_{0}(u)$ (here $I_0$ and ${\mathcal{N}}_0$ are the functional and the Nehari manifold of the problem at infinity) is used and the Ambrosetti-Rabinowitz condition plays a fundamental role. Actually, when we do not assume such a type of growth condition on $g$, these arguments do not work any more. However, a similar study can be done by replacing the Nehari manifold with a more suitable one.\
Indeed, if we define by $$\label{eq:P}
{{\cal P}}_0:=\left\{ u\in {H^1({{\mathbb{R}^N}})}\setminus\{0\} \; \Big{|}\; \frac{N-2}2 \irn
|{\nabla }u|^2 = N {\int_{{{\mathbb{R}^N}}}}G(u)\right\},$$ the Pohozaev manifold related to , and we set $${\mathcal S_0}:=\{u\in{H^1({{\mathbb{R}^N}})}\setminus\{0\}\mid u\hbox{ is a solution of \eqref{eq}}\},$$ it is well known that ${\mathcal S_0}\subset {{\cal P}}_0.$ Moreover, in [@S] it has been proved that ${{\cal P}}_0$ is a natural constraint for the functional related to $$I_0(u)=\frac 12 {\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 - {\int_{{{\mathbb{R}^N}}}}G(u),\quad u\in{H^1({{\mathbb{R}^N}})}.$$ In order to look for a ground state solution, a very natural question arises: is the infimum $I_0|_{{{\cal P}}_0}$ achieved?

Following two different ways, Jeanjean & Tanaka [@JT] and Shatah [@S] have given a positive answer to this question, showing that $$\label{eq:SP}
\min_{u \in {\cal S}_0}I_0(u)=\min_{u \in {\cal P}_0}I_0(u).$$ Inspired by these papers and observed that each solution of satisfies the following Pohozaev identity: $$\label{eq:poho}
\frac{N-2}{2}{\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 + \frac N2 {\int_{{{\mathbb{R}^N}}}}V(x)u^2 +\frac 12 \irn
({\nabla }V(x)\mid x)u^2 = N {\int_{{{\mathbb{R}^N}}}}G(u),$$ we indicate with ${{\cal P}}$ the Pohozaev manifold related to : $$\label{eq:PV}
{{\cal P}}=\left\{ u\in {H^1({{\mathbb{R}^N}})}\setminus\{0\} \mid u \hbox{ satisfies
\eqref{eq:poho}} \right\}$$ and we wonder if there exists a minimizer for $I|_{{\cal P}}$.

Proceeding in analogy with [@BGM], we get the following result

\[non\] If we assume ([**g1-4**]{}), ([**V1**]{}), ([**V3**]{}) and

-   $({\nabla }V(x)\mid x){\leqslant}0,$ for all $x\in{{\mathbb{R}^N}};$

-   $NV(x)+({\nabla }V(x)\mid x){\geqslant}0$ for all $x\in {{\mathbb{R}^N}}$, and the inequality is strict somewhere,

then $b:=\inf_{u\in{{\cal P}}}I(u)$ is not a critical level for the functional $I$.

Requiring something more, instead of ([**V6**]{}), we have a further information on the level $b$:

\[co:non\] Assuming the same hypotheses of Theorem \[non\], if in ([**V6**]{}) we have the strict inequality, then the level $b$ is not achieved as a minimum on the Pohozaev manifold ${{\cal P}}$.

[**Acknowledgement**]{} The authors express their deep gratitude to Prof. L. Jeanjean for stimulating discussions and useful suggestions.

[**NOTATION**]{}

-   For any $1{\leqslant}s< +\infty$, $L^s({{\mathbb{R}^3}})$ is the usual Lebesgue space endowed with the norm $$\|u\|_s^s:={\int_{{{\mathbb{R}^3}}}}|u|^s;$$

-   ${H^1({{\mathbb{R}^N}})}$ is the usual Sobolev space endowed with the norm $$\|u\|^2:={\int_{{{\mathbb{R}^3}}}}|{\nabla }u|^2+ u^2;$$

-   $\D$ is completion of $C_0^\infty({{\mathbb{R}^3}})$ with respect to the norm $$\|u\|_{\D}^2:={\int_{{{\mathbb{R}^3}}}}|{\nabla }u|^2;$$

-   for any $r>0,$ $x\in{{\mathbb{R}^3}}$ and $A\subset {{\mathbb{R}^3}}$ $$\begin{aligned}
    B_r(x) &:=\{y\in{{\mathbb{R}^3}}\mid |y-x|{\leqslant}r\},\\
    B_r    &:=\{y\in{{\mathbb{R}^3}}\mid |y|{\leqslant}r\},\\
    A^c    &:= {{\mathbb{R}^3}}\setminus A.\end{aligned}$$

The existence result
====================

The aim of this section is to prove Theorems \[main\] and \[th:radial\].

Set $$H^1_r({\mathbb{R}}^N):=\{u\in{H^1({{\mathbb{R}^N}})}\mid u \hbox{ is radial}\}$$ and, following [@BL1], define $s_0:=\min\{s\in
[\zeta,+\infty[\;\mid g(s)=0\}$ ($s_0=+\infty$ if $g(s)\neq 0$ for any $s{\geqslant}\zeta$) and set $\tilde g:{\mathbb{R}}\to{\mathbb{R}}$ the function such that $$\label{eq:tilde}
      \tilde g(s)=\left\{
        \begin{array}{ll}
                g(s) &\hbox{ on } [0,s_0];
                \\
                0 &\hbox{ on } {\mathbb{R}}_+\setminus [0,s_0];
                \\
                -\tilde g(-s) &\hbox{ on } {\mathbb{R}}_-.
      \end{array}
      \right.$$ By the strong maximum principle, a solution of with $\tilde g$ in the place of $g$ is a solution of . So we can suppose that $g$ is defined as in , so that ([**g1**]{}), ([**g2**]{}), ([**g4**]{}) and then the following limit $$\label{eq:limg}
        \lim_{s\to\infty} \frac{|g(s)|}{|s|^{2^*-1}}=0$$ hold. Moreover, we set for any $s{\geqslant}0,$ $$\begin{aligned}
        g_1(s) & :=(g(s)+ms)^+,
        \\
        g_2(s) & :=g_1(s)-g(s),
    \end{aligned}$$ and we extend them as odd functions.\
Since $$\begin{aligned}
        \lim_{s\to 0}\frac{g_1(s)}{s} &= 0,\label{eq:lim1}\\
        \lim_{s\to\infty}\frac{g_1(s)}{|s|^{2^*-1}}&=0,\label{eq:lim2}
    \end{aligned}$$ and $$g_2(s) {\geqslant}ms,\quad  \forall s{\geqslant}0,\label{eq:g2}$$ by some computations, we have that for any ${\varepsilon}>0$ there exists $C_{\varepsilon}>0$ such that $$g_1(s) {\leqslant}C_{\varepsilon}s^{2^*-1}+{\varepsilon}g_2(s),\quad  \forall
        s{\geqslant}0\label{eq:g1g2}.$$ If we set $$G_i(t):=\int^t_0g_i(s)\,ds,\quad i=1,2,$$ then, by and , we have $$G_2(s) {\geqslant}\frac m 2 s^2,\quad  \forall s\in{\mathbb{R}}\label{eq:G2}$$ and for any ${\varepsilon}>0$ there exists $C_{\varepsilon}>0$ such that $$G_1(s) {\leqslant}\frac {C_{\varepsilon}} {2^*} |s|^{2^*}+{\varepsilon}G_2(s),\quad  \forall
        s\in{\mathbb{R}}\label{eq:G1G2}.$$

Using an idea from [@J], we look for bounded Palais-Smale sequences of the following perturbed functionals $$I_\l(u)=\frac 12 {\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 + V(x)u^2 + 
{\int_{{{\mathbb{R}^N}}}}G_2(u) - \l{\int_{{{\mathbb{R}^N}}}}G_1(u),$$ for almost all $\l$ near $1$. Then we will deduce the existence of a non-trivial critical point $v_\l$ of the functional $I_\l$ at the mountain pass level. Afterward, we study the convergence of the sequence $(v_\l)_\l$, as $\l$ goes to 1 (observe that $I_1=I$).

We will apply the following slight modified version of [@J Theorem 1.1] (see [@J2]).

\[th:J\] Let $\big(X,\|\cdot\|\big)$ be a Banach space and $J\subset{\mathbb{R}}^+$ an interval. Consider the family of $C^1$ functionals on $X$ $$I_\l(u)=A(u)- {\lambda}B(u),\quad\forall\l\in J,$$ with $B$ nonnegative and either $A(u)\to + \infty$ or $B(u)\to+\infty$ as $\|u\|\to\infty.$\
For any $\l\in J$ we set $$\label{eq:gamma}
        \Gamma_\l:=\{\gamma\in C([0,1],X)\mid \gamma(0)=0\neq\gamma(1),
        I_\l(\gamma(1))< 0,\}.$$ If for every $\l\in J$ the set ${\Gamma}_\l$ is nonempty and $$\label{eq:cl}
        c_\l:=\inf_{\gamma\in\Gamma_\l}\max_{t\in[0,1]}
        I_\l(\gamma(t)) > I_\l(v),$$ then for almost every $\l\in J$ there is a sequence $(v_n)_n
\subset X$ such that

-   $(v_n)_n$ is bounded;

-   $I_\l(v_n)\to c_\l$;

-   $(I_\l)'(v_n)\to 0$ in the dual $X^{-1}$ of $X$.

In our case, $X={H^1_r({{\mathbb{R}^N}})}$ and $$\begin{aligned}
        A(u) &:=\frac 12 {\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 + V(x)u^2 
        + {\int_{{{\mathbb{R}^N}}}}G_2(u),\\
        B(u) &:={\int_{{{\mathbb{R}^N}}}}G_1(u).
    \end{aligned}$$

In order to apply Theorem \[th:J\], we have just to define a suitable interval $J$ such that $\Gamma_\l\neq\emptyset$, for any $\l\in J$, and holds.\
Observe that, according to [@BL1], there exists a function $z\in{H^1_r({{\mathbb{R}^N}})}$ such that $${\int_{{{\mathbb{R}^N}}}}G_1(z)-{\int_{{{\mathbb{R}^N}}}}G_2(z)={\int_{{{\mathbb{R}^N}}}}G(z)>0.$$ Then there exists $0<\bar\d<1$ such that $$\label{eq:d}
        \bar\d{\int_{{{\mathbb{R}^N}}}}G_1(z)-{\int_{{{\mathbb{R}^N}}}}G_2(z)>0.$$ We define $J$ as the interval $[\bar \delta , 1].$

\[le:Gamma\] $\Gamma_\l\neq\emptyset$, for any $\l\in J$.

Let $\l\in J$. Set $\bar{\theta}>0$ sufficiently large and $\bar
z=z(\cdot /
\bar{\theta})$.\
Define $\g:[0,1]\to{H^1_r({{\mathbb{R}^N}})}$ in the following way $$\g(t)=\left\{\begin{array}{ll}
            0,& \hbox{if }t=0,
            \\
            \bar z^t=\bar z(\cdot/t), & \hbox{if }0<t{\leqslant}1.
        \end{array}
        \right.$$ It is easy to see that $\g$ is a continuous path from $0$ to $\bar
z.$ Moreover, we have that $$\begin{gathered}
        I_\l(\g(1)) {\leqslant}\frac {\bar{\theta}^{N-2}} 2 {\int_{{{\mathbb{R}^N}}}}|{\nabla }z|^2 
        +\frac{\bar{\theta}^N} 2 {\int_{{{\mathbb{R}^N}}}}V\left(\bar{\theta}x\right) z^2 
        \\
        +\bar{\theta}^N
        \left({\int_{{{\mathbb{R}^N}}}}G_2(z) - \bar\d{\int_{{{\mathbb{R}^N}}}}G_1(z)\right)
    \end{gathered}$$ and then, by , ([**V3**]{}) and the Lebesgue theorem, for a suitable choice of $\bar{\theta}$, certainly $\g\in{\Gamma}_\l.$

\[le:cl\] $c_\l>0$ for all $\l\in J.$

Observe that for any $u\in{H^1_r({{\mathbb{R}^N}})}$ and $\l\in J$, using and for ${\varepsilon}< 1$, we have $$\begin{aligned}
        I_\l(u) & {\geqslant}\frac 1 2 {\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 + V(x)u^2
        +{\int_{{{\mathbb{R}^N}}}}G_2 (u) -{\int_{{{\mathbb{R}^N}}}}G_1 (u)\\
                  & {\geqslant}\frac 1 2 {\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2  + (1-{\varepsilon}) \frac
                  m 2 {\int_{{{\mathbb{R}^N}}}}u^2 - \frac{C_{\varepsilon}}{2^*} \irn
                  |u|^{2^*}
    \end{aligned}$$ and then, by Sobolev embeddings, we conclude that there exists $\rho >0$ such that for any $\l\in J$ and $u\in{H^1_r({{\mathbb{R}^N}})}$ with $u\neq 0$ and $\|u\|{\leqslant}\rho,$ it results $I_\l(u)>0.$ In particular, for any $\|u\|=\rho,$ we have $I_\l(u) {\geqslant}\tilde c >0.$ Now fix $\l\in J$ and $\g\in{\Gamma}_\l.$ Since $\g(0)=0$ and $I_\l(\g(1))< 0$, certainly $\|\g(1)\|
> \rho.$ By continuity, we deduce that there exists $t_\g\in
]0,1[$ such that $\|\g(t_\g)\|=\rho.$ Therefore, for any $\l\in
J,$ $$\label{eq:cll}
c_\l{\geqslant}\inf_{\g\in{\Gamma}_\l} I_\l(\g(t_\g)) {\geqslant}\tilde c >0.$$

We present a variant of the Strauss’ compactness lemma [@Str] (see also [@BL1 Theorem A.1]), whose proof is similar to that contained in [@BL1]. It will be a fundamental tool in our arguments:

\[le:str\] Let $P$ and $Q:{\mathbb{R}}\to{\mathbb{R}}$ be two continuous functions satisfying $$\lim_{s\to\infty}\frac{P(s)}{Q(s)}=0,$$ $(v_n)_n,$ $v$ and $z$ be measurable functions from ${{\mathbb{R}^N}}$ to ${\mathbb{R}}$ such that $$\begin{aligned}
&\sup_n{\int_{{{\mathbb{R}^N}}}}| Q(v_n(x))z|\,dx <+\infty,
\\ &P(v_n(x))\to v(x) \:\hbox{a.e. in }{{\mathbb{R}^N}}. $$ Then $\|(P(v_n)-v)z\|_{L^1(B)}\to 0$, for any bounded Borel set $B$.

Moreover, if we have also $$\begin{aligned}
\lim_{s\to 0}\frac{P(s)}{Q(s)} &=0,\\ \lim_{x\to\infty}\sup_n |v_n(x)| &= 0, $$ then $\|(P(v_n)-v)z\|_{L^1({{\mathbb{R}^N}})}\to 0.$

In analogy with the well-known compactness result in [@BL2], we state the following result

\[le:PS\] For any ${\lambda}\in J$, each bounded Palais-Smale sequence for the functional $I_\l$ admits a convergent subsequence.

Let $\l\in J$ and $(u_n)_n$ be a bounded (PS) sequence for $I_\l$, namely $$\begin{aligned}
\label{eq:PS}
        &(I_\l(u_n))_n \hbox{ is bounded },\nonumber\\
        &\lim_n (I_\l)'(u_n)= 0 \hbox{ in } ({H^1_r({{\mathbb{R}^N}})})'.
    \end{aligned}$$ Up to a subsequence, we can suppose that there exists $u\in{H^1_r({{\mathbb{R}^N}})}$ such that $$\label{eq:weak}
        u_n\rightharpoonup u\;\hbox{weakly in }{H^1_r({{\mathbb{R}^N}})}$$ and $$\label{eq:aeconv}
        u_n(x)\to u(x)\;\hbox{a.e. in }{{\mathbb{R}^N}}.$$ By weak lower semicontinuity we have: $$\begin{aligned}
{\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 {\leqslant}&\liminf_n {\int_{{{\mathbb{R}^N}}}}|{\nabla }u_n|^2; \label{eq:semi1}
\\
{\int_{{{\mathbb{R}^N}}}}V(x) u^2 {\leqslant}&\liminf_n {\int_{{{\mathbb{R}^N}}}}V(x)u_n^2. \label{eq:semi2}\end{aligned}$$ If we apply Lemma \[le:str\] for $P(s)=g_i(s)$, $i=1,2,$ $Q(s)=
|s|^{2^*-1},$ $(v_n)_n=(u_n)_n,$ $v=g_i(u),$ $i=1,2$ and $z\in
C^\infty_0({{\mathbb{R}^N}}),$ by , and we deduce that $${\int_{{{\mathbb{R}^N}}}}g_i(u_n)z\to{\int_{{{\mathbb{R}^N}}}}g_i(u)z\quad i=1,2.$$ As a consequence, by and we deduce $(I_\l)'(u)=0$ and hence $$\label{eq:solo}
{\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2+ V(x)u^2= {\int_{{{\mathbb{R}^N}}}}{\lambda}g_1(u)u - g_2(u)u.$$\
If we apply Lemma \[le:str\] for $P(s)=g_1(s)s,$ $Q(s)= s^2+
|s|^{2^*},$ $(v_n)_n=(u_n)_n,$ $v=g_1(u)u,$ and $z=1,$ by , , and the well known Strauss’ radial lemma (see [@Str]) we deduce that $$\begin{aligned}
        {\int_{{{\mathbb{R}^N}}}}g_1(u_n)u_n \to {\int_{{{\mathbb{R}^N}}}}g_1(u)u.\label{eq:convg}
    \end{aligned}$$ Moreover, by and Fatou’s lemma $$\begin{aligned}
        {\int_{{{\mathbb{R}^N}}}}g_2(u)u{\leqslant}&\liminf_n {\int_{{{\mathbb{R}^N}}}}g_2(u_n)u_n.\label{eq:convg2}
    \end{aligned}$$ By , and , and since $\langle (I_\l)'(u_n),u_n\rangle\to0$ $$\begin{aligned}
\limsup_n {\int_{{{\mathbb{R}^N}}}}|{\nabla }u_n|^2 + V(x)u_n^2 &=\limsup_n \left[{\lambda}\irn
g_1(u_n)u_n -{\int_{{{\mathbb{R}^N}}}}g_2(u_n)u_n\right] \nonumber
\\
&{\leqslant}{\lambda}{\int_{{{\mathbb{R}^N}}}}g_1(u)u - {\int_{{{\mathbb{R}^N}}}}g_2(u)u \nonumber\\
&={\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 + V(x)u^2. \label{eq:limsup}\end{aligned}$$ By , and , we get $$\begin{aligned}
\lim_n {\int_{{{\mathbb{R}^N}}}}|{\nabla }u_n|^2= &{\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 \label{eq:sf1},
\\
\lim_n {\int_{{{\mathbb{R}^N}}}}V(x) u_n^2= &{\int_{{{\mathbb{R}^N}}}}V(x) u^2, \nonumber $$ hence $$\label{eq:sfg2}
\lim_n {\int_{{{\mathbb{R}^N}}}}g_2(u_n)u_n= {\int_{{{\mathbb{R}^N}}}}g_2(u)u.$$ Since $g_2(s)s=ms^2 +q(s)$, with $q$ a positive and continuous function, by Fatou’s Lemma we have $$\begin{aligned}
{\int_{{{\mathbb{R}^N}}}}q(u) {\leqslant}&\liminf_n {\int_{{{\mathbb{R}^N}}}}q(u_n);
\\
{\int_{{{\mathbb{R}^N}}}}u^2 {\leqslant}&\liminf_n {\int_{{{\mathbb{R}^N}}}}u_n^2.\end{aligned}$$ These last two inequalities and imply that, up to a subsequence, $${\int_{{{\mathbb{R}^N}}}}u^2 =\lim_n {\int_{{{\mathbb{R}^N}}}}u_n^2,$$ which, together with , shows that $u_n \to u$ strongly in ${H^1_r({{\mathbb{R}^N}})}$.

\[le:ul\] For almost every $\l\in J$, there exists $u^\l\in {H^1_r({{\mathbb{R}^N}})}$, $u^\l\neq
0$, such that $(I_\l)'(u^\l)=0$ and $I_\l(u^\l)= c_\l$.

By Theorem \[th:J\], for almost every $\l\in J$, there exists a bounded sequence $(u^\l_n)_n\subset{H^1_r({{\mathbb{R}^N}})}$ such that $$\begin{aligned}
        I_\l(u^\l_n)&\to c_\l;\label{eq:conv}\\
        (I_\l)'(u^\l_n)&\to 0\;\hbox{in } ({H^1_r({{\mathbb{R}^N}})})'.\label{eq:p-s}
    \end{aligned}$$ Up to a subsequence, by Lemma \[le:PS\], we can suppose that there exists $u^\l\in{H^1_r({{\mathbb{R}^N}})}$ such that $u^\l_n \to u^\l$ in ${H^1_r({{\mathbb{R}^N}})}$. By Lemma \[le:cl\], and we conclude.

Now we are able to provide the proof of our main result:

By Lemma \[le:ul\], we are allowed to consider a suitable $\l_n\nearrow 1$ such that for any $n{\geqslant}1$ there exists $v_n\in{H^1_r({{\mathbb{R}^N}})}\setminus\{0\}$ satisfying $$\begin{aligned}
I_{\l_n}(v_n)&= c_{\l_n},\label{eq:mp}
\\
(I_{\l_n})'(v_n)&=0\;\hbox{in }({H^1_r({{\mathbb{R}^N}})})'.      \label{eq:sol}\end{aligned}$$ We want to prove that $(v_n)_n$ is a bounded Palais-Smale sequence for $I$ at the level $c:=c_1$.\
By standard argument, since ${H^1_r({{\mathbb{R}^N}})}$ is a natural constraint, we have that $v_n$ is a weak solution of the problem $$-\Delta w + V(x) w + g_2(w) - \l_n g_1(w) = 0$$ and it satisfies the Pohozaev equality $$\begin{gathered}
\label{eq:Poho}
        {\int_{{{\mathbb{R}^N}}}}|{\nabla }v_n|^2 + \frac N {N-2} {\int_{{{\mathbb{R}^N}}}}V(x) v_n^2
        + \frac 1 {N-2}{\int_{{{\mathbb{R}^N}}}}({\nabla }V(x)\mid x) v_n^2
        \\
        +\frac {2N} {N-2}
        {\int_{{{\mathbb{R}^N}}}}G_2(v_n)-\l_n G_1(v_n)=0.
    \end{gathered}$$ Therefore, by , and we have that the following system holds $$\left\{
            \begin{array}{l}
                \frac 1 2 (\alpha_n + \beta_n) + \gamma_{2,n} - \l_n
                \gamma_{1,n} = c_{\l_n},\label{eq:sys1}
                \\
                \alpha_n + \beta_n + \d_{2,n} - \l_n \d_{1,n} = 0,
                \\
                \alpha_n + \frac N {N-2} \beta_n + \frac 1 {N-2}\eta_n +\frac {2N} {N-2}
                \gamma_{2,n} - \frac {2N} {N-2}\l_n \gamma_{1,n}=0,
            \end{array}
        \right.$$ where $$\begin{aligned}
        \alpha_n = {\int_{{{\mathbb{R}^N}}}}|{\nabla }v_n|^2, & \displaystyle\beta_n={\int_{{{\mathbb{R}^N}}}}V(x) v_n^2,& \eta_n={\int_{{{\mathbb{R}^N}}}}(\n
        V(x)\mid x) v_n^2,
        \\
        \gamma_{i,n}={\int_{{{\mathbb{R}^N}}}}G_i(v_n),   & \displaystyle\d_{i,n} ={\int_{{{\mathbb{R}^N}}}}g_i(v_n)v_n, &
        i=1,2.
    \end{aligned}$$ By the first and the third of the system we get $$\frac 1 N \alpha_n - \frac 1 {2N} \eta_n = c_{\l_n}$$ and then, using Holder inequality, by $({\bf V2})$ and the boundedness of $(c_{\l_n})_n$ (indeed the map ${\lambda}\mapsto c_\l$ is non-increasing), we have $$\label{eq:bound}
        \alpha_n{\leqslant}C \hbox{ for all }n{\geqslant}1.$$ By the second of the system we have $$\d_{2,n} - \l_n \d_{1,n} = - \alpha_n  - \beta_n{\leqslant}0$$ and then by , there exists $0<{\varepsilon}<1$ and $C_{\varepsilon}>0$ such that $$\d_{2,n} {\leqslant}\d_{1,n} {\leqslant}C_{\varepsilon}{\int_{{{\mathbb{R}^N}}}}|v_n|^{2^*}+{\varepsilon}\d_{2,n}.$$ Therefore, by the Sobolev embedding $\D\hookrightarrow
L^{2^*}({{\mathbb{R}^N}})$ $$(1-{\varepsilon}) \d_{2,n} {\leqslant}C_{\varepsilon}{\int_{{{\mathbb{R}^N}}}}|v_n|^{2^*}{\leqslant}C
        \alpha_n^{2^*}$$ and then, by , $\d_{2,n}$ is bounded. By and we deduce that $(v_n)_n$ is bounded in ${H^1_r({{\mathbb{R}^N}})}$.\
Up to a subsequence, there exists $v\in{H^1_r({{\mathbb{R}^N}})}$ such that $$\label{eq:weak2}
        v_n\rightharpoonup v\;\hbox{weakly in }{H^1_r({{\mathbb{R}^N}})}.$$ By , we have that $$I'(v_n)=(I_{\l_n})'(v_n) + (\l_n -1) g_1(v_n)= (\l_n-1)
        g_1(v_n)$$ so $$\label{eq:nome}
I'(v_n)\to 0 \quad \hbox{in }({H^1_r({{\mathbb{R}^N}})})',$$ if $\left(g_1(v_n)\right)_n$ is bounded in $\big({H^1_r({{\mathbb{R}^N}})}\big)'.$ But this is true by the Banach-Steinhaus theorem, since Lemma \[le:str\] implies that for any $z\in{H^1_r({{\mathbb{R}^N}})}$ $${\int_{{{\mathbb{R}^N}}}}g_1(v_n)z \to {\int_{{{\mathbb{R}^N}}}}g_1(v)z.$$ Moreover, from and the boundedness of $(v_n)_n$, we deduce that $$\label{eq:mou}
        I(v_n)=I_{\l_n}(v_n) + (\l_n -1) {\int_{{{\mathbb{R}^N}}}}G_1(v_n)\to c.$$ Therefore, by and , $(v_n)_n$ is a Palais-Smale sequence for the functional $I$ and so, by Lemma \[le:PS\], $v$ is a nontrivial mountain pass type solution for .\
To conclude, observe that, by standard arguments, we can use the strong maximum principle to get $v>0.$

In order to prove Theorem \[th:radial\] we set $$\begin{aligned}
        {\mathcal S_r}&:=\left\{u\in{H^1_r({{\mathbb{R}^N}})}\setminus\{0\}\mid
        I'(u)=0\right\},\\
        \sigma_r&:=\inf_{u\in {\mathcal S_r}} I(u).
    \end{aligned}$$

\[le:s\] We have $\sigma_r>0.$

By and Sobolev embedding we have that for any $u\in{\mathcal S_r}$ $$\begin{aligned}
            \|{\nabla }u\|_2^2&{\leqslant}{\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 + V(x) u^2 +
            (1-{\varepsilon}){\int_{{{\mathbb{R}^N}}}}g_2(u)u\\
            &{\leqslant}C_{\varepsilon}\|u\|_{2^*}^{2^*}{\leqslant}C \|{\nabla }u\|_2^{2^*}
        \end{aligned}$$ where $0<{\varepsilon}<1$ and $C_{\varepsilon},C>0.$ So we deduce that $$\inf_{u\in{\mathcal S_r}} \|{\nabla }u\|_2>0.$$ Now, since ${\mathcal S_r}\subset {{\cal P}}$ (see ), for any $u\in{\mathcal S_r}$ by ([**V2**]{}) we have $$I(u)= \frac 1N{\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 -\frac 1{2N} {\int_{{{\mathbb{R}^N}}}}({\nabla }V(x)\mid x) u^2{\geqslant}C >0 .$$

Finally we provide the following

Let $(u_n)_n\in{\mathcal S_r}$ such that $I(u_n)\to \sigma_r.$ Arguing as in the proof of Theorem \[main\] we have that the sequence is bounded. By Lemma \[le:PS\], there exists $u\in{H^1_r({{\mathbb{R}^N}})}$ such that $u_n\to u$ in ${H^1_r({{\mathbb{R}^N}})}$ and then the conclusion follows by Lemma \[le:s\].

The nonexistence result
=======================

In this section we give the proof of Theorem \[non\] and we will assume that $V$ satisfies hypotheses ([**V1**]{}), ([**V3**]{}), ([**V5-6**]{}).

Let us show that the functional $I$ is bounded below on the manifold ${{\cal P}}$:

\[le:PV\] For all $u \in {{\cal P}}$, $I(u)>0$.

It is easy to see that for any $u\in {{\cal P}}$, by ([**V2**]{}) we get $$\label{eq:iv}
I(u)= \frac 1N{\int_{{{\mathbb{R}^N}}}}|{\nabla }u|^2 -\frac 1{2N} {\int_{{{\mathbb{R}^N}}}}({\nabla }V(x)\mid x) u^2>
0.$$

By Lemma \[le:PV\], we can define $$\label{eq:mv}
b=\inf_{u \in {{\cal P}}}I(u){\geqslant}0.$$

\[le:p-pv\] Let $w\in {H^1({{\mathbb{R}^N}})}$ be such that ${\int_{{{\mathbb{R}^N}}}}G(w)>0$. Then there exists $\bar{\theta}>0$ such that $w^{\bar{\theta}}=w(\cdot/\bar{\theta})\in {{\cal P}}$. In particular this result is true for any $w\in {{\cal P}}_0$ (see ).

For any ${\theta}>0$, we set $$f({\theta}):=I(w^{\theta}) =\frac{{\theta}^{N-2}}{2} {\int_{{{\mathbb{R}^N}}}}|{\nabla }w|^2 +\frac{{\theta}^N}{2}
{\int_{{{\mathbb{R}^N}}}}V({\theta}x) w^2 - {\theta}^N {\int_{{{\mathbb{R}^N}}}}G(w).$$ By the Lebesgue theorem and ([**V3**]{}), we get $$\lim_{{\theta}\to +\infty}{\int_{{{\mathbb{R}^N}}}}V({\theta}x) w^2=0,$$ and so $$\lim_{{\theta}\to +\infty}f({\theta})=-\infty.$$ We argue that there exists $\bar{\theta}>0$ such that $f'(\bar{\theta})=0$: hence $w^{\bar{\theta}}\in {{\cal P}}$.

Let $w\in {{\cal P}}_0$. For any $y\in {{\mathbb{R}^N}}$, we set $w_y:=w(\cdot
-y)\in{\mathcal P_0}$. Set ${\theta}_y>0$ such that $\tilde
w_y=w_y(\cdot/{\theta}_y)\in {{\cal P}}$.

\[le:t\] We have $\lim_{|y|\to \infty}{\theta}_y=1$.

[Step 1]{}: $\limsup_{|y|\to\infty}{\theta}_y<+\infty.$\
Suppose, by contradiction, that ${\theta}_{y_n} \to +\infty$, for $|y_n|\to \infty$.\
For any $y\in {{\mathbb{R}^N}}$, we have $$\label{eq:iwt}
I(\tilde w_y) =\frac{{\theta}_y^{N-2}}{2} {\int_{{{\mathbb{R}^N}}}}|{\nabla }w|^2
+\frac{{\theta}^N_y}{2} {\int_{{{\mathbb{R}^N}}}}V({\theta}_y x) \,w^2\!\left(\!x-\frac{y}{{\theta}_y}
\!\right) - {\theta}^N_y {\int_{{{\mathbb{R}^N}}}}G(w).$$ Let us show that $$\label{eq:wt}
\lim_{|y|\to \infty}{\int_{{{\mathbb{R}^N}}}}V({\theta}_y x) \,w^2\!\left(\!x-\frac{y}{{\theta}_y}
\!\right)=0.$$ Indeed we have $$\begin{aligned}
{\int_{{{\mathbb{R}^N}}}}V({\theta}_y x) \,w^2\!\left(\!x-\frac{y}{{\theta}_y} \!\right)
&=\int_{B_r} V({\theta}_y x) \,w^2\!\left(\!x-\frac{y}{{\theta}_y} \!\right)
+\int_{B_r^c} V({\theta}_y x) \,w^2\!\left(\!x-\frac{y}{{\theta}_y} \!\right) \nonumber
\\
&{\leqslant}\sup_{x\in{{\mathbb{R}^N}}}V(x)\int_{B_r(-y/{\theta}_y)}w^2 
+\sup_{x\in B_r^c}V({\theta}_y x)\,\|w\|_2^2. \label{eq:dai}\end{aligned}$$ By the absolute continuity of the Lebesgue integral, for any ${\varepsilon}>0$ there exists $\bar r>0$ such that, for any $r<\bar r$ and for any $y\in {{\mathbb{R}^N}}$, we get $$\label{eq:ihih}
\sup_{x\in{{\mathbb{R}^N}}}V(x) \cdot \int_{B_r(-y/{\theta}_y)}w^2 {\leqslant}{\varepsilon}.$$ Therefore, since we are supposing that ${\theta}_{y_n} \to +\infty$, as $|y_n|\to \infty$, by , and ([**V2**]{}), we get . As a consequence, by and , we infer that $I(\tilde
w_{y_n})\to -\infty$, as $|y_n|\to \infty$, and we get a contradiction with Lemma \[le:PV\].\
\
[Step 2]{}: $\lim_{|y|\to \infty}{\theta}_y=1$.\
Since $w\in {{\cal P}}_0$ and $\tilde w_y\in {{\cal P}}$, we get $$\begin{gathered}
\label{eq:ty}
N({\theta}_y^2-1){\int_{{{\mathbb{R}^N}}}}\! G(w)\\
=\frac{{\theta}_y^2}{2} {\int_{{{\mathbb{R}^N}}}}\big[N V({\theta}_y
x+y) +({\nabla }V({\theta}_y x+y)\mid({\theta}_y x+y))\big]w^2.\end{gathered}$$ By ([**V3**]{}), ([**V5**]{}) and ([**V6**]{}), using the dominated convergence and the conclusion of the Step 1, the right hand side in goes to zero as $|y|\to \infty$, and so the lemma is proved.

We set (see ) $$b_0:=\min_{u\in{\cal S}_0}I_0(u)=\min_{u\in {{\cal P}}_0}I_0(u).$$

\[le:&lt;=\] $b{\leqslant}b_0$.

Let $w\in {H^1({{\mathbb{R}^N}})}$ be a ground state solution of . Then $w\in
{{\cal P}}_0$ and $I_0(w)=b_0$. For any $y\in {{\mathbb{R}^N}}$, we set $w_y=w(\cdot
-y)$. By the invariance by translations of , we have that $w_y\in {{\cal P}}_0$ and $I_0(w_y)=b_0$. By Lemma \[le:p-pv\], for any $y\in {{\mathbb{R}^N}}$ there exists ${\theta}_y>0$ such that $\tilde w_y=
w_y(\cdot/{\theta}_y)\in {{\cal P}}$. We get $$\begin{aligned}
|I(\tilde w_y)-b_0| &=|I(\tilde w_y)-I_0(w_y)|
\\
&{\leqslant}\frac{|{\theta}_y^{N-2}-1|}{2} {\int_{{{\mathbb{R}^N}}}}|{\nabla }w|^2 + \frac{{\theta}_y^N}{2}\irn
V({\theta}_y x+y) w^2
\\
&\quad + |{\theta}_y^N -1| {\int_{{{\mathbb{R}^N}}}}G(w).\end{aligned}$$ Therefore, by Lemma \[le:t\], we infer that $$\lim_{|y|\to \infty} I(\tilde w_y)=b_0,$$ hence $b{\leqslant}b_0$.

\[le:pv-p\] Let $z\in {H^1({{\mathbb{R}^N}})}$ be such that ${\int_{{{\mathbb{R}^N}}}}G(z)>0$. Then there exists $\bar{\theta}>0$ such that $z^{\bar{\theta}}=z(\cdot/\bar{\theta})\in {{\cal P}}_0$. In particular, by ([**V6**]{}) this result is true for any $z\in {{\cal P}}$ with $\bar{\theta}{\leqslant}1$.

The first part of the statement follows from the fact that, for any $z\in {H^1({{\mathbb{R}^N}})}$ such that ${\int_{{{\mathbb{R}^N}}}}G(z)>0$, certainly there exists $\bar\theta>0$ such that $$\label{eq:po}
        \frac{N-2}{2}{\int_{{{\mathbb{R}^N}}}}|{\nabla }z|^2 = N\bar{\theta}^2
        {\int_{{{\mathbb{R}^N}}}}G(z).$$ Consider now the case of $z\in {{\cal P}}$. Since $$\label{eq:po2}
        \frac{N-2}{2}{\int_{{{\mathbb{R}^N}}}}|{\nabla }z|^2 + \frac N2 {\int_{{{\mathbb{R}^N}}}}V(x)z^2 +\frac 12 \irn
        ({\nabla }V(x)\mid x)z^2 = N {\int_{{{\mathbb{R}^N}}}}G(z),$$ by ([**V6**]{}) we have ${\int_{{{\mathbb{R}^N}}}}G(z)>0.$ Let $\bar\theta>0$ such that holds. Combining and we get $$\label{eq:1-t}
\frac 12 {\int_{{{\mathbb{R}^N}}}}\big[ NV(x)+({\nabla }V(x)\mid x) \big] z^2 = N
(1-\bar{\theta}^2){\int_{{{\mathbb{R}^N}}}}G(z).$$ By ([**V6**]{}), we get the conclusion.

Now we can prove Theorem \[non\]:

Suppose by contradiction that there exists $z\in {H^1({{\mathbb{R}^N}})}$ critical point of the functional $I$ at level $b$: in particular, $z\in {{\cal P}}$ and $I(z)=b$. Let ${\theta}\in (0,1]$ be such that $z^{\theta}\in {{\cal P}}_0$. Let us show that ${\theta}<1$.\
By standard arguments and using the strong maximum principle, we infer that $z$ does not change sign and so we can assume that $z>0$. Therefore, by ([**V6**]{}) and , we get that ${\theta}<1$.\
By and ([**V5**]{}), we infer that $$\begin{aligned}
b&=I(z)= \frac 1N {\int_{{{\mathbb{R}^N}}}}|{\nabla }z|^2 -\frac 1{2N} {\int_{{{\mathbb{R}^N}}}}({\nabla }V(x)\mid x)
z^2  \nonumber
\\
&>  \frac {{\theta}^{N-2}}N  {\int_{{{\mathbb{R}^N}}}}|{\nabla }z|^2=I_0(z^{\theta}){\geqslant}b_0,
$$ and we get a contradiction with Lemma \[le:&lt;=\].

If the strict inequality in ([**V6**]{}) is satisfied almost everywhere, then for any $z\in {{\cal P}}$ there exists ${\theta}\in (0,1)$ such that $z^{\theta}\in {{\cal P}}_0$. Arguing as in the proof of Theorem \[non\] we conclude.

In view of Theorem \[non\], the proof of Corollary \[co:non\] would follow immediately if ${{\cal P}}$ was a natural constraint for the functional $I$.

[99]{}

V. Benci, C.R. Grisanti, A.M. Micheletti, [*Existence and non existence of the ground state solution for the nonlinear Schrödinger equations with $V(\infty)=0$*]{}, Topol. Methods Nonlinear Anal., [**26**]{}, (2005), 203–219.

H. Berestycki, P.L. Lions, [*Nonlinear scalar field equations. I. Existence of a ground state*]{}, Arch. Rational Mech. Anal., [**82**]{}, (1983), 313–345.

H. Berestycki, P.L. Lions, [*Nonlinear scalar field equations. II. Existence of infinitely many solutions*]{}, Arch. Rational Mech. Anal., [**82**]{}, (1983), 347–375.

L. Jeanjean, [*On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${{\mathbb{R}^N}}$*]{}, Proc. R. Soc. Edinb., Sect. A, Math., [**129**]{}, (1999), 787–809.

L. Jeanjean, [*Local condition insuring bifurcation from the continuous spectrum*]{}, Math. Z., [**232**]{}, (1999), 651–664.

L. Jeanjean, K. Tanaka, [*A remark on least energy solutions in ${{\mathbb{R}^N}}$*]{}, Proc. Am. Math. Soc., [**131**]{},, (2003) 2399–2408.

L. Jeanjean, K. Tanaka, [*A positive solution for a nonlinear Schrödinger equation on ${{\mathbb{R}^N}}$*]{}, Indiana Univ. Math. J., [**54**]{}, (2005), 443–464.

P.H. Rabinowitz, *On a class of nonlinear Schrödinger equations*, Z. Angew. Math. Phys., [**43**]{}, (1992), 270–291.

J. Shatah, [*Unstable ground state of nonlinear Klein-Gordon equations*]{}, Trans. Amer. Math. Soc., [**290**]{}, (1985), 701–710.

W.A. Strauss, [*Existence of solitary waves in higher dimensions*]{}, Comm. Math. Phys., [**55**]{}, (1977), 149–162.

[^1]: Dipartimento di Matematica, Università degli Studi di Bari, Via E. Orabona 4, I-70125 Bari, Italy, e-mail: [azzollini@dm.uniba.it]{}

[^2]: Dipartimento di Matematica, Politecnico di Bari, Via E. Orabona 4, I-70125 Bari, Italy, e-mail: [a.pomponio@poliba.it]{}
