\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small Two nonlinear days in Urbino 2017,\newline \emph{Electronic Journal of Differential Equations}, Conference 25 (2018), pp. 221--233.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \setcounter{page}{221} \title[\hfilneg EJDE-2018/conf/25\hfil Ground state solutions] {Ground state solutions for Bessel fractional equations with irregular nonlinearities} \author[S. Secchi \hfil EJDE-2018/conf/25\hfilneg] {Simone Secchi} \address{Simone Secchi \newline Dipartimento di Matematica e Applicazioni, Universit\`a degli Studi di Milano Bicocca, Italy} \email{simone.secchi@unimib.it} \thanks{Published September 15, 2018} \subjclass[2010]{35KJ60, 35Q55, 35S05} \keywords{Bessel fractional operator, fractional Laplacian} \begin{abstract} We consider the semilinear fractional equation \[ (I-\Delta)^s u = a(x) |u|^{p-2}u \quad\text{in }\mathbb{R}^N, \] where \(N \geq 3\), \(02\), \(00$ we introduce the \emph{Bessel function space} \[ L^{s,2}(\mathbb{R}^N) = \{ f \in L^2(\mathbb{R}^N): f=G_s \star g \ \text{for some $g \in L^2(\mathbb{R}^N)$} \}, \] where the Bessel convolution kernel is defined by \[ G_s (x) = \frac{1}{(4 \pi )^{s /2}\Gamma(s/2)} \int_0^\infty \exp \Big( -\frac{\pi}{t} |x|^2 \Big) \exp \Big( -\frac{t}{4\pi} \Big) t^{\frac{s - N}{2}-1} \, dt. \] The Bessel space is endowed with the norm~$\|f\| = \|g\|_2$ if $f=G_s \star g$. The operator $(I-\Delta)^{-s} u = G_{2s}\star u$ is usually called Bessel operator of order $s$. In Fourier variables the same operator reads \[ G_s = \mathcal{F}^{-1} \circ \Big( \big(1+|\xi|^2 \big)^{-s /2} \circ \mathcal{F} \Big), \] so that \[ \|f\| = \| (I-\Delta)^{s /2} f \|_2. \] For more detailed information, see \cite{Adams, Stein} and the references therein. In \cite{Fall} the pointwise formula \[ (I-\Delta)^s u(x) = c_{N,s} \operatorname{P.V.} \int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{\frac{N+2s}{2}}} K_{\frac{N+2s}{2}}(|x-y|) \, dy + u(x) \] was derived for functions $u \in C_c^2(\mathbb{R}^N)$. Here $c_{N,s}$ is a positive constant depending only on $N$ and $s$, P.V. denotes the principal value of the singular integral, and $K_\nu$ is the modified Bessel function of the second kind with order $\nu$ (see \cite[Remark 7.3]{Fall} for more details). However a closed formula for $K_\nu$ is not known. We summarize the main properties of Bessel spaces. For the proofs we refer to \cite[Theorem 3.1]{Felmer}, \cite[Chapter V, Section 3]{Stein}. \begin{theorem} \label{th:sobolev} \begin{enumerate} \item $L^{s,2}(\mathbb{R}^N) = W^{s,2}(\mathbb{R}^N) = H^s (\mathbb{R}^N)$, where the sign of equality must be understood in the sense of an isomorphism. \item If $s \geq 0$ and $2 \leq q \leq 2_s^*=2N/(N-2s)$, then $L^{s,2}(\mathbb{R}^N)$ is continuously embedded into $L^q(\mathbb{R}^N)$; if $2 \leq q < 2_s^*$ then the embedding is locally compact. \item Assume that $0 \leq s \leq 2$ and $s > N/2$. If $s -N/2 >1$ and $0< \mu \leq s - N/2-1$, then $L^{s,2}(\mathbb{R}^N)$ is continuously embedded into $C^{1,\mu}(\mathbb{R}^N)$. If $s -N/2 <1$ and $0 < \mu \leq s -N/2$, then $L^{s,2}(\mathbb{R}^N)$ is continuously embedded into $C^{0,\mu}(\mathbb{R}^N)$. \end{enumerate} \end{theorem} \begin{remark} \rm According to Theorem \ref{th:sobolev}, the Bessel space $L^{s,2}(\mathbb{R}^N)$ is topologically undistinguishable from the Sobolev fractional space $H^s(\mathbb{R}^N)$. Since our equation involves the Bessel norm, we will not exploit this characterization. \end{remark} Going back to \eqref{eq:1.1}, it must be said that in the case $s \in (0,1)$ less is known than in the \emph{local} case $s=1$. Equation~\eqref{eq:1.1} arises from the more general Schr\"{o}dinger-Klein-Gordon equation \[ \mathrm{i}\frac{\partial \psi}{\partial t} = (I-\Delta)^s \psi -\psi -f(x,\psi) \] describing the the behaviour of bosons, spin-0 particles in relativistic fields. We refer to \cite{FelmerVergara, Secchi17,Secchi17-1,Secchi17-2} for very recent results about the existence of variational solutions. When $s=1/2$, the operator $(I-\Delta)^{1/2}=\sqrt{\strut I-\Delta}$ is also called \emph{pseudorelativistic} or \emph{semirelativistic}, and it is very important in the study of several physical phenomena. The interested reader can refer to \cite{CingolaniSecchi15,CingolaniSecchi18} and to the references therein for more information. \begin{remark} \rm The identity operator $I$ is often replaced by a multiple $m^2 I$, for some real number~$m \neq 0$. The operator reads then $(-\Delta+m^2)^s$, but for our purposes this generality does not give any advantage. \end{remark} A common feature in the current literature is that the existence of solutions to \eqref{eq:1.1} is related to the behavior of the potential function $a$ at infinity. This is a very useful tool for applying concentration-compactness methods or for working in weighted Lebesgue spaces. In the present paper, following \cite{AckerChagoya}, we investigate \eqref{eq:1.1} under much weaker assumptions on $a$, see Section 2. The first existence results for semilinear elliptic equations with \emph{irregular} potentials appeared, as far as we know, in \cite{CeramiMolle}. \section{Variational setting} We introduce some tools that will be used systematically in the rest of this article. \begin{definition} \rm \begin{itemize} \item For any $y \in \mathbb{R}^N$, we define the translation operator $\tau_y$ acting on a (suitably regular) function $f$ as $\tau_y f\colon x \mapsto f(x-y)$. \item In a normed space $X$, we denote by $B(x,r)$ the ball centered at $x \in X$ with radius~$r >0$, and by $\overline{B}(x,r)$ its closure. The boundary of~$B(0,1)$ will be denoted by $S(X)$. \item For any $a \in L^\infty(\mathbb{R}^N)$, we define \[ \mathscr{P} = \overline{B}(0,|a|_\infty) \subset L^\infty(\mathbb{R}^N). \] Looking at $L^\infty(\mathbb{R}^N)$ as the dual space of $L^1(\mathbb{R}^N)$, the set $\mathscr{P}$ will be endowed with the weak* topology. It is well-known that $\mathscr{P}$ becomes a compact metrizable space, see \cite[Theorems 3.15 and 3.16]{Rudin}. \item For any $a \in L^\infty(\mathbb{R}^N)$, we define the subset $\mathscr{A} = \left\{ \tau_y a: y \in \mathbb{R}^N \right\}$ of $\mathscr{P}$, endowed with the relative topology. Finally, we introduce $\mathscr{B} = \overline{\mathscr{A}} \setminus \mathscr{A}$. \item For any $a \in L^\infty(\mathbb{R}^N)$, we define \begin{equation} \label{eq:a-bar} \bar{a} = \sup \left\{ \operatorname{ess\,sup} u: u \in \mathscr{B} \right\}. \end{equation} If $\mathscr{B}=\emptyset$, we agree that $\bar{a} = -\infty$. \end{itemize} \end{definition} The following is the main assumption of this article. \begin{itemize} \item[(A1)] The function $a \in L^\infty(\mathbb{R}^N)$ is such that $a^{+} = \max\{a,0\}$ is not identically zero, and either (i) $\bar{a} \leq 0$ or (ii) $\bar{a} \leq a$. \end{itemize} Weak solutions to \eqref{eq:1.1} are critical points of the functional $I_a \colon L^{s,2}(\mathbb{R}^N) \to \mathbb{R}^N$ defined by \[ I_a(u) = \frac{1}{2} \| u \|_{L^{s,2}}^2 - \frac{1}{p} \int_{\mathbb{R}^N} a |u|^p. \] \begin{definition} \rm A solution $u \in L^{s,2}(\mathbb{R}^N)$ is called a ground-state solution to \eqref{eq:1.1} if $I_a$ attains at $u$ the infimum over the set of all solutions to \eqref{eq:1.1}, namely \[ I_a(u) = \min \big\{ I_a(v): \text{$v \in L^{s,2}(\mathbb{R}^N)$ solves \eqref{eq:1.1}} \big\}. \] \end{definition} We now state the main result of our paper. \begin{theorem} \label{th:main} Equation \eqref{eq:1.1} has (at least) a positive ground state provided that $2 b\), then for some \(\delta>0\) we can say that the set \(\Omega = \{ x \in \mathbb{R}^N: a(x) \geq b + \delta \}\) has positive measure. Let us define \(\varphi = \chi_\Omega / \mathcal{L}^N(\Omega)\), so that \[ \int_{\mathbb{R}^N} a \varphi = \frac{1}{\mathcal{L}^N(\Omega)} \int_\Omega a \geq b+\delta, \] contrary to \eqref{eq:3.2}. This completes the proof. \end{proof} Recall from assumption (A1) that $a^{+} \neq 0$ as an element of $L^\infty(\mathbb{R}^N)$. Therefore Lemma \ref{lem:3.1} yields a function $\varphi \in S(L^1(\mathbb{R}^N))$ such that $\varphi \geq 0$ and $\int_{\mathbb{R}^N} a \varphi >0$. By a standard mollification argument, we can assume without loss of generality that $\varphi \in C_c^\infty(\mathbb{R}^N)$. Since $L^{s,2}(\mathbb{R}^N)$ is continuously embedded into $L^p(\mathbb{R}^N)$ for every $20 \Big\} \] and \[ \mathscr{S}_a^{+} = \mathscr{B}_a^{+} \cap S(L^{s,2}(\mathbb{R}^N)). \] \begin{lemma} The set $\mathscr{B}_a^{+}$ is non-empty and open in $L^{s,2}(\mathbb{R}^N)$. \end{lemma} \begin{proof} We already know that $\varphi \in \mathscr{B}_a^{+}$. Furthermore, the map $u \mapsto \int_{\mathbb{R}^N} a|u|^p$ is continuous from $L^{s,2}(\mathbb{R}^N)$ to $\mathbb{R}$, since $a \in L^\infty(\mathbb{R}^N)$ and $20$ of $h$. By direct computation, $ t u \in \mathscr{N}_a$ if, and only if, $t=\bar{t}(u)$. Explicitly, \[ \bar{t}(u) = \frac{\|u\|_{L^{s,2}}^2}{\int_{\mathbb{R}^N} a|u|^p}. \] This shows that the map $u \mapsto \bar{t}(u)$ is continuous from $\mathscr{B}_a^{+}$ to $(0,+\infty)$. The rest of the proof follows easily. \end{proof} \begin{lemma} The set $\mathscr{N}_a$ is closed in $L^{s,2}(\mathbb{R}^N)$. \end{lemma} \begin{proof} If $u \in \mathscr{N}_a$, then \[ \|u\|_{L^{s,2}}^2 = \int_{\mathbb{R}^N} a |u|^p \leq \int_{\mathbb{R}^N} a^{+} |u|^p \leq S_p |a^{+}|_\infty \|u\|_{L^{s,2}}^p. \] It follows that \begin{equation} \label{eq:3.3} \inf_{u \in \mathscr{N}_a} \|u\|_{L^{s,2}} \geq \frac{1}{S_p |a^{+}|_\infty^{1/(p-2)}}. \end{equation} As a consequence, $0$ is not a cluster point of $\mathscr{N}_a$, which turns out to be closed. \end{proof} It is now standard to invoke the Implicit Function Theorem to prove that $\mathscr{N}_a$ is a $C^2$-submanifold of $L^{s,2}(\mathbb{R}^N)$ and that \eqref{eq:3.3} implies \[ \inf_{u \in \mathscr{N}_a} I_a(u) \geq \Big( \frac{1}{2} - \frac{1}{p} \Big) \frac{1}{S_p^2 |a^{+}|_\infty^{2/(p-2)}}. \] More importantly, $\mathscr{N}_a$ is a \emph{natural constraint} for $I_a$, i.e. every critical point of the restriction $\bar{I}_a$ of $I_a$ to $\mathscr{N}_a$ is a nontrivial critical point of $I_a$. The following result was proved in \cite[Proposition 3.2]{FelmerVergara}, and allows us to consider only positive ground states. \begin{proposition} Any weak solution to \eqref{eq:1.1} is strictly positive. \end{proposition} \begin{proposition} \label{prop:3.7} Let $\bar{I}_a$ be the restriction of the functional $I_a$ to the manifold $\mathscr{N}_a$. Every Palais-Smale sequence at level $c$ for $\bar{I}_a$ is also a Palais-Smale sequence at level $c$ for $I_a$. \end{proposition} \begin{proof} Assume that $\{u_n\}_n \subset \mathscr{N}_a$ is a Palais-Smale sequence at level $c$ for $\bar{I}_a$, namely \[ \lim_{n \to +\infty} \bar{I}_a(u_n) =c \] and \[ \lim_{n \to +\infty} D\bar{I}_a(u_n) =0 \] in the norm topology. It suffices to show that the sequence $\{ \nabla I_a(u_n)\}_n$ converges to zero in $L^{s,2}(\mathbb{R}^N)$. Let us abbreviate $\psi(u) = D I_a(u)[u]$, so that $\mathscr{N}_a = \psi^{-1}(\{0\}) \setminus \{0\}$. From the fact that $u_n \in \mathscr{N}_a$, we deduce that $I_a(u_n) = (1/2-1/p)\|u_n\|_{L^{s,2}}^2$, and hence the sequence $\{u_n\}_n$ is bounded. This implies that \begin{equation} \label{eq:3.4} \sup_n \frac{\| \nabla \psi (u_n) \|_{L^{s,2}}}{\|u_n\|_{L^{s,2}}} < +\infty. \end{equation} Explicitly, we have that, for every $n \in \mathbb{N}$, \begin{equation} \label{eq:3.5} \langle \nabla \psi(u_n)\mid u_n \rangle = (2-p) \|u_n\|_{L^{s,2}}^2 <0 \end{equation} and \begin{equation} \label{eq:3.6} \nabla \bar{I}_a(u_n) = \nabla I_a(u_n) - \frac{\langle \nabla I_a(u_n)\mid \nabla \psi (u_n) \rangle}{\|\nabla \psi(u_n)\|_{L^{s,2}}^2} \nabla \psi(u_n). \end{equation} Observe that $\nabla I_a(u_n) \perp u_n$ because $u_n \in \mathscr{N}_a$. If we consider the quantity \[ \| \nabla \psi (u_n)\|_{L^{s,2}}^2 - \Big( \frac{\langle \nabla I_a(u_n)\mid \nabla \psi(u_n) \rangle}{\|\nabla I_a(u_n)\|_{L^{s,2}}^2} \Big)^2, \] we immediately see that it equals the square of the norm of the projection of the vector $\nabla \psi(u_n)$ onto the subspace of $L^{s,2}(\mathbb{R}^N)$ orthogonal to the unit vector $\nabla I_a(u_n)/\|\nabla I_a(u_n)\|$. Since this subspace contains in particular the vector $u_n/\|u_n\|_{L^{s,2}}$, it follows from the Pythagorean Theorem that \begin{equation} \| \nabla \psi (u_n) \|_{L^{s,2}}^2 - \Big( \frac{\langle \nabla I_a(u_n)\mid \nabla \psi(u_n) \rangle}{\|\nabla I_a(u_n)\|_{L^{s,2}}^2} \Big)^2 \geq \Big( \frac{\langle \nabla \psi(u_n)\mid u_n \rangle}{\|u_n\|_{L^{s,2}}} \Big)^2. \end{equation} This yields, recalling \eqref{eq:3.6}, \eqref{eq:3.5} and \eqref{eq:3.4}, \begin{align*} &\| \nabla \bar{I}_a(u_n) \|_{L^{s,2}} \| \nabla I_a(u_n)\|_{L^{s,2}} \\ &\geq \langle \nabla \bar{I}_a(u_n)\mid \nabla I_a(u_n) \rangle \\ &= \frac{\|\nabla I_a(u_n)\|_{L^{s,2}}^2}{\|\nabla \psi(u_n)\|_{L^{s,2}}^2} \bigg( \| \nabla \psi(u_n) \|_{L^{s,2}}^2 - \Big( \frac{\langle \nabla I_a(u_n)\mid \nabla \psi(u_n)\rangle}{\|\nabla I_a(u_n)\|_{L^{s,2}}^2} \Big)^2 \bigg)^2 \\ &\geq \frac{\|\nabla I_a(u_n)\|_{L^{s,2}}^2}{\|\nabla \psi(u_n)\|_{L^{s,2}}^2} \Big( \frac{\langle \nabla \psi(u_n)\mid u_n \rangle}{\|u_n\|_{L^{s,2}}} \Big)^2 \\ &= \frac{\|\nabla I_a(u_n)\|_{L^{s,2}}^2}{\|\nabla \psi(u_n)\|_{L^{s,2}}^2} (2-p)^2 \|u_n\|_{L^{s,2}}^2 \\ &\geq C \|\nabla I_a(u_n)\|_{L^{s,2}}^2. \end{align*} This argument proves that $\lim_{n \to +\infty} \|\nabla I_a(u_n)\|_{L^{s,2}}=0$, and we complete the proof. \end{proof} \section{Splitting and vanishing sequences} The analysis of Palais-Smale sequences can be harder than in the more familiar case of a potential function $a$ that has a precise asymptotic behavior at infinity. For this reason, we recall a language taken from \cite{AckerChagoya}. \begin{definition} \rm A map $F \colon X \to Y$ between two Banach spaces splits in the BL sense (BL stands for Brezis and Lieb.) if for any sequence $\{u_n\}_n \subset X$ such that $u_n \rightharpoonup u$ in $X$ there results \[ F(u_n-u) = F(u_n)-F(u)+o(1) \] in the norm topology of~$Y$. \end{definition} \begin{lemma} \label{lem:4.2} Suppose that $\{u_n\}_n \subset L^{s,2}(\mathbb{R}^N)$ and $\{y_n\}_n \subset \mathbb{R}^N$ are such that $\tau_{-y_n}u_n \rightharpoonup u_0$ in $L^{s,2}(\mathbb{R}^N)$. Then \[ I_{\tau_{-y_n}a}(\tau_{-y_n}u_n) - I_{\tau_{-y_n}a} (\tau_{-y_n}u_n-u_0) -I_{\tau_{-y_n}a}(u_0)=o(1) \] and \[ DI_{\tau_{-y_n}a}(\tau_{-y_n}u_n) - DI_{\tau_{-y_n}a} (\tau_{-y_n}u_n-u_0) - DI_{\tau_{-y_n}a}(u_0) =o(1). \] \end{lemma} \begin{proof} Since both $F(u)=p^{-1}|u|^p$ and $F'(u)=|u|^{p-2}u$ split from $L^{s,2}(\mathbb{R}^N)$ into $L^1(\mathbb{R}^N)$, see \cite[Lemma 4.4]{Secchi17}, we can write \begin{align*} &\int_{\mathbb{R}^N} | ( \tau_{-y_n} a ) ( F( \tau_{-y_n}u_n) - F(\tau_{-y_n}u_n-u_0)-F(u_0) ) | \\ & \leq |a|_\infty \int_{\mathbb{R}^N} \left| F( \tau_{-y_n}u_n) - F(\tau_{-y_n}u_n-u_0)-F(u_0) \right| =o(1) \end{align*} and \begin{align*} &\int_{\mathbb{R}^N} \left| \left( \tau_{-y_n} a \right) \left( F'(\tau_{-y_n} u_n) - F'(\tau_{-y_n} u_n-u_0) - F'(u_0) \right) \right|^{p/(p-1)} \\ &\leq |a|_\infty^{p/(p-1)} \int_{\mathbb{R}^N} \left| F'(\tau_{-y_n} u_n) - F'(\tau_{-y_n} u_n-u_0) - F'(u_0) \right|^{p/(p-1)}. \end{align*} Recalling that the squared norm splits in the BL sense, the proof is complete. \end{proof} \begin{definition} \label{def:4.3} \rm A sequence $\{u_n\}_n \subset L^{s,2}(\mathbb{R}^N)$ vanishes if $\tau_{x_n} u_n \rightharpoonup 0$ in $L^{s,2}(\mathbb{R}^N)$ for any sequence $\{x_n\}_n$ of points in $\mathbb{R}^N$. \end{definition} \begin{remark} \label{rem:4.4} \rm Any vanishing sequence is necessarily bounded in $L^{s,2}(\mathbb{R}^N)$, and by the Rellich-Kondratchev theorem (see \cite[Corollary 7.2]{DiNezza}) $\tau_{x_n} u_n \to 0$ strongly in $L_{\mathrm{loc}}^2(\mathbb{R}^N)$ for every sequence $\{x_n\}_n \subset \mathbb{R}^N$. This yields that, for every $R>0$, \[ \lim_{n \to +\infty} \sup \Big\{ \int_{B(x,R)} |u_n|^2: x \in \mathbb{R}^N \Big\} =0. \] By the fractional version of Lions' vanishing lemma \cite[Proposition II.4]{Secchi12}, we deduce that $u_n \to 0$ strongly in $L^q(\mathbb{R}^N)$ for every $20$, independent of $i$, such that $\|u^i\|_{L^{s,2}} \geq C$. For every $j$ we also have \[ 0 \leq \|u_{n}^{j+1}\|_{L^{s,2}}^2 = \|u_n\|^2_{L^{s,2}} - \sum_{i=1}^j \|u^i\|_{L^{s,2}}^2 + o(1), \] which implies that the iteration must stop after finitely many steps. Therefore there exists a positive integer $k$ such that $\{u_n^{k+1}\}_n$ vanishes, $\{u_n^{k+1}\}_n$ converges to zero strongly in $L^p(\mathbb{R}^N)$ and \eqref{eq:4.7} holds true. Similarly, \[ -\int_{\mathbb{R}^N} a | u_n^{k+1} |^p \leq I_a(u_n^{k+1}) = I_a(u_n) - \sum_{i=1}^k I_{a^i} (u^i) + o(1), \] and also \eqref{eq:4.8} follows from $c = \lim_{n \to +\infty} I_a(u_n)$. The proof is complete. \end{proof} \section{Existence of a ground state} The proof of the following comparison lemma is probably known, but we reproduce here for the reader's convenience. \begin{lemma} \label{lem:5.1} Suppose that $a_1$, $a_2 \in L^\infty(\mathbb{R}^N)$. If $a_1 \geq a_2$, then $c_{a_1} \leq c_{a_2}$. If, in addition, $a_1 \neq a_2$ and $I_{a_2}$ possesses a ground state, then $c_{a_1}0. \] We can therefore define \begin{equation} \label{eq:5.1} t = \Big( \frac{\int_{\mathbb{R}^N} a_2 |u|^p}{\int_{\mathbb{R}^N} a_1 |u|^p} \Big)^{1/(p-2)} \leq 1. \end{equation} Then we have \[ DI_{a_1}(tu)[tu] = t^2 \Big( \|u\|_{L^{s,2}}^2 - t^{p-2} \int_{\mathbb{R}^N} a_1 |u|^p \Big) = t^2 D I_{a_2}(u)[u]=0, \] and hence $tu \in \mathscr{N}_{a_1}$. Since \begin{align*} I_{a_2}(u) &= \frac{1}{2} \|u\|_{L^{s,2}}^2 - \frac{1}{p} \int_{\mathbb{R}^N} a_2 |u|^p = \Big( \frac{1}{2} - \frac{1}{p} \Big) \|u\|_{L^{s,2}}^2 \\ &\geq \Big( \frac{1}{2} - \frac{1}{p} \Big) \|tu\|_{L^{s,2}}^2 = J_{a_1}(u) \geq c_{a_1}, \end{align*} we conclude that $c_{a_2} = \inf_{u \in \mathscr{N}_{a_2}}I_{a_2}(u) \geq c_{a_1}$. Furthermore, if $a_1 \neq a_2$ (as elements of $L^\infty(\mathbb{R}^N)$) and $u$ is a ground state of $I_{a_2}$, then $|u|>0$. In \eqref{eq:5.1} we then have $t<1$, and it follows that $c_{a_2}=I_{a_2}(u) > I_{a_1}(tu) \geq c_{a_1}$. \end{proof} Recall the definition \eqref{eq:a-bar} of \(\bar{a}\). Then we have the following result. \begin{proposition} \label{prop:5.3} It results \begin{align*} c_{a} < c_{\bar{a}}. \end{align*} \end{proposition} \begin{proof} We first consider (i) of assumption (A1). Since \(\bar{a}\leq 0\), we have \(c_{\bar{a}}=\infty\). But \(c_{a} \in \mathbb{R}\) because \(a^{+} \neq 0\), and there is nothing more to prove. We can assume that \(\bar{a}>0\) in the rest of the proof. If (ii) of assumption (A1) holds, recalling that \(\bar{a}>-\infty\) entails \(\mathscr{B}\neq \emptyset\) we can conclude that \(a \neq \bar{a}\). Now Lemma \ref{lem:5.1} implies that \(c_a < c_{\bar{a}}\), since \(I_{\bar{a}}\) has a ground state by the arguments of \cite[Theorem 1.1]{Ambrosio}. \end{proof} We are now ready to prove our main existence result. \begin{proof}[Proof of Theorem \ref{th:main}] We have \(\mathscr{N}_{a} \neq \emptyset\) and \(c_{a}<\infty\) because \(a^{+} \neq 0\). From \eqref{eq:3.3} we get \(c_{a}>0\). An application of Ekeland's Principle yields in a standard way a mimnimizing sequence \(\{u_n\}_n \subset \mathscr{N}_{a}\) for the functional \(\bar{I}_{a}\) defined as the restriction of \(I_{a}\) to \(\mathscr{N}_{a}\). This sequence is also a (PS)-sequence for \(\bar{I}_{a}\) at the level \(c_{a}\). By Proposition \ref{prop:3.7} \(\{u_n\}_n\) is a (PS)-sequence for \(I_{a}\) at the level \(c_{a}\). The strong convergence of \(\{u_n\}_n\) to zero is easily ruled out, since \(I_{a}(u_n) \to c_{a}>0\). Proposition \ref{prop:4.8} yields then a number \(k \in \mathbb{N}\), functions \(a^i \in \overline{\mathscr{A}}\) and non-trivial critical points \(u^i\) of \(I_{a^i}\) such that \begin{align*} c_{a} \geq \sum_{i=1}^k I_{a^i}(u^i). \end{align*} From the knowledge that each \(u^i\) is a non-trivial critical point of \(I_{a^i}\) we deduce \((a^i)^{+}\neq 0\) for every \(i=1,\ldots,k\). Again by \eqref{eq:3.3} we get \(I_{a^i}(u^i)>0\) for every \(i=1,\ldots,k\). Suppose that for \emph{some} index \(i\) there results \(a^i \in \mathscr{B}\). Then \(a^i \leq \bar{a}\), and Lemma \ref{lem:5.1} together with Proposition \ref{prop:5.3} yield \(I_{a^i}(u^i) \geq c_{a^i} \geq c_{\bar{a}}>c_{a}\). This is a contrdiction. Therefore each \(a^i\) is a translation of \(a\), and \(I_{a^i}(u^i) \geq c_{a}\) for every \(i=1,\ldots,k\). This forces \(k=1\), and a translation of \(u^1\) is a ground state of \(I_a\). \end{proof} \section{An example} Assumption (A1) can be rephrased in a more familiar way for continuous bounded potentials. \begin{proposition} For any $a \in L^\infty(\mathbb{R}^N)$, define \begin{align*} \hat{a} = \lim_{R \to +\infty} \operatorname*{ess\,sup}_{x \in \mathbb{R}^N \setminus B(0,R)} a(x). \end{align*} If {\rm (A1)} holds with $\bar{a}$ replaced by $\hat{a}$, then {\rm (A1)} holds with $\bar{a}$. \end{proposition} \begin{proof} If \(\mathscr{B} = \emptyset\), then $\bar{a}=-\infty$ and (A1) holds. We may assume that \(\mathscr{B} \neq \emptyset\), so that \(a\) cannot be constant. Let us prove that \begin{align} \bar{a} \leq \hat{a}. \label{eq:6.1} \end{align} Pick $b \in \mathscr{B}$. There is a sequence $\{x_n\}_n \subset \mathbb{R}^N$ such that $\tau_{x_n}a \rightharpoonup^\star b$. Translations are continuous in the weak$^\star$ topology of $L^\infty(\mathbb{R}^N)$, since they are continuous in $L^1(\mathbb{R}^N)$. For the sake of contradiction, suppose that $\{x_n\}_n$ contains a bounded subsequence. Up to a further subsequence, there must exist a point $\xi\in\mathbb{R}^N$ such that $x_n \to \xi$ and $\tau_{x_n} a \rightharpoonup^\star \tau_{\xi}a$. Since $\mathscr{P}$ is metrizable, $\tau_{\xi} a = b \notin \mathscr{A}$, a contradiction. Therefore $\lim_{n \to +\infty} |x_n|=+\infty$. Let $\varepsilon>0$ be given, and apply Lemma \ref{lem:3.1}: there exists $\varphi \in L^1(\mathbb{R}^N)$ with $\varphi \geq 0$ and $\|\varphi\|_{L^1}=1$ such that \[ \int_{\mathbb{R}^N} b \varphi \geq \operatorname{ess\,sup}b - \frac{\varepsilon}{2}. \] Choose $\tilde{\psi} \in C_c^\infty(\mathbb{R}^N)$ such that $\tilde{\psi} \geq 0$ and \[ \| \varphi - \tilde{\psi} \|_{L^1} \leq \frac{\varepsilon}{4 \|b\|_{L^\infty}}. \] Now $\psi = \tilde{\psi}/\|\tilde{\psi}\|_{L^1} \in C_c^\infty(\mathbb{R}^N)$ satisfies \[ \| \varphi - \psi \|_{L^1} \leq \frac{\varepsilon}{2 \|b\|_{L^\infty}}, \] $\psi \geq 0$ and $\|\psi\|_{L^1} = 1$. This implies \[ \int_{\mathbb{R}^N} b \psi = \int_{\mathbb{R}^N} b \varphi - \int_{\mathbb{R}^N} b (\varphi-\psi) \geq \int_{\mathbb{R}^N} b \varphi - \|b\|_{L^\infty} \|\psi - \varphi \|_{L^1} \geq \operatorname{ess\,sup}b - \varepsilon. \] Suppose that $\operatorname{supp}\psi \subset B(0,R)$: then \begin{align*} \operatorname{ess\,sup}b - \varepsilon &\leq \int_{\mathbb{R}^N} b \psi = \lim_{n \to +\infty} \int_{\mathbb{R}^N} (\tau_{x_n}a)\psi \\ &\leq \lim_{n \to +\infty} \operatorname*{ess\,sup}_{x \in B(-x_n,R)} a(x) \int_{\mathbb{R}^N} \psi \\ &\leq \lim_{n \to +\infty} \operatorname*{ess\,sup}_{x \in \mathbb{R}^N \setminus B(0,|x_n|-R)} a(x) = \hat{a}. \end{align*} Since $\varepsilon>0$ is arbitrary, we conclude that $\operatorname{ess\,sup}b \leq \hat{a}$. If (i) of assumption (A1) holds, then \eqref{eq:6.1} yields $\bar{a} \leq \hat{a} \leq 0$. If (ii) holds, then \eqref{eq:6.1} yields $\bar{a} \leq \hat{a} \leq a$, and the proof is complete. \end{proof} The following corollary is an immediate consequence of Theorem \ref{th:main}. \begin{corollary} If \(a\) is a bounded continuous function such that either \[ \limsup_{|x| \to +\infty} a(x) \leq 0 \] or \[ \limsup_{|x| \to +\infty} a(x) \leq a, \] then equation \eqref{eq:1.1} has (at least) a positive ground state as soon as $2