\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 153, pp. 1--18.\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} \title[\hfilneg EJDE-2018/153\hfil Critical magnetic fractional problems] {Bifurcation and multiplicity results for critical magnetic fractional problems} \author[A. Fiscella, E. Vecchi \hfil EJDE-2018/153\hfilneg] {Alessio Fiscella, Eugenio Vecchi} \address{Alessio Fiscella \newline Departamento de Matem\'atica, Universidade Estadual de Campinas, IMECC, Rua S\'ergio Buarque de Holanda 651, Campinas, SP CEP 13083-859 Brazil} \email{fiscella@ime.unicamp.br} \address{Eugenio Vecchi \newline Dipartimento di Matematica , Sapienza Universit\`a di Roma, P.le Aldo Moro 5, 00185, Roma, Italy} \email{vecchi@mat.uniroma1.it} \thanks{Submitted April 6, 2018. Published August 13, 2018.} \subjclass[2010]{35R11, 35Q60, 35A15, 35B33} \keywords{Fractional magnetic operators; critical nonlinearities; \hfill\break\indent variational methods} \begin{abstract} This article concerns the bifurcation phenomena and the existence of multiple solutions for a non-local boundary value problem driven by the magnetic fractional Laplacian $(-\Delta)_{A}^{s}$. In particular, we consider $$ (-\Delta)_{A}^{s}u =\lambda u + |u|^{2^{\ast}_s -2} u \quad\text{in }\Omega, \quad u=0\quad \text{in }\mathbb{R}^{n}\setminus \Omega, $$ where $\lambda$ is a real parameter and $\Omega \subset \mathbb{R}^n$ is an open and bounded set with Lipschitz boundary. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \allowdisplaybreaks \section{Introduction}\label{Intro} The aim of this article is to study a critical non-local boundary value problem that could be considered as the magnetic fractional version of the famous Br\'ezis-Nirenberg problem \cite{BrNi}. More precisely, we deal with the following problem \begin{equation}\label{Pb} \begin{gathered} (-\Delta)_{A}^{s} u = \lambda u + |u|^{2^{\ast}_s-2} u, \quad \text{in } \Omega,\\ u= 0, \quad \text{in } \mathbb{R}^n \setminus \Omega, \end{gathered} \end{equation} where $\Omega \subset \mathbb{R}^n$ is an open and bounded set with Lipschitz boundary $\partial \Omega$, dimension $n > 2s$ with $s \in (0,1)$, $2^{\ast}_s := 2n/(n-2s)$ is the fractional critical Sobolev exponent and $\lambda\in\mathbb R$ is a parameter. In particular, we are interested in the extension to the magnetic setting of a classical result due to Cerami, Fortunato and Struwe \cite{CFS}, that has recently been generalized to the diamagnetic fractional case in \cite{FMS16}. A first motivation for the study of \eqref{Pb} is the increasing interest in the non-local magnetic PDEs, driven by the so called \emph{magnetic fractional Laplacian} $(-\Delta)^s_A$; see e.g.\ \cite{Ambrosio, AmdA,BSX, d'AS,FPV17, LRZ18, MPSZ, MRZ, WX}. This operator has been introduced in \cite{d'AS} through the following representation, when acting on smooth complex-valued functions $u\in C^{\infty}_0(\mathbb{R}^n, \mathbb{C})$, \begin{equation}\label{FrLap} (-\Delta)_{A}^{s}u(x) =2\lim_{\varepsilon\to 0^+}\int_{\mathbb{R}^n\setminus B(x,\varepsilon)}\frac{u(x)-e^{\mathrm{i}(x-y)\cdot A(\frac{x+y}{2})}u(y)} {|x-y|^{n+2s}} dy, \quad x\in\mathbb{R}^n, \end{equation} where $B(x,\varepsilon)$ denotes the ball of center $x$ and radius $\varepsilon$. We can consider $(-\Delta)_{A}^{s}$ as a fractional counterpart of the magnetic Laplacian $(\nabla - \mathrm{i} A)^2$, with $A: \mathbb{R}^n \to \mathbb{R}^n$ being a $L^{\infty}_{\rm loc}$-vector potential; see e.g.\ \cite[Chapter 7]{LL}. In this context, when $n=3$, the curl of $A$ represents a magnetic field acting on a charged particle. While, it is clear that for $A=0$ and $u$ smooth real-valued function, $(-\Delta)_{A}^{s}$ coincides with the standard fractional Laplacian defined as principal value integral \[ (-\Delta)^s u(x)=\lim_{\varepsilon \to 0^+}\int_{\mathbb{R}^n\setminus B(x, \varepsilon)} \frac{u(x)-u(y)}{|x-y|^{n+2s}}dy, \quad x\in\mathbb{R}^n. \] We refer to \cite{Hitch, MBRS} and the references therein for further details on the fractional Laplacian. Furthermore, $(-\Delta)^s_A$ appears quite naturally in the definition of \emph{magnetic fractional Sobolev spaces} and some non--local functionals that has been recently used in \cite{NPSV,NPSV2, PSV1,PSV2, BM} to study characterizations of magnetic Sobolev spaces in the spirit of the works by Bourgain, Br\'ezis and Mironescu \cite{BBM}, Maz'ya and Shaposhnikova \cite{mazia}, and Nguyen \cite{Nguyen}. We also want to mention that, for $s=1/2$, the definition of the fractional operator in \eqref{FrLap} goes back to the '80's, and is related to the proper definition of a quantized operator corresponding to the symbol of the classical relativistic Hamiltonian, namely $$ \sqrt{(\xi - A(x))^2 + m^2} + V(x), \quad (\xi,x) \in \mathbb{R}^n \times \mathbb{R}^n. $$ More precisely, it is related to the kinetic part of the above symbol. We point out that in the literature there are at least three definitions for such a quantized operator. Two of them are given in terms of pseudo-differential operators, and the third one as the square root of a non--negative operator, see e.g. \cite{I13,FLS08}. The survey \cite{I13} also shows that these three non--local operators are in general different, but they coincide when the vector potential $A$ is assumed to be linear, an assumption which is equivalent to require a constant magnetic field when we are in the physically relevant situation of $\mathbb{R}^3$. Another reason to study critical magnetic fractional problems like \eqref{Pb} is provided by the rich background of results concerning the diamagnetic fractional version of the Br\'ezis-Nirenberg problem, namely \begin{equation}\label{sPb} \begin{gathered} (-\Delta)^s u = \lambda u + |u|^{2^{\ast}_s -2}u, \quad \text{in } \Omega,\\ u=0, \quad \text{in } \mathbb{R}^n \setminus \Omega. \end{gathered} \end{equation} In \cite{SeVa15}, they prove the existence of a non-trivial weak solution of \eqref{sPb}, whenever $n \geq 4s$ and $\lambda \in (0,\lambda_{1,s})$, with $\lambda_{1,s}$ the first Dirichlet eigenvalue of $(-\Delta)^s$. In \cite{Se}, a generalization of a classical result due to Capozzi, Fortunato and Palmieri \cite{CFP} is given, stating the existence of a solution of \eqref{sPb}, for every $\lambda >0$ which is not a variational Dirichlet eigenvalue of $(-\Delta)^s$. Another result in the same spirit is proved in \cite{SeVa13b}, with the requirement that $n \in (2s,4s)$. One of the key points of the aforementioned papers is the introduction of a suitable functional space which allows them to encode in a proper way the non--local boundary condition of \eqref{sPb}. Also, a crucial ingredient is the knowledge of the family of functions attaining the best fractional Sobolev constant. This latter aspect is however not necessary to prove the fractional counterpart of \cite{CFS} in \cite{FMS16}. We mention here \cite{BCDS12,Tan11} for other results concerning fractional Br\'ezis-Nirenberg-type problems. Motivated by the above papers, we study a bifurcation phenomena for problem \eqref{Pb} by a variational approach. However, because of the presence of a critical term in \eqref{Pb} and the lack of compactness at critical level $L^{2^{\ast}_s}$, the related Euler--Lagrange functional does not satisfy a global Palais-Smale condition. For this, as in the classical and in the fractional diamagnetic cases, we provide the Palais-Smale condition in a certain range strongly depending on the best {\em magnetic fractional Sobolev constant} given by \begin{equation}\label{MaSb} \mathcal{S}_A:=\inf_{v\in X_{0,A}\setminus\{0\}} \frac{\displaystyle \iint_{\mathbb{R}^{2n}}\frac{|v(x)- e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}v(y)|^2}{|x-y|^{n+2s}}\, dx \,dy } {\displaystyle\Big(\int_\Omega |v(x)|^{2^*}\,dx\Big)^{2/2^*}}, \end{equation} where $X_{0,A}$ denotes the suitable functional space, where finding solutions of \eqref{Pb}. See Section \ref{funct} for a detailed description of $X_{0,A}$. Of course by the continuous embedding $X_{0,A} \hookrightarrow L^{2^{\ast}_s}(\Omega, \mathbb C)$, given in \cite[Lemma 2.2]{FPV17}, the constant $\mathcal{S}_A$ is well defined and strictly positive. Considering the presence of a linear term in \eqref{Pb}, we also study the following auxiliary eigenvalue problem \begin{equation}\label{EigPb} \begin{gathered} (-\Delta)_A^s u = \lambda u, \quad \text{in } \Omega,\\ u= 0, \quad \text{in } \mathbb{R}^n \setminus \Omega. \end{gathered} \end{equation} Inspired by the classical case, we call \emph{variational Dirichlet eigenvalues}, the values of the real $\lambda$'s for which \eqref{EigPb} admits a non-trivial weak solution $u \in X_{0,A}$, that will be called \emph{eigenfunction}. Among many properties, we will show that there is a sequence of isolated variational Dirichlet eigenvalues whose associated eigenfuctions form an orthogonal basis of the space $X_{0,A}$. To the best of our knowledge, a systematic study and variational characterizations (akin to the classical one) of the variational Dirichlet eigenvalues of the magnetic fractional Laplacian $(-\Delta)^s_A$ are not explicitly stated in the literature. We refer to Section \ref{Spectrum} for more details. In the diamagnetic case, we recall that the variational Dirichlet eigenvalues of $(-\Delta)^s$ are introduced in \cite{SeVa13}. We are now ready to state our main result. \begin{theorem}\label{Bif} Let $s \in (0,1)$, $n>2s$, $\Omega \subset \mathbb{R}^n$ be an open and bounded set with Lipschitz continuous boundary $\partial \Omega$. Let $\lambda \in \mathbb{R}$ and let $\lambda^{\ast}$ be the variational Dirichlet eigenvalue of problem \eqref{EigPb} given by \begin{equation}\label{lambdastella} \lambda^{\ast}:= \min \{ \lambda_k : \lambda < \lambda_k \}. \end{equation} Call $m \in \mathbb{N}$ its multiplicity. Assume that \begin{equation}\label{condlambda*} \lambda \in \Big( \lambda^{\ast}- \frac{\mathcal{S}_A}{|\Omega|^{2s/n}}, \lambda^{\ast}\Big), \end{equation} with $\mathcal{S}_A$ defined in \eqref{MaSb}. Then problem \eqref{Pb} admits at least $m$ pairs of non-trivial weak solutions $\{ -u_{\lambda,i}, u_{\lambda,i}\}$, whose functional norm satisfies $$ \|u_{\lambda,i}\|_{X_{0,A}} \to 0, \quad \text{as } \lambda \to \lambda^{\ast}, $$ for every $i=1,\ldots,m$. \end{theorem} The proof of Theorem \ref{Bif} relies on an abstract result due to Bartolo, Benci and Fortunato \cite{BBF} and on the study of the variational Dirichlet eigenvalues performed in Section \ref{Spectrum}. This paper is organized as follows. In Section \ref{funct}, we introduce the necessary functional and variational setup to study the boundary value problem \eqref{Pb}. In Section \ref{Spectrum}, we study the variational Dirichlet eigenvalues of $(-\Delta)^s_A$. In Section \ref{ps}, we provide the Palais--Smale condition on a suitable range. In the last Section \ref{PROOF}, we finally prove Theorem \ref{Bif}. \section{Functional and variational setup}\label{funct} Throughout this article, we indicate with $|\Omega|$ the $n$-dimensional Lebesgue measure of a measurable set $\Omega \subset \mathbb{R}^n$. Moreover, for every $z \in \mathbb{C}$ we denote by $\Re z$ its real part, and by $\overline{z}$ its complex conjugate. Let $\Omega \subset \mathbb{R}^n$ be an open set. We denote by $L^2(\Omega,\mathbb{C})$ the space of measurable functions $u:\Omega\to\mathbb{C}$ such that $$ \|u\|_{L^2(\Omega)}=\left(\int_{\Omega}|u(x)|^2 \, dx\right)^{1/2}<\infty, $$ where $|\cdot|$ is the Euclidean norm in $\mathbb{C}$. For $s\in (0,1)$, we define the magnetic Gagliardo semi-norm as $$ [u]_{H^{s}_A(\Omega)}:=\Big(\iint_{\Omega\times\Omega}\frac{|u(x)-e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{n+2s}}\,dx\,dy\Big)^{1/2}. $$ We denote by $H^{s}_A(\Omega)$ the space of functions $u\in L^2(\Omega,\mathbb{C})$ such that $[u]_{H^{s}_A(\Omega)}<\infty$, normed with $$ \|u\|_{H^{s}_A(\Omega)}:=\Big(\|u\|_{L^2(\Omega)}^2+[u]_{H^{s}_A(\Omega)}^2\Big)^{1/2}. $$ For $A=0$, this definition is consistent with the usual fractional space $H^{s}(\Omega)$. We stress out that $C^{\infty}_0(\mathbb{R}^n, \mathbb{C})\subseteq H^{s}_{A} (\mathbb{R}^n)$, see \cite[Proposition 2.2]{d'AS}. However, to encode the boundary condition $u=0$ in $\mathbb{R}^n\setminus\Omega$, the natural functional space introduced in \cite{FPV17} to deal with weak solutions of problem \eqref{Pb} is $$ X_{0,A} := \big\{ u \in H^{s}_{A}(\mathbb{R}^n): u = 0\ \text{a.e. in } \mathbb{R}^n \setminus \Omega \big\}, $$ which generalizes to the magnetic framework the space introduced in \cite{SeVa12}. As in \cite{d'AS}, we define the following real scalar product on $X_{0,A}$ $$ \langle u,v\rangle_{X_{0,A}}:= \Re \iint_{\mathbb{R}^{2n}}\frac{\big(u(x)-e^{\mathrm{i} (x-y) \cdot A(\frac{x+y}{2})}u(y)\big) \overline{\big(v(x)-e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}v(y)\big)}} {|x-y|^{n+2s}}\, dx dy, $$ which induces the norm $$ \|u\|_{X_{0,A}} := \Big( \iint_{\mathbb{R}^{2n}}\frac{|u(x)- e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{n+2s}}\, dx dy \Big)^{1/2}. $$ By arguing as in \cite[Lemma 7]{SeVa12}, we see that $(X_{0,A}, \langle \cdot,\cdot\rangle_{X_{0,A}})$ is a Hilbert space and hence reflexive. We can now describe the variational formulation of problem \eqref{Pb}. We will say that a function $u \in X_{0,A}$ is a weak solution of \eqref{Pb} if $$ \langle u,\varphi \rangle_{X_{0,A}}= \lambda \Re \int_{\Omega}u(x) \overline{\varphi(x)}\, dx + \Re \int_{\Omega}|u(x)|^{2^{\ast}_s -2}u(x) \overline{\varphi(x)}\, dx, $$ for every $\varphi \in X_{0,A}$. Clearly, the weak solutions of \eqref{Pb} are the critical points of the Euler--Lagrange functional $\mathcal{J}_{A,\lambda}:X_{0,A} \to \mathbb{R}$, associated with~\eqref{Pb}, that is \begin{equation}\label{Jlam} \mathcal{J}_{A,\lambda}(u):= \|u\|_{X_{0,A}}^2 - \frac{\lambda}{2}\|u\|_{L^2(\Omega)}^2 - \frac{1}{2^{\ast}_s}\|u\|_{L^{2^*_s}(\Omega)}^{2^*_s}. \end{equation} It is easy to see that $\mathcal{J}_{A,\lambda}$ is well-defined and of class $C^1(X_{0,A}, \mathbb{R})$. Hence, to prove Theorem \ref{Bif}, we apply the following abstract critical point theorem to our functional $\mathcal{J}_{A,\lambda}$. \begin{theorem}[{\cite[Theorem 2.4]{BBF}}] \label{bbf} Let $(H, \|\cdot\|_H)$ be a real Hilbert space. Let $\mathcal{J}:H \to \mathbb{R}$ be a $C^1$ functional satisfying the following conditions: \begin{itemize} \item[(A1)] $\mathcal{J}(u) = \mathcal{J}(-u)$ and $\mathcal{J}(0)=0$; \item[(A2)] there exists a positive real constant $c>0$ such that $\mathcal{J}$ satisfies the Palais-Smale condition at the level $c$; \item[(A3)] there exist two closed subspaces $V$, $W \subset H$, and there exist positive real constants $\rho$, $\delta$, $\beta >0$, with $\delta <\beta k$. Therefore, $$f_j \in \mathbb{E}_j= \left(\operatorname{span}_{\mathbb{R}}\{f_1, \ldots, f_{j-1}\}\right)^{\bot} \subseteq \left( \operatorname{span}_{\mathbb{R}} \{f_k\}\right)^{\bot},$$ which implies $$ \langle f_j,f_k\rangle_{X_{0,A}}=0. $$ Now, since $f_j$ is an eigenfunction, by \eqref{autovalore} we obtain $$ 0 = \langle f_j,f_k\rangle_{X_{0,A}}= \lambda_j \, \Re \int_{\Omega}f_j(x)\overline{f_{k}(x)}\, dx, $$ as desired. \end{proof} \end{proposition} \begin{proposition}\label{SuccLambda} The eigenvalues of problem \eqref{EigPb} form a sequence of real positive numbers $\{\lambda_{k}\}_{k}$ with \begin{equation}\label{monotonia} 0 < \lambda_{1} \leq \lambda_2 \leq \ldots \leq \lambda_k \leq \ldots, \end{equation} and \begin{equation} \label{divergenza} \lambda_k \to \infty, \quad \text{as } k\to \infty. \end{equation} \end{proposition} \begin{proof} By Proposition \ref{prima parte} we know that $\lambda_1>0$, while by \eqref{inclusioni} follows \eqref{monotonia}. To prove \eqref{divergenza}, assume by contradiction that there exists a positive real constant $l>0$ such that $\lambda_{k} \to l$ as $k \to \infty$. This implies that the sequence $\{\lambda_k\}_{k}$ is bounded in $\mathbb{R}$. Moreover, since by Proposition \ref{prima parte} $f_k$ attains the minimum in \eqref{k+1}, we have $\|f_k\|^2_{X_{0,A}} =\lambda_k$. Thus, by \cite[Lemma 2.2]{FPV17} there exists a subsequence $\{f_{k_j}\}_j\subset X_{0,A}$ such that $$ f_{k_j} \to f, \quad \text{in } L^{2}(\Omega,\mathbb{C}), $$ for some $f\in L^{2}(\Omega,\mathbb{C})$. This implies that $\{f_{k_j}\}_j$ is a Cauchy sequence in $L^{2}(\Omega,\mathbb{C})$. On the other hand, when $i\neq j$ by Proposition \ref{L2Orto} we have that $f_{k_j}$ and $f_{k_i}$ are orthogonal in $L^{2}(\Omega,\mathbb{C})$, hence $$ \|f_{k_j} - f_{k_i}\|^{2}_{L^{2}(\Omega)} = \|f_{k_j}\|^{2}_{L^{2}(\Omega)} + \|f_{k_i}\|^{2}_{L^{2}(\Omega)} =2, $$ which yields a contradiction. We have to prove that the sequence $\{\lambda_k\}_{k}$ provides all the eigenvalues of \eqref{EigPb}. Suppose that there exists at least one eigenvalue $\lambda \notin \{\lambda_{k}\}_{k}$. Let us denote by $f \in X_{0,A}$ its corresponding eigenfunction. Without loss of generality, we can also assume that $\|f\|_{L^{2}(\Omega)}=1$. Now, by \eqref{autovalore} with $\varphi=f$, we obtain \begin{equation}\label{bah} 2 \Phi(f) = \iint_{\mathbb{R}^{2n}} \frac{|f(x)-e^{\mathrm{i} (x-y) \cdot A( \frac{x+y}{2})}f(y)|}{|x-y|^{n+2s}} \,dx\,dy = \lambda. \end{equation} Since $\lambda_1$ is minimal in \eqref{Primo}, it holds that $$ \lambda_1 = \Phi(f_1) \leq \Phi(f)=\lambda. $$ For this and \eqref{monotonia}, we have that there exists $j\in \mathbb{N}$ such that $$ \lambda_j < \lambda < \lambda_{j+1}. $$ We claim now that $f \notin \mathbb{E}_{j+1}$. Indeed, if $f \in \mathbb{E}_{j+1}$, then by \eqref{k+1} and \eqref{bah}, $$ \lambda_{j+1} \leq 2 \, \Phi(f) = \lambda, $$ which yields a contradiction. Thus, since $f \notin \mathbb{E}_{j+1}$, there exists $i \in \{1, \ldots,j\}$ such that $\langle f,f_i\rangle_{X_{0,A}}\neq 0$, which is in contrast with Lemma \ref{chiamalo}. This concludes the proof. \end{proof} \begin{lemma}\label{vnulla} If $v \in X_{0,A}$ is such that $\langle v,f_k\rangle_{X_{0,A}}=0$ for every $k \in \mathbb{N}$, then $v \equiv 0$. \end{lemma} \begin{proof} By contradiction, suppose that there exists $v\in X_{0,A}$ with $v\not \equiv 0$ and such that $\langle v,f_k\rangle_{X_{0,A}}=0$ for every $k \in \mathbb{N}$. Without loss of generality, we can assume that $\|v\|_{L^{2}(\Omega)}=1$. By \eqref{divergenza}, we know that there exists $k \in \mathbb{N}$ such that $$ 2 \Phi(v) < \lambda_{k+1}. $$ Therefore $v \notin \mathbb{E}_{k+1}$, and this implies that there exists $j \in \mathbb{N}$ such that $\langle v,f_j\rangle_{X_{0,A}} \neq 0$, which is impossible. This completes the proof. \end{proof} \begin{proposition}\label{innominata} The sequence of eigenfunctions $\{f_k\}_{k}$ corresponding to $\{ \lambda_k \}_{k}$ is an orthonormal basis of $L^{2}(\Omega, \mathbb{C})$ and an orthogonal basis of $X_{0,A}$. \end{proposition} \begin{proof} The orthogonality follows from Proposition \ref{L2Orto}. Let us prove that $\{f_k\}_{k}$ is a basis of $X_{0,A}$. To this aim, let us define $F_k := f_k / \|f_k\|_{X_{0,A}}$. Moreover, for any $g \in X_{0,A}$, we consider \begin{equation*} g_j := \sum_{k=1}^{j} \langle g,F_k\rangle_{X_{0,A}} F_k, \end{equation*} which yields that $g_j \in \operatorname{span}_{\mathbb{R}}\{f_1, \ldots, f_j \}$ for any $j \in \mathbb{N}$. We want to show that $g_j \to g$ in $X_{0,A}$ as $j \to \infty$. We define $v_j := g - g_j$ and, since by Proposition \ref{L2Orto}, the $\{f_k\}_{k}$ are orthogonal with respect to $\langle \cdot,\cdot\rangle_{X_{0,A}}$, we have \begin{align*} 0 &\leq \|v_j\|^2_{X_{0,A}} = \|g\|^2_{X_{0,A}} + \|g_j\|^2_{X_{0,A}} - 2\langle g,g_j\rangle_{X_{0,A}}\\ &=\|g\|^2_{X_{0,A}} + \langle g_j,g_j\rangle_{X_{0,A}} -2 \sum_{k=1}^{j}\langle g,F_k\rangle_{X_{0,A}}^2 \\ &=\|g\|^2_{X_{0,A}} - \sum_{k=1}^{j}\langle g,F_k\rangle_{X_{0,A}}^2, \end{align*} which implies that $$ \sum_{k=1}^{j}\langle g,F_k\rangle_{X_{0,A}}^2 \leq \|g\|^2_{X_{0,A}}, $$ and hence $\displaystyle \sum_{k=1}^{\infty}\langle g,F_k\rangle_{X_{0,A}}^2$ is a convergent series. We can then consider the partial sum $$ \tau_j := \sum_{k=1}^{j}\langle g,F_k\rangle_{X_{0,A}}^2, $$ which is a Cauchy sequence in $\mathbb{R}$. Moreover, due to the orthogonality of $\{f_k\}_{k}$ with respect to $\langle \cdot,\cdot\rangle_{X_{0,A}}$, we have \begin{equation}\label{vJvj} \|v_h - v_j\|^{2}_{X_{0,A}} = \big\| \sum_{k=j+1}^{h}\langle g,F_k\rangle_{X_{0,A}}F_k \big\|^2_{X_{0,A}} = \tau_h - \tau_j, \end{equation} for $h>j$. Since $\{\tau_j\}_j$ is a Cauchy sequence in $\mathbb{R}$, \eqref{vJvj} implies that $\{v_j\}_j$ is a Cauchy sequence in $X_{0,A}$. Hence, there exists $v \in X_{0,A}$ such that $v_j \to v$ in $X_{0,A}$ as $j \to \infty$. We now note that, if $j \geq k$, then $\langle g_j,F_k\rangle_{X_{0,A}} = \langle g,F_k\rangle_{X_{0,A}}$. Therefore, for $j \geq k$, \begin{equation*} \langle v_j,F_k\rangle_{X_{0,A}} = \langle g,F_k\rangle_{X_{0,A}} - \langle g_j,F_k\rangle_{X_{0,A}} = 0. \end{equation*} On the other hand, since $v_j \to v$ in $X_{0,A}$, we obtain \begin{equation*} \langle v,F_k\rangle_{X_{0,A}} = 0, \quad \text{for every } k \in \mathbb{N}. \end{equation*} By Lemma \ref{vnulla}, we know that $v \equiv 0$, hence $$ g_j = g-v_j\,\, \to\,\, g-v = g \in X_{0,A}, \quad \text{as } j \to \infty, $$ and this shows that $\{f_k\}_{k}$ is a basis in $X_{0,A}$, since $g_j\in \operatorname{span}_{\mathbb{R}}\{f_1, \ldots, f_j \}$ for any $j\in\mathbb N$. Let us conclude by showing that $\{f_k\}_{k}$ is a basis in $L^2(\Omega,\mathbb{C})$. For this, let $v \in L^{2}(\Omega,\mathbb{C})$ and $v_j \in C^{\infty}_{0}(\Omega,\mathbb{C})$ be such that $\|v_j - v\|_{L^{2}(\Omega)} \leq 1/j$. By combining \cite[Proposition 2.2]{d'AS} and \cite[Lemma 2.1]{FPV17}, we have $C^{\infty}_{0}(\Omega,\mathbb{C}) \subset X_{0,A}$, implying $v_j \in X_{0,A}$. We also know that $\{f_k\}_{k}$ is a basis of $X_{0,A}$, therefore there exists $k_j \in \mathbb{N}$ and a function $w_j \in \operatorname{span}_{\mathbb{R}}\{f_1, \ldots, f_{k_j}\}$ such that $$ \|v_j - w_j\|_{X_{0,A}} \leq \frac{1}{j}. $$ Now, by \cite[Lemma 2.1]{FPV17} we have $$ \|v_j - w_j\|_{L^{2}(\Omega)} \leq \|v_j -w_j\|_{H_A^s(\mathbb R^n)} \leq C \|v_j - w_j \|_{X_{0,A}} \leq \frac{C}{j}. $$ By triangle inequality, we obtain that $$ \|v-w_j\|_{L^{2}(\Omega)} \leq \|v-v_j\|_{L^{2}(\Omega)} + \|v_j - w_j\|_{L^{2}(\Omega)} \leq \frac{C+1}{j}, $$ which shows that $\{f_k\}_k$ is a basis of $L^{2}(\Omega,\mathbb{C})$. \end{proof} We conclude this section with a proposition showing that for any $k\in\mathbb N$ each eigenvalue $\lambda_k$ has finite multiplicity. \begin{proposition} Let $h \geq 0$ and $k\in\mathbb N$. If $\lambda_k$ has multiplicity $h+1$, namely \[ %\label{Multi} \lambda_{k-1} < \lambda_k = \ldots = \lambda_{k+h} < \lambda_{k+h+1}, \] then the set of the eigenfunctions corresponding to $\lambda_k$ is given by $$ \operatorname{span}_{\mathbb{R}}\{ f_k, \ldots, f_{k+h}\}. $$ \end{proposition} \begin{proof} By Proposition \ref{prima parte}, we know that every element $g \in \operatorname{span}_{\mathbb{R}}\{ f_k, \ldots, f_{k+h}\}$ is an eigenfunction of \eqref{EigPb} corresponding to the eigenvalue $\lambda_k= \ldots = \lambda_{k+h}$. Conversely, we need to show that any eigenfunction $\psi \in X_{0,A}$ corresponding to $\lambda_k$ belongs to $\operatorname{span}_{\mathbb{R}}\{ f_k, \ldots, f_{k+h}\}$. First, we consider the orthogonal decomposition $$ X_{0,A} = \operatorname{span}_{\mathbb{R}}\{f_k, \ldots, f_{k+h}\} \oplus \left(\operatorname{span}_{\mathbb{R}}\{f_k, \ldots, f_{k+h}\}\right)^{\bot}. $$ Thus, we can write $\psi = \psi_1 + \psi_2$, where \begin{equation}\label{psi1} \psi_1 \in \operatorname{span}_{\mathbb{R}}\{f_k, \ldots, f_{k+h}\} \quad \text{and} \quad \psi_2 \in \left(\operatorname{span}_{\mathbb{R}}\{f_k, \ldots, f_{k+h}\}\right)^{\bot}, \end{equation} which implies by Proposition \ref{L2Orto}, \begin{equation}\label{psi12} \langle \psi_1,\psi_2\rangle_{X_{0,A}}=0. \end{equation} Being $\psi$ an eigenfunction corresponding to $\lambda_k$, by \eqref{autovalore} and \eqref{psi12} we obtain \begin{equation}\label{a.35} \lambda_k \|\psi\|^2_{L^{2}(\Omega)} = \|\psi\|^{2}_{X_{0,A}} = \|\psi_1\|^{2}_{X_{0,A}} + \|\psi_2\|^{2}_{X_{0,A}}. \end{equation} Since $\{f_k, \ldots, f_{k+h}\}$ are eigenfunctions corresponding to $\lambda_k$ by Proposition \ref{prima parte}, $\psi_1$ is an eigenfunction corresponding to $\lambda_k$ as well. Therefore, by also \eqref{psi12} we obtain \begin{equation*} \lambda_k \Re \int_{\Omega}\psi_1(x) \overline{\psi_2 (x)} \, dx =\langle \psi_1,\psi_2\rangle_{X_{0,A}}=0, \end{equation*} which implies \begin{equation}\label{a.37} \|\psi\|^2_{L^{2}(\Omega)} = \|\psi_1 + \psi_2\|^2_{L^{2}(\Omega)} = \|\psi_1\|^2_{L^{2}(\Omega)} + \|\psi_{2}\|^{2}_{L^{2}(\Omega)}. \end{equation} Now, by definition of $\psi_1$, there exist $c_k, \ldots, c_{k+h}$ real constants, such that $$ \psi_1 = \sum_{i=k}^{k+h}c_i \, f_i. $$ Byo the orthogonality of $\{f_i\}_{i}$ in Proposition \ref{L2Orto} and considering each $f_i$ attains the minimum in \eqref{k+1} by Proposition \ref{prima parte}, we have \begin{equation}\label{a.38} \|\psi_1\|^2_{X_{0,A}} = \sum_{i=k}^{k+h}c_i^2 \|f_i\|^2_{X_{0,A}} = \sum_{i=k}^{k+h}c_i^2 \lambda_i = \lambda_k \, \sum_{i=k}^{k+h}c_i^2 = \lambda_k \|\psi_1\|^{2}_{L^{2}(\Omega)}. \end{equation} Now, since $\psi$ and $\psi_1$ are eigenfunctions corresponding to $\lambda_k$, and $\psi_2 = \psi - \psi_1$, it follows that $\psi_2$ must be an eigenfunction corresponding to $\lambda_k$ as well. Therefore, by Lemma \ref{chiamalo} $$ \langle \psi_2,f_1\rangle_{X_{0,A}}= \ldots = \langle \psi_2,f_{k-1}\rangle_{X_{0,A}} = 0, $$ which, together with \eqref{psi1}, implies \begin{equation}\label{psi2} \psi_2 \in \big( \operatorname{span}_{\mathbb{R}}\{f_1, \ldots, f_{k+h} \} \big)^{\bot} = \mathbb{E}_{k+h+1}. \end{equation} Now, we claim that $\psi_2 \equiv 0$. Assume by contradiction that this is not the case. Then, by \eqref{k+1} and \eqref{psi2} \begin{equation}\label{a.41} \lambda_k < \lambda_{k+h+1} = \min_{u \in \mathbb{E}_{k+h+1}} \frac{\|u\|^2_{X_{0,A}}}{\|u\|^2_{L^{2}(\Omega)}} \leq \frac{\|\psi_2\|^2_{X_{0,A}}}{\|\psi_2\|^2_{L^{2}(\Omega)}}. \end{equation} We can now compute by \eqref{a.35}, \eqref{a.37}, \eqref{a.38} and \eqref{a.41} \begin{equation*} \lambda_k \|\psi\|^2_{L^{2}(\Omega)} = \|\psi_1\|^2_{X_{0,A}} + \|\psi_2\|^2_{X_{0,A}} > \lambda_k \|\psi_1\|^2_{L^{2}(\Omega)} + \lambda_k \|\psi_2\|^2_{L^{2}(\Omega)} =\lambda_k \|\psi\|^2_{L^{2}(\Omega)}, \end{equation*} which yields a contradiction, hence proving that $\psi_2 \equiv 0$. In particular, this and \eqref{psi1} imply that $$ \psi = \psi_1 \in \operatorname{span}_{\mathbb{R}} \{f_k, \ldots, f_{k+h}\}, $$ as desired. \end{proof} \section{Palais-Smale condition} \label{ps} In this section we verify that the functional $\mathcal{J}_{A,\lambda}$ satisfies the $(PS)_c$ condition under a suitable level, depending on the magnetic Sobolev constant given in \eqref{MaSb}. For this, we recall that $\{u_j\}_j\subset X_{0,A}$ is a \emph{Palais-Smale sequence for $\mathcal{J}_{A,\lambda}$ at level $c\in\mathbb{R}$} (in short $(PS)_c$ sequence) if \begin{equation}\label{ps1} \mathcal{J}_{A,\lambda}(u_j)\to c\quad\text{and}\quad\mathcal{J}'_{A,\lambda}(u_j)\to0\quad \text{as }j\to \infty. \end{equation} We say that $\mathcal{J}_{A,\lambda}$ \emph{satisfies the Palais-Smale condition at level} $c$ if any Palais--Smale sequence $\{u_j\}_j$ at level $c$ admits a convergent subsequence in $X_{0,A}$. \begin{proposition}\label{palais} Let $c \in (-\infty, s\,\mathcal{S}_A^{n/(2s)}/n)$. Then the functional $\mathcal{J}_{A,\lambda}$ satisfies the Palais-Smale condition at any level $c$. \end{proposition} \begin{proof} The proof follows its diamagnetic counterpart in \cite{FMS16}. Let $\{u_j\}_j\subset X_{0,A}$ be any sequence verifying \eqref{ps1}. As usual, we first need to prove the boundedness of $\{u_j\}_j$ in $X_{0,A}$. To this aim, since $u_j \not\equiv 0$ and by \eqref{ps1}, there exists a positive constant $k>0$, independent of j, such that \begin{equation}\label{kpos} |\mathcal{J}_{A,\lambda}(u_j)| \leq k \quad \text{and}\quad \Big| \Big\langle \mathcal{J}'_{A,\lambda}(u_j), \frac{u_j}{\|u_j\|_{X_{0,A}}} \Big\rangle \Big| \leq k, \end{equation} for every $j \in \mathbb{N}$. Now, a direct computation combined with \eqref{kpos}, shows that \begin{equation}\label{stella} k \left(1 + \|u_j\|_{X_0,A}\right) \geq\mathcal{J}_{A,\lambda}(u_j) - \frac{1}{2} \langle \mathcal{J}'_{A,\lambda}(u_j),u_j \rangle = \frac{s}{n} \|u_j\|^{2^{\ast}_s}_{L^{2^{\ast}_s}(\Omega)}, \quad \text{for any } j \in \mathbb{N}. \end{equation} From this, by H\"older inequality and considering that $2/2^{\ast}_s<1$, we have \begin{equation}\label{due} \|u_j\|^2_{L^2(\Omega)} \leq k_* \big( 1+ \|u_j\|_{X_{0,A}}\big), \quad \text{for any } j \in \mathbb{N}, \end{equation} with $k_*>0$ another suitable constant independent of $j$. By \eqref{kpos}, \eqref{stella} and \eqref{due}, it follows that there exists a constant $\widetilde{k}>0$, independent of $j$, such that $$ k \geq \mathcal{J}_{A,\lambda}(u_j) \geq \frac{1}{2} \|u_j\|^2_{X_{0,A}} - \widetilde{k} \left( 1 + \|u_j\|_{X_{0,A}}\right), \quad \text{for any } j \in \mathbb{N}, $$ and this is sufficient to conclude that the sequence $\{u_j\}_j \subset X_{0,A}$ is bounded. Since the space $(X_{0,A}, \langle \cdot,\cdot\rangle_{X_{0,A}})$ is a Hilbert space, there exists $u_{\infty} \in X_{0,A}$ such that $u_j \rightharpoonup u_{\infty}$ in $X_{0,A}$ as $j \to \infty$, that is \begin{equation}\label{convdebole} \begin{aligned} &\lim_{j\to\infty}\Re \iint_{\mathbb{R}^{2n}} \frac{\big(u_j(x)-e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}u_j(y) \big) \overline{\big( \varphi(x)-e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}\varphi(y) \big)}} {|x-y|^{n+2s}}\, dx\,dy\\ &=\Re \iint_{\mathbb{R}^{2n}} \frac{\big(u_{\infty}(x)-e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}u_{\infty}(y) \big) \overline{\big( \varphi(x)-e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}\varphi(y) \big)}}{|x-y|^{n+2s}}\, dx\,dy, \end{aligned} \end{equation} for any $\varphi\in X_{0,A}$. Also, by \cite[Lemma 2.2]{FPV17} and \cite[Theorem 4.9]{brezis}, up to a subsequence still labeled by $\{u_j\}_j$, we have \begin{equation}\label{convergenze} u_j \rightharpoonup u_{\infty} \text{ in } L^{2^{\ast}_s}(\Omega,\mathbb C),\quad u_j \to u_{\infty} \text{ in } L^{2}(\Omega,\mathbb C),\quad u_j \to u_{\infty} \text{ a.e in } \Omega, \end{equation} as $j\to \infty$. By \eqref{stella} and the boundedness of $\{u_j\}_j$ in $X_0$, we have that $\|u_j\|_{L^{2^{\ast}_s}(\Omega)}$ is uniformly bounded in $j$, and therefore the sequence $\{ |u_j|^{2^{\ast}_{s} -2}u_j \}_j$ is uniformly bounded in $L^{2^{\ast}_s / (2^{\ast}_s -1)}(\Omega)$. Thus, by \eqref{convergenze} \begin{equation}\label{ast} |u_j|^{2^{\ast}_s -2} u_j \rightharpoonup |u_{\infty}|^{2^{\ast}_s -2} u_{\infty}, \quad \text{ in } L^{2^{\ast}_s / (2^{\ast}_s -1)}(\Omega,\mathbb C), \end{equation} as $j\to \infty$. Now, by passing to the limit as $j\to\infty$ on \eqref{ps1} and considering \eqref{convdebole}, \eqref{convergenze} and \eqref{ast}, we have \begin{equation}\label{debole} \begin{aligned} &\Re \iint_{\mathbb{R}^{2n}}\frac{\big(u_\infty(x) - e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}u_\infty(y)\big) \overline{\big(\varphi(x)- e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}\varphi(y) \big)}} {|x-y|^{n+2s}} \, dx dy \\ &= \lambda \Re \int_{\Omega}u(x) \overline{\varphi(x)}\, dx + \Re\int_{\Omega}|u(x)|^{2^{\ast}_s -2}u(x) \overline{\varphi(x)}\, dx, \end{aligned} \end{equation} for any $\varphi\in X_{0,A}$. This proves that $u_{\infty} \in X_{0,A}$ is a weak solution of \eqref{Pb}. By \eqref{convdebole} and since $(X_{0,A}, \langle \cdot,\cdot\rangle_{X_{0,A}})$ is a Hilbert space, we have \begin{equation}\label{blx0} \|u_j\|^2_{X_{0,A}} = \|u_j - u_{\infty}\|^2_{X_{0,A}} + \|u_{\infty}\|^2_{X_{0,A}} + o(1), \quad \text{as } j\to \infty. \end{equation} While, by \eqref{convergenze} and \cite[Theorem 1]{BrLi} we have \begin{equation}\label{blast} \|u_j\|^{2^{\ast}_s}_{L^{2^{\ast}_s}(\Omega)} = \|u_j - u_{\infty}\|^{2^{\ast}_s}_{L^{2^{\ast}_s}(\Omega)} + \|u_{\infty}\|^{2^{\ast}_s}_{L^{2^{\ast}_s}(\Omega)} + o(1), \quad \text{as } j\to \infty. \end{equation} Therefore, by \eqref{convergenze}, \eqref{blx0} and \eqref{blast}, we obtain \begin{equation}\label{Rel1} \begin{aligned} \mathcal{J}_{A,\lambda}(u_j) &= \frac{1}{2}\|u_j - u_{\infty}\|^2_{X_{0,A}} + \frac{1}{2}\|u_{\infty}\|^2_{X_{0,A}} -\frac{\lambda}{2} \|u_{\infty}\|^2_{L^{2}(\Omega)}\\ &\quad- \frac{1}{2^{\ast}_s}\|u_j-u_{\infty}\|^{2^{\ast}_s}_{L^{2^{\ast}_s}(\Omega)} - \frac{1}{2^{\ast}_s}\|u_{\infty}\|^{2^{\ast}_s}_{L^{2^{\ast}_s}(\Omega)} + o(1)\\ &=\mathcal{J}_{A,\lambda}(u_{\infty}) + \frac{1}{2} \|u_j - u_{\infty}\|^{2}_{X_{0,A}} - \frac{1}{2^{\ast}_s}\|u_j - u_{\infty}\|^{2^{\ast}_s}_{L^{2^{\ast}_s}(\Omega)} + o(1), \end{aligned} \end{equation} as $j\to \infty$. Furthermore, by \eqref{ps1}, \eqref{convdebole}, \eqref{convergenze}, \eqref{ast} and considering $u_\infty$ as solution of \eqref{Pb}, we obtain \begin{align*} o(1) &=\langle\mathcal J'_{A,\lambda}(u_j) - \mathcal J'_{A,\lambda} (u_\infty), u_j -u_\infty\rangle \\ &=\iint_{\mathbb{R}^{2n}}\frac{|u_j(x) - e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}u_j(y)|^2}{|x-y|^{n+2s}}dx dy \\ &\quad -2\Re \iint_{\mathbb{R}^{2n}} \frac{\big(u_j(x) -e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}u_j(y) \big) \overline{\big(u_\infty(x)-e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}u_\infty(y) \big)}}{|x-y|^{n+2s}} dx\,dy\\ &\quad +\iint_{\mathbb{R}^{2n}}\frac{|u_\infty(x) - e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}u_\infty(y)|^2}{|x-y|^{n+2s}}dx dy \\ &\quad -\Re\int_\Omega\left(|u_j(x)|^{2^{\ast}_s-2}u_j(x) -|u_\infty(x)|^{2^{\ast}_s-2}u_\infty(x)\right) \left(\overline{u_j(x)}-\overline{u_\infty(x)}\right)dx+o(1)\\ &= \iint_{\mathbb{R}^{2n}}\frac{|u_j(x) - e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}u_j(y)|^2}{|x-y|^{n+2s}}dx dy \\ &\quad -\iint_{\mathbb{R}^{2n}}\frac{|u_\infty(x) - e^{\mathrm{i} (x-y)\cdot A(\frac{x+y}{2})}u_\infty(y)|^2}{|x-y|^{n+2s}}dx dy\\ &-\int_\Omega|u_j(x)|^{2^{\ast}_s}dx+\int_\Omega|u_\infty(x)|^{2^{\ast}_s}dx +o(1)\quad\text{as }j\to\infty, \end{align*} from which, by \eqref{blx0} and \eqref{blast}, we obtain the formula \begin{equation}\label{Rel2} \|u_j - u_{\infty}\|^2_{X_{0,A}} = \|u_j -u_{\infty}\|^{2^{\ast}_s}_{L^{2^{\ast}_s}(\Omega)} + o(1), \end{equation} as $j\to\infty$. Hence, combining \eqref{Rel1} and \eqref{Rel2}, we obtain \begin{equation*} \mathcal{J}_{A,\lambda}(u_j) = \mathcal{J}_{A,\lambda}(u_{\infty}) + \frac{s}{n}\|u_j -u_{\infty}\|^2_{X_{0,A}} +o(1), \quad \text{as } j \to \infty. \end{equation*} From this, recalling \eqref{ps1}, combining \eqref{Jlam} and \eqref{debole} with $\varphi=u_\infty$, we obtain as $j \to \infty$ \begin{equation}\label{varie} \begin{aligned} c&= \mathcal{J}_{A,\lambda}(u_{\infty}) + \frac{s}{n} \|u_j - u_{\infty}\|^{2}_{X_{0,A}} + o(1)\\ &=\frac{s}{n}\Big(\|u_{\infty}\|^{2}_{X_{0,A}} +\|u_j - u_{\infty}\|^{2}_{X_{0,A}}\Big)+o(1). \end{aligned} \end{equation} By the boundedness of $\{u_j\}_j$ in $X_0$, up to a subsequence, we also have \begin{equation}\label{LimL1} \|u_j - u_{\infty}\|^2_{X_{0,A}} \to L \in [0,\infty), \quad \text{as } j \to\infty. \end{equation} Combining formula \eqref{Rel2} with \eqref{LimL1}, and recalling \eqref{MaSb}, we obtain $$ L^{2/2^{\ast}_s} \mathcal{S}_A \leq L $$ which implies that either $L=0$ or $L \geq \mathcal{S}_A^{n/(2s)}$. Let us assume by contradiction $L \geq \mathcal{S}_A^{n/(2s)}$. Then, by \eqref{varie} and \eqref{LimL1} we obtain $$ c =\frac{s}{n}(\|u_\infty\|_{X_{0,A}}^2+L) \geq \frac{s}{n}L\geq\frac{s}{n}\mathcal{S}_A^{n/(2s)}, $$ in contrast with our standing assumption on $c$. Therefore $L=0$ and from \eqref{LimL1} we conclude the proof. \end{proof} \section{Proof of Theorem \ref{Bif}}\label{PROOF} In this section we prove our main result given in Theorem \ref{Bif}. To state the number of solutions of problem \eqref{Pb}, we must explicitly describe $\lambda^\ast$ with its multiplicity, as in \eqref{lambdastella}. Since, $\lambda^\ast=\lambda_k$ for some $k\in \mathbb{N}$, with multiplicity $m\in \mathbb{N}$ by assumption, we have that \begin{equation}\label{add3} \begin{gathered} \lambda^\ast=\lambda_1<\lambda_2 \quad \text{if } k=1\\ \lambda_{k-1}<\lambda^\ast=\lambda_k=\dots =\lambda_{k+m-1}<\lambda_{k+m} \quad\text{if } k\geq 2\,. \end{gathered} \end{equation} Also, we have to observe that, under condition~\eqref{condlambda*}, the parameter $\lambda>0$. Indeed, by definition of $\lambda^\ast$ and taking into account Proposition \ref{SuccLambda}, it is easily seen that \begin{equation}\label{lambdalambda*} \lambda^\ast\geq \lambda_1. \end{equation} Now, by H\"older inequality, $$ \|u\|_{L^2(\Omega)}^2\le |\Omega|^{2s/n}\|u\|_{L^{2^*_s}(\Omega)}^2. $$ From this, \eqref{MaSb} and \eqref{Primo}, we obtain $$ \lambda_1\geq \mathcal{S}_A |\Omega|^{-2s/n}\,, $$ which combined with \eqref{lambdalambda*} yields $$ \lambda^\star\geq \mathcal{S}_A|\Omega|^{-2s/n}\,. $$ Hence, as a consequence of this and of \eqref{condlambda*}, we obtain $\lambda^{\star}>0$. \begin{lemma}\label{geo1} Let $\lambda\in\mathbb R$ satisfy \eqref{condlambda*}, with $\lambda^\star$ in \eqref{lambdastella} counted with multiplicity $m\in\mathbb{N}$. Then, for any $w \in \operatorname{span}_{\mathbb{R}}\left\{f_{1},\ldots,f_{k+m-1}\right\}$, we have $$ \mathcal{J}_{A,\lambda}(w)\leq\frac{s}{n}(\lambda^\star-\lambda)^{n/(2s)}|\Omega|. $$ \end{lemma} \begin{proof} Let $u\in \operatorname{span}_{\mathbb{R}}\{f_{1},\ldots,f_{k+m-1}\}$. Then, we have $$ u(x)=\sum_{i=1}^{k+m-1}c_{i}f_{i}(x) $$ with $c_{i}\in\mathbb{R}$, for $i=1,\ldots,k+m-1$. By Proposition \ref{innominata}, and taking into account \eqref{add3}, we obtain \begin{align*} \|u\|_{X_{0,A}}^2 &=\sum_{i=1}^{k+m-1}c^{2}_{i}\|f_{i}\|_{X_{0,A}}^2 =\sum_{i=1}^{k+m-1}\lambda_{i}c^{2}_{i} \\ &\leq\lambda_{k}\sum_{i=1}^{k+m-1}c^{2}_{i} =\lambda_{k}\|u\|_{L^2(\Omega)}^2\\ &=\lambda^\star\|u\|_{L^2(\Omega)}^2\,, \end{align*} so that, by using H\"older inequality, we have \begin{equation}\label{add22} \begin{aligned} \mathcal{J}_{A,\lambda}(u) &=\frac{1}{2}\|u\|_{X_{0,A}}^2-\frac{\lambda}{2}\|u\|_{L^2(\Omega)}^2 -\frac{1}{2^\ast_s}\|u\|_{L^{2^\ast_s}(\Omega)}^{2^\ast_s} \\ &\leq\frac{1}{2}(\lambda^\star-\lambda)\|u\|_{L^2(\Omega)}^2-\frac{1}{2^\ast_s}\|u\|_{L^{2^\ast_s}(\Omega)}^{2^\ast_s}\\ &\leq\frac{1}{2}(\lambda^\star-\lambda)|\Omega|^{2s/n} \|u\|_{L^{2^\ast_s}(\Omega)}^2-\frac{1}{2^\ast_s}\|u\|_{L^{2^\ast_s} (\Omega)}^{2^\ast_s}. \end{aligned} \end{equation} Now, for $t\geq 0$ we define the function $$ g(t):=\frac{1}{2}(\lambda^\star-\lambda)|\Omega|^{2s/n}t^2 -\frac{1}{2^\ast_s}t^{2^\ast_s}\,. $$ Note that the function $g$ is differentiable in $(0, \infty)$ and $$ g'(t)=(\lambda^\star-\lambda)|\Omega|^{2s/n}t-t^{2^\ast_s-1}, $$ with $(\lambda^\star-\lambda)|\Omega|^{2s/n}>0$ since $\lambda<\lambda^\star$ by \eqref{condlambda*}. Thus, we have $g'(t)\geq 0$ if and only if $$ t\leq \widetilde t :=\big[(\lambda^\star-\lambda)|\Omega|^{2s/n}\big]^{1/(2^\ast_s-2)}\,. $$ As a consequence of this, $\widetilde t$ is a maximum point for $g$ and so for any $t\geq 0$ \begin{equation}\label{ADD111} g(t)\leq \max_{t\geq 0}g(t)=g(\widetilde t) =\frac{s}{n}(\lambda^\star-\lambda)^{n/(2s)}|\Omega|. \end{equation} By \eqref{add22} and \eqref{ADD111} we obtain \[ \label{max} \sup_{u\in\operatorname{span}_{\mathbb{R}}\left\{f_{1},\ldots,f_{k+m-1}\right\}}\mathcal{J}_{A,\lambda}(u)\leq\max_{t\geq 0}g(t)=\frac{s}{n}(\lambda^\star-\lambda)^{n/(2s)}|\Omega|, \] concluding the proof. \end{proof} \begin{lemma}\label{geo2} Let $\lambda\in\mathbb R$ satisfy \eqref{condlambda*}, with $\lambda^\star$ in \eqref{lambdastella} counted with multiplicity $m\in\mathbb{N}$. Then, there exists $\delta>0$, with $$ \delta<\frac{s}{n}(\lambda^\star-\lambda)^{n/(2s)}|\Omega|, $$ and $\rho>0$ such that $\mathcal{J}_{A,\lambda}(v) \geq \delta$, for any $v \in V$ with $\|v\|_{X_{0,A}} = \rho$, where \begin{equation}\label{vu} V=\begin{cases} X_0 & \text{if } k=1\\ \mathbb E_k & \text{if } k\geq 2\,, \end{cases} \end{equation} with $\mathbb E_k$ given in \eqref{E_k1}. \end{lemma} \begin{proof} Let $u\in V$. Then \begin{equation}\label{ADD123} \|u\|_{X_{0,A}}^2\geq\lambda^\star\|u\|_{L^2(\Omega)}^2 \end{equation} Indeed, if $u\equiv 0$, then the assertion is trivial, while if $u\in V\setminus\{0\}$ it follows from the variational characterization of $\lambda^\star=\lambda_k$, given by \eqref{Primo} and \eqref{k+1}. Thus, by \eqref{MaSb}, \eqref{ADD123} and taking into account that $\lambda>0$, as seen above, it follows that \begin{equation}\label{add333} \begin{aligned} \mathcal{J}_{A,\lambda}(u) & \geq\frac{1}{2}\Big(1-\frac{\lambda}{\lambda^\star}\Big)\|u\|^2_{X_{0,A}} -\frac{1}{2^\ast_s\mathcal{S}_A^{1/2}}\|u\|^{2^\ast_s}_{X_{0,A}}\\ & =\|u\|^2_{X_{0,A}}\Big(\frac{1}{2}\Big(1-\frac{\lambda}{\lambda^\star}\Big)- \frac{1}{2^\ast_s\mathcal{S}_A^{1/2}}\|u\|^{2^\ast_s-2}_{X_{0,A}}\Big)\,. \end{aligned} \end{equation} Thus, let $u\in V$ be such that $\|u\|_{X_{0,A}}=\rho>0$. Since $2^\ast_s>2$, to conclude the proof it is enough to choose $\rho$ sufficiently small so that \begin{equation}\label{add33bis} \frac{1}{2}\Big(1-\frac{\lambda}{\lambda^\star}\Big)- \frac{1}{2^\ast_s\mathcal{S}_A^{1/2}}\rho^{2^\ast_s-2}>0, \end{equation} and \begin{equation}\label{add33} \rho^2\Big(\frac{1}{2}\Big(1-\frac{\lambda}{\lambda^\star}\Big) -\frac{1}{2^\ast_s\mathcal{S}_A^{1/2}}\rho^{2^\ast_s-2}\Big)) <\frac{\rho^2}{2}\Big(1-\frac{\lambda}{\lambda^\star}\Big) <\frac{s}{n}(\lambda^\star-\lambda)^{n/(2s)}|\Omega|. \end{equation} \end{proof} \begin{proof}[Proof of Theorem \ref{Bif}] Let $\lambda\in\mathbb R$ satisfy \eqref{condlambda*}, with $\lambda^\star$ in \eqref{lambdastella} counted with multiplicity $m\in\mathbb{N}$. By \eqref{Jlam} it is immediate to see that $\mathcal{J}_{A,\lambda}$ satisfies (A1) of Theorem \ref{bbf}. While (A2) and (A3) hold true thanks to Proposition \ref{palais}, Lemmas \ref{geo1} and \ref{geo2}, considering $W=\operatorname{span}_{\mathbb{R}}\left\{e_{1},\ldots,e_{k+m-1}\right\}$, $V$ given in \eqref{vu} and that, by \eqref{condlambda*}, $$ \delta<\frac{s}{n}(\lambda^\star-\lambda)^{n/(2s)}|\Omega|<\frac{s}{n} \mathcal{S}_A^{n/(2s)}. $$ Thus, since $\text{dim}~W=k+m-1$ and $\operatorname{codim}V=k-1$, by Theorem \ref{bbf} the functional $\mathcal{J}_{A,\lambda}$ has $m$ pairs $\{-u_{\lambda, i},\,u_{\lambda, i}\}$ of critical points whose critical values $\mathcal{J}_{A,\lambda}(\pm u_{\lambda, i})$ are such that \begin{equation}\label{add55} 0<\delta\leq \mathcal{J}_{A,\lambda}(\pm u_{\lambda, i}) \leq \frac{s}{n}(\lambda^\star-\lambda)^{n/(2s)}|\Omega| \end{equation} for any $i=1, \dots, m$. Since $\mathcal{J}_{A,\lambda}(0)=0$ and by \eqref{add55}, it is immediate to see that these critical points are non-trivial. Hence, problem~\eqref{Pb} admits $m$ pairs of non-trivial weak solutions. Now, fix $i\in \{1, \dots, m\}$. By \eqref{add55} we obtain \begin{equation}\label{2star} \begin{aligned} \frac{s}{n}(\lambda^\ast-\lambda)^{n/(2s)}|\Omega| &\geq\mathcal{J}_{A,\lambda}(u_{\lambda, i}) =\mathcal{J}_{A,\lambda}(u_{\lambda, i})-\frac{1}{2} \langle \mathcal{J}'_{A,\lambda}(u_{\lambda, i}),u_{\lambda, i}\rangle \\ &=\frac s n\,\|u_{\lambda, i}\|^{2^\ast_s}_{L^{2^\ast_s}(\Omega)}, \end{aligned} \end{equation} so that, passing to the limit as $\lambda\to\lambda^\ast$ in \eqref{2star}, it follows that \begin{equation}\label{add66} \|u_{\lambda, i}\|_{L^{2^\ast_s}(\Omega)}\to0\quad\text{as }\lambda\to\lambda^\ast. \end{equation} Also, by \eqref{add66} and \cite[Lemma 2.2]{FPV17}, we also get \begin{equation}\label{add77} \|u_{\lambda, i}\|_{L^2(\Omega)}\to0\quad\text{as }\lambda\to\lambda^\ast. \end{equation} Thus, arguing as in \eqref{2star}, we have $$ \frac{s}{n}(\lambda^\star-\lambda)^{n/(2s)}|\Omega| \geq \mathcal{J}_{A,\lambda}(u_{\lambda, i}) =\frac{1}{2}\|u_{\lambda, i}\|^2_{X_{0,A}} -\frac{\lambda}{2}\|u_{\lambda, i}\|^2_{L^2(\Omega)} -\frac{1}{2^\ast_s}\|u_{\lambda, i}\|^{2^\ast_s}_{L^{2^\ast_s}(\Omega)}, $$ which combined with \eqref{add66} and \eqref{add77} gives $$ \|u_{\lambda, i}\|_{X_{0,A}}\to0\quad\text{as }\lambda\to\lambda^\ast. $$ This completes the proof. \end{proof} \subsection*{Acknowledgments} We thank the anonymous referee for his/her comments that improved the readability of this paper. 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