Abstract
In this paper, we deal with the following class of fractional (p, q)-Laplacian Kirchhoff type problem:
where \(\varepsilon >0\), \(s\in (0, 1)\), \(1<p<q<\frac{N}{s}<2q\), \((-\Delta )_{t}^{s}\), with \(t\in \{p, q\}\), is the fractional t-Laplacian operator, \(V:\mathbb {R}^{N}\rightarrow \mathbb {R}\) is a positive continuous potential such that \(\inf _{\partial \Lambda }V>\inf _{\Lambda } V\) for some bounded open set \(\Lambda \subset \mathbb {R}^{N}\), and \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a superlinear continuous nonlinearity with subcritical growth at infinity. By combining the method of Nehari manifold, a penalization technique, and the Lusternik–Schnirelman category theory, we study the multiplicity and concentration properties of solutions for the above problem when \(\varepsilon \rightarrow 0\).
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1 Introduction
In this paper, we investigate the multiplicity and concentration phenomenon of solutions for the following fractional (p, q)-Laplacian Kirchhoff type problem:
where \(\varepsilon >0\) is a small parameter, \(s\in (0, 1)\), \(1< p<q<\frac{N}{s}<2q\), \(V: \mathbb {R}^{N}\rightarrow \mathbb {R}\) is a bounded and continuous potential fulfilling the following conditions [18]:
- \((V_1)\):
-
there exists \(V_{0}>0\) such that \(V_{0}=\inf _{x\in \mathbb {R}^{N}} V(x)\),
- \((V_2)\):
-
there exists a bounded open set \(\Lambda \subset \mathbb {R}^{N}\) such that
$$\begin{aligned} V_{0}< \min _{\partial \Lambda } V \quad \text{ and } \quad 0\in M=\{x\in \Lambda : V(x)=V_{0}\}, \end{aligned}$$
and \(f:\mathbb {R}\rightarrow \mathbb {R}\) is a continuous nonlinearity such that \(f(t)=0\) for \(t\le 0\) and satisfying the following hypotheses:
- \((f_{1})\):
-
\(\displaystyle {\lim _{|t|\rightarrow 0} \frac{|f(t)|}{|t|^{2p-1}}=0}\),
- \((f_{2})\):
-
there exists \(\nu \in (2q, q^{*}_{s})\) such that \(\displaystyle {\lim _{|t|\rightarrow \infty } \frac{|f(t)|}{|t|^{\nu -1}}=0}\), where \(q^{*}_{s}= \frac{Nq}{N-sq}\),
- \((f_{3})\):
-
there exists \(\vartheta \in (2q, \nu )\) such that \(\displaystyle {0<\vartheta F(t)= \vartheta \int _{0}^{t} f(\tau ) \, d\tau \le t f(t)}\) for all \(t>0\),
- \((f_{4})\):
-
the map \(t\mapsto \displaystyle {\frac{f(t)}{t^{2q-1}}}\) is increasing in \((0, \infty )\).
The symbol \((-\Delta )^{s}_{t}\), with \(t\in \{p, q\}\), stands for the fractional t-Laplacian operator defined, up to a normalization constant depending on N, s and t, by setting
for any function \(u: \mathbb {R}^{N}\rightarrow \mathbb {R}\) sufficiently smooth. We recall that the recent years have seen a surge of interest in nonlocal and fractional problems involving the fractional t-Laplacian operator because of the presence of two features: the nonlinearity of the operator and its nonlocal character. For this reason, several existence, multiplicity and regularity results have been established by many authors; see for instance [4, 8, 10, 20, 24, 28, 38].
When \(s=1\), the study of (1.1) is strictly related to the following (p, q)-Laplacian equation
which comes from a general reaction–diffusion system
This system has a wide range of applications in physics and related sciences, such as biophysics, plasma physics, and chemical reaction design. In such applications, the function u describes a concentration, \({{\,\mathrm{div}\,}}(D(u) \nabla u)\) corresponds to the diffusion with diffusion coefficient D(u), and the reaction term c(x, u) relates to source and loss processes. Typically, in chemical and biological applications, the reaction term c(x, u) is a polynomial of u with variable coefficients; see [17]. Some classical results for (p, q)-Laplacian problems in bounded or unbounded domains can be found in [2, 22, 26, 27, 31, 33, 34] and the references therein. We also mention [15, 30] in which the authors discussed Kirchhoff type problems with the (p, q)-Laplacian operator \(-\Delta _{p}-\Delta _{q}\).
For what concerns the nonlocal framework, only few papers studied fractional (p, q)-Laplacian problems. Such problems involve the sum of two nonlocal nonlinear operators with different scaling properties and so some nontrivial additional technical difficulties arise with respect to the local case \(s=1\) and \(p\ne q\), and the fractional case \(s\in (0, 1)\) and \(p=q\).
In [16], the authors obtained existence, nonexistence, and multiplicity of solutions for a subcritical fractional (p, q)-Laplacian problem. In [5], the author proved an existence result for a critical fractional (p, q)-Laplacian problem, by using a concentration-compactness lemma and the mountain pass theorem. Multiplicity results for a class of fractional (p, q)-Laplacian problems in bounded domains and with critical nonlinearities have been established in [12]. The multiplicity of concentrating solutions for a fractional (p, q)-Laplacian problem of Schrödinger type has been recently demonstrated in [11]. For other contributions devoted to this class of problems, we refer to [1, 7, 9, 12, 25, 29].
To our knowledge, no results for Kirchhoff type problems driven by the fractional (p, q)-Laplacian operator \((-\Delta )^{s}_{p}+(-\Delta )^{s}_{q}\) appear in the current literature. Particularly motivated by this fact and the above-mentioned works, in this paper, we examine the multiplicity and concentration properties of solutions for (1.1). More precisely, our main result can be stated as follows:
Theorem 1.1
Assume that \((V_{1})\)-\((V_{2})\) and \((f_{1})\)-\((f_{4})\) hold. Then, for any \(\delta >0\) such that
there exists \(\varepsilon _{\delta }>0\) such that, for any \(\varepsilon \in (0, \varepsilon _{\delta })\), problem (1.1) has at least \(cat_{M_{\delta }}(M)\) positive solutions. Moreover, if \(u_{\varepsilon }\) denotes one of these solutions and \(x_{\varepsilon }\in \mathbb {R}^{N}\) is a global maximum point of \(u_{\varepsilon }\), then
The proof of Theorem 1.1 is based on the generalized Nehari manifold method, a penalization technique, and the Lusternik–Schnirelman category theory. Firstly, inspired by [18], we modify the nonlinearity f in a suitable way and we consider an auxiliary problem whose advantage with respect to (1.1) is that the corresponding energy functional \(\mathcal {J}_{\varepsilon }\) possesses a mountain pass geometry [3]. Moreover, an accurate analysis allows us to verify that \(\mathcal {J}_{\varepsilon }\) satisfies the Palais–Smale condition at any level \(c\in \mathbb {R}\) (\((PS)_{c}\) condition for short). Secondly, since we are interested in providing a multiplicity result for (1.1), and our nonlinearity f is only continuous, we implement the barycenter machinery and adapt some abstract critical point results found in [36]. This kind of argument also appears in [23] to analyze a Schrödinger–Kirchhoff elliptic equation, in [6] to handle various fractional Laplacian elliptic problems, and in [11] to deal with a fractional (p, q)-Schrödinger equation. However, with respect to [6, 11, 23], the mixture of Kirchhoff terms and two different nonhomogeneous nonlocal operators makes the study of (1.1) rather tough and an appropriate investigation will be done to circumvent some significant technical complications; see for instance the proofs of Lemmas 2.4, 2.5, 2.7 and Theorem 3.1. Finally, we show that the solutions of the modified problem are solutions to (1.1) for \(\varepsilon >0\) small enough, by using a Moser type iteration [32] and the Hölder regularity result in [11]. As far as we know, this is the first time that the penalization approach and the Lusternik–Schnirelman category theory are combined to treat fractional (p, q)-Laplacian problems like (1.1).
The paper is organized as follows. In Sect. 2, we collect some basic results for fractional Sobolev spaces and we introduce the modified problem. In Sect. 3, we tackle the limiting Kirchhoff problem. In Sect. 4, we present a multiplicity result for the modified problem. The last section is dedicated to the proof of Theorem 1.1.
2 The Modified Problem
2.1 Notations and Some Useful Lemmas
Let \(p\in [1, \infty ]\) and \(A\subset \mathbb {R}^{N}\) be a measurable set. We will denote by \(|\cdot |_{L^{p}(A)}\) the norm in \(L^{p}(A)\), and we will simply use the notation \(|\cdot |_{p}\) when \(A=\mathbb {R}^{N}\).
Let \(s\in (0, 1)\), \(p\in (1, \infty )\) and \(N>sp\). The fractional Sobolev space \(W^{s, p}(\mathbb {R}^{N})\) is defined by
which is a Banach space with the norm
For \(u, v\in W^{s,p}(\mathbb {R}^{N})\), we put
The following embeddings are well known in the literature.
Theorem 2.1
[19] Let \(s\in (0, 1)\), \(p\in (1, \infty )\) and \(N>sp\). Then, \(W^{s, p}(\mathbb {R}^{N})\) is continuously embedded in \(L^{t}(\mathbb {R}^{N})\) for any \(t\in [p, p^{*}_{s}]\) and compactly embedded in \(L^{t}_{loc}(\mathbb {R}^{N})\) for any \(t\in [1, p^{*}_{s})\).
For the reader’s convenience, we also recall some useful lemmas.
Lemma 2.1
[8] Let \(s\in (0, 1)\), \(p\in (1, \infty )\) and \(N>sp\). Let \(r\in [p, p^{*}_{s})\). If \(\{u_{n}\}_{n\in \mathbb {N}}\) is a bounded sequence in \(W^{s, p}(\mathbb {R}^{N})\) and if
where \(R>0\), then \(u_{n}\rightarrow 0\) in \(L^{t}(\mathbb {R}^{N})\) for all \(t\in (p, p^{*}_{s})\).
Lemma 2.2
[8] Let \(s\in (0, 1)\), \(t\in (1, \infty )\) and \(N>st\). Let \(\{u_{n}\}_{n\in \mathbb {N}}\subset W^{s, t}(\mathbb {R}^{N})\) be a bounded sequence in \(W^{s, t}(\mathbb {R}^{N})\), and let \(\phi \in C^{\infty }(\mathbb {R}^{N})\) be a function such that \(0\le \phi \le 1\) in \(\mathbb {R}^{N}\), \(\phi =0\) in \(B_{1}(0)\) and \(\phi =1\) in \(B^{c}_{2}(0)\). For each \(\rho >0\) let \(\phi _{\rho }(x)=\phi (\frac{x}{\rho })\). Then
Proof
The proof of this result can be found in [8], but here we give a more direct proof. Using the definition of \(\phi _{\rho }\), polar coordinates and the boundedness of \(\{u_{n}\}_{n\in \mathbb {N}}\) in \(W^{s, t}(\mathbb {R}^{N})\), we can see that
and letting first \(n\rightarrow \infty \) and then \(\rho \rightarrow \infty \), we get the thesis. \(\square \)
Let \(s\in (0, 1)\) and \(p, q\in (1, \infty )\). Consider the space
endowed with the norm
Since \(W^{s,r}(\mathbb {R}^{N})\), with \(r\in (1, \infty )\), is a separable reflexive Banach space (this can be proved by using the operator \(T: W^{s, r}(\mathbb {R}^{N})\rightarrow L^{r}(\mathbb {R}^{N})\times L^{r}(\mathbb {R}^{2N})\) defined by \(Tu=(u, (u(x)-u(y))|x-y|^{-\frac{N}{r}-s})\) and arguing as in the proof of Proposition 8.1 in [13]), we obtain that \(\mathcal {W}\) is also a separable reflexive Banach space.
For any \(\varepsilon >0\), we introduce the space
equipped with the norm
where
2.2 The Penalization Approach
We adapt in a suitable way the del Pino–Felmer penalization approach [18] to attack (1.1). First, we observe that the map \(t\mapsto \frac{f(t)}{t^{p-1}+t^{q-1}}\) is increasing in \((0, \infty )\). Indeed,
and noting that \(t\mapsto \frac{f(t)}{t^{2q-1}}\) is increasing in \((0, \infty )\) (by \((f_4)\)), and that \(t\mapsto \frac{t^{2q-1}}{t^{p-1}+t^{q-1}}\) is increasing in \((0, \infty )\) (because \(2q>p\)), we deduce the desired result.
Now, let us fix
and let \(a>0\) be such that
We define
and
where \(\chi _{A}\) denotes the characteristic function of \(A\subset \mathbb {R}^{N}\). By \((f_1)\)-\((f_4)\), we infer that \(g: \mathbb {R}^{N}\times \mathbb {R}\rightarrow \mathbb {R}\) is a Carathéodory function that fulfills the following assumptions:
- \((g_1)\):
-
\(\displaystyle \lim _{t\rightarrow 0} \frac{g(x,t)}{t^{2p-1}}=0\) uniformly with respect to \(x\in \mathbb {R}^{N}\),
- \((g_2)\):
-
\(g(x,t)\le f(t)\) for all \(x\in \mathbb {R}^{N}\) and \(t>0\),
- \((g_3)\):
-
(i) \(0< \vartheta G(x,t)\le g(x,t)t\) for all \(x\in \Lambda \) and \(t>0\), (ii) \(0\le pG(x,t)\le g(x,t)t\le \frac{V_{0}}{K} (t^{p}+t^{q})\) for all \(x\in \Lambda ^{c}\) and \(t>0\),
- \((g_4)\):
-
for each \(x\in \Lambda \), the function \(t\mapsto \displaystyle \frac{g(x,t)}{t^{p-1}+t^{q-1}}\) is increasing in \((0, \infty )\), and for each \(x\in \Lambda ^{c}\), the function \(t\mapsto \displaystyle \frac{g(x,t)}{t^{p-1}+t^{q-1}}\) is increasing in (0, a).
Let us introduce the auxiliary problem
We stress that if \(u_{\varepsilon }\) is a solution to (2.1) such that \(u_{\varepsilon }(x)\le a\) for all \(x\in \Lambda _{\varepsilon }^{c}\), where \(\Lambda _{\varepsilon }=\{x\in \mathbb {R}^{N} \, : \, \varepsilon x\in \Lambda \}\), then \(u_{\varepsilon }\) is also a solution to (1.1). Then we consider the functional \(\mathcal {J}_{\varepsilon }: \mathbb {X}_{\varepsilon }\rightarrow \mathbb {R}\) associated with (2.1), that is
Clearly, \(\mathcal {J}_{\varepsilon }\in C^{1}(\mathbb {X}_{\varepsilon }, \mathbb {R})\) and it holds
for any \(u, \varphi \in \mathbb {X}_{\varepsilon }\). We denote by \(\mathcal {N}_{\varepsilon }\) the Nehari manifold associated with \(\mathcal {J}_{\varepsilon }\), namely
and we set
Let \(\mathbb {X}_{\varepsilon }^{+}\) be the open set given by
and \(\mathbb {S}_{\varepsilon }^{+}=\mathbb {S}_{\varepsilon }\cap \mathbb {X}_{\varepsilon }^{+}\), where \(\mathbb {S}_{\varepsilon }=\{u\in \mathbb {X}_{\varepsilon }: \Vert u\Vert _{\mathbb {X}_{\varepsilon }}=1\}\) is the unit sphere in \(\mathbb {X}_{\varepsilon }\). Note that \(\mathbb {S}_{\varepsilon }^{+}\) is an incomplete \(C^{1,1}\)-manifold of codimension one. Hence, \(\mathbb {X}_{\varepsilon }=T_{u}\mathbb {S}_{\varepsilon }^{+}\oplus \mathbb {R}u\) for all \(u\in \mathbb {S}_{\varepsilon }^{+}\), where
The next lemma ensures that \(\mathcal {J}_{\varepsilon }\) possesses a mountain pass geometry [3].
Lemma 2.3
The functional \(\mathcal {J}_{\varepsilon }\) satisfies the following properties:
- (i):
-
There exist \(\alpha , \rho >0\) such that \(\mathcal {J}_{\varepsilon }(u) \ge \alpha \) for any \(u\in \mathbb {X}_{\varepsilon }\) with \(\Vert u\Vert _{\mathbb {X}_{\varepsilon }}= \rho \).
- (ii):
-
There exists \(e\in \mathbb {X}_{\varepsilon }\) such that \(\Vert e\Vert _{\mathbb {X}_{\varepsilon }}>\rho \) and \(\mathcal {J}_{\varepsilon }(e)<0\).
Proof
(i) Pick \(\zeta \in (0, V_{0})\). From \((g_{1})\), \((g_{2})\), \((f_{1})\), and \((f_{2})\), we can find \(C_{\zeta }>0\) such that
Taking into account the above estimate and applying Theorem 2.1, we have
Choosing \(\Vert u\Vert _{\mathbb {X}_{\varepsilon }}=\rho \in (0, 1)\) and recalling that \(1<p<q\), we get \(\Vert u\Vert _{V_{\varepsilon },p}<1\) and thus \(\Vert u\Vert ^{p}_{V_{\varepsilon },p}\ge \Vert u\Vert ^{q}_{V_{\varepsilon },p}\). Using
and Theorem 2.1, we can see that
Since \(\nu >q\), there exists \(\alpha >0\) such that \(\mathcal {J}_{\varepsilon }(u)\ge \alpha \) for any \(u\in \mathbb {X}_{\varepsilon }\) with \(\Vert u\Vert _{\mathbb {X}_{\varepsilon }}= \rho \).
(ii) It follows from \((f_{3})\) that, for some constants \(A, B>0\),
Then, for all \(u\in \mathbb {X}^{+}_{\varepsilon }\) and \(t>0\), we obtain
which combined with the fact that \(\vartheta>2q>2p\) implies that \(\mathcal {J}_{\varepsilon }(tu)\rightarrow -\infty \) as \(t\rightarrow \infty \). Hence, for large \(t>1\), we can take \(e=tu\) such that \(\Vert e\Vert _{\mathbb {X}_{\varepsilon }}>\rho \) and \(\mathcal {J}_{\varepsilon }(e)<0\). \(\square \)
In view of Lemma 2.3, we can define the minimax level
Exploiting a version of the mountain pass theorem without the Palais–Smale condition (see [37]), we can find a Palais–Smale sequence \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {X}_{\varepsilon }\) at the level \(c'_{\varepsilon }\) (\((PS)_{c'_{\varepsilon }}\) sequence for short).
Remark 2.1
We may always assume that any \((PS)_{c}\) sequence \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {X}_{\varepsilon }\) of \(\mathcal {J}_{\varepsilon }\) is nonnegative. Indeed, noting that \(\langle \mathcal {J}'_{\varepsilon }(u_{n}), u^{-}_{n}\rangle =o_{n}(1)\), where \(u_{n}^{-}=\min \{u_{n}, 0\}\), and using \(g(\varepsilon \cdot , t)=0\) for \(t\le 0\), we have
Recalling that
we arrive at
that is \(u_{n}^{-}\rightarrow 0\) in \(\mathbb {X}_{\varepsilon }\). Moreover, \(\{u_{n}^{+}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {X}_{\varepsilon }\). Since \([u_{n}]_{s, t}^{t}=[u^{+}_{n}]_{s, t}^{t}+o_{n}(1)\) and \(\Vert u_{n}\Vert _{V_{\varepsilon }, t}=\Vert u_{n}^{+}\Vert _{V_{\varepsilon }, t}+o_{n}(1)\) for \(t\in \{p, q\}\), we can easily deduce that \(\mathcal {J}_{\varepsilon }(u_{n})=\mathcal {J}_{\varepsilon }(u^{+}_{n})+o_{n}(1)\) and \(\mathcal {J}'_{\varepsilon }(u_{n})=\mathcal {J}'_{\varepsilon }(u^{+}_{n})+o_{n}(1)\). Therefore, \(\mathcal {J}_{\varepsilon }(u^{+}_{n})\rightarrow c\) and \(\mathcal {J}'_{\varepsilon }(u^{+}_{n})\rightarrow 0\).
The next two results are very important because they allow us to overcome the nondifferentiability of \(\mathcal {N}_{\varepsilon }\) and the incompleteness of \(\mathbb {S}_{\varepsilon }^{+}\).
Lemma 2.4
Assume that \((V_1)\)-\((V_2)\) and \((f_1)\)-\((f_4)\) hold. Then we have the following properties:
- (i):
-
For each \(u\in \mathbb {X}_{\varepsilon }^{+}\), let \(h_{u}:\mathbb {R}^{+}\rightarrow \mathbb {R}\) be defined by \(h_{u}(t)= \mathcal {J}_{\varepsilon }(tu)\). Then, there is a unique \(t_{u}>0\) such that
$$\begin{aligned}&h'_{u}(t)>0 \, \text{ for } \text{ all } t\in (0, t_{u}),\\&h'_{u}(t)<0 \, \text{ for } \text{ all } t\in (t_{u}, \infty ). \end{aligned}$$ - (ii):
-
There exists \(\tau >0\), independent of u, such that \(t_{u}\ge \tau \) for any \(u\in \mathbb {S}_{\varepsilon }^{+}\). Moreover, for each compact set \(\mathbb {K}\subset \mathbb {S}_{\varepsilon }^{+}\), there is a constant \(C_{\mathbb {K}}>0\) such that \(t_{u}\le C_{\mathbb {K}}\) for any \(u\in \mathbb {K}\).
- (iii):
-
The map \({\hat{m}}_{\varepsilon }: \mathbb {X}_{\varepsilon }^{+}\rightarrow \mathcal {N}_{\varepsilon }\) given by \({\hat{m}}_{\varepsilon }(u)= t_{u}u\) is continuous and \(m_{\varepsilon }= {\hat{m}}_{\varepsilon }|_{\mathbb {S}_{\varepsilon }^{+}}\) is a homeomorphism between \(\mathbb {S}_{\varepsilon }^{+}\) and \(\mathcal {N}_{\varepsilon }\). Moreover, \(m_{\varepsilon }^{-1}(u)=\frac{u}{\Vert u\Vert _{\mathbb {X}_{\varepsilon }}}\).
- (iv):
-
If there is a sequence \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {S}_{\varepsilon }^{+}\) such that \(\mathrm{dist}(u_{n}, \partial \mathbb {S}_{\varepsilon }^{+})\rightarrow 0\), then \(\Vert m_{\varepsilon }(u_{n})\Vert _{\mathbb {X}_{\varepsilon }}\rightarrow \infty \) and \(\mathcal {J}_{\varepsilon }(m_{\varepsilon }(u_{n}))\rightarrow \infty \).
Proof
(i) From the proof of Lemma 2.3, we derive that \(h_{u}(0)=0\), \(h_{u}(t)>0\) for \(t>0\) small enough and \(h_{u}(t)<0\) for \(t>0\) sufficiently large. Then there exists a global maximum point \(t_{u}>0\) for \(h_{u}\) in \([0, \infty )\) such that \(h_{u}'(t_{u})=0\) and \(t_{u}u \in \mathcal {N}_{\varepsilon }\). We claim that \(t_{u}>0\) is the unique number such that \(h'_{u}(t_{u})=0\). Arguing by contradiction, we assume that there exists \(t_{1}> t_{2}>0\) such that \(h_{u}'(t_{1})=h_{u}'(t_{2})=0\), or equivalently
Hence,
and
Using the definition of g, \((g_4)\) and \((f_4)\), we have
Multiplying both sides by \(\frac{(t_{1}t_{2})^{2q-p}}{t_{2}^{2q-p}- t_{1}^{2q-p}}<0\) (recall that \(2q>p\) and \(t_{1}>t_{2}\)), we get
where we used the fact that \((f_4)\) and our choice of the constant a produce
Therefore,
which is inconsistent with \(u\ne 0\) and \(K>1\).
(ii) Fix \(u\in \mathbb {S}_{\varepsilon }^{+}\). By (i), there exists \(t_{u}>0\) such that \(h_{u}'(t_{u})=0\), that is
Pick \(\xi >0\). From \((g_1)\)-\((g_2)\) and Theorem 2.1, we derive
Choosing \(\xi >0\) sufficiently small, we have
Now, if \(t_{u}\le 1\), then \(t_{u}^{q-1}\le t_{u}^{p-1}\), and using the facts that \(1=\Vert u\Vert _{\mathbb {X}_{\varepsilon }}\ge \Vert u\Vert _{V_{\varepsilon },p}\) and that \(q>p\) imply that \(\Vert u\Vert _{V_{\varepsilon },p}^{p}\ge \Vert u\Vert ^{q}_{V_{\varepsilon },p}\), we get
Since \(\nu >q\), there exists \(\tau >0\), independent of u, such that \(t_{u}\ge \tau \).
When \(t_{u}>1\), then \(t_{u}^{q-1}>t_{u}^{p-1}\), and observing that \(1=\Vert u\Vert _{\mathbb {X}_{\varepsilon }}\ge \Vert u\Vert _{V_{\varepsilon },p}\) and that \(q>p\) yield \(\Vert u\Vert _{V_{\varepsilon },p}^{p}\ge \Vert u\Vert ^{q}_{V_{\varepsilon },p}\), we obtain
As \(\nu>q>p\), we can find \(\tau >0\), independent of u, such that \(t_{u}\ge \tau \).
Now, let \(\mathbb {K}\subset \mathbb {S}_{\varepsilon }^{+}\) be a compact set, and suppose, by contradiction, that there exists \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {K}\) such that \(t_{n}=t_{u_{n}}\rightarrow \infty \). Since \(\mathbb {K}\) is compact, there is \(u\in \mathbb {K}\) such that \(u_{n}\rightarrow u\) in \(\mathbb {X}_{\varepsilon }\). By the proof of (ii) of Lemma 2.3, we see that
On the other hand, if \(v\in \mathcal {N}_{\varepsilon }\), by \(\langle \mathcal {J}_{\varepsilon }'(v), v \rangle =0\) and \((g_{3})\), we get
Taking \(v_{n}=t_{u_{n}}u_{n}\in \mathcal {N}_{\varepsilon }\) in the above inequality, we arrive at
Since \(\Vert v_{n}\Vert _{\mathbb {X}_{\varepsilon }}=t_{n}\rightarrow \infty \) and \(\Vert v_{n}\Vert _{\mathbb {X}_{\varepsilon }}=\Vert v_{n}\Vert _{V_{\varepsilon },p}+\Vert v_{n}\Vert _{V_{\varepsilon },q}\), we can use (2.3) to reach a contradiction.
(iii) First we note that \({\hat{m}}_{\varepsilon }\), \(m_{\varepsilon }\) and \(m_{\varepsilon }^{-1}\) are well defined. Indeed, by (i), for each \(u\in \mathbb {X}_{\varepsilon }^{+}\), there is a unique \(m_{\varepsilon }(u)\in \mathcal {N}_{\varepsilon }\). On the other hand, if \(u\in \mathcal {N}_{\varepsilon }\) then \(u\in \mathbb {X}^{+}_{\varepsilon }\). Otherwise, we would have
and by \((g_3)\)-(ii) we infer that
which gives a contradiction because \(K>1\) and \(u\ne 0\). Consequently, \(m_{\varepsilon }^{-1}(u)= \frac{u}{\Vert u\Vert _{\mathbb {X}_{\varepsilon }}}\in \mathbb {S}_{\varepsilon }^{+}\) is well defined and continuous. Since
we deduce that \(m_{\varepsilon }\) is a bijection. Now we prove that \({\hat{m}}_{\varepsilon }: \mathbb {X}_{\varepsilon }^{+}\rightarrow \mathcal {N}_{\varepsilon }\) is continuous. Let \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {X}_{\varepsilon }^{+}\) and \(u\in \mathbb {X}^{+}_{\varepsilon }\) be such that \(u_{n}\rightarrow u\) in \(\mathbb {X}_{\varepsilon }\). By (ii), there exists \(t_{0}>0\) such that \(t_{n}=t_{\frac{u_{n}}{\Vert u_{n}\Vert _{\mathbb {X}_{\varepsilon }}}}\rightarrow t_{0}\). Using \(t_{n}\frac{u_{n}}{\Vert u_{n}\Vert _{\mathbb {X}_{\varepsilon }}}\in \mathcal {N}_{\varepsilon }\), that is
and letting \(n\rightarrow \infty \) we find
which implies that \(t_{0}\frac{u}{\Vert u\Vert _{\mathbb {X}_{\varepsilon }}}\in \mathcal {N}_{\varepsilon }\). From (i), \(t_{\frac{u}{\Vert u\Vert _{\mathbb {X}_{\varepsilon }}}}= t_{0}\) and this assures that \({\hat{m}}_{\varepsilon }(u_{n})\rightarrow {\hat{m}}_{\varepsilon }(u)\) in \(\mathbb {X}_{\varepsilon }^{+}\). Therefore, \({\hat{m}}_{\varepsilon }\) and \(m_{\varepsilon }\) are continuous functions.
(iv) Let \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {S}_{\varepsilon }^{+}\) be a sequence such that \(\mathrm{dist}(u_{n}, \partial \mathbb {S}_{\varepsilon }^{+})\rightarrow 0\). Then, for each \(v\in \partial \mathbb {S}_{\varepsilon }^{+}\) and \(n\in \mathbb {N}\), we have \(u_{n}^{+}\le |u_{n}-v|\) a.e. in \(\Lambda _{\varepsilon }\). Hence, by \((V_1)\), \((V_2)\) and Theorem 2.1, we can see that for each \(r\in [p, q^{*}_{s}]\), there exists \(C_{r}>0\) such that
Combining \((g_{1})\), \((g_{2})\), \((g_{3})\)-(ii) and \(q>p\), we get, for all \(t>0\),
Thus, for all \(t>0\),
Now, we recall that \(K>\frac{q}{p}>1\), and that \(1=\Vert u_{n}\Vert _{\mathbb {X}_{\varepsilon }}\ge \Vert u_{n}\Vert _{V_{\varepsilon },p}\) implies \(\Vert u_{n}\Vert ^{p}_{V_{\varepsilon },p}\ge \Vert u_{n}\Vert ^{q}_{V_{\varepsilon },p}\). Then, for all \(t>1\), we obtain
By using the definition of \(m_{\varepsilon }(u_{n})\), (2.4) and (2.5), we have
Letting \(t\rightarrow \infty \) we deduce that \(\mathcal {J}_{\varepsilon }(m_{\varepsilon }(u_{n}))\rightarrow \infty \) as \(n\rightarrow \infty \). Furthermore, by the definition of \(\mathcal {J}_{\varepsilon }\), we can see that for all \(n\in \mathbb {N}\)
and this yields \(\Vert m_{\varepsilon }(u_{n})\Vert _{\mathbb {X}_{\varepsilon }}\rightarrow \infty \) as \(n\rightarrow \infty \). \(\square \)
Remark 2.2
There exists \(\kappa >0\), independent of \(\varepsilon \), such that \(\Vert u\Vert _{\mathbb {X}_{\varepsilon }}\ge \kappa \) for all \(u\in \mathcal {N}_{\varepsilon }\). Indeed, if \(u\in \mathcal {N}_{\varepsilon }\), we can use \((g_1)\), \((g_2)\) and Theorem 2.1 to see that
Choosing \(\zeta \in (0, V_{0})\), we get \(\Vert u\Vert _{V_{\varepsilon },q}\ge \kappa =(C'_{\zeta })^{-\frac{1}{q^{*}_{s}-q}}\) and thus \(\Vert u\Vert _{\mathbb {X}_{\varepsilon }}\ge \Vert u\Vert _{V_{\varepsilon },q}\ge \kappa \).
Now we define the maps
by setting \({\hat{\psi }}_{\varepsilon }(u)= \mathcal {J}_{\varepsilon }({\hat{m}}_{\varepsilon }(u))\) and \(\psi _{\varepsilon }={\hat{\psi }}_{\varepsilon }|_{\mathbb {S}_{\varepsilon }^{+}}\). From Lemma 2.4 and arguing as in the proofs of Proposition 9 and Corollary 10 in [36], we may obtain the result below.
Proposition 2.1
Assume that \((V_{1})\)-\((V_{2})\) and \((f_{1})\)-\((f_{4})\) hold. Then we have the following properties:
- (a):
-
\({\hat{\psi }}_{\varepsilon } \in C^{1}(\mathbb {X}_{\varepsilon }^{+}, \mathbb {R})\) and
$$\begin{aligned} \langle {\hat{\psi }}_{\varepsilon }'(u), v\rangle = \frac{\Vert {\hat{m}}_{\varepsilon }(u)\Vert _{\mathbb {X}_{\varepsilon }}}{\Vert u\Vert _{\mathbb {X}_{\varepsilon }}} \langle \mathcal {J}_{\varepsilon }'({\hat{m}}_{\varepsilon }(u)), v\rangle \quad {{ \text{ for } \text{ all } u\in \mathbb {X}_{\varepsilon }^{+} \text{ and } v\in \mathbb {X}_{\varepsilon }.}} \end{aligned}$$ - (b):
-
\(\psi _{\varepsilon } \in C^{1}(\mathbb {S}_{\varepsilon }^{+}, \mathbb {R})\) and
$$\begin{aligned} \langle \psi _{\varepsilon }'(u), v \rangle = \Vert m_{\varepsilon }(u)\Vert _{\mathbb {X}_{\varepsilon }} \langle \mathcal {J}_{\varepsilon }'(m_{\varepsilon }(u)), v\rangle \quad {{ \text{ for } \text{ all } v\in T_{u}\mathbb {S}_{\varepsilon }^{+}.}} \end{aligned}$$ - (c):
-
If \(\{u_{n}\}_{n\in \mathbb {N}}\) is a \((PS)_{c}\) sequence for \(\psi _{\varepsilon }\), then \(\{m_{\varepsilon }(u_{n})\}_{n\in \mathbb {N}}\) is a \((PS)_{c}\) sequence for \(\mathcal {J}_{\varepsilon }\). If \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathcal {N}_{\varepsilon }\) is a bounded \((PS)_{c}\) sequence for \(\mathcal {J}_{\varepsilon }\), then \(\{m_{\varepsilon }^{-1}(u_{n})\}_{n\in \mathbb {N}}\) is a \((PS)_{c}\) sequence for \(\psi _{\varepsilon }\).
- (d):
-
u is a critical point of \(\psi _{\varepsilon }\) if and only if \(m_{\varepsilon }(u)\) is a critical point for \(\mathcal {J}_{\varepsilon }\). Moreover, the corresponding critical values coincide and
$$\begin{aligned} \inf _{u\in \mathbb {S}_{\varepsilon }^{+}} \psi _{\varepsilon }(u)= \inf _{u\in \mathcal {N}_{\varepsilon }} \mathcal {J}_{\varepsilon }(u). \end{aligned}$$
Remark 2.3
As in [36], we have the following minimax characterization of the infimum of \(\mathcal {J}_{\varepsilon }\) over \(\mathcal {N}_{\varepsilon }\):
Moreover, arguing as in [37], we can prove that \(c_{\varepsilon }=c'_{\varepsilon }\).
In the remainder of this section, we check that the modified functional satisfies the Palais–Smale condition. We start by showing the boundedness of Palais–Smale sequences.
Lemma 2.5
Let \(c\in \mathbb {R}\) and let \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {X}_{\varepsilon }\) be a \((PS)_{c}\) sequence for \(\mathcal {J}_{\varepsilon }\). Then \(\{u_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {X}_{\varepsilon }\).
Proof
Using \((g_{3})\), \(q>p\) and \(\vartheta >2q\), we see that
where \({\tilde{C}}=\left[ \left( \frac{1}{q}- \frac{1}{\vartheta }\right) - \left( \frac{1}{p}- \frac{1}{\vartheta }\right) \frac{1}{K} \right] >0\) since \(K>\left( \frac{\vartheta -p}{\vartheta -q}\right) \frac{q}{p}\). Suppose, by contradiction, that \(\Vert u_{n}\Vert _{\mathbb {X}_{\varepsilon }}\rightarrow \infty \). Then we discuss the following cases:
Case 1 \(\Vert u_{n}\Vert _{V_{\varepsilon },p}\rightarrow \infty \) and \(\Vert u_{n}\Vert _{V_{\varepsilon }, q}\rightarrow \infty \).
For n large, we get \(\Vert u_{n}\Vert _{V_{\varepsilon }, q}^{q-p}\ge 1\), that is \(\Vert u_{n}\Vert _{V_{\varepsilon }, q}^{q}\ge \Vert u_{n}\Vert _{V_{\varepsilon }, q}^{p}\). Therefore, from (2.6),
which is a contradiction.
Case 2 \(\Vert u_{n}\Vert _{V_{\varepsilon },p}\rightarrow \infty \) and \(\Vert u_{n}\Vert _{V_{\varepsilon }, q}\) is bounded.
We have
and thus
Since \(p>1\) and letting \(n\rightarrow \infty \), we find \(0< {\tilde{C}}\le 0\), that is a contradiction.
Case 3 \(\Vert u_{n}\Vert _{V_{\varepsilon },q}\rightarrow \infty \) and \(\Vert u_{n}\Vert _{V_{\varepsilon }, p}\) is bounded.
This case is similar to the case 2, so we skip the details.
In conclusion, \(\{u_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {X}_{\varepsilon }\). \(\square \)
Lemma 2.6
Let \(c\in \mathbb {R}\) and let \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {X}_{\varepsilon }\) be a \((PS)_{c}\) sequence for \(\mathcal {J}_{\varepsilon }\). Then for any \(\eta >0\) there exists \(R=R(\eta )>0\) such that
Proof
Let \(\psi \in C^{\infty }(\mathbb {R}^{N})\) be such that \(0\le \psi \le 1\), \(\psi =0\) in \(B_{\frac{1}{2}}(0)\), \(\psi _{R}=1\) in \(B_{1}^{c}(0)\), and \(|\nabla \psi |_{\infty }\le C\), for some \(C>0\). For \(R>0\), define \(\psi _{R}(x)=\psi (\frac{x}{R})\). Then, \(0\le \psi _{R}\le 1\), \(\psi _{R}=0\) in \(B_{\frac{R}{2}}(0)\), \(\psi _{R}=1\) in \(B_{R}^{c}(0)\), and \(|\nabla \psi _{R}|_{\infty }\le \frac{C}{R}\) with \(C>0\) independent of R. Since \(\{\psi _{R}u_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {X}_{\varepsilon }\), it holds \(\langle \mathcal {J}_{\varepsilon }'(u_{n}), \psi _{R}u_{n}\rangle =o_{n}(1)\), that is
Pick \(R>0\) such that \(\Lambda _{\varepsilon }\subset B_{\frac{R}{2}}(0)\). By the definition of \(\psi _{R}\) and using \((g_{3})\)-(ii), we obtain that
Now, from the Hölder inequality and the boundedness of \(\{u_{n}\}_{n\in \mathbb {N}}\) in \(\mathbb {X}_{\varepsilon }\), we get, for \(t\in \{p, q\}\),
An inspection of the proof of Lemma 2.2 shows that, for \(t\in \{p, q\}\),
Combining (2.8), (2.9) and (2.10), and recalling the definition of \(\psi _{R}\), for some \(C>0\), we can take \(R=R(\eta )>(\frac{C}{\eta })^{\frac{1}{s}}\) so that (2.7) is satisfied. \(\square \)
Since we are working with a Kirchhoff type problem, the next lemma will be fundamental to obtain the strong convergence of bounded Palais–Smale sequences.
Lemma 2.7
Let \(c\in \mathbb {R}\) and let \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {X}_{\varepsilon }\) be a \((PS)_{c}\) sequence for \(\mathcal {J}_{\varepsilon }\). Let \(R>0\). Then
Proof
Let \(\eta \in C^{\infty }(\mathbb {R}^{N})\) be such that \(0\le \eta \le 1\), \(\eta =1\) in \(B_{1}(0)\), \(\eta =0\) in \(B^{c}_{2}(0)\) and \(|\nabla \eta |_{\infty }\le 2\). For \(\rho >0\), put \(\eta _{\rho }(x)=\eta (\frac{x}{\rho })\). Then \(0\le \eta _{\rho }\le 1\), \(\eta =1\) in \(B_{\rho }(0)\), \(\eta =0\) in \(B^{c}_{2\rho }(0)\) and \(|\nabla \eta |_{\infty }\le \frac{2}{\rho }\). Since \(\{u_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {X}_{\varepsilon }\) (by Lemma 2.5), we may suppose that \([u_{n}]_{s, p}^{p}\rightarrow \ell _{p}\) and \([u_{n}]_{s, q}^{q}\rightarrow \ell _{q}\) as \(n\rightarrow \infty \).
Fix \(R>0\) and take \(\rho >R\). We recall the following well-known elementary inequalities [35]: for any \(\xi , \eta \in \mathbb {R}^{N}\) we have
for some constants \(c_{1}, c_{2}>0\). Note that, when \(1<r<2\), using (2.13) and the elementary inequality
we deduce that there exists \(c_{3}>0\) such that, for any \(\xi , \eta \in \mathbb {R}^{N}\), the following relation is satisfied
For \(t\in \{p, q\}\) and \(n\in \mathbb {N}\), we set
Note that, for \(t\in \{p, q\}\) and \(n\in \mathbb {N}\), we have
Define
and
Then it holds
Since
and \(\{u_{n}\eta _{\rho }\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {X}_{\varepsilon }\), we see that \(\langle \mathcal {J}'_{\varepsilon }(u_{n}), u_{n}\eta _{\rho }\rangle =o_{n}(1)\). Using the Hölder inequality and the boundedness of \(\{u_{n}\}_{n\in \mathbb {N}}\) in \(\mathbb {X}_{\varepsilon }\), we have
which combined with Lemma in 2.2 (applied with \(\phi _{\rho }=1-\eta _{\rho }\)) yields
Consequently, recalling that \([u_{n}]_{s, t}^{t}\rightarrow \ell _{t}\) for \(t\in \{p, q\}\), we get
We also observe that
and using \(\langle \mathcal {J}'_{\varepsilon }(u_{n}), u\eta _{\rho }\rangle =o_{n}(1)\), we can argue as before to achieve that
Next we prove that
From the weak convergence, we have
Notice that, for \(t\in \{p, q\}\),
By \(u_{n}\rightharpoonup u\) in \(\mathbb {X}_{\varepsilon }\) and \([u_{n}]^{t}_{s, t}\rightarrow \ell _{t}\) for \(t\in \{p, q\}\), we deduce that
On the other hand, using the boundedness of \(\{u_{n}\}_{n\in \mathbb {N}}\) in \(\mathbb {X}_{\varepsilon }\) and applying the Hölder inequality, we see that
Since \(\eta _{\rho }\rightarrow 1\) a.e. in \(\mathbb {R}^N\) as \(\rho \rightarrow \infty \) and \(u\in W^{s, t}(\mathbb {R}^{N})\), it follows from the dominated convergence theorem that
The validity of (2.18) is now an immediate consequence of the definition of \(I^{2}_{n, \rho }\) and of the above relations.
Finally, exploiting \(u_{n}\rightarrow u\) in \(L^{r}_{loc}(\mathbb {R}^{N})\) for all \(r\in [p, q^{*}_{s})\) and the growth assumptions on g, we obtain
Combining (2.15) with (2.16)–(2.19), we find
whence
Assume first that \(t\ge 2\). Using (2.12), the boundedness of \(\{u_{n}\}_{n\in \mathbb {N}}\) in \(\mathbb {X}_{\varepsilon }\) and (2.20), we get
In a similar fashion,
Suppose now that \(1<t<2\). From (2.14), the boundedness of \(\{u_{n}\}_{n\in \mathbb {N}}\) in \(\mathbb {X}_{\varepsilon }\), Hölder’s inequality, and (2.20), we derive
Analogously,
Consequently, for \(t\in \{p, q\}\),
which implies (2.11). This completes the proof. \(\square \)
Now we are ready to prove the following compactness result.
Lemma 2.8
\(\mathcal {J}_{\varepsilon }\) satisfies the \((PS)_{c}\) condition at any level \(c\in \mathbb {R}\).
Proof
Let \(c\in \mathbb {R}\) and let \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {X}_{\varepsilon }\) be a \((PS)_{c}\) sequence for \(\mathcal {J}_{\varepsilon }\). By Lemma 2.5, we know that \(\{u_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {X}_{\varepsilon }\). Up to a subsequence, we may suppose that \(u_{n}\rightharpoonup u\) in \(\mathbb {X}_{\varepsilon }\) and \(u_{n}\rightarrow u\) in \(L^{r}_{loc}(\mathbb {R}^{N})\) for all \(r\in [1, q^{*}_{s})\). In view of Lemma 2.6, for each \(\eta >0\), there exists \(R=R(\eta )>(\frac{C}{\eta })^{\frac{1}{s}}\), with \(C>0\) independent of \(\eta \), such that (2.11) holds. This fact combined with Lemma 2.7 yields
Letting \(\eta \rightarrow 0\), we have \(R\rightarrow \infty \) and then
whence
Since the Brezis–Lieb lemma [14] gives
we infer that
This last fact implies that \(u_{n}\rightarrow u\) in \(\mathbb {X}_{\varepsilon }\) as \(n\rightarrow \infty \). \(\square \)
Corollary 2.1
The functional \(\psi _{\varepsilon }\) satisfies the \((PS)_{c}\) condition on \(\mathbb {S}_{\varepsilon }^{+}\) at any level \(c\in \mathbb {R}\).
Proof
Let \(c\in \mathbb {R}\) and let \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {S}^{+}_{\varepsilon }\) be a \((PS)_{c}\) sequence for \(\psi _{\varepsilon }\). Hence,
By Proposition 2.1-(c), we know that \(\{m_{\varepsilon }(u_{n})\}_{n\in \mathbb {N}}\subset \mathbb {X}_{\varepsilon }\) is a \((PS)_{c}\) sequence for \(\mathcal {J}_{\varepsilon }\). Then, by Lemma 2.8, we deduce that \(\mathcal {J}_{\varepsilon }\) satisfies the \((PS)_{c}\) condition in \(\mathbb {X}_{\varepsilon }\), and thus there exists \(u\in \mathbb {S}_{\varepsilon }^{+}\) such that, up to a subsequence,
By Lemma 2.4-(iii), we conclude that \(u_{n}\rightarrow u\) in \(\mathbb {S}_{\varepsilon }^{+}\). \(\square \)
We conclude this section by establishing an existence result for (2.1).
Theorem 2.2
Assume that \((V_1)\)–\((V_2)\) and \((f_1)\)–\((f_4)\) hold. Then, for all \(\varepsilon >0\), there exists a positive ground state solution to (2.1).
Proof
In light of Lemmas 2.3 and 2.8, we can apply the mountain pass theorem [3] to see that for all \(\varepsilon >0\) there exists a nontrivial critical point \(u_{\varepsilon }\in \mathbb {X}_{\varepsilon }\) of \(\mathcal {J}_{\varepsilon }\). By Remark 2.3, we deduce that \(u_{\varepsilon }\) is a ground state solution to (2.1). Using \(\langle \mathcal {J}'_{\varepsilon }(u_{\varepsilon }), u^{-}_{\varepsilon }\rangle =0\), where \(u^{-}=\min \{u, 0\}\), \((V_1)\), \(g(\cdot , t)=0\) for \(t\le 0\) and (2.2), we have
which gives \(u^{-}_{\varepsilon }=0\), that is \(u_{\varepsilon }\ge 0\) in \(\mathbb {R}^{N}\). Arguing as in the proof of Lemma 5.1 below (see also Lemma 4.1 and Theorem 2.2 in [11]), we obtain that \(u_{\varepsilon }\in L^{\infty }(\mathbb {R}^{N})\cap C(\mathbb {R}^{N})\), and applying the strong maximum principle [7] we infer that \(u_{\varepsilon }>0\) in \(\mathbb {R}^{N}\). \(\square \)
3 The Limiting Kirchhoff Problem
Since we are interested in providing a multiplicity result for the auxiliary problem (2.1), it is important to analyze the limiting problem associated with (1.1), namely
Let \(\mathbb {Y}_{V_{0}}=W^{s, p}(\mathbb {R}^{N})\cap W^{s, q}(\mathbb {R}^{N})\) equipped with the norm
where
The energy functional \(\mathcal {L}_{V_{0}}: \mathbb {Y}_{V_{0}}\rightarrow \mathbb {R}\) associated with (3.1) is given by
Standard arguments show that \(\mathcal {L}_{V_{0}}\in C^{1}(\mathbb {Y}_{V_{0}}, \mathbb {R})\) and that
for any \(u, \varphi \in \mathbb {Y}_{V_{0}}\). We also consider the Nehari manifold \(\mathcal {M}_{V_{0}}\) associated with \(\mathcal {L}_{V_{0}}\), that is
and we set \(d_{V_{0}}=\inf _{u\in \mathcal {M}_{V_{0}}} \mathcal {L}_{V_{0}}(u)\). Now we define
and \(\mathbb {S}_{V_{0}}^{+}= \mathbb {S}_{V_{0}}\cap \mathbb {Y}_{V_{0}}^{+}\), where \(\mathbb {S}_{V_{0}}\) is the unit sphere of \(\mathbb {Y}_{V_{0}}\). As in Sect. 2, \(\mathbb {S}_{V_{0}}^{+}\) is an incomplete \(C^{1,1}\)-manifold of codimension one and contained in \(\mathbb {Y}_{V_{0}}^{+}\). Thus, \(\mathbb {Y}_{V_{0}}= T_{u}\mathbb {S}_{V_{0}}^{+}\oplus \mathbb {R}u\) for each \(u\in \mathbb {S}_{V_{0}}^{+}\), where
In the sequel, we state without proofs the following results which can be obtained arguing as in Sect. 2.
Lemma 3.1
Assume that \((f_1)\)–\((f_4)\) hold. Then we have the following properties:
- (i):
-
For each \(u\in \mathbb {Y}_{V_{0}}^{+}\), let \(h:\mathbb {R}^{+}\rightarrow \mathbb {R}\) be defined by \(h_{u}(t)= \mathcal {L}_{V_{0}}(tu)\). Then, there is a unique \(t_{u}>0\) such that
$$\begin{aligned}&h'_{u}(t)>0 \, \text{ for } \text{ all } t\in (0, t_{u}),\\&h'_{u}(t)<0 \, \text{ for } \text{ all } t\in (t_{u}, \infty ). \end{aligned}$$ - (ii):
-
There exists \(\tau >0\), independent of u, such that \(t_{u}\ge \tau \) for any \(u\in \mathbb {S}_{V_{0}}^{+}\). Moreover, for each compact set \(\mathbb {K}\subset \mathbb {S}_{V_{0}}^{+}\), there is a constant \(C_{\mathbb {K}}>0\) such that \(t_{u}\le C_{\mathbb {K}}\) for any \(u\in \mathbb {K}\).
- (iii):
-
The map \({\hat{m}}_{V_{0}}: \mathbb {Y}_{V_{0}}^{+}\rightarrow \mathcal {M}_{V_{0}}\) given by \({\hat{m}}_{V_{0}}(u)= t_{u}u\) is continuous and \(m_{V_{0}}= {\hat{m}}_{V_{0}}|_{\mathbb {S}_{V_{0}}^{+}}\) is a homeomorphism between \(\mathbb {S}_{V_{0}}^{+}\) and \(\mathcal {M}_{V_{0}}\). Moreover, \(m_{V_{0}}^{-1}(u)=\frac{u}{\Vert u\Vert _{\mathbb {Y}_{V_{0}}}}\).
- (iv):
-
If there is a sequence \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {S}_{V_{0}}^{+}\) such that \(\mathrm{dist}(u_{n}, \partial \mathbb {S}_{V_{0}}^{+})\rightarrow 0\), then \(\Vert m_{V_{0}}(u_{n})\Vert _{\mathbb {Y}_{V_{0}}}\rightarrow \infty \) and \(\mathcal {L}_{V_{0}}(m_{V_{0}}(u_{n}))\rightarrow \infty \).
Let us consider the maps
defined by \({\hat{\psi }}_{V_{0}}(u)= \mathcal {L}_{V_{0}}({\hat{m}}_{V_{0}}(u))\) and \(\psi _{V_{0}}={\hat{\psi }}_{V_{0}}|_{\mathbb {S}_{V_{0}}^{+}}\).
Proposition 3.1
Assume that \((f_{1})\)-\((f_{4})\) hold. Then we have the following properties:
- (a):
-
\({\hat{\psi }}_{V_{0}} \in C^{1}(\mathbb {Y}_{V_{0}}^{+}, \mathbb {R})\) and
$$\begin{aligned} \langle {\hat{\psi }}_{V_{0}}'(u), v\rangle = \frac{\Vert {\hat{m}}_{V_{0}}(u)\Vert _{\mathbb {Y}_{V_{0}}}}{\Vert u\Vert _{\mathbb {Y}_{V_{0}}}} \langle \mathcal {L}_{V_{0}}'({\hat{m}}_{V_{0}}(u)), v\rangle \quad \text{ for } \text{ all } u\in \mathbb {Y}_{V_{0}}^{+} \text{ and } v\in \mathbb {Y}_{V_{0}}. \end{aligned}$$ - (b):
-
\(\psi _{V_{0}} \in C^{1}(\mathbb {S}_{V_{0}}^{+}, \mathbb {R})\) and
$$\begin{aligned} \langle \psi _{V_{0}}'(u), v \rangle = \Vert m_{V_{0}}(u)\Vert _{\mathbb {Y}_{V_{0}}} \langle \mathcal {L}_{V_{0}}'(m_{V_{0}}(u)), v\rangle \quad \text{ for } \text{ all } v\in T_{u}\mathbb {S}_{V_{0}}^{+}. \end{aligned}$$ - (c):
-
If \(\{u_{n}\}_{n\in \mathbb {N}}\) is a \((PS)_{d}\) sequence for \(\psi _{V_{0}}\), then \(\{m_{V_{0}}(u_{n})\}_{n\in \mathbb {N}}\) is a \((PS)_{d}\) sequence for \(\mathcal {L}_{V_{0}}\). If \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathcal {M}_{V_{0}}\) is a bounded \((PS)_{d}\) sequence for \(\mathcal {L}_{V_{0}}\), then \(\{m_{V_{0}}^{-1}(u_{n})\}_{n\in \mathbb {N}}\) is a \((PS)_{d}\) sequence for \(\psi _{V_{0}}\).
- (d):
-
u is a critical point of \(\psi _{V_{0}}\) if and only if \(m_{V_{0}}(u)\) is a nontrivial critical point for \(\mathcal {L}_{V_{0}}\). Moreover, the corresponding critical values coincide and
$$\begin{aligned} \inf _{u\in \mathbb {S}_{V_{0}}^{+}} \psi _{V_{0}}(u)= \inf _{u\in \mathcal {M}_{V_{0}}} \mathcal {L}_{V_{0}}(u). \end{aligned}$$
Remark 3.1
As in Sect. 2, we have the following minimax characterization of the infimum of \(\mathcal {L}_{V_{0}}\) over \(\mathcal {M}_{V_{0}}\):
The lemma below allows us to assume that the weak limit of a \((PS)_{d_{V_{0}}}\) sequence of \(\mathcal {L}_{V_{0}}\) is nontrivial.
Lemma 3.2
Let \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {Y}_{V_{0}}\) be a \((PS)_{d_{V_{0}}}\) sequence for \(\mathcal {L}_{V_{0}}\) such that \(u_{n}\rightharpoonup 0\) in \(\mathbb {Y}_{V_{0}}\). Then we have either
- (a):
-
\(u_{n}\rightarrow 0\) in \(\mathbb {Y}_{V_{0}}\), or
- (b):
-
there exists a sequence \(\{y_{n}\}_{n\in \mathbb {N}}\subset \mathbb {R}^{N}\) and constants \(R, \beta >0\) such that
$$\begin{aligned} \liminf _{n\rightarrow \infty }\int _{B_{R}(y_{n})} |u_{n}|^{q} \, dx \ge \beta . \end{aligned}$$
Proof
Suppose that (b) is false. Since \(\{u_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {Y}_{V_{0}}\), we can use Lemma 2.1 to see that
Moreover, by \((f_{1})\) and \((f_{2})\), we have that
Since \(\langle \mathcal {L}'_{V_{0}}(u_{n}), u_{n}\rangle =o_{n}(1)\), we get
that is \(\Vert u_{n}\Vert _{\mathbb {Y}_{V_{0}}}\rightarrow 0\) as \(n\rightarrow \infty \). Then, (a) is true. \(\square \)
Remark 3.2
As it has been mentioned earlier, if \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {Y}_{V_{0}}\) is a \((PS)_{d_{V_{0}}}\) sequence for \(\mathcal {L}_{V_{0}}\) such that \(u_{n}\rightharpoonup u\) in \(\mathbb {Y}_{V_{0}}\), then we may assume that \(u\ne 0\). Otherwise, if \(u_{n}\rightharpoonup 0\) in \(\mathbb {Y}_{V_{0}}\) and, if \(u_{n}\nrightarrow 0\) in \(\mathbb {Y}_{V_{0}}\), it follows from Lemma 3.2 that there are \(\{y_{n}\}_{n\in \mathbb {N}}\subset \mathbb {R}^{N}\) and \(R, \beta >0\) such that
Define \(v_{n}(x)=u_{n}(x+y_{n})\). Then, using the invariance of \(\mathbb {R}^N\) by translation, we see that \(\{v_{n}\}_{n\in \mathbb {N}}\) is a bounded \((PS)_{d_{V_{0}}}\) sequence for \(\mathcal {L}_{V_{0}}\) such that \(v_{n}\rightharpoonup v\) in \(\mathbb {Y}_{V_{0}}\) with \(v\ne 0\).
In the following lemma, we obtain a positive ground state solution for the autonomous problem (3.1).
Theorem 3.1
Let \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {Y}_{V_{0}}\) be a \((PS)_{d_{V_{0}}}\) sequence of \(\mathcal {L}_{V_{0}}\). Then there exists \(u\in \mathbb {Y}_{V_{0}}\setminus \{0\}\), with \(u\ge 0\), such that, up to a subsequence, \(u_{n}\rightarrow u\) in \(\mathbb {Y}_{V_{0}}\). Moreover, u is a positive ground state solution to (3.1).
Proof
Proceeding as in the proof of Lemma 2.5, we can verify that \(\{u_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {Y}_{V_{0}}\). By passing to a subsequence if necessary, we may assume that
From Remark 3.2, we may suppose that \(u\ne 0\). Moreover, we may assume that \([u_{n}]^{p}_{s, p}\rightarrow t_{1}\) and \([u_{n}]^{q}_{s, q}\rightarrow t_{2}\). Our aim is to prove that \([u_{n}]_{s, t}\rightarrow [u]_{s, t}\) for \(t\in \{p, q\}\). By Fatou’s lemma, we know that \([u]_{s, p}^{p}\le t_{1}\) and \([u]_{s, q}^{q}\le t_{2}\). Now we show that \([u]_{s, p}^{p}= t_{1}\) and \([u]_{s, q}^{q}= t_{2}\). Assume, by contradiction, that \([u]_{s, p}^{p}<t_{1}\) and \([u]_{s, q}^{q}\le t_{2}\). Since \(\langle \mathcal {L}'_{V_{0}}(u_{n}), \varphi \rangle \rightarrow 0\) for all \(\varphi \in C^{\infty }_{c}(\mathbb {R}^{N})\), and \(C^{\infty }_{c}(\mathbb {R}^{N})\) is dense in \(\mathbb {Y}_{V_{0}}\) (see [19]), we can deduce that
Therefore,
that is \(\langle \mathcal {L}'_{V_{0}}(u), u\rangle <0\). From \((f_{1})\) and \((f_{2})\), we have \(\langle \mathcal {L}'_{V_{0}}(t_{0}u), t_{0}u\rangle >0\) for some \(0<t_{0}\ll 1\). Hence, there exists \(\tau \in (t_{0}, 1)\) such that \(\langle \mathcal {L}'_{V_{0}}(\tau u), \tau u\rangle =0\). Combining this fact with the characterization of \(d_{V_{0}}\) and using the fact that \(t\mapsto \frac{1}{2q} f(t)t-F(t)\) is increasing (thanks to \((f_3)\) and \((f_4)\)), by Fatou’s lemma, we get
and we arrive at a contradiction. Hence, \([u_{n}]_{s, t}\rightarrow [u]_{s, t}\) for \(t\in \{p, q\}\), and we obtain \(\mathcal {L}'_{V_{0}}(u)=0\). Finally, we prove that u is positive in \(\mathbb {R}^N\). Since \(\langle \mathcal {L}'_{V_{0}}(u), u^{-}\rangle =0\), where \(u^{-}=\min \{u, 0\}\), and \(f(t)=0\) for \(t\le 0\), we have
which implies that \(u^{-}=0\), that is \(u\ge 0\) in \(\mathbb {R}^{N}\). Thus, \(u\ge 0\) and \(u\not \equiv 0\) in \(\mathbb {R}^{N}\). Using a Moser iteration argument [32] (see the proof of Lemma 5.1 below), we obtain that \(u\in L^{\infty }(\mathbb {R}^{N})\). Since u solves
where \(\alpha _{u}=1+[u]_{s,p}^{p}\) and \(\beta _{u}=1+[u]_{s, q}^{q}\) are bounded quantities, we can argue as in the proof of Theorem 2.2 in [11] to infer that \(u \in C^{0, \alpha }(\mathbb {R}^{N})\). In particular, \(u(x)\rightarrow 0\) as \(|x|\rightarrow \infty \). By using the strong maximum principle [7], we deduce that \(u>0\) in \(\mathbb {R}^{N}\). \(\square \)
The next lemma is a compactness result for the autonomous problem (3.1).
Lemma 3.3
Let \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathcal {M}_{V_{0}}\) be a sequence such that \(\mathcal {L}_{V_{0}}(u_{n})\rightarrow d_{V_{0}}\). Then, \(\{u_{n}\}_{n\in \mathbb {N}}\) has a convergent subsequence in \(\mathbb {Y}_{V_{0}}\).
Proof
Since \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathcal {M}_{V_{0}}\) and \(\mathcal {L}_{V_{0}}(u_{n})\rightarrow d_{V_{0}}\), it follows from Lemma 3.1-(iii), Proposition 3.1-(d) and the definition of \(d_{V_{0}}\) that
and
Let us define \(\mathcal {G}: \overline{\mathbb {S}}_{V_{0}}^{+}\rightarrow \mathbb {R}\cup \{\infty \}\) by
We observe that the following properties hold:
-
\((\overline{\mathbb {S}}_{V_{0}}^{+}, \delta _{V_{0}})\), where \(\delta _{V_{0}}(u, v)=\Vert u-v\Vert _{\mathbb {Y}_{V_{0}}}\), is a complete metric space.
-
\(\mathcal {G}\in C(\overline{\mathbb {S}}_{V_{0}}^{+}, \mathbb {R}\cup \{\infty \})\), by Lemma 3.1-(iv).
-
\(\mathcal {G}\) is bounded below, by Proposition 3.1-(d).
By using the Ekeland variational principle [21], there exists \(\{{\hat{v}}_{n}\}_{n\in \mathbb {N}}\subset \mathbb {S}_{V_{0}}^{+}\) such that \(\{{\hat{v}}_{n}\}_{n\in \mathbb {N}}\) is a \((PS)_{d_{V_{0}}}\) sequence for \(\psi _{V_{0}}\) and \(\Vert {\hat{v}}_{n}-v_{n}\Vert _{\mathbb {Y}_{V_{0}}}=o_{n}(1)\). Now the remainder of the proof follows from Proposition 3.1, Theorem 3.1, and arguing as in the proof of Corollary 2.1. \(\square \)
We conclude this section by showing a useful relation between the minimax levels \(c_{\varepsilon }\) and \(d_{V_{0}}\).
Lemma 3.4
It holds \(\lim _{\varepsilon \rightarrow 0} c_{\varepsilon }=d_{V_{0}}\).
Proof
For \(\varepsilon >0\), let \(\omega _{\varepsilon }(x)= \psi _{\varepsilon }(x)\omega (x)\), where \(\omega \) is a positive ground state of (3.1) (whose existence is guaranteed by Theorem 3.1), and \(\psi _{\varepsilon }(x)= \psi (\varepsilon x)\) with \( \psi \in C^{\infty }_{c}(\mathbb {R}^{N})\) such that \(0\le \psi \le 1\), \( \psi (x)=1\) if \(|x|\le 1\) and \( \psi (x)=0\) if \(|x|\ge 2\). For simplicity, we assume that \(\mathrm{supp}( \psi )\subset B_{2}\subset \Lambda \). Using the dominated convergence theorem, we see that
as \(\varepsilon \rightarrow 0\). Now, for each \(\varepsilon >0\), there exists \(t_{\varepsilon }>0\) such that
Therefore, \(\langle \mathcal {J}_{\varepsilon }'(t_{\varepsilon } \omega _{\varepsilon }), \omega _{\varepsilon }\rangle =0\) and this implies that
If \(t_{\varepsilon }\rightarrow \infty \), then
and using (3.3), \(p<2q\) and \((f_3)\), we obtain that \([\omega ]^{2q}_{s, q}=\infty \), which is impossible. Then, \(t_{\varepsilon }\rightarrow t_{0}\in [0, \infty )\). If \(t_{0}=0\), using \((f_1)\) and \((f_2)\), we see that, for \(\zeta \in (0, V_{0})\), it holds
This together with \(q>p\) yields \(\Vert \omega \Vert ^{p}_{s, p}=0\), that is a contradiction. Hence, \(t_{\varepsilon }\rightarrow t_{0}\in (0, \infty )\).
Taking the limit as \(\varepsilon \rightarrow 0\) in (3.4), we get
which combined with \(2q>q>p\), \((f_{4})\) and \(\omega \in \mathcal {M}_{V_{0}}\), implies that \(t_{0}=1\).
Now, we note that
Since \(V(\varepsilon \cdot )\) is bounded on the support of \(\omega _{\varepsilon }\), we can use the dominated convergence theorem, (3.3) and the above inequality to deduce that \(\limsup _{\varepsilon \rightarrow 0}c_{\varepsilon }\le d_{V_{0}}\). By \((V_1)\), we obtain that \(\liminf _{\varepsilon \rightarrow 0}c_{\varepsilon }\ge d_{V_{0}}\), and thus \(\lim _{\varepsilon \rightarrow 0}c_{\varepsilon }= d_{V_{0}}\). This completes the proof. \(\square \)
4 A Multiplicity Result for (2.1)
In this section, we deal with the multiplicity of solutions to (2.1). Let \(\delta >0\) be such that
and let \(w\in \mathbb {Y}_{V_{0}}\) be a positive ground state solution to (3.1) (by virtue of Theorem 3.1).
Consider a nonincreasing function \(\eta \in C^{\infty }([0, \infty ), [0, 1])\) such that \(\eta (t)=1\) if \(0\le t\le \frac{\delta }{2}\), \(\eta (t)=0\) if \(t\ge \delta \) and \(|\eta '(t)|\le c\) for some \(c>0\). For any \(y\in M\), we define
Let \(\Phi _{\varepsilon }: M\rightarrow \mathcal {N}_{\varepsilon }\) be given by
where \(t_{\varepsilon }>0\) satisfies
By construction, \(\Phi _{\varepsilon }(y)\) has compact support for any \(y\in M\).
Lemma 4.1
The function \(\Phi _{\varepsilon }\) has the following property:
Proof
Assume, by contradiction, that there exist \(\delta _{0}>0\), \(\{y_{n}\}_{n\in \mathbb {N}}\subset M\) and \(\varepsilon _{n}\rightarrow 0\) such that
For each \(n\in \mathbb {N}\) and for all \(z\in B_{\frac{\delta }{\varepsilon _{n}}}(0)\), we have \(\varepsilon _{n} z\in B_{\delta }(0)\), and thus
Using the change of variable \(z=\frac{\varepsilon _{n}x-y_{n}}{\varepsilon _{n}}\) and the fact that \(G=F\) in \(\Lambda \times \mathbb {R}\), we can write
We claim that \(t_{\varepsilon _{n}}\rightarrow 1\) as \(n\rightarrow \infty \). We start by proving that \(t_{\varepsilon _{n}}\rightarrow t_{0}\in [0, \infty )\). Since \(\Phi _{\varepsilon _{n}}(y_{n})\in \mathcal {N}_{\varepsilon _{n}}\) and \(g=f\) on \(\Lambda \times \mathbb {R}\), we have
Observing that \(\eta (|x|)=1\) for \(x\in B_{\frac{\delta }{2}}(0)\) and that \(B_{\frac{\delta }{2}}(0)\subset B_{\frac{\delta }{\varepsilon _{n}}}(0)\) for all n large enough, the identity (4.3) yields
which together with \((f_4)\) gives
where
(we recall that w is continuous and positive in \(\mathbb {R}^N\)). If \(t_{\varepsilon _{n}}\rightarrow \infty \), the dominated convergence theorem results in
and recalling that \(2q>q>p\), we also have
On the other hand, by \((f_3)\), we get
Combining (4.4), (4.6) and (4.7), we achieve a contradiction. Consequently, \(\{t_{\varepsilon _{n}}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {R}\) and, up to a subsequence, we may assume that \(t_{\varepsilon _{n}}\rightarrow t_{0}\) for some \(t_{0}\in [0, \infty )\). From (4.3), (4.5), \((f_1)\), \((f_2)\), we can see that \(t_{0}\in (0, \infty )\). Now we prove that \(t_{0}=1\). Letting \(n\rightarrow \infty \) in (4.3), and using (4.5) and the dominated convergence theorem, we have that
Since \(w\in \mathcal {M}_{V_0}\), it holds
Then we obtain
Using \(2q>q>p\) and assumption \((f_4)\), we conclude that \(t_{0}=1\). Therefore, passing to the limit as \(n\rightarrow \infty \) in (4.2), we deduce that
which contradicts (4.1). \(\square \)
Let \(\rho =\rho (\delta )>0\) be such that \(M_{\delta }\subset B_{\rho }(0)\). Define \(\varUpsilon : \mathbb {R}^{N}\rightarrow \mathbb {R}^{N}\) by setting
Let us consider the barycenter map \(\beta _{\varepsilon }: \mathcal {N}_{\varepsilon }\rightarrow \mathbb {R}^{N}\) given by
Arguing as in the proof of Lemma 3.6 in [11], we can prove the following result.
Lemma 4.2
The function \(\beta _{\varepsilon }\) satisfies the following limit
The next compactness result plays an important role in showing that the solutions of the modified problem are also solutions of the original one.
Lemma 4.3
Let \(\varepsilon _{n}\rightarrow 0\) and \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathcal {N}_{\varepsilon _{n}}\) be such that \(\mathcal {J}_{\varepsilon _{n}}(u_{n})\rightarrow d_{V_0}\). Then there exists \(\{{\tilde{y}}_{n}\}_{n\in \mathbb {N}}\subset \mathbb {R}^{N}\) such that \(v_{n}(x)=u_{n}(x+ {\tilde{y}}_{n})\) has a convergent subsequence in \(\mathbb {Y}_{V_0}\). Moreover, up to a subsequence, \(\{y_{n}\}_{n\in \mathbb {N}}=\{\varepsilon _{n}{\tilde{y}}_{n}\}_{n\in \mathbb {N}}\) is such that \(y_{n}\rightarrow y_{0}\in M\).
Proof
Since \(\langle \mathcal {J}'_{\varepsilon _{n}}(u_{n}), u_{n} \rangle =0\) and \(\mathcal {J}_{\varepsilon _{n}}(u_{n})\rightarrow d_{V_0}\), we can argue as in the proof of Lemma 2.5 to verify that \(\{u_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {Y}_{V_{0}}\). According to \(d_{V_0}>0\), \(\Vert u_{n}\Vert _{\mathbb {X}_{\varepsilon _{n}}}\nrightarrow 0\). Then, proceeding as in the proof of Lemma 3.2, we obtain a sequence \(\{{\tilde{y}}_{n}\}_{n\in \mathbb {N}}\subset \mathbb {R}^{N}\) and constants \(R, \beta >0\) such that
Set \(v_{n}(x)=u_{n}(x+ {\tilde{y}}_{n})\). Thus, \(\{v_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {Y}_{V_0}\), and, up to a subsequence, we may assume that \(v_{n}\rightharpoonup v\not \equiv 0\) in \(\mathbb {Y}_{V_0}\). Let \(t_{n}\in (0, \infty )\) be such that \({\tilde{v}}_{n}=t_{n}v_{n} \in \mathcal {M}_{V_0}\), and set \(y_{n}=\varepsilon _{n}{\tilde{y}}_{n}\). From the definition of \(d_{V_{0}}\), \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathcal {N}_{\varepsilon _{n}}\), \((g_2)\) and \(\mathcal {J}_{\varepsilon _{n}}(u_{n})\rightarrow d_{V_{0}}\), we have
which implies that
In particular, \(\{{\tilde{v}}_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {Y}_{V_0}\) and, by extracting a subsequence if necessary, we may assume that \({\tilde{v}}_{n}\rightharpoonup {\tilde{v}}\) in \(\mathbb {Y}_{V_0}\). Since \(\{v_{n}\}_{n\in \mathbb {N}}\) and \(\{{\tilde{v}}_{n}\}_{n\in \mathbb {N}}\) are bounded in \(\mathbb {Y}_{V_0}\), and \(v_{n}\nrightarrow 0\) in \(\mathbb {Y}_{V_0}\), we deduce that \(\{t_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {R}\) and, up to a subsequence, we may assume that \(t_{n}\rightarrow t_{0}\ge 0\). If \(t_{0}=0\), then \({\tilde{v}}_{n}\rightarrow 0\) in \(\mathbb {Y}_{V_0}\) (because \(\{v_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {Y}_{V_0}\)), and thus \(\mathcal {L}_{V_0}({\tilde{v}}_{n})\rightarrow 0\), which contradicts \(d_{V_{0}}>0\). Hence, \(t_{0}\in (0, \infty )\). From the uniqueness of the weak limit, we see that \({\tilde{v}}=t_{0} v\not \equiv 0\). This fact combined with Lemma 3.3 yields \({\tilde{v}}_{n}\rightarrow {\tilde{v}}\) in \(\mathbb {Y}_{V_0}\), and so \(\displaystyle {v_{n}\rightarrow v}\) in \(\mathbb {Y}_{V_0}\). Furthermore,
In what follows, we show that \(\{y_{n}\}_{n\in \mathbb {N}}\) admits a subsequence, still denoted by itself, such that \(y_{n}\rightarrow y_{0}\in M\). We begin by proving that \(\{y_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {R}^{N}\). Suppose, by contradiction, that there exists a subsequence of \(\{y_{n}\}_{n\in \mathbb {N}}\), still denoted by itself, such that \(|y_{n}|\rightarrow \infty \). Choose \(R>0\) such that \(\Lambda \subset B_{R}(0)\). For n large enough, we may assume that \(|y_{n}|>2R\). Then, for each \(x\in B_{R/\varepsilon _{n}}(0)\),
Using \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathcal {N}_{\varepsilon _{n}}\), a change of variable, the definition of g and the above relation, we have
Since \(v_{n}\rightarrow v\) in \(\mathbb {Y}_{V_0}\) and \(|B_{R/\varepsilon _{n}}^{c}(0)|\rightarrow 0\), it follows from the dominated convergence theorem that
On the other hand, \({\tilde{f}}(v_{n})v_{n}\le \frac{V_{0}}{K} (|v_{n}|^{p}+|v_{n}|^{q})\), and so
Consequently,
and we reach a contradiction because \(v_{n}\rightarrow v\not \equiv 0\) in \(\mathbb {Y}_{V_0}\). Thus, \(\{y_{n}\}_{n\in \mathbb {N}}\) is bounded in \(\mathbb {R}^{N}\) and, up to a subsequence, we may assume that \(y_{n}\rightarrow y_{0}\in \mathbb {R}^{N}\). If \(y_{0}\notin {\overline{\Lambda }}\), then we can argue as before to get \(v_{n}\rightarrow 0\) in \(\mathbb {Y}_{V_0}\), that is a contradiction. Hence, \(y_{0}\in {\overline{\Lambda }}\). Let us note that if \(V(y_{0})=V_{0}\), then \(y_{0}\notin \partial \Lambda \) in view of (V2). Therefore, it suffices to prove that \(V(y_{0})=V_{0}\) to deduce that \(y_{0}\in M\). To accomplish this, we assume, by contradiction, that \(V(y_{0})>V_{0}\). Using this fact, \({\tilde{v}}_{n}\rightarrow {\tilde{v}}\) in \(\mathbb {Y}_{V_0}\), Fatou’s lemma and the invariance of \(\mathbb {R}^{N}\) by translation, we see that
which is a contradiction. The proof is now complete. \(\square \)
Let us define
where \(\pi (\varepsilon )=\sup _{y\in M}|\mathcal {J}_{\varepsilon }(\Phi _{\varepsilon }(y))-d_{V_0}|\rightarrow 0\) as \(\varepsilon \rightarrow 0\), according to Lemma 4.1. By the definition of \(\pi (\varepsilon )\), we have that, for all \(y\in M\) and \(\varepsilon >0\), \(\Phi _{\varepsilon }(y)\in {\widetilde{\mathcal {N}}}_{\varepsilon }\) and thus \({\widetilde{\mathcal {N}}}_{\varepsilon }\ne \emptyset \). Arguing as in the proof of Lemma 3.7 in [11], we deduce the following result.
Lemma 4.4
For any \(\delta >0\), we have
We conclude the section by presenting a relation between the topology of M and the number of solutions of the modified problem (2.1). Since \(\mathbb {S}^{+}_{\varepsilon }\) is not a complete metric space, we invoke the abstract category result in [36] to achieve our purpose.
Theorem 4.1
Assume that \((V_1)\)–\((V_2)\) and \((f_1)\)–\((f_4)\) hold. Then, for any \(\delta >0\) such that \(M_{\delta }\subset \Lambda \), there exists \({\bar{\varepsilon }}_\delta >0\) such that, for any \(\varepsilon \in (0, {\bar{\varepsilon }}_\delta )\), problem (2.1) has at least \(cat_{M_{\delta }}(M)\) positive solutions.
Proof
For each \(\varepsilon >0\), we define the map \(\alpha _{\varepsilon } : M \rightarrow \mathbb {S}_{\varepsilon }^{+}\) by setting \(\alpha _{\varepsilon }(y)= m_{\varepsilon }^{-1}(\Phi _{\varepsilon }(y))\). By Lemma 4.1, we see that
Hence, there is a number \({\hat{\varepsilon }}>0\) such that the set \( \widetilde{\mathcal {S}}^{+}_{\varepsilon }=\{ w\in \mathbb {S}_{\varepsilon }^{+} : \psi _{\varepsilon }(w) \le d_{V_0} + \pi (\varepsilon )\} \) is nonempty for all \(\varepsilon \in (0, {\hat{\varepsilon }})\), since \(\psi _{\varepsilon }(M)\subset \widetilde{\mathcal {S}}^{+}_{\varepsilon }\). Here \(\pi (\varepsilon )=\sup _{y\in M}|\psi _{\varepsilon }(\alpha _{\varepsilon }(y))-d_{V_0}|\rightarrow 0\) as \(\varepsilon \rightarrow 0\). From the above considerations, and taking into account Lemma 4.1, Lemma 2.4-(iii), Lemmas 4.4 and 4.2, we see that there exists \({\bar{\varepsilon }}= {\bar{\varepsilon }}_{\delta }>0\) such that, for any \(\varepsilon \in (0, {\bar{\varepsilon }})\), the diagram
is well defined. According to Lemma 4.2, for \(\varepsilon >0\) small, we can write \(\beta _{\varepsilon }(\Phi _{\varepsilon }(y))= y+ \theta (\varepsilon , y)\) for \(y\in M\), where \(|\theta (\varepsilon , y)|<\frac{\delta }{2}\) uniformly in \(y\in M\). Define \(H(t, y)= y+ (1-t)\theta (\varepsilon , y)\) for \((t, y)\in [0,1]\times M\). Clearly, \(H: [0,1]\times M\rightarrow M_{\delta }\) is continuous, \(H(0, y)=\beta _{\varepsilon }(\Phi _{\varepsilon }(y))\) and \(H(1, y)=y\) for all \(y\in M\). Then H(t, y) is a homotopy between \(\beta _{\varepsilon } \circ \Phi _{\varepsilon } = (\beta _{\varepsilon } \circ m_{\varepsilon }) \circ (m_{\varepsilon }^{-1}\circ \Phi _{\varepsilon })\) and the inclusion map \(id: M \rightarrow M_{\delta }\). This fact implies that
It follows from Corollary 2.1, Lemma 3.4, and Theorem 27 in [36], with \(c= c_{\varepsilon }\le d_{V_0}+\pi (\varepsilon ) =d\) and \(K= \alpha _{\varepsilon }(M)\), that \(\Psi _{\varepsilon }\) has at least \(cat_{\alpha _{\varepsilon }(M)} \alpha _{\varepsilon }(M)\) critical points on \(\widetilde{\mathcal {S}}^{+}_{\varepsilon }\). Therefore, by Proposition 2.1-(d) and (4.9), we conclude that \(\mathcal {J}_{\varepsilon }\) admits at least \(cat_{M_{\delta }}(M)\) critical points in \(\widetilde{\mathcal {N}}_{\varepsilon }\). \(\square \)
5 Proof of Theorem 1.1
This section is devoted to the proof of the main result of this paper. The idea is to show that the solutions obtained in Theorem 4.1 satisfy, for \(\varepsilon >0\) small enough, the estimate \(u_{\varepsilon }(x)\le a\) for any \(x\in \Lambda ^{c}_{\varepsilon }\). This fact implies that these solutions are indeed solutions of the original problem (1.1). We start with the following lemma which plays a key role in studying the behavior of the maximum points of solutions to (1.1), whose proof is related to the Moser iteration method [32].
Lemma 5.1
Let \(\varepsilon _{n}\rightarrow 0\) and \(\{u_{n}\}_{n\in \mathbb {N}}\subset \widetilde{\mathcal {N}}_{\varepsilon _{n}}\) be a sequence of solutions to (2.1). Then \(\mathcal {J}_{\varepsilon _{n}}(u_{n})\rightarrow d_{V_{0}}\), and there exists \(\{{\tilde{y}}_{n}\}_{n\in \mathbb {N}}\subset \mathbb {R}^{N}\) such that \(v_{n}=u_{n}(\cdot +{\tilde{y}}_{n})\in L^{\infty }(\mathbb {R}^{N})\) and for some \(C>0\) it holds
Moreover,
Proof
Since \(\mathcal {J}_{\varepsilon _{n}}(u_{n})\le d_{V_{0}}+\pi (\varepsilon _{n})\), with \(\pi (\varepsilon _{n})\rightarrow 0\) as \(n\rightarrow \infty \), we can argue as at the beginning of the proof of Lemma 4.3 to deduce that \(\mathcal {J}_{\varepsilon _{n}}(u_{n})\rightarrow d_{V_{0}}\). Then, using Lemma 4.3, we can find \(\{{\tilde{y}}_{n}\}_{n\in \mathbb {N}}\subset \mathbb {R}^{N}\) such that \(v_{n}=u_{n}(\cdot +{\tilde{y}}_{n})\rightarrow v\) in \(\mathbb {Y}_{V_{0}}\) for some \(v\in \mathbb {Y}_{V_{0}}\setminus \{0\}\) and \(\varepsilon _{n}{\tilde{y}}_{n}\rightarrow y_{0}\in M\).
Now we examine the boundedness of \(\{v_{n}\}_{n\in \mathbb {N}}\) in \(L^{\infty }(\mathbb {R}^{N})\). For each \(n\in \mathbb {N}\) and \(L>0\), we define
where \(v_{n,L}= \min \{v_{n}, L\}\), and \(\beta >1\) will be chosen later. Taking \(\gamma (v_{n})\) as test function in the problem solved by \(v_{n}\), we have
In light of the growth assumptions on g, we know that for all \(\xi \in (0, V_{0})\), there exists \(C_{\xi }>0\) such that
From the above facts and \((V_{1})\), we obtain
Observing that, for \(t\in \{p, q\}\), \(1\le 1+[v_{n}]_{s, t}^{t}\le C\) for all \(n\in \mathbb {N}\), we can reproduce the Moser iteration argument carried out in the proof of Lemma 4.1 in [11] to derive that \(|v_{n}|_{\infty }\le C\) for all \(n\in \mathbb {N}\). Since \(\{v_{n}\}_{n\in \mathbb {N}}\) is uniformly bounded in \(L^{\infty }(\mathbb {R}^{N})\cap \mathbb {Y}_{V_{0}}\), we can argue as in the proof of Theorem 2.2 in [11] to deduce that \(\Vert v_{n}\Vert _{C^{0, \alpha }(\mathbb {R}^{N})}\le C\) for all \(n\in \mathbb {N}\). This fact combined with \(v_{n}\rightarrow v\) in \(\mathbb {Y}_{V_{0}}\) implies that \(v_{n}(x)\rightarrow 0\) as \(|x|\rightarrow \infty \) uniformly in \(n\in \mathbb {N}\). The proof of Lemma 5.1 is complete. \(\square \)
We now have all ingredients to prove Theorem 1.1.
Proof of Theorem 1.1
Let \(\delta >0\) be a number satisfying \(M_{\delta } \subset \Lambda \). We first show that there exists \({\tilde{\varepsilon }}_{\delta }>0\) such that, for any \(\varepsilon \in (0, {\tilde{\varepsilon }}_{\delta })\) and any solution \(u_{\varepsilon } \in \widetilde{\mathcal {N}}_{\varepsilon }\) of (2.1), it holds
Assume, by contradiction, that there exists a subsequence \(\varepsilon _{n}\rightarrow 0\), \(u_{n}=u_{\varepsilon _{n}}\in \widetilde{\mathcal {N}}_{\varepsilon _{n}}\) such that \(\mathcal {J}'_{\varepsilon _{n}}(u_{\varepsilon _{n}})=0\) and
As in the proof of Lemma 5.1, we can verify that \(\mathcal {J}_{\varepsilon _{n}}(u_{n}) \rightarrow d_{V_0}\). Then, applying Lemma 4.3, we obtain a sequence \(\{{\tilde{y}}_{n}\}_{n\in \mathbb {N}}\subset \mathbb {R}^{N}\) such that \(v_{n}=u_{n}(\cdot +{\tilde{y}}_{n})\rightarrow v\) in \(\mathbb {Y}_{V_{0}}\) and \(\varepsilon _{n}{\tilde{y}}_{n}\rightarrow y_{0} \in M\).
Pick \(r>0\) such that \(B_{r}(y_{0})\subset B_{2r}(y_{0})\subset \Lambda \). Thus, \(B_{\frac{r}{\varepsilon _{n}}}(\frac{y_{0}}{\varepsilon _{n}})\subset \Lambda _{\varepsilon _{n}}\) for all \(n\in \mathbb {N}\). Moreover, for any \(y\in B_{\frac{r}{\varepsilon _{n}}}({\tilde{y}}_{n})\), we see that
for n large enough. For these values of n, we have
Using (5.1), we can find \(R>0\) such that \(v_{n}(x)<a\) for any \(|x|\ge R\) and \(n\in \mathbb {N}\), and so \(u_{n}(x)<a\) for any \(x\in B_{R}^{c}({\tilde{y}}_{n})\) and \(n\in \mathbb {N}\). On the other hand, there exists \(n_{0} \in \mathbb {N}\) such that, for any \(n\ge n_{0}\),
Hence, \(u_{n}(x)<a\) for any \(x\in \Lambda ^{c}_{\varepsilon _{n}}\) and \(n\ge n_{0}\), which is in contrast with (5.4). This proves our claim.
Let \({\bar{\varepsilon }}_{\delta }>0\) be given by Theorem 4.1 and set \(\varepsilon _{\delta }= \min \{{\tilde{\varepsilon }}_{\delta }, {\bar{\varepsilon }}_{\delta }\}\). Fix \(\varepsilon \in (0, \varepsilon _{\delta })\). Applying Theorem 4.1, we get at least \(cat_{M_{\delta }}(M)\) positive solutions to (2.1). If \(u_{\varepsilon }\) denotes one of these solutions, we have that \(u_{\varepsilon }\in \widetilde{\mathcal {N}}_{\varepsilon }\), and using (5.3) and the definition of g, we deduce that \(u_{\varepsilon }\) is also a solution to (1.1). Consequently, (1.1) admits at least \(cat_{M_{\delta }}(M)\) positive solutions.
Now we investigate the behavior of the maximum points of solutions to (1.1). Take \(\varepsilon _{n}\rightarrow 0\) and consider a sequence \(\{u_{n}\}_{n\in \mathbb {N}}\subset \mathbb {X}_{\varepsilon _{n}}\) of solutions to (1.1) as above. Let us observe that \((g_{1})\) implies that there exists \(\sigma \in (0, a)\) such that
Arguing as before, we can choose \(R>0\) such that
Moreover, up to a subsequence, we may assume that
Indeed, if (5.7) does not hold, then, in view of (5.6), we have that \(|u_{n}|_{\infty }<\sigma \). Hence, using \(\langle \mathcal {J}'_{\varepsilon _{n}}(u_{n}), u_{n}\rangle =0\) and (5.5), we get
which leads to a contradiction. Therefore, (5.7) is satisfied.
Let \(p_{n}\in \mathbb {R}^{N}\) be a global maximum point of \(u_{n}\). Combining (5.6) and (5.7), we infer that \(p_{n}={\tilde{y}}_{n}+q_{n}\), for some \(q_{n}\in B_{R}(0)\). Since \(\varepsilon _{n}{\tilde{y}}_{n}\rightarrow y_{0}\in M\) and \(|q_{n}|<R\) for all \(n\in \mathbb {N}\), we have that \(\varepsilon _{n}p_{n}\rightarrow y_{0}\), and using the continuity of V we obtain
The proof of Theorem 1.1 is now complete. \(\square \)
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Ambrosio, V. A Kirchhoff Type Equation in \(\pmb {\mathbb {R}}^{N}\) Involving the fractional (p, q)-Laplacian. J Geom Anal 32, 135 (2022). https://doi.org/10.1007/s12220-022-00876-5
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DOI: https://doi.org/10.1007/s12220-022-00876-5
Keywords
- Fractional (p, q)-Laplacian problem
- Kirchhoff type problem
- Penalization technique
- Lusternik–Schnirelman theory