Abstract
We prove the existence of multiple positive solutions for a fractional Laplace problem with critical growth and sign-changing weight in non-contractible domains.
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1 Introduction
In this paper we consider the following critical problem involving fractional Laplacian:
where \(s\in (0,1)\) is fixed and \((-\Delta )^{s}\) is the fractional Laplace operator, \(\varOmega \subset \mathbb{R}^{N}\) (\(N>2s\)) is a smooth bounded domain, \(1< p<2\), \(2^{*}_{s}:=\frac{2N}{N-2s}\), and \(a\in C(\bar{ \varOmega })\) changes sign in Ω.
During the last years there has been an increasing interest in the study of the fractional Laplacian, motivated by great applications and by important advances in the theory of nonlinear partial differential equations, see [3, 7, 11, 14, 15, 17, 20, 21, 24, 25, 35, 36] for details. Nonlinear equations involving fractional Laplacian are currently actively studied. The fractional Laplace operator \((-\Delta )^{s}\) (up to normalization factors) may be defined as
where \(K(x)=\vert x\vert ^{-(N+2s)}\), \(x\in \mathbb{R}^{N}\). We will denote by \(H^{s}(\mathbb{R}^{N})\) the usual fractional Sobolev space endowed with the so-called Gagliardo norm
while \(X_{0}\) is the function space defined as
We refer to [22, 29, 30] for a general definition of \(X_{0}\) and its properties. The embedding \(X_{0}\hookrightarrow L^{q}( \varOmega )\) is continuous for any \(q\in [1,2^{*}_{s}]\) and compact for any \(q\in [1,2^{*}_{s})\). The space \(X_{0}\) is endowed with the norm defined as
By Lemma 5.1 in [29] we have \(C^{2}_{0}(\varOmega )\subset X_{0}\). Thus \(X_{0}\) is nonempty. Note that \((X_{0}, \Vert \cdot \Vert _{X_{0}})\) is a Hilbert space with scalar product
It is well known that the following critical problem
has no positive solution if Ω is a star-shaped domain, where \(2^{*}=\frac{2N}{N-2}\). For a non-contractible domain Ω, Coron [12] proved that (1.2) has a positive solution. Later, Bahri and Coron [4] improved Coron’s existence result by showing, via topological arguments based upon homology theory, that (1.2) admits a positive solution provided that \(H_{m}(\varOmega ,\mathbb{Z} _{2})\ne \{0\}\) for some \(m>0\). After that, many papers have studied the existence and multiplicity of positive solutions of the problem similar to (1.2), see [16, 18, 37, 39].
It is natural to think that, as in the local case, by assuming suitable geometrical or topological conditions on Ω, one can get the existence of nontrivial solutions for the nonlocal fractional problem. In a recent work, Secchi et al. [28] consider the following nonlocal fractional problem:
They proved that (1.3) admits at least a positive solution if there is a point \(x_{0}\in \mathbb{R}^{N}\) and radii \(R_{2}>R_{1}>0\) such that
and \(R_{2}/R_{1}\) is sufficiently large.
Motivated by the works mentioned above, we study problem (1.1), which involves the critical exponent, the effect of the coefficient \(a(x)\), and the domain with “rich topology”. We try to extend some important results, which are well known for the classical case of the Laplacian (see, e.g., Theorem 1.1 in [39]), to a nonlocal setting.
Taking into account that we are looking for positive solutions, we consider the energy functional associated with (1.1)
where \(u^{+}=\max \{u,0\}\) denotes the positive part of u. By the maximum principle (Proposition 2.2.8 in [33]), it is easy to check that critical points of I are the positive solutions of (1.1).
We make the following assumptions:
-
(H1)
There exist three constants \(\rho _{2}>\rho _{1}>\rho _{0}>0\) such that \(\bar{B}_{\rho _{2}}(0)\setminus B_{\rho _{1}}(0)\subset \varOmega \) and \(B_{\rho _{0}}(0)\cap \varOmega =\emptyset \), where \(B_{\rho }(0)=\{x \in \mathbb{R}^{N}: \vert x\vert <\rho \}\) for any \(\rho >0\);
-
(H2)
There exists a domain \(\bar{B}_{\rho _{2}}(0)\setminus B_{\rho _{1}}(0)\subset \mathcal{D}\subset \varOmega \) such that \(a(x)>0\) for \(x\in \mathcal{D}\) and \(a(x)\le 0\) for \(x\in \varOmega \setminus \mathcal{D}\).
Theorem 1.1
Assume that (H1), (H2) hold. Then there exists \(\sigma _{0}>0\) such that if \(\vert a^{+}\vert _{q}<\sigma _{0}\), where \(a^{+}(x)=\max \{a(x),0\}\), \(q=\frac{2^{*} _{s}}{2^{*}_{s}-p}\), (1.1) has three positive solutions \(\tilde{u}_{i}(1\le i\le 3)\) such that
We should remark that \(\tilde{u}_{2}\) and \(\tilde{u}_{3}\) satisfy \(I(\tilde{u}_{i})< I(\tilde{u}_{1})+\frac{s}{N}S_{s}^{\frac{N}{2s}}\) (\(i=2,3\)), where \(S_{s}\) is the Sobolev constant. It is an interesting task to find the fourth positive solution \(\tilde{u}_{4}\) with \(I(\tilde{u}_{4})>I( \tilde{u}_{1})+\frac{s}{N}S_{s}^{\frac{N}{2s}}\) provided \(\rho _{2}/ \rho _{1}\) is sufficiently large, although we shall not undertake it here.
This paper is organized as follows. In Sect. 2 we introduce Nehari manifold and state technical and elementary lemmas useful along the paper. In Sect. 3 we prove the existence of the first solution of (1.1). In Sect. 4 we establish some essential estimates of energy. In Sect. 5 we prove the existence of the other two solutions by Lusternik–Schnirelmann category. We denote by \(\vert \cdot \vert _{r}\) the \(L^{r}(\varOmega )\)-norm for any \(r>1\), respectively.
2 Preliminaries
Recall that I is unbounded from below; we can get rid of this problem once we restrict I to the Nehari manifold
Notice that \(u^{+}\not \equiv 0\) for any \(u\in \mathcal{N}\), and on \(\mathcal{N}\) the functional I reads
Set
In our context, the Sobolev constant is given by
Lemma 2.1
I is coercive and bounded from below on \(\mathcal{N}\).
Proof
If \(u\in \mathcal{N}\), by (2.1) and the Sobolev inequality,
Since \(1< p<2\), we get that I is coercive and bounded from below on \(\mathcal{N}\). □
Define
Then, for \(u\in \mathcal{N}\), we have
Adopting a method similar to that used in [34], we split \(\mathcal{N}\) into three parts:
Lemma 2.2
Assume that u is a minimizer for I on \(\mathcal{N}\) and \(u\notin \mathcal{N}^{0}\). Then \(\langle I'(u),v\rangle =0\) for any \(v\in X_{0}\).
The proof is similar to that of Theorem 2.3 in [9], we omit it.
Set
Lemma 2.3
\(\mathcal{N}^{0}=\emptyset \) if \(\vert a^{+}\vert _{q}<\sigma _{1}\).
Proof
Assume by contradiction that there exists \(a\in C(\bar{\varOmega })\) with \(\vert a^{+}\vert _{q}<\sigma _{1}\) such that \(\mathcal{N}\ne \emptyset \). By (2.4) and (2.2), we have
Consequently,
Similarly, by (2.5), we have
and so
Thus, we get that \(\vert a^{+}\vert _{q}\ge \sigma _{1}\), which is impossible. □
Define
Lemma 2.4
For each \(u\in X_{0}^{+}\), we have
-
(i)
if \(\int _{\varOmega }a(x)(u^{+})^{p}\,dx\le 0\), then there exists unique \(t^{-}(u)>t_{\max }\) such that \(t^{-}(u)u\in \mathcal{N}^{-}\) and \(\varphi (t):=I(tu)\) is increasing on \((0,t^{-}(u))\) and decreasing on \((t^{-}(u),+\infty )\), where
$$ t_{\max }= \biggl(\frac{(2-p) \Vert u \Vert _{X_{0}}^{2}}{(2^{*}_{s}-p)\int _{ \varOmega }(u^{+})^{2^{*}_{s}}\,dx} \biggr)^{\frac{N-2s}{4s}}. $$Furthermore,
$$ \varphi \bigl(t^{-}(u)\bigr)=\sup_{t\ge 0} \varphi (t). $$(2.6) -
(ii)
If \(\int _{\varOmega }a(x)(u^{+})^{p}\,dx>0\), then there exist unique \(0< t^{+}(u)< t_{\max }< t^{-}(u)\) such that \(t^{+}(u)u\in \mathcal{N} ^{+}\), \(t^{-}(u)u \in \mathcal{N}^{-}\), and \(\varphi (t)\) is decreasing on \((0,t^{+}(u))\cup (t^{-}(u),+\infty )\) and increasing on \((t^{+}(u),t^{-}(u))\). Furthermore,
$$ \varphi \bigl(t^{+}(u)\bigr)=\inf_{0\le t\le t^{-}(u)} \varphi (t), \quad\quad \varphi \bigl(t^{-}(u)\bigr)=\sup _{t\ge t^{+}(u)}\varphi (t). $$(2.7) -
(iii)
\(t^{-}(u)\) is a continuous function for \(u\in X_{0}^{+}\).
-
(iv)
\(\mathcal{N}^{-}= \{u\in X_{0}^{+}: \frac{1}{\Vert u\Vert _{X_{0}}}t ^{-} (\frac{u}{\Vert u\Vert _{X_{0}}} )=1 \}\).
Proof
Fix \(u\in X_{0}^{+}\). We consider the following function:
Clearly, \(tu\in \mathcal{N}\) if and only if \(\gamma (t)=\int _{\varOmega }a(x)(u^{+})^{p}\,dx\). Moreover,
So, it is easy to see that \(tu\in \mathcal{N}^{+}\) (or \(\mathcal{N} ^{-}\)) if and only if \(\gamma '(t)>0\) (or <0). Notice that γ is increasing on \((0,t_{\max })\) and decreasing on \((t_{\max },+\infty )\) and \(\gamma (t)\to -\infty \) as \(t\to +\infty \).
(i) If \(\int _{\varOmega }a(x)(u^{+})^{p}\,dx\le 0\), then \(\gamma (t)= \int _{\varOmega }a(x)(u^{+})^{p}\,dx\) has a unique solution \(t^{-}(u)>t _{\max }\) and \(\gamma '(t^{-}(u))<0\). Thus, \(t^{-}(u)u\in \mathcal{N} ^{-}\). Since
we get that (2.6) holds.
(ii) Assume that \(\int _{\varOmega }a(x)\vert u\vert ^{p}\,dx>0\). Direct computation yields that
since \(\vert a^{+}\vert _{q}<\sigma _{1}\). Thus, \(\gamma (t)=\int _{\varOmega }a(x)(u ^{+})^{p}\,dx\) has exactly two solutions \(t^{+}(u)< t_{\max }< t^{-}(u)\) such that \(\gamma '(t^{+}(u))>0\) and \(\gamma '(t^{-}(u))<0\), and \(\varphi (t)\) is decreasing on \((0,t^{+}(u))\cup (t^{-}(u),+\infty )\) and increasing on \((t^{+}(u),t^{-}(u))\). Consequently, \(t^{+}(u)u \in \mathcal{N}^{+}\) and \(t^{-}(u)u\in \mathcal{N}^{-}\), and (2.7) holds.
(iii) The uniqueness of \(t^{-}(u)\) and its extremal property give that \(t^{-}(u)\) is a continuous function of u.
(iv) Set
Let \(v=\frac{u}{\Vert u\Vert _{X_{0}}}\) for any \(u\in \mathcal{N}^{-}\). By (i) and (ii), there exists \(t^{-}(v)>0\) such that \(t^{-}(v)v\in \mathcal{N}^{-}\), that is, \(\frac{t^{-}(v)}{\Vert u\Vert _{X_{0}}}u\in \mathcal{N}^{-}\). Since \(u\in \mathcal{N}^{-}\), we have \(t^{-}(v)=\Vert u \Vert _{X_{0}}\). Hence, we get \(\mathcal{N}^{-}\subset \mathcal{S}\). On the other hand, let \(u\in \mathcal{S}\). Then,
Thus, \(\mathcal{S}\subset \mathcal{N}^{-}\). □
3 Existence of the first solution
Define
Set
Lemma 3.1
-
(i)
\(m^{+}<0\) if function a satisfies \(\vert a^{+}\vert _{q}\in (0,\sigma _{1})\);
-
(ii)
there exists positive constant \(c_{0}\) such that \(m^{-}\ge c_{0}\) if \(\vert a^{+}\vert _{q}<\sigma _{2}\). In particular, \(m^{+}=\inf_{u\in \mathcal{N}}I(u)\) if function a satisfies \(\vert a^{+}\vert _{q}\in (0,\sigma _{2})\).
Proof
(i) If \(u\in \mathcal{N}^{+}\), then by (2.5) we get that
Thus, by (2.1),
and so \(m^{+}<0\).
(ii) If \(u\in \mathcal{N}^{-}\), then by (2.4),
Consequently,
By (2.3) and \(\vert a^{+}\vert _{q}<\sigma _{2}\), we have
□
From now on, we assume that \(\vert a^{+}\vert _{q}\in (0,\sigma _{2})\).
Lemma 3.2
I satisfies the \((\mathit{PS})_{\beta }\) condition in \(X_{0}\) for \(\beta < m ^{+}+\frac{s}{N}S_{s}^{\frac{N}{2s}}\).
Proof
Let \(\{u_{n}\}\) be a \((\mathit{PS})_{\beta }\) sequence for I such that
Then, for n big enough, we have
It follows that \(\Vert u_{n}\Vert _{X_{0}}\) is bounded. Going if necessary to a subsequence, we can assume that
We derive from (3.1) that \(\langle I'(u_{0}),v\rangle =0\), \(\forall v\in X_{0}\), i.e., \(u_{0}\) is a solution of (1.1). In particular, \(u_{0}\in \mathcal{N}\). Thus, by Lemma 3.1, we have \(I(u_{0})\ge m^{+}\). Since \(X_{0}\) is a Hilbert space, we have
By Brézis–Lieb’s lemma [8], we get
Since \((u_{n}^{+})^{2^{*}_{s}-1}\) is bounded in \(L^{p'}(\varOmega )\) with \(p'=2^{*}_{s}/(2^{*}_{s}-1)\) and \(L^{p'}(\varOmega )\) is a reflexible space, we get \((u_{n}^{+})^{2^{*}_{s}-1}\rightharpoonup (u_{0}^{+})^{2^{*} _{s}-1}\) in \(L^{p'}(\varOmega )\), and so
Similarly, since \(u_{n}\rightharpoonup u_{0}\) in \(L^{2^{*}_{s}}( \varOmega )\) and \((u_{0}^{+})^{2^{*}_{s}-1}\in L^{p'}(\varOmega )\), we get
and
Thus, there exists a positive constant \(\sigma >0\) such that
for n large enough. By (3.7), (3.8), and Sobolev inequality, we get
This implies \(\Vert u_{n}-u_{0}\Vert _{X_{0}}\to 0\) in \(X_{0}\). □
Theorem 3.3
There exists a minimizer \(\tilde{u}_{1}\) of the critical problem (1.1), and it satisfies
-
(i)
\(\tilde{u}_{1}\in \mathcal{N}^{+}\) and \(I(\tilde{u}_{1})=m^{+}\);
-
(ii)
\(\tilde{u}_{1}\in C^{0,s}(\mathbb{R}^{N})\) is a positive solution of (1.1);
-
(iii)
\(I(\tilde{u}_{1})\to 0\) as \(\vert a^{+}\vert _{q}\to 0\).
Proof
Applying Ekeland’s variational principle [13] and using the similar argument as the proof of Theorem 1 in [34], we get that there exists \(\{u_{n}\}\subset \mathcal{N}^{+}\) such that
By Lemma 3.2, there exist a subsequence (still denoted by \(\{u_{n}\}\)) and \(\tilde{u}_{1}\in \mathcal{N}^{+}\), a solution of (1.1), such that \(u_{n}\to \tilde{u}_{1}\) in \(X_{0}\) and \(m^{+}=I(\tilde{u}_{1})\). By the maximum principle (Proposition 2.2.8 in [33]), \(\tilde{u}_{1}\) is strictly positive in Ω. By Proposition 2.2 in [6], \(u\in L^{\infty }(\varOmega )\). Furthermore, by Proposition 1.1 in [26] (or Proposition 5 in [31]), \(u\in C^{0,s}(\mathbb{R}^{N})\).
By (2.6),
This implies \(\Vert \tilde{u}_{1}\Vert _{X_{0}}\to 0\) as \(\vert a^{+}\vert _{q}\to 0\), and so \(I(\tilde{u}_{1})\to 0\) as \(\vert a^{+}\vert _{q}\to 0\). □
4 Estimates of energy
Recall that \(S_{s}\) is defined as
It is well known from [32] that the infimum in the formula above is attained at ũ, where
with \(\kappa \in \mathbb{R}\setminus \{0\}\), \(\mu >0\) and \(x_{0}\in \mathbb{R}^{N}\) fixed constants. We suppose \(\kappa >0\) for our convenience. Equivalently, the function ū defined as
is such that
The function
is a solution of
Now, we consider the family of functions \(U_{\varepsilon }\) defined as
for any \(\varepsilon >0\). The function \(U_{\varepsilon }\) is a solution of problem (4.2) and satisfies
Let us fix \(\rho _{a}\), \(\rho _{b}\), ρ̃, \(\rho _{c}\), \(\rho _{d}\) such that
Let \(\eta \in C^{\infty }_{0}(\mathbb{R}^{N})\) be a radially symmetric function such that \(0\le \eta \le 1\) in \(\mathbb{R}^{N}\) and
For every \(\varepsilon \in (0,1)\) and \(\mathbf{e}\in \mathbb{S}^{N-1}:= \{x\in \mathbb{R}^{N}:\vert x\vert =1\}\), we denote by \(u_{\varepsilon , \mathbf{e}}\) the following function:
Lemma 4.1
There hold
-
(i)
\(\int _{\mathbb{R}^{N}}\vert u_{\varepsilon , \mathbf{e}}\vert ^{2^{*}_{s}}=S _{s}^{\frac{N}{2s}}+O(\varepsilon ^{N})\) uniformly in \(\mathbf{e}\in \mathbb{S}^{N-1}\);
-
(ii)
\(\Vert u_{\varepsilon ,\mathbf{e}}\Vert _{X_{0}}^{2} =S_{s}^{ \frac{N}{2s}}+O(\varepsilon ^{N-2s})\) uniformly in \(\mathbf{e} \in \mathbb{S}^{N-1}\).
Proof
(i) By Proposition 22 in [32], we have
Direct computation yields that
Thus, by (4.8), we prove (i).
(ii) Set \(\delta =\frac{1}{2}\min \{\tilde{\rho }-\rho _{b},\rho _{c}- \tilde{\rho }\}\). Define
We have
We consider the following four cases:
(i) Assume \((x,y)\in A_{4}\). Then \(\vert x-\tilde{\rho }\mathbf{e}\vert \ge \tilde{\rho }-\rho _{b}\) or \(\vert x-\tilde{\rho }\mathbf{e}\vert \ge \rho _{c}- \tilde{\rho }\). Thus, there exists constant \(C>0\) such that
Consequently,
Moreover, if \((x,y)\in A_{4}\) and \(\vert x-y\vert \le \frac{1}{2}(\rho _{c}-\rho _{b})\), then \((x,y)\in B_{1}\cup B_{2}\), and so \(\vert \xi -\tilde{\rho } \mathbf{e}\vert \ge \rho _{c}-\tilde{\rho }>0\) or \(\vert \xi -\tilde{\rho } \mathbf{e}\vert \ge \tilde{\rho }-\rho _{b}>0\) for any ξ on the segment joining x and y. By the mean value theorem, there exists ξ̄ on the segment joining x and y such that
Hence, by (4.11) and the inequality above, we get
or
Consequently, by the definition of \(u_{\varepsilon ,\mathbf{e}}\) and (4.13),
(ii) Assume \((x,y)\in A_{3}\). Let \(\xi =tx+(1-t)y=y+t(x-y)\) for any \(t\in [0,1]\). If \(\vert y\vert \ge \rho _{c}\), then
and so
If \(\vert y\vert \le \rho _{b}\), then
and so
Thus, by the mean value theorem, there exists ξ̄ on the segment joining x and y such that
Consequently,
where \(\tilde{A}_{3}= \{(x,y)\in \mathbb{R}^{N}\times \mathbb{R} ^{N}: x\in \mathcal{D}_{1}, y\in \mathbb{R}^{N}, \vert x-y\vert \le \delta \}\).
(iii) Assume \((x,y)\in A_{2}\). Since \(x\in \mathcal{D}_{1}\), we have
Direct computation yields
where \(\tilde{A}_{2}=\{(x,y)\in \mathbb{R}^{N}\times \mathbb{R}^{N}: x\in \mathcal{D}_{1}, y\in \mathbb{R}^{N}, \vert x-y\vert >\delta \}\).
For any \((x,y)\in A_{2}\),
Therefore, using the change of variable \(\xi =x\), \(\zeta =x-y\), we have that
Similar to (4.17), we have
By (4.10), (4.14), (4.15), and (4.20), we have
Using the change of variable and (4.13) in [32], we have
On the other hand, by the definition of \(S_{s}\) and (i), we have
Combining (4.23) and (4.24), we prove (ii). □
Lemma 4.2
Assume that \(a\in C(\bar{\varOmega })\) with \(\vert a^{+}\vert _{q}\in (0,\sigma _{2})\). There exists \(\varepsilon _{0}>0\) such that, for \(\varepsilon <\varepsilon _{0}\),
uniformly in \(\mathbf{e}\in \mathbb{S}^{N-1}\), where \(\tilde{u}_{1}\) is a minimizer of I in Theorem 3.3.
Proof
Since I is continuous in \(X_{0}\) and \(u_{\varepsilon ,\mathbf{e}}\) is uniformly bounded in \(X_{0}\) for ε small enough, there exists \(t_{1}>0\) such that, for \(t\in [0,t_{1}]\),
Since \(u_{\varepsilon ,\mathbf{e}}(x)=0\) for any \(x\in \{x\in \varOmega : a(x)<0\}\), we have
It is easy to get from Lemma 4.1 that
for ε small enough. Note that the last term in (4.25) satisfies
Thus, \(I(\tilde{u}_{1}+tu_{\varepsilon ,\mathbf{e}})\to -\infty \) as \(t\to +\infty \) uniformly in ε and e. Consequently, there exists \(t_{2}>t_{1}\) such that \(I(\tilde{u}_{1}+tu _{\varepsilon ,\mathbf{e}})< m^{+}+\frac{s}{N}S_{s}^{\frac{N}{2s}}\) for \(t\ge t_{2}\). Then, we only need to verify the inequality
for ε small enough.
From now on, we assume that \(t\in [t_{1}, t_{2}]\).
There exists a constant \(C>0\) such that
We have used the following inequality (see [5, 40] for example): for \(r>2\), there exists a constant \(C_{r}\) (depending on r) such that
Combining (4.25) and (4.26), and using the fact that \(\tilde{u}_{1}\) is a positive solution of (1.1), we have
Here we have used the elementary inequality: \((\alpha +\beta )^{p} \ge \alpha ^{p}+p\alpha ^{p-1}\beta \), \(\forall \alpha ,\beta >0\).
Now, we estimate the last but one term in (4.27). By Theorem 3.3, there exists a constant \(C_{1}>0\) such that \(\tilde{u}_{1}(x) \ge C_{1}\) for \(x\in E:=\{x\in \mathbb{R}^{N}:\rho _{b}\le \vert x\vert \le \rho _{c}\}\). Thus,
for ε small enough, where
Direct computation yields that
where
Hence, by (4.28) and (4.29), we have
for \(\varepsilon >0\) small enough. Consequently, by (4.27), we have
for \(\varepsilon >0\) small enough. □
Let
Lemma 4.3
Assume that \(a\in C(\bar{\varOmega })\) with \(\vert a^{+}\vert _{q}\in (0,\sigma _{2})\). We have
-
(i)
\(X_{0}^{+}=\mathcal{A}_{1}\cup \mathcal{A}_{2}\cup \mathcal{N} ^{-}\);
-
(ii)
\(\mathcal{N}^{+}\subset \mathcal{A}_{1}\);
-
(iii)
for each \(\varepsilon <\varepsilon _{0}\) (\(\varepsilon _{0}\) is defined in Lemma 4.2), there exists \(t_{0}>1\) such that \(\tilde{u}_{1}+t_{0}u_{\varepsilon ,\mathbf{e}}\in \mathcal{A}_{2}\) for all \(\mathbf{e}\in \mathbb{S}^{N-1}\);
-
(iv)
for each \(\varepsilon <\varepsilon _{0}\), there exists \(s_{0} \in (0,1)\) such that \(\tilde{u}_{1}+s_{0}t_{0}u_{\varepsilon , \mathbf{e}}\in \mathcal{N}^{-}\) for all \(\mathbf{e}\in \mathbb{S}^{N-1}\);
-
(v)
\(m^{-}< m^{+}+\frac{s}{N}S_{s}^{\frac{N}{2s}}\).
Proof
(i) By Lemma 2.4(iv) we prove (i).
(ii) For any \(u\in \mathcal{N}^{+}\), by (2.6), we get that \(\int _{\varOmega }a(x)(u^{+})^{p}\,dx>0\). Let \(v=\frac{u}{\Vert u\Vert _{X_{0}}}\). By Lemma 2.4, there exists \(t^{-}(v)>0\) such that \(t^{-}(v)v \in \mathcal{N}^{-}\), that is,
Hence,
By Lemma 2.4, we have
Thus, we get \(\mathcal{N}^{+}\subset \mathcal{A}_{1}\).
(iii) We claim that there exists \(C>0\) such that \(\sup_{t \ge 0}t^{-} (\frac{\tilde{u}_{1}+tu_{\varepsilon ,\mathbf{e}}}{ \Vert \tilde{u}_{1}+tu_{\varepsilon ,\mathbf{e}}\Vert _{X_{0}}} )< C\). Assume by contradiction that there exists a sequence \(\{t_{n}\}\) such that \(t_{n}\to +\infty \) and \(t^{-}(v_{n})\to +\infty \) as \(n\to \infty \), where \(v_{n}:=\frac{\tilde{u}_{1}+t_{n}u_{\varepsilon , \mathbf{e}}}{\Vert \tilde{u}_{1}+t_{n}u_{\varepsilon ,\mathbf{e}}\Vert _{X _{0}}}\). Since \(t^{-}(v_{n})v_{n}\in \mathcal{N}^{-}\), by Lebesgue’s dominated convergence theorem, we have
as \(n\to \infty \). Thus,
as \(n\to \infty \), which is impossible since I is bounded from below on \(\mathcal{N}\) by Lemma 2.1. Set
Then
Hence, we get \(\tilde{u}_{1}+t_{0}u_{\varepsilon ,\mathbf{e}}\in \mathcal{A}_{2}\).
(iv) Define \(\gamma : [0,1]\to \mathbb{R}\) as
By Lemma 2.4(iii), \(\gamma (s)\) is a continuous function of s. Since \(\gamma (0)>1\) and \(\gamma (1)<1\) there exists \(s_{0} \in (0,1)\) such that \(\gamma (s_{0})=1\), that is, \(\tilde{u}_{1}+s _{0}t_{0}u_{\varepsilon ,\mathbf{e}}\in \mathcal{N}^{-}\).
(v) By Lemma 4.2 and (iv), we have \(m^{-}< m^{+}+\frac{s}{N}S _{s}^{\frac{N}{2s}}\). □
Consider the following critical problem:
We define the energy functional \(J: X_{0}\to \mathbb{R}\) associated with the critical problem (4.31) as
Set
and
Similarly, we define \(J_{\infty }: \dot{H}^{s}(\mathbb{R}^{N})\to \mathbb{R}\) as
where \(\dot{H}^{s}(\mathbb{R}^{N})\) denotes the space of functions \(u\in L^{p}(\mathbb{R}^{N})\) such that \(\int _{\mathbb{R}^{2N}}(u(x)-u(y))^{2}K(x-y)\,dx\,dy< \infty \). Set
and
It is easy to see that \(\gamma (\mathbb{R}^{N})=\frac{s}{N}S_{s}^{ \frac{N}{2s}}\).
Lemma 4.4
\(\gamma (\varOmega )=\gamma (\mathbb{R}^{N})\) and \(\gamma (\varOmega )\) is never achieved except when \(\varOmega =\mathbb{R}^{N}\).
The proof of Lemma 4.4 can be found in [19], and we give a proof for the reader’s’ convenience although these results are known.
Proof
Since \(\mathcal{M}(\varOmega )\subset \mathcal{M}(\mathbb{R}^{N})\), we have \(\gamma (\mathbb{R}^{N})\le \gamma (\varOmega )\). Conversely, let \(\{u_{n}\}\subset \dot{H}^{s}(\mathbb{R}^{N})\) be a minimizing sequence for \(\gamma (\mathbb{R}^{N})\). By density of \(C^{\infty }_{0}( \mathbb{R}^{N})\) in \(\dot{H}^{s}(\mathbb{R}^{N})\) we may assume that \(u_{n}\in C^{\infty }_{0}(\mathbb{R}^{N})\). We can choose \(y_{n} \in \mathbb{R}^{N}\) and \(\lambda _{n}>0\) such that
Since
we get \(\gamma (\varOmega )\le \gamma (\mathbb{R}^{N})\). Thus, \(\gamma (\varOmega )=\gamma (\mathbb{R}^{N})\).
Assume by contradiction that \(\varOmega \ne \mathbb{R}^{N}\) and \(u\in X_{0}\) is a minimizer for \(\gamma (\varOmega )\). Let \(t>0\) such that \(t\vert u\vert \in \mathcal{M}(\varOmega )\). Then
Consequently,
Thus, \(t=1\) and \(\vert u\vert \in \mathcal{M}(\varOmega )\) is another minimizer for \(\gamma (\varOmega )\). For this reason we assume straight away that \(u\ge 0\). Clearly, \(u\in \mathcal{\mathbb{R}^{N}}\) is a minimizer for \(J_{\infty }\). Therefore, we get that \(J'_{\infty }(u)=0\). So that u is a solution of
By the maximum principle (Proposition 2.2.8 in [33]), \(u>0\) in \(\mathbb{R}^{N}\). This is a contradiction. □
Lemma 4.5
If \(u\in \mathcal{N}^{-}\) satisfies \(I(u)\le m^{+}+\frac{s}{N}S_{s} ^{\frac{N}{2s}}\), then \(\int _{\varOmega }a(x)(u^{+})^{p}\,dx>0\).
Proof
Let \(u\in \mathcal{N}^{-}\) with \(I(u)\le m^{+}+\frac{s}{N}S_{s}^{ \frac{N}{2s}}\). Then there exists unique \(t(u)>0\) such that \(t(u)u\in \mathcal{M}(\varOmega )\). Assume by contradiction that \(\int _{\varOmega }a(x)(u^{+})^{p}\,dx\le 0\). By Lemmas 2.4 and 4.4,
Hence, by Lemma 3.1,
We get a contradiction. □
5 Existence of the other two solutions
For \(\mu >0\), we define
Lemma 5.1
For each \(u\in \mathcal{N}^{-}\), we have
-
(i)
there exists unique \(t_{\mu }(u)>0\) such that \(t_{\mu }(u)u\in \mathcal{N}_{\mu }\), and
$$ \sup_{t\ge 0}I_{\mu }(tu)=I_{\mu } \bigl(t_{\mu }(u)u\bigr)=\frac{s}{N} \biggl(\frac{ \Vert u \Vert _{X_{0}}^{2^{*}_{s}}}{\mu \int _{\varOmega } \vert u \vert ^{2^{*}_{s}}\,dx} \biggr) ^{\frac{N-2s}{2s}}; $$ -
(ii)
there exists unique \(t(u)>0\) such that \(t(u)u\in \mathcal{M}( \varOmega )\), and for \(c\in (0,1)\),
$$ J\bigl(t(u)u\bigr)\le (1-c)^{-\frac{N}{2s}} \biggl(I(u)+ \frac{2-p}{2p}c^{ \frac{p}{p-2}} \bigl( \bigl\vert a^{+} \bigr\vert _{q}S_{s}^{-\frac{p}{2}} \bigr)^{ \frac{2}{2-p}} \biggr). $$(5.1)
Proof
(i) The proof is standard, and we omit it.
(ii) Let \(\mu =(1-c)^{-1}\). Then, by Young’s inequality,
By Lemmas 3.1 and 2.4, we have \(I(u)\ge m^{-}>0\) and \(I(u)=\sup_{t\ge 0}I(tu)\). By (i), we have
Thus, we get (5.1). □
Lemma 5.2
There exists \(\delta _{0}>0\) such that, for \(u\in \mathcal{M}(\varOmega )\) with \(J(u)\le \frac{s}{N}S_{s}^{\frac{N}{2s}}+\delta _{0}\), we have
Proof
Assume by contradiction that there exists a sequence \(\{u_{n}\}\subset \mathcal{M}(\varOmega )\) such that
Without loss of generality, we can assume that \(\{u_{n}\}\) is a \((\mathit{PS})_{\gamma (\varOmega )}\)-sequence (for example, see Lemma 7 in [38]) for J. Since J is coercive on \(\mathcal{M}(\varOmega )\), there exists a subsequence of \(\{u_{n}\}\) (still denoted by \(\{u_{n}\}\)) and \(u_{0}\in X_{0}\) such that \(u_{n}\rightharpoonup u _{0}\) in \(X_{0}\). Since Ω is a bounded domain, we have \(u_{0}\equiv 0\). By Theorem 1.1 in [23] and Lemma 4.4, there exist ℓ nontrivial solutions \(v^{1},\ldots,v^{\ell } \in \dot{H}^{s}(\mathbb{R}^{N})\) to
or
where \(\ell \in \mathbb{N}\), sequences of points \(x^{1}_{n},\ldots, x^{\ell }_{n}\subset \varOmega \) and finitely many sequences of numbers \(r^{1}_{n},\ldots,r^{\ell }_{n}\subset (0,+\infty )\) converging to zero such that, up to a subsequence,
and
If \(\ell >1\), then by (5.6) we have \(J(u_{n})\to \sum^{\ell }_{j=1}J_{\infty }(v^{j})>\gamma (\varOmega )\), which is a contradiction. Thus, by (5.5),
By (H1), \(\vert x_{n}^{1}\vert \) is bounded from below. Hence, we may assume \(\frac{x_{n}^{1}}{\vert x_{n}^{1}\vert }\to \mathbf{e}\) as \(n\to \infty \), where \(\vert \mathbf{e}\vert =1\). By Lebesgue’s dominated convergence theorem, we have
which is impossible. □
Lemma 5.3
There exists \(\sigma _{0}\in (0,\sigma _{2})\) such that, for \(\vert a^{+}\vert _{q} \in (0,\sigma _{0})\), we have
for all \(u\in \mathcal{N}^{-}\) with \(I(u)< m^{+}+\frac{s}{N}S_{s}^{ \frac{N}{2s}}\).
Proof
For \(u\in \mathcal{N}^{-}\) with \(I(u)< m^{+}+\frac{s}{N}S_{s}^{ \frac{N}{2s}}\), there exists \(t(u)>0\) such that \(t(u)u\in \mathcal{M}( \varOmega )\). By Lemma 5.1(ii), for any \(c\in (0,1)\), we have
since \(m^{+}<0\) by Lemma 3.1. Thus, there exists \(\sigma _{0} \in (0,\sigma _{2})\) such that, for \(a\in C(\bar{\varOmega })\) with \(\vert a^{+}\vert _{q}\in (0,\sigma _{0})\),
where \(\delta _{0}\) is given in Lemma 5.2. Consequently, by Lemma 5.2,
Hence, we complete the proof. □
Now, we use Lusternik and Schnirelmann’s theory in order to obtain multiplicity results. The notion of category was introduced by Lusternik and Schnirelmann. It is a topological tool used in the estimate of the lower bounded of the number of critical points of a functional.
Definition 5.4
Let \(\mathfrak{X}\) be a topological space. A closed subset A of \(\mathfrak{X}\) is contractible in \(\mathfrak{X}\) if there exists \(h\in C([0,1]\times A,\mathfrak{X})\) and \(v\in \mathfrak{X}\) such that, for every \(u\in A\),
Definition 5.5
The (L–S) category of A with respect to \(\mathfrak{X}\) (or simply the category of A with respect to \(\mathfrak{X}\)), denoted by \(\operatorname{cat}_{ \mathfrak{X}}(A)\), is the least integer k such that \(A\subset A_{1}\cup \cdots \cup A_{k}\), with \(A_{i}\) (\(i=1,\ldots,k\)) closed and contractible in \(\mathfrak{X}\).
We set \(\operatorname{cat}_{\mathfrak{X}}(\emptyset )=0\) and \(\operatorname{cat}_{\mathfrak{X}}(A)=+ \infty \) if there are no integers with the above property. We will use the notation \(\operatorname{cat}(\mathfrak{X})\) for \(\operatorname{cat}_{\mathfrak{X}}(\mathfrak{X})\). For fundamental properties of Lusternik–Schnirelmann category, we refer to Ambrosetti [2], Schwartz [27], and Chang [10].
Theorem 5.6
(Lusternik–Schnirelmann theorem)
Let M be a smooth Banach–Finsler manifold. Suppose that \(f\in C ^{1}(M,\mathbb{R})\) is a functional bounded from below, satisfying the \((\mathit{PS})\) condition. Then f has at least \(\operatorname{cat}(M)\) critical points.
We say f satisfies the \((\mathit{PS})\) condition if any sequence \(\{u_{n}\} \subset M\), such that
has a converging subsequence.
The following lemma is from [1].
Lemma 5.7
Let \(\mathfrak{X}\) be a topological space. Suppose that there exist two continuous maps
such that \(G\circ F\) is homotopic to identity map of \(\mathbb{S}^{N-1}\), that is, there exists \(\xi \in C([0,1]\times \mathbb{S}^{N-1}, \mathbb{S}^{N-1})\) such that
Then
For \(\varepsilon <\varepsilon _{0}\) (\(\varepsilon _{0}\) is defined in Lemma 4.2), we define a map \(\varPhi : \mathbb{S}^{N-1}\to X_{0}\) by
where \(s_{0}\), \(t_{0}\) are given in Lemma 4.3.
Lemma 5.8
\(\varPhi (\mathbb{S}^{N-1})\) is compact.
Proof
Let \(\{\mathbf{e}_{n}\}\subset \mathbb{S}^{N-1}\) be a sequence such that \(\mathbf{e}_{n}\to \mathbf{e}_{0}\) as \(n\to \infty \). Using a similar argument as that in the proof of Lemma 4.1 and Lebesgue’s dominated convergence theorem, we obtain \(\Vert u_{\varepsilon , \mathbf{e}_{n}}\Vert _{X_{0}}\to \Vert u_{\varepsilon ,\mathbf{e}_{0}}\Vert _{X _{0}}\) as \(n\to \infty \). Since \(X_{0}\) is a Hilbert space and \(u_{\varepsilon ,\mathbf{e}_{n}}\rightharpoonup u_{\varepsilon , \mathbf{e}_{0}}\), we get \(\Vert u_{\varepsilon ,\mathbf{e}_{n}}-u_{\varepsilon ,\mathbf{e}_{0}}\Vert _{X_{0}}\to 0\). Consequently, \(\varPhi (\mathbf{e}_{n}) \to \varPhi (\mathbf{e}_{0})\). □
For \(c\in \mathbb{R}\), we define
Lemma 5.9
There exists \(d_{\varepsilon }\in (0,m^{+}+\frac{s}{N}S_{s}^{ \frac{N}{2s}} )\) such that \(\varPhi (\mathbb{S}^{N-1})\subset I ^{d_{\varepsilon }}\) for each \(\varepsilon \in (0, \varepsilon _{0})\).
Proof
By Lemmas 4.2 and 4.3(iii), for each \(\varepsilon \in (0,\varepsilon _{0})\), we have \(\tilde{u}_{1}+s_{0}t_{0}u_{\varepsilon ,\mathbf{e}}\in \mathcal{N}^{-}\) and
uniformly in \(\mathbf{e}\in \mathbb{S}^{N-1}\). Since \(\varPhi ( \mathbb{S}^{N-1})\) is compact by Lemma 5.8, there exists \(d_{\varepsilon }\in (0, m^{+}+\frac{S}{N}S_{s}^{\frac{N}{2s}} )\) such that \(\varPhi ( \mathbb{S}^{N-1})\subset I^{d_{\varepsilon }}\). □
Set \(\beta =m^{+}+\frac{s}{N}S_{s}^{\frac{N}{2s}}\) and define \(\varPsi : I^{\beta }\to \mathbb{S}^{N-1}\) by
By Lemma 5.3, Ψ is well-defined. Let
We define \(\widetilde{\varPsi }: \varSigma \to \mathbb{S}^{N-1}\) by
Clearly, Ψ̃ is an extension of Ψ.
Lemma 5.10
\(u_{\varepsilon ,\mathbf{e}}\in \varSigma \) for all \(\mathbf{e}\in \mathbb{S}^{N-1}\) and for ε small enough.
Proof
For every \(u_{\varepsilon ,\mathbf{e}}\), one sees immediately that there exists \(t(\varepsilon ,\mathbf{e})>0\) such that \(t(\varepsilon , \mathbf{e})u_{\varepsilon ,\mathbf{e}}\in \mathcal{M}(\varOmega )\). Indeed, \(t(\varepsilon ,\mathbf{e})u_{\varepsilon ,\mathbf{e}}\in \mathcal{M}( \varOmega )\) is equivalent to
which is solved by
By Lemma 4.1, we have
uniformly in \(\mathbf{e}\in \mathbb{S}^{N-1}\). Thus,
uniformly in \(\mathbf{e}\in \mathbb{S}^{N-1}\). By Lemma 5.2, we get \(t(\varepsilon ,\mathbf{e})u_{\varepsilon ,\mathbf{e}}\in \varSigma \) for \(\varepsilon >0\) small enough. Consequently, \(u_{\varepsilon , \mathbf{e}}\in \varSigma \). □
Lemma 5.11
\(\varPsi \circ \varPhi : \mathbb{S}^{N-1}\to \mathbb{S}^{N-1}\) is homotopic to the identity.
Proof
By Lemma 5.10, there exists \(\varepsilon ^{*}\in (0,\varepsilon _{0})\) such that, for \(\varepsilon \in (0,\varepsilon ^{*})\), \(u_{\varepsilon ,\mathbf{e}}\in \varSigma \) and \(u_{2(1-\theta )\varepsilon ,\mathbf{e}}\in \varSigma \) for all \(\mathbf{e}\in \mathbb{S}^{N-1}\) and \(\theta \in [\frac{1}{2},1)\). Let \(\gamma : [s_{1},s_{2}] \to \mathbb{S}^{N-1}\) be a regular geodesic between \(\widetilde{\varPsi }(u_{\varepsilon ,\mathbf{e}})\) and \(\widetilde{\varPsi }(\varPhi (\mathbf{e}))\) such that
Define \(\xi : [0,1]\times \mathbb{S}^{N-1}\to \mathbb{S}^{N-1}\) by
Set \(\tilde{x}=(x-\tilde{\rho }\mathbf{e})/(2(1-\theta )\varepsilon )\), \(\tilde{y}=(y-\tilde{\rho }\mathbf{e})/(2(1-\theta )\varepsilon )\). Then
as \(\theta \to 1^{-}\) by (4.4) and Lebesgue’s dominated convergence theorem. Consequently,
Clearly, \(\xi (\theta ,\mathbf{e})\to \gamma (s-1)=\widetilde{\varPsi }(u _{\varepsilon ,\mathbf{e}})\) as \(\theta \to \frac{1}{2}^{-}\). Thus, \(\xi \in C([0,1]\times \mathbb{S}^{N-1},\mathbb{S}^{N-1})\), and
for all \(\mathbf{e}\in \mathbb{S}^{N-1}\). □
Proof of Theorem 1.1
By Lemmas 5.7, 5.9, and 5.11, there exists \(d_{\varepsilon }\in (0,m^{+}+\frac{s}{N}S_{s}^{\frac{N}{2s}} )\) such that
By Lemma 3.2 and Theorem 5.6, I has at least two critical points \(\tilde{u}_{2}\) and \(\tilde{u}_{3}\) in \(\{u\in \mathcal{N}^{-}: I(u)< m^{+}+\frac{s}{N}S_{s}^{\frac{N}{2s}}\}\). By the maximum principle (Proposition 2.2.8 in [33]), \(\tilde{u}_{2}\) and \(\tilde{u} _{3}\) are strictly positive in Ω. By Theorem 3.3, we get three positive solutions \(\tilde{u}_{i}\) (\(i=1,2,3\)) of (1.1). By (2.5) and Lemma 4.5, we have \(\int _{\varOmega }a(x) \tilde{u}^{p}_{i}>0\), \(i=1,2,3\). □
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Pang, L., Li, X. & Zhang, Y. Existence of multiple positive solutions for fractional Laplace problems with critical growth and sign-changing weight in non-contractible domains. Bound Value Probl 2019, 81 (2019). https://doi.org/10.1186/s13661-019-1193-1
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DOI: https://doi.org/10.1186/s13661-019-1193-1