Abstract
In this paper, we consider the existence of eigenvalues and relative eigenfunctions for Carrier equations and present spectral asymptotics and bifurcation concerning the eigenvalues of some related elliptic linear problem.
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1 Introduction
In this paper, we consider the following nonlocal elliptic problem
where \(\varOmega \subseteq {\mathbb {R}}^N\) (\(N\ge 1\)) is a smooth and bounded domain, and \(a>0\), \(b>0\).
Problem (1.1) is related to the stationary analogue of the equation
proposed by Carrier [6] which describes the vibration of the elastic string when the change of the tension is not very little.
For the case \(b=0\), problem (1.1) is changed as
and some authors considered the spectral asymptotics, bifurcation and the normalized solutions for problem (1.2) via variational method, see [5, 7, 8, 18,19,20,21, 23,24,25].
Since \(-\left( a+ b\int _{\varOmega }|u(x)|^2dx\right) \varDelta u\) is lack of variational structure, it is difficult to study problem (1.1) via variational method. Some authors focus on the existence of positive solutions for problem (1.1) or some generalized cases only via the theory of topological theory, the method of lower and upper solutions and pseudomontone operators theory when \(\lambda\) is fixed, see [1,2,3, 9,10,11,12,13, 26,27,28]. For examples in [26] and [27], authors considered the following problem
where \(\gamma \ge 1\), \(0<q\le 1\), \(p>1\), \(a:{\mathbb {R}}\rightarrow (0,+\infty )\) is a continuous function with \(\inf _{t\in {\mathbb {R}}}a(t)=a(0)>0\); using the theory of fixed point index on cone, the authors proved that there exist \(0<\lambda _1\le \lambda _2\) such that (1.3) has no positive solutions for \(\lambda >\lambda _2\), at least a positive solution for \(\lambda =\lambda _1\) and \(\lambda _2\) and at least two positive solutions for \(\lambda \in (0,\lambda _1)\); in [14], combing sub-super and bifurcation methods, the authors showed that there exists a drastic change on the structure of the set of positive solutions when the non-local coefficient grows fast enough to infinity for problem (1.3).
Our aim is to present some results on spectral asymptotics and bifurcation for problem (1.1).
This paper is organized as follows. In Sect. 2, using the Liusternik–Schnirelmann (LS) theory, we obtain, given any \(r>0\), the existence of infinitely many eigenvalues \(\mu _{n,r}\)( \(n=1, 2, \cdots\)) for problem (1.1) associated with eigenfunctions \(u_{n,r}\) satisfying \(\int _{\varOmega }u_{n,r}^2(x)dx=r^2\). And then Sect. 3 presents bifurcation and comparison results concerning the eigenvalues of some related linear problems \((2.1)_{\lambda }\). In Sect. 4, we discuss the asymptotic laws of the eigenvalues \(\mu _{n,r}\) of problem (1.1) as \(n\rightarrow +\infty\) when f is superlinear at \(+\infty\). Our paper was motivated in part by the papers [7, 8, 15, 16, 18, 21, 22].
2 Existence of the eigenvalues of problem (1.1)
It is easy to see that problem (1.1) is equivalent to its weak formulation, namely that of finding \(u\in W_0^{1,2}(\varOmega )\) and \(\lambda \in R\) such that
for all \(v\in W_0^{1,2}(\varOmega )\), where \(W_0^{1,2}(\varOmega )\) denote the closure of \(C_0^{\infty }(\varOmega )\) in the Sobolev space \(W^{1,2}(\varOmega )\) with the scalar product \((u,u) =\int _{\varOmega }\nabla u\cdot \nabla u dx\) and the corresponding norm \(\Vert u\Vert =(\int _{\varOmega }|\nabla u|^2dx)^{\frac{1}{2}}\), while \(\Vert u\Vert _p\) denotes the norm of \(u\in L^p(\varOmega )\).
For \(r>0\), let
and for each \(n = 1\), 2, \(\ldots\), set
where \(\gamma (K)\) denotes the genus of K. For fixed \(r>0\) and for \(u\in W_0^{1,2}(\varOmega )\), define
and
where
It is well known that the linear elliptic problem
has eigenvalues \(\lambda _1<\lambda _2\le \cdots \le \lambda _n\le \cdots\) and the corresponding eigenfunction to \(\lambda _n\) is \(u_n\) with \(u_n\in M_r\), see [7]. For each eigenvalue \(\lambda _n\), multiplying \(u_n\) and integrating on \(\varOmega\) for \((2.1)_{\lambda }\), we have
Since the set of all eigenfunctions corresponding to \(\lambda _n\) is a linear space, if we choose \(v_n\) is a eigenfunction of \(\lambda _n\) with \(\int _{\varOmega }|v_n|^2dx=1\), then the eigenfunction \(u_n\) of \(\lambda _n\) with \(u_n\in M_r\) can be written as \(u_n=l_nv_n\). From
we get \(l_n=\pm r\), i.e.,
which together with (2.2) gives
and so
Now, we introduce (see [4]) the “LS critical levels”
The following lemma is needed in our proof.
Lemma 2.1
(See [8]) Let \(p:1\le p\le p_0=(N + 2)/(N - 2)\) (so that \(2\le p+1\le 2^*\)) and let \(\beta =(N/2^*)(2^*-(p+1))\). Then, for each \(\gamma : 0\le \gamma \le \beta\), there exists \(c > 0\) such that
for all \(u\in W_0^{1,2}(\varOmega )\). (Here and henceforth \(\Vert u\Vert _p\) denotes the norm of u in \(L^p(\varOmega )\).)
We will consider the following condition:
\((A_1) f:\varOmega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) is continuous, \(f(x,-u)=-f(x,u)\) and satisfies
for some c, \(d\ge 0\) and some \(0\le p<{\overline{p}}=\min \{2^*-1,1+4/N\}\).
From the LS theory, we have the following existence result.
Theorem 2.1
Assume \((A_1)\) holds. Then, for given \(r>0\), there exists a sequence \(\{u_{n,r}\}\) of (weak) eigenfunctions of (1.1) belonging to \(M_r\), and such that
where \(c_{n,r}\) is as in (2.4); the eigenvalue \(\mu _{n,r}\) corresponding to \(u_{n,r}\) satisfies
Proof
The proof is divided into three steps.
Step 1. We show that
First, \((A_1)\) and Schwarz’s inequality imply that
for some new constants c, \(d>0\).
Moreover, we use the inequality (2.5) with \(\gamma =\beta\): on setting \(2\alpha = p + 1 -\beta =( p - 1)N/2\), (2.6) becomes
Next, from (2.6) and (2.7), for \(u\in M_r\), we have
which together with the compactmess of \(K\subset K_{n,r}\) implies that
Finally, from (2.6) and (2.7), for \(u\in M_r\), we have also
The assumption \(p<\min \{2^*-1,1+4/N\}\) is equivalent to \(2\alpha <2\), which implies that I is bounded below on \(M_r\) (for each r).
Consequently,
(2) We show that I satisfies the Palais-Smale condition (PS) on \(M_r\), i.e., for \(c\not =0\), \(\varepsilon >0\) small enough, \(u_n\in I^{-1}[c-\varepsilon ,c+\varepsilon ]\cap M_r\) and \(\Vert I_{M_r}'(u_n)\Vert \rightarrow 0\), then there is a \(u\in M_r\) and a subsequence \(\{u_{n_j}\}\) such that
Now (2.9) and the boundedness of \(\{I(u_n)\}\) with \(\{u_n\}\subseteq M_r\) guarantees that \(\{u_n\}\) is bounded \(W_0^{1,2}(\varOmega )\), which implies that there exist \(u^*\in W_0^{1,2}(\varOmega )\) and subsequence \(\{u_{n_j}\}\) of \(\{u_n\}\) such that \(u_{n_j}\rightharpoonup u^*\), as \(j\rightarrow +\infty\). Since
we have
Hence
(3) We show that \(c_{n,r}\) is a critical value of I(u) in \(M_r\), i.e., there exists a \(u_{n,r}\in M_r\) such that \(c_{n,r} =2I(u_{n,r})\) and \(I|_{M_r}'(u_{n,r})=0\).
First, we show that \(\forall \varepsilon _k\downarrow 0^+\), there exists \(u_{k}\in 2I^{-1}[c_{n,r}-\varepsilon _k,c_{n,r}+\varepsilon _k]\) such that \(I'_{M_r}(u_{k})=0\).
On the contrary, suppose that there is a \(\varepsilon _0>0\) such that \(2I^{-1}[c_{n,r}-\varepsilon _0,c_{n,r}+\varepsilon _0]\cap K=\emptyset\), where \(K=\{u\in M_r|I|_{M_r}'(u)=0\}\). Let \(A_c=\{u|2I(u)\le c\}\) and \(K_c=\{u|2I(u)=c, I|_{M_r}'(u)=\theta \}\). From [17], let N be a neighourhood of \(K_c\), there exists a \(\eta (t,u)=\eta _t(u)\in C([0,1]\times W_0^{1,2}(\varOmega ),W_0^{1,2}(\varOmega ))\) and \(\varepsilon _0>\varepsilon >0\) such that
-
(a)
\(\eta _0(u)=u\) for all \(u\in W_0^{1,2}(\varOmega )\);
-
(b)
\(\eta _t(u)=u\) for all \(u\in 2I^{-1}[c_{n,r}-\varepsilon _0,c_{n,r}+\varepsilon _0]\) and for all \(t\in [0,1]\);
-
(c)
\(\eta _t(u)\) is a homeomorphism from \(W_0^{1,2}(\varOmega )\) onto \(W_0^{1,2}(\varOmega )\) for all \(t\in [0,1]\);
-
(d)
\(I(\eta _t(u))\le I(u)\) for all \(u\in W_0^{1,2}(\varOmega )\), for all \(t\in [0,1]\);
-
(e)
\(\eta _1(A_{c+\varepsilon }-N)\subset A_{c-\varepsilon }\);
-
(f)
If \(K_c=\emptyset\), \(\eta _1(A_{c+\varepsilon })\subset A_{c-\varepsilon }\);
-
(g)
If f is even, \(\eta _t\) is odd in u.
Since \(c_{n,r}=\inf _{K_{n,r}}\sup _K2I<+\infty\), for \(0<\varepsilon <\varepsilon _0\), there is a \(A_n\subseteq M_r\) such that \(c_{n,r}\le \sup _{u\in A_n}2I(u)\le c_{n,r}+\varepsilon\). Let c be replaced by \(c_{n,r}+\varepsilon\) in the above (a)-(g). It infers from (b) that \(\gamma (A_n)=n\) and \(\gamma (\eta _1(A_n))=\gamma (A_n)=n\). Since \(2I^{-1}[c_{n,r}-\varepsilon _0,c_{n,r}+\varepsilon _0]\cap K=\emptyset\) and \(\varepsilon <\varepsilon _0\), from (f), we have \(\eta _1(A_{c_{n,r}+\varepsilon })\subset A_{c_{n,r}-\varepsilon }\), which together with \(A_n\subset 2I^{-1}[c_{n,r}-\varepsilon ,c_{n,r}+\varepsilon ]\subseteq A_{c_{n,r}+\varepsilon }\) guarantees that \(\eta _1(A_{n})\subset A_{c_{n,r}-\varepsilon }\) also. Hence,
This is contradiction.
Second, obviously, \(\{I(u_k)\}\) is bounded and \(\{I'_{M_r}(u_k)=0\}\). The Palais-Smale condition implies that \(\{u_k\}\) has a convergent subsequence. Without loss of generality, we assume that
It is easy to see that \(u_{n,r}\in M_r\) such that
and
Let \(\mu _{n,r}=r^{-2}I'(u_{n,r})(u_{n,r})\). Note one has
By \(u_{n,r}\in M_r\), (2.6) becomes
i.e. problem (1.1) has a sequence eigenvalues \(\{\mu _{n,r}\}\) with corresponding eigenfunctions \(\{u_{n,r}\}\). Let \(v=u_{n,r}\). Then (2.10) becomes
The proof is completed. \(\square\)
Corollary 2.1
Let \(f\equiv 0\) and equation (1.1) becomes
Then, \((2.11)_{\lambda }\) has branches
Proof
From the L-S procedure in Theorem 2.1, \((2.11)_{\lambda }\) has exactly the eigenvalues \(\mu ^0_{n,r}\) with the corresponding eigenfunction \(u^0_{n,r}\)(\(\Vert u^0_{n,r}\Vert _2=r\)) which satisfies
Comparing \((2.11)_{\lambda }\) with \((2.1)_{\lambda }\), we get
and \(u^0_{n,r}=k_nu_n\), where \(u_n\) is the corresponding eigenvalue function to \(\lambda _n\) of \((2.1)_{\lambda }\) with \(\Vert u_n\Vert _2=r\). Moreover,
Since \(u^0_{n,r}=k_nu_n\), one has
which implies \(k_n=\pm 1\) and \(u^0_{n,r}=\pm u_n\). Hence, (2.12) becomes
From (2.2), we have
and so
which together with (2.3) implies that \((2.11)_{\lambda }\) has branches
The proof is completed. \(\square\)
3 Bifurcation results concerning the eigenvalues of some related linear problem to (1.1)
In Sect. 2, we obtained the branches of solutions of (1.1) when \(f\equiv 0\). Now we consider the case \(f\not \equiv 0\).
Theorem 3.1
Let the assumptions of Theorem 2.1 be satisfied with \(p > 1\) and \(d =0\) in the growth assumption \((A_1)\). Then each \(a\lambda _n\) is a bifurcation point (in \(W_0^{1,2}(\varOmega )\)) for (1.1); more precisely, for each \(n = 1\), 2, \(\cdots\), the eigenvalue-eigenfunction pairs \((\mu _{n,r}, u_{n,r})\) given by Theorem 2.1 satisfy \(\mu _{n,r}=a\lambda _n+b\lambda _nr^2+O(r^{\min \{2,p-1\}})\) as \(r\rightarrow 0\).
Proof
Let \(\gamma =p-1\). Then (see Lemma 2.1) we have
Note (\(d=0\) in \((A_1)\))
Since
from (3.1), we have
Hence,
It infers from (3.2) that
and so
i.e.
which implies that
Now (2.9) guarantees that
and so
It deduces from Theorem 2.1 and (3.2) that
which together with (3.3) implies that
From Theorem 2.1 and (3.1), (3.4), one has
Then
which implies that
Consequently,
The proof is completed. \(\square\)
4 The asymptotic distribution of the eigenvalue \(\mu _{n,r}\) of (1.1)
In this section, we consider the asymptotic laws of the eigenvalue \(\mu _{n,r}\) of (1.1).
Lemma 4.1
Assume \((A_1)\) holds. For \(r>0\) and \(n = 1,2, . . .\), let \(\mu _{n,r}\), \(c_{n,r}\) be as in Theorem 2.1 , and let \(\lambda _n\) be the eigenvalues of the linear problem \((2.1)_{\lambda }\). Then
and
where \(\alpha =(p-1)N/4\) and \(\beta =(p+1)-(p-1)N/2\); here and henceforth c, d denote some, but not always the same, positive constants.
Proof
First notice that the growth assumption \((A_1)\) implies
and similarly
Next, as \(1\le p<{\overline{p}}\), from Lemma 2.1, if \(\int _{\varOmega }u^2dx=r^2\), we have
and similarly
with \(\alpha\) and \(\beta\) as in the statement of Lemma 4.1.
To prove (4.1), observe that (4.3) implies
holds ( c instead of \(c (\frac{1}{a})^{\alpha }\)). In other words, we have
where \(g: R^+\rightarrow R^+\) is defined by
As g is continuous and nondecreasing, we get
Now Theorem 2.1 implies that
for some new constants c and \(d>0\). Therefore,
which shows (4.1) is true.
Since
we have
It deduces from Theorem 2.1 and (4.3)-(4.4) that
which completes the proof of the lemma. \(\square\)
Lemma 4.2
(Theorem 2, [5]) The eigenvalues \(\lambda _{n}\) of \((2.1)_{\lambda }\) satisfy, as \(n\rightarrow +\infty\)
where
and V is the value of \(B(\theta ,1)\).
Theorem 4.1
Assume that \((A_1)\) holds. Then given any \(r > 0\), (1.1) has infinitely many eigenfunctions \(u_{n,r} (n = 1,2, . . .)\) with \(\int _{\varOmega }u^2_{n,r}dx=r^2\), whose corresponding eigenvalues \(\mu _{n,r}\) satisfy, as \(n\rightarrow +\infty\) and with k as in (4.7),
where \({\overline{p}}\) is defined in \((A_1)\).
Proof
Since condition \((A_1)\) is true, Theorem 2.1 guarantees that for given any \(r > 0\), (1.1) has infinitely many eigenfunctions \(u_{n,r} (n = 1,2, . . .)\) with \(\int _{\varOmega }u^2_{n,r}dx=r^2\).
Now \(p<{\overline{p}}=\min \{2^*-1,1+4/N\}\) guarantees that \(\alpha =(p-1)(N/4)<1\). Thus, (4.3) guarantees that
and
which together (4.1) and (4.2) implies that
and so
Consequently
Since
we have
The proof is completed. \(\square\)
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This article is part of the topical collection dedicated to Prof. Dajun Guo for his 85th birthday, edited by Yihong Du, Zhaoli Liu, Xingbin Pan, and Zhitao Zhang.
This work is supported by the National Natural Science Foundation of China(61603226) and the Fund of Natural Science of Shandong Province (ZR2018MA022).
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Zhang, Y., Yan, B. On spectral asymptotics and bifurcation for Carrier equations with odd superlinear term. SN Partial Differ. Equ. Appl. 1, 30 (2020). https://doi.org/10.1007/s42985-020-00026-y
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DOI: https://doi.org/10.1007/s42985-020-00026-y