1 Introduction

In this paper, we consider the following nonlocal elliptic problem

$$\begin{aligned} \displaystyle -\left( a+b\int _{\varOmega }|u(x)|^2dx\right) \varDelta u+f(x,u)=\lambda u,\ \ x \hbox { in } \varOmega ,\\ u(x)=0,\ \ x\hbox { on }\partial \varOmega, \end{aligned}$$
(1.1)

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

$$\begin{aligned} u_{tt}-\left( a+b\int _{0}^{\pi }|u|^2dx\right) u_{xx}=0 \end{aligned}$$

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

$$\begin{aligned} \displaystyle - a\varDelta u+f(x,u)=\lambda u,\ \ x \hbox { in } \varOmega ,\\ u(x)=0,\ \ x\hbox { on }\partial \varOmega . \end{aligned}$$
(1.2)

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

$$\begin{aligned} \displaystyle -a\left( \int _{\varOmega }|u(x)|^{\gamma }pdx\right) \varDelta u=\lambda u^q+u^p,\ \ x \hbox { in } \varOmega ,\\ u(x)=0,\ \ x\hbox { on }\partial \varOmega , \end{aligned}$$
(1.3)

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

$$\begin{aligned} \left( a+b\int _{\varOmega }u^2(x)dx\right) \int _{\varOmega }\nabla u\cdot \nabla vdx+\int _{\varOmega }f(x,u)vdx=\lambda \int _{\varOmega }uvdx \end{aligned}$$

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

$$\begin{aligned} M_r:=\left\{ u\in W_0^{1,2}(\varOmega )|\int _{\varOmega }u^2dx=r^2\right\} \end{aligned}$$

and for each \(n = 1\), 2, \(\ldots\), set

$$\begin{aligned} K_{n,r} = \{K \subseteq M_r: K \hbox { compact, symmetric}, \gamma (K) = n \} \end{aligned}$$

where \(\gamma (K)\) denotes the genus of K. For fixed \(r>0\) and for \(u\in W_0^{1,2}(\varOmega )\), define

$$\begin{aligned} \varPhi (u):=(a+br^2)\frac{1}{2}\Vert \nabla u\Vert _2^2,\ \ \varPsi (u):=\int _{\varOmega }F(x,u(x))dx \end{aligned}$$

and

$$\begin{aligned} I(u):=\varPhi (u)+\varPsi (u), \end{aligned}$$

where

$$\begin{aligned} F(x,u(x))=\int _0^{u(x)}f(x,s)ds. \end{aligned}$$

It is well known that the linear elliptic problem

$$\begin{aligned} {-}\varDelta u=\lambda u,\ \ &{} x \hbox { in } \varOmega ,\\ u(x)=0,\ \ &{} x\hbox { on }\partial \varOmega , \quad (2.1)_{\lambda } \end{aligned}$$

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

$$\begin{aligned} r^2\lambda _n=\int _{\varOmega }u^2_ndx\lambda _n=\int _{\varOmega }|\nabla u_n|^2dx. \end{aligned}$$
(2.2)

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

$$\begin{aligned} r^2=\int _{\varOmega }|u_n|^2dx=\int _{\varOmega }|l_nv_n|^2dx=l_n^2\int _{\varOmega }|v_n|^2dx, \end{aligned}$$

we get \(l_n=\pm r\), i.e.,

$$\begin{aligned} u_n=\pm r v_n, \ \ n=1,2,\ldots , \end{aligned}$$
(2.3)

which together with (2.2) gives

$$\begin{aligned} r^2\lambda _n=\int _{\varOmega }|\nabla u_n|^2dx=\int _{\varOmega }|\nabla (\pm r v_n)|^2dx=r^2\int _{\varOmega }|\nabla v_n|^2dx, \end{aligned}$$

and so

$$\begin{aligned} \lambda _n=\int _{\varOmega }|\nabla v_n|^2dx. \end{aligned}$$

Now, we introduce (see [4]) the “LS critical levels”

$$\begin{aligned} c_{n,r}: =\inf _{K_{n,r}}\sup _K2I. \end{aligned}$$
(2.4)

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

$$\begin{aligned} \Vert u\Vert _{p+1}^{p+1}\le c\Vert \nabla u\Vert _2^{p+1-\gamma }\Vert u\Vert _2^{\gamma } \end{aligned}$$
(2.5)

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

$$\begin{aligned} |f(x,u)|\le c|u|^p+d \end{aligned}$$

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

$$\begin{aligned} 2I(u_{n,r}) =c_{n,r} \end{aligned}$$

where \(c_{n,r}\) is as in (2.4); the eigenvalue \(\mu _{n,r}\) corresponding to \(u_{n,r}\) satisfies

$$\begin{aligned}r^2 \mu _{n,r}=(a+br^2)\Vert \nabla u_{n,r}\Vert _2^2+\int _{\varOmega }f(x,u_{n,r})u_{n,r}dx. \end{aligned}$$

Proof

The proof is divided into three steps.

Step 1. We show that

$$\begin{aligned} -\infty<c_{n,r} =\inf _{K_{n,r}}\sup _K2I<+\infty . \end{aligned}$$

First, \((A_1)\) and Schwarz’s inequality imply that

$$\begin{aligned} \int _{\varOmega }|F(x,u(x))|dx\le c\int _{\varOmega }|u|^{p+1}dx+d\left( \int _{\varOmega }|u|^2dx\right) ^{\frac{1}{2}} \end{aligned}$$
(2.6)

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

$$\begin{aligned} \int _{\varOmega }|u|^{p+1}dx\le c'\Vert \nabla u\Vert _2^{2\alpha }\left( \int _{\varOmega }u^2dx\right) ^{\frac{\beta }{2}}. \end{aligned}$$
(2.7)

Next, from (2.6) and (2.7), for \(u\in M_r\), we have

$$\begin{aligned} I(u)&{}e \le (a+br^2)\frac{1}{2}\Vert \nabla u\Vert _2^2+\int _{\varOmega }F(x,u(x))dx\\ &{} \le (a+br^2)\frac{1}{2}\Vert \nabla u\Vert _2^2+c\int _{\varOmega }|u|^{p+1}dx+d\left( \int _{\varOmega }|u|^2dx\right) ^{\frac{1}{2}}\\ &{} \le (a+br^2)\frac{1}{2}\Vert \nabla u\Vert _2^2+c'\Vert \nabla u\Vert _2^{2\alpha }(r^\beta )+dr,\\ \end{aligned}$$

which together with the compactmess of \(K\subset K_{n,r}\) implies that

$$\begin{aligned} \sup _{u\in K}2I(u)<+\infty . \end{aligned}$$
(2.8)

Finally, from (2.6) and (2.7), for \(u\in M_r\), we have also

$$\begin{aligned} I(u)& \ge (a+br^2)\frac{1}{2}\Vert \nabla u\Vert _2^2-\int _{\varOmega }|F(x,u(x))|dx\\ & \ge (a+br^2) \frac{1}{2}\Vert \nabla u\Vert _2^2 -[cc'r^{\beta }\Vert \nabla u\Vert _2^{2\alpha }+dr]. \end{aligned}$$
(2.9)

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,

$$\begin{aligned} -\infty<c_{n,r} =\inf _{K_{n,r}}\sup _K2I<+\infty . \end{aligned}$$

(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

$$\begin{aligned} \Vert \nabla (u_{n_j}-u)\Vert _2\rightarrow 0. \end{aligned}$$

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

$$\begin{aligned} I_{M_r}'(u)(v)& =I'(u)(v)-r^{-2}I'(u)(u)\displaystyle\int _{\varOmega }uvdx\\ & =(a+br^2)\displaystyle\int _{\varOmega }\nabla u\nabla vdx+\displaystyle\int _{\varOmega }f(x,u)vdx\\ &\ \ \ \ -r^{-2}\left( (a+br^2)\Vert \nabla u\Vert _2^2+\displaystyle\int _{\varOmega }f(x,u)udx\right) \displaystyle\int _{\varOmega }uvdx, \ \ u,v\in W_0^{1,2}(\varOmega ),\\ \end{aligned}$$

we have

$$\begin{aligned} &(a+br^2)\displaystyle\int _{\varOmega }\nabla u_{n_j}\nabla (u_{n_j}-u^*)dx\\ & =I'_{M_r}(u_{n_j})(u_{n_j}-u^*) -\displaystyle\int _{\varOmega }f(x,u_{n_j})(u_{n_j}-u^*)dx\\ & \ \ \ \ +r^{-2}\left( (a+br^2)\Vert \nabla u_{n_j}\Vert _2^2+\displaystyle\int _{\varOmega }f(x,u_{n_j})u_{n_j}dx\right) \displaystyle\int _{\varOmega }u_{n_j}(u_{n_j}-u^*)dx\\ & \ \ \ \ \rightarrow 0. \end{aligned}$$

Hence

$$\begin{aligned} \Vert \nabla (u_{n_j}-u^*)\Vert _2\rightarrow 0, \hbox { as }j\rightarrow +\infty . \end{aligned}$$

(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

  1. (a)

    \(\eta _0(u)=u\) for all \(u\in W_0^{1,2}(\varOmega )\);

  2. (b)

    \(\eta _t(u)=u\) for all \(u\in 2I^{-1}[c_{n,r}-\varepsilon _0,c_{n,r}+\varepsilon _0]\) and for all \(t\in [0,1]\);

  3. (c)

    \(\eta _t(u)\) is a homeomorphism from \(W_0^{1,2}(\varOmega )\) onto \(W_0^{1,2}(\varOmega )\) for all \(t\in [0,1]\);

  4. (d)

    \(I(\eta _t(u))\le I(u)\) for all \(u\in W_0^{1,2}(\varOmega )\), for all \(t\in [0,1]\);

  5. (e)

    \(\eta _1(A_{c+\varepsilon }-N)\subset A_{c-\varepsilon }\);

  6. (f)

    If \(K_c=\emptyset\), \(\eta _1(A_{c+\varepsilon })\subset A_{c-\varepsilon }\);

  7. (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,

$$\begin{aligned} c_{n,r}=\inf _{K_{n,r}}\sup _K2I\le \sup _{u\in \eta _1(A_n)}2I(u)\le c_{n,r}-\varepsilon . \end{aligned}$$

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

$$\begin{aligned} u_k\rightarrow u_{n,r},\ \ \ \ k\rightarrow +\infty . \end{aligned}$$

It is easy to see that \(u_{n,r}\in M_r\) such that

$$\begin{aligned} c_{n,r}=2I(u_{n,r}) \end{aligned}$$

and

$$\begin{aligned} I'(u_{n,r})(v)=r^{-2}I'(u_{n,r})(u_{n,r})\cdot u_{n,r}(v), \forall v\in W_0^{1,2}(\varOmega ). \end{aligned}$$

Let \(\mu _{n,r}=r^{-2}I'(u_{n,r})(u_{n,r})\). Note one has

$$\begin{aligned} (a+br^2)\int _{\varOmega }\nabla u_{n,r}\nabla vdx+\int _{\varOmega }f(x,u_{n,r})v(x)dx=\mu _{n,r}\int _{\varOmega }u_{n,r}vdx,\forall v\in W_0^{1,2}. \end{aligned}$$
(2.10)

By \(u_{n,r}\in M_r\), (2.6) becomes

$$\begin{aligned} \left( a+b\int _{\varOmega }u_{n,r}^2dx\right) \int _{\varOmega }\nabla u_{n,r}\nabla vdx+\int _{\varOmega }f(x,u_{n,r})v(x)dx=\mu _{n,r}\int _{\varOmega }u_{n,r}vdx,\forall v\in W_0^{1,2}, \end{aligned}$$
(2.11)

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

$$\begin{aligned} r^2 \mu _{n,r}=(a+br^2)\Vert \nabla u_{n,r}\Vert _2^2+\int _{\varOmega }f(x,u_{n,r})u_{n,r}dx. \end{aligned}$$

The proof is completed. \(\square\)

Corollary 2.1

Let \(f\equiv 0\) and equation (1.1) becomes

$$\begin{aligned} -(a+b\Vert u\Vert _2^2)\varDelta u=\lambda u,\ \ x \hbox { in } \varOmega ,\\ u(x)=0,\ \ x\hbox { on }\partial \varOmega . \quad (2.11)_{\lambda } \end{aligned}$$

Then, \((2.11)_{\lambda }\) has branches

$$\begin{aligned} C_n=\{(a+br^2)\lambda _n,\pm r v_{n})| r>0\},\ \ n=1,2,\ldots . \end{aligned}$$

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

$$\left\{ \begin{aligned} -\varDelta u^0_{n,r}=\mu ^0_{n,r}\frac{1}{a+b\Vert u^0_{n,r}\Vert _2^2}u^0_{n,r}= \mu ^0_{n,r}\frac{1}{a+br^2}u^0_{n,r},\ \ x \hbox { in } \varOmega ,\\ u^0_{n,r}(x)=0,\ \ x\hbox { on }\partial \varOmega . \end{aligned} \right.$$

Comparing \((2.11)_{\lambda }\) with \((2.1)_{\lambda }\), we get

$$\begin{aligned} \mu ^0_{n,r}\frac{1}{a+br^2}=\lambda _n \end{aligned}$$

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,

$$\begin{aligned} c^0_{n,r}=2\varPhi (u^0_{n,r})=(a+br^2)\Vert \nabla u^0_{n,r}\Vert _2^2,\ \ \mu ^0_{n,r}=(a+br^2)\lambda _n. \end{aligned}$$
(2.12)

Since \(u^0_{n,r}=k_nu_n\), one has

$$\begin{aligned} r=\Vert u^0_{n,r}\Vert _2=\Vert k_nu_n\Vert _2=|k_n|r, \end{aligned}$$

which implies \(k_n=\pm 1\) and \(u^0_{n,r}=\pm u_n\). Hence, (2.12) becomes

$$\begin{aligned} c^0_{n,r}=2\varPhi (u^0_{n,r})=(a+br^2)\Vert \nabla u_{n}\Vert _2^2, \ \ \mu ^0_{n,r}=(a+br^2)\lambda _n. \end{aligned}$$

From (2.2), we have

$$\begin{aligned} r^2\lambda _n=\int _{\varOmega }|\nabla u_n|^2dx=\Vert \nabla u_n\Vert _2^2=\Vert \nabla u^0_{n,r}\Vert _2^2, \end{aligned}$$

and so

$$\begin{aligned} c^0_{n,r}=2\varPhi (u^0_{n,r})=(a+br^2)r^2\lambda _n,\ \ \mu ^0_{n,r}=(a+br^2)\lambda _n, \end{aligned}$$
(2.12)

which together with (2.3) implies that \((2.11)_{\lambda }\) has branches

$$\begin{aligned} C_n=\{(a+br^2)\lambda _n,\pm r v_{n})| r>0\},\ \ n=1,2,\ldots . \end{aligned}$$

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

$$\begin{aligned} \Vert u\Vert _{p+1}^{p+1}\le c\Vert \nabla u\Vert _2^2\Vert u\Vert _2^{p-1}, u\in W_0^{1,2}(\varOmega ). \end{aligned}$$
(3.1)

Note (\(d=0\) in \((A_1)\))

$$\begin{aligned} |I(u)-\varPhi (u)|=|\int _{\varOmega }F(x,u)dx|\le c\int _{\varOmega }|u|^{p+1}dx. \end{aligned}$$
(3.2)

Since

$$\begin{aligned} \varPhi (u)=(a+br^2)\Vert u\Vert ^2, \end{aligned}$$

from (3.1), we have

$$\begin{aligned} \int _{\varOmega }|u|^{p+1}dx\le c\frac{1}{a+br^2}\varPhi (u)\Vert u\Vert _2^{p-1}\le \frac{c}{a}\varPhi (u)\Vert u\Vert _2^{p-1}. \end{aligned}$$

Hence,

$$\begin{aligned} \int _{\varOmega }|u|^{p+1}dx\le \frac{c}{a}\varPhi (u)r^{p-1},\ \ \forall u\in M_r. \end{aligned}$$

It infers from (3.2) that

$$\begin{aligned} \left( 1-\frac{c}{a}r^{p-1}\right) \varPhi (u)\le I(u)\le \left( 1+\frac{c}{a}r^{p-1}\right) \varPhi (u), \end{aligned}$$

and so

$$\begin{aligned} \left( 1-\frac{c}{a}r^{p-1}\right) \inf _{K_{n,r}}\sup _K2\varPhi (u)\le \inf _{K_{n,r}}\sup _K2I(u)\le \left( 1+\frac{c}{a}r^{p-1}\right) \inf _{K_{n,r}}\sup _K2\varPhi (u), \end{aligned}$$

i.e.

$$\begin{aligned} \left( 1-\frac{c}{a}r^{p-1}\right) c^0_{n,r}\le c_{n,r}\le \left( 1+\frac{c}{a}r^{p-1}\right) c^0_{n,r}, \end{aligned}$$

which implies that

$$\begin{aligned} |c_{n,r}-c^0_{n,r}|\le \frac{c}{a} c_{n,r}^0r^{p-1}. \end{aligned}$$

Now (2.9) guarantees that

$$\begin{aligned} |c^0_{n,r}|\le cr^{2} \end{aligned}$$
(3.3)

and so

$$\begin{aligned} |c_{n,r}-c^0_{n,r}|\le cr^{p+1}. \end{aligned}$$

It deduces from Theorem 2.1 and (3.2) that

$$\begin{aligned} c_{n,r}&=(a+br^2)\Vert \nabla u_{n,r}\Vert _2^2+2\int _{\varOmega }F(x,u_{n,r})dx\\ & \ge (a+br^2)\Vert \nabla u_{n,r}\Vert _2^2-2cr^{p-1}\Vert \nabla u_{n,r}\Vert _2^2\\ &=(a+br^2-2cr^{p-1})\Vert \nabla u_{n,r}\Vert _2^2, \end{aligned}$$

which together with (3.3) implies that

$$\begin{aligned} \Vert \nabla u_{n,r}\Vert _2^2&{}\displaystyle \le \frac{c_{n,r}}{a+br^2-2cr^{p-1}}\\ & \le \frac{c^0_{n,r}+cr^{p+1}}{a+br^2-2cr^{p-1}}\\ &{}\le cr^{2}. \end{aligned}$$
(3.4)

From Theorem 2.1 and (3.1), (3.4), one has

$$\begin{aligned} |c_{n,r}-r^2\mu _{n,r}|&{}\displaystyle = \left| 2\int _{\varOmega }F(x,u_{n,r})dx-\int _{\varOmega }f(x,u_{n,r})u_{n,r}dx\right| \\ &{}\displaystyle \le c\Vert \nabla u_{n,r}\Vert _2^{2\alpha }r^{\beta }+dr\\ &{}\le c(c_{n,r})^{\alpha }+dr\\ &{}\le c(c_{n,r}^0+cr^{\beta }(c^0_{n,r})^{\alpha }+dr)^{\alpha }+dr,\\ \end{aligned}$$

Then

$$\begin{aligned} |r^2\mu _{n,r}-r^2\mu ^0_{n,r}|& = |r^2\mu _{n,r}-c_{n,r}+c_{n,r}-c^0_{n,r}+c^0_{n,r}-r^2\mu ^0_{n,r}|\\ & \le |r^2\mu _{n,r}-c_{n,r}|+|c_{n,r}-c^0_{n,r}|+|c^0_{n,r}-r^2\mu ^0_{n,r}|\\ & \le c_1r^{p+1}+c_2r^{p+1}+c_3r^{4}\\ &{}\le cr^{\min \{4,p+1\}}, \end{aligned}$$

which implies that

$$\begin{aligned} |\mu _{n,r}-\mu ^0_{n,r}|\le cr^{\min \{2,p-1\}}. \end{aligned}$$

Consequently,

$$\begin{aligned} \mu _{n,r}=a\lambda _n+b\lambda _nr^2+O(r^{\min \{2,p-1\}}). \end{aligned}$$

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

$$\begin{aligned} |c_{n,r}-c^0_{n,r}|\le cr^{\beta }(c^0_{n,r})^{\alpha }+dr \end{aligned}$$
(4.1)

and

$$\begin{aligned} |c_{n,r}-r^2\mu _{n,r}|\le c(c_{n,r}^0+cr^{\beta }(c^0_{n,r})^{\alpha }+dr)^{\alpha }+dr, \end{aligned}$$
(4.2)

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

$$\begin{aligned} \left| \int _{\varOmega }F(x,u)dx\right| \le c\int _{\varOmega }|u|^{p+1}dx+d\int _{\varOmega }|u|dx, \end{aligned}$$

and similarly

$$\begin{aligned} \left| \int _{\varOmega }f(x,u)udx\right| \le c\int _{\varOmega }|u|^{p+1}dx+d\int _{\varOmega }|u|dx. \end{aligned}$$

Next, as \(1\le p<{\overline{p}}\), from Lemma 2.1, if \(\int _{\varOmega }u^2dx=r^2\), we have

$$\begin{aligned}\left| \int _{\varOmega }F(x,u)dx\right| & \le c\Vert \nabla u\Vert _2^{2\alpha }r^{\beta }+dr\\ &e =c\left( \frac{1}{a+br^2}(a+br^2)\Vert \nabla u\Vert _2^{2}\right) ^{\alpha }r^{\beta }+dr\\ & \le c \left( \frac{1}{a}\right) ^{\alpha }(\varPhi (u))^{\alpha }r^{\beta }+dr\\ \end{aligned}$$
(4.3)

and similarly

$$\begin{aligned} \left| \int _{\varOmega }f(x,u)udx\right| \le c \left( \frac{1}{a}\right) ^{\alpha }(\varPhi (u))^{\alpha }r^{\beta }+dr, \end{aligned}$$
(4.4)

with \(\alpha\) and \(\beta\) as in the statement of Lemma 4.1.

To prove (4.1), observe that (4.3) implies

$$\begin{aligned} I(u)&{}\displaystyle =\varPhi (u)+\int _{\varOmega }F(x,u)dx\\ &{}\displaystyle \le \varPhi (u)+cr^{\beta }(\varPhi (u))^{\alpha }+dr\\ \end{aligned}$$

holds ( c instead of \(c (\frac{1}{a})^{\alpha }\)). In other words, we have

$$\begin{aligned} I(u)\le g(\varPhi (u)) \end{aligned}$$

where \(g: R^+\rightarrow R^+\) is defined by

$$\begin{aligned} g(t) = t + cr^{\beta }t^{\alpha } + dr. \end{aligned}$$

As g is continuous and nondecreasing, we get

$$\begin{aligned} \inf _{K_{n,r}}\sup _{K\in K_{n,r}}I(u)\le \inf _{K_{n,r}}\sup _{K\in K_{n,r}}g(\varPhi (u))=g(\inf _{K_{n,r}}\sup _{K\in K_{n,r}}\varPhi (u)). \end{aligned}$$

Now Theorem 2.1 implies that

$$\begin{aligned} c_{n,r}\le 2g(c^0_{n,r})=c_{n,r}^0+cr^{\beta }(c^0_{n,r})^{\alpha }+dr \end{aligned}$$

for some new constants c and \(d>0\). Therefore,

$$\begin{aligned} |c_{n,r}-c^0_{n,r}|\le cr^{\beta }(c^0_{n,r})^{\alpha }+dr, \end{aligned}$$
(4.5)

which shows (4.1) is true.

Since

$$\begin{aligned} c_{n,r}=(a+br^2)\Vert u_{n,r}\Vert _2^2+ 2\int _{\varOmega }F(x,u)dx, \end{aligned}$$

we have

$$\begin{aligned} (a+br^2)\Vert \nabla u_{n,r}\Vert _2^2=c_{n,r}-2\int _{\varOmega }F(x,u_{n,r})dx. \end{aligned}$$

It deduces from Theorem 2.1 and (4.3)-(4.4) that

$$\begin{aligned} |c_{n,r}-r^2\mu _{n,r}|&= \left| 2\int _{\varOmega }F(x,u_{n,r})dx-\int _{\varOmega }f(x,u_{n,r})u_{n,r}dx\right| \\ & \le c\Vert \nabla u_{n,r}\Vert _2^{2\alpha }r^{\beta }+dr\\ &\le c(c_{n,r})^{\alpha }+dr\\ &\le c(c_{n,r}^0+cr^{\beta }(c^0_{n,r})^{\alpha }+dr)^{\alpha }+dr,\\ \end{aligned}$$

which completes the proof of the lemma. \(\square\)

Lemma 4.2

(Theorem 2, [5]) The eigenvalues \(\lambda _{n}\) of \((2.1)_{\lambda }\) satisfy, as \(n\rightarrow +\infty\)

$$\begin{aligned} \lambda _{n}=kn^{2/N}+O(n^{1/N}\log n),\ \ n=1,2,\cdots , \end{aligned}$$
(4.6)

where

$$\begin{aligned} k=(2\pi )^2(V)^{-2/N} \end{aligned}$$
(4.7)

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),

$$\begin{aligned} \mu _{n,r}=(a+br^2)kn^{2/N}+O(n^{1/N}\log n),\ \end{aligned}$$

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

$$\begin{aligned} c_{n,r}& =c^0_{n,r}+O((c_{n,r}^0)^{\alpha })\\ & =(a+br^2)r^2\lambda _n+O((c_{n,r}^0)^{\alpha })\\ & =(a+br^2)r^2\lambda _n+O(\lambda _n^{\alpha })\\ \end{aligned}$$

and

$$\begin{aligned}&& (c_{n,r})^{\alpha } =((a+br^2)r^2\lambda _n+O(\lambda _n^{\alpha }))^{\alpha }\\ &&=O(\lambda _n^{\alpha }),\\&&(c_{n,r})^{\frac{1}{2}}=O(\lambda _n^{\frac{1}{2}}), \end{aligned}$$

which together (4.1) and (4.2) implies that

$$\begin{aligned} |r^2\mu _{n,r}-c^0_{n,r}| &{}= |r^2\mu _{n,r}-c_{n,r}+c_{n,r}-c^0_{n,r}|\\ & \le |r^2\mu _{n,r}-c_{n,r}|+|c_{n,r}-c^0_{n,r}|\\ & \le c(c^0_{n,r})^{\alpha }+ cr^{\beta }(c^0_{n,r})^{\alpha }\\ & =O(\lambda _n^{\alpha }), \end{aligned}$$

and so

$$\begin{aligned} r^2\mu _{n,r}=c^0_{n,r}+ O(\lambda _n^{\alpha })=r^2(a+br^2)\lambda _n+O(\lambda _n^{\alpha }). \end{aligned}$$

Consequently

$$\begin{aligned} \mu _{n,r}=(a+br^2)\lambda _n+O(\lambda _n^{\alpha }). \end{aligned}$$

Since

$$\begin{aligned} \lambda _n=kn^{2/N}+O(n^{1/N}\log n), \end{aligned}$$

we have

$$\begin{aligned} \mu _{n,r}&{}= (a+br^2)kn^{2/N}+O(n^{1/N}\log n)+O((kn^{2/N}+O(n^{1/N}\log n))^{\alpha })\\ & =(a+br^2)kn^{2/N}+O(n^{1/N}\log n).\\ \end{aligned}$$

The proof is completed. \(\square\)