Eigenvalue problem for p-Laplacian with mixed boundary conditions

Abstract

Eigenvalue problem for p-Laplacian with mixed boundary conditions is concerned on a bounded domain. The existence of nonnegative eigenvalues are obtained by using the Lusternik-Schnirelman principle. Boundedness of eigenfunctions is obtained by using the Moser iteration. The simplicity and isolation of the first eigenvalue are proved. The existence of the second eigenvalue is also illustrated.

Introduction

In this paper, we study the following eigenvalue problem

$$\left\{\begin{array}{ll}-{△}_{p}u\left(x\right)=\lambda |u{|}^{p-2}u,& \text{in}\Omega ,\\ u=0,& \text{on}\sigma ,\\ |\nabla u(x){|}^{p-2}\frac{\mathrm{\partial u}}{\mathrm{\partial n}}=\lambda |u{|}^{p-2}u,& \text{on}\Gamma ,\end{array}\right.$$

(1.1)

where △ _p u=d i v(|∇u|^p−2|∇u|) is the p-Laplacian operator, 1<p<+∞,Ω is a bounded domain in ${\mathbb{R}}^N$ with C¹ boundary ∂ Ω, $\sigma \cup \Gamma =\mathrm{\partial \Omega }$ and $\sigma \cap \Gamma =\varnothing$, Γ is a sufficiently smooth (N−1)-dimensional manifold, and n is the outward normal vector on ∂ Ω.

Throughout the paper we define X:={u∈W^1,p(Ω):u| _σ =0} is a closed subspace of W^1,p(Ω) with the norm $\parallel u{\parallel }^p={\int }_{\Omega }\left(\right|\nabla u{|}^p+\left|u{|}^p\right)\mathit{\text{dx}}$. Eigenvalue problems for the p-Laplacian and p(x)-Laplacian have been studied extensively for many years and many interesting results have been obtained. These results are on the structure of the spectrum of Dirichlet, no-flux, Niemann, Robin, and Steklov problems as demonstrated in [1–8]. Problem (1.1) is a mixed boundary value problem, and is different from the classical ones. References [9, 10] studied the following problem

$$\left\{\begin{array}{ll}-△u=\mathrm{\lambda u},& \text{in}\Omega ,\\ u=0,& \text{on}\sigma ,\\ \frac{\mathrm{\partial u}}{\mathrm{\partial n}}=\mathrm{\lambda \beta u},& \text{on}\Gamma ,\end{array}\right.$$

(1.2)

where $\beta =\frac{a^2}{b^2}$. Problem (1.1) is a generalization of (1.2) as p=2 and β=1. In this paper, we extend their results and study the complete character of eigenvalue problem (1.1) which is an abstract one and has never been known.

Methods

Since our methods of proofs of the theorem are different from the others, we must consider the boundary σ and Γ. We use the multiplicative inequality in [11, 12] to proof the boundedness of eigenfunctions. For example, problem (1.1) includes the following classical problem

$$\left\{\begin{array}{ll}{X}^{\mathrm{\prime \prime }}\left(x\right)+\mathrm{\lambda X}\left(x\right)=0,& 0<x<l,\\ X\left(0\right)=0,\\ X^{\prime }\left(l\right)-\mathrm{\lambda X}\left(l\right)=0,\end{array}\right.$$

(1.3)

as its special case, which leads to the equation for eigenvalue λ>0,

$$\text{tan}\sqrt{\lambda }l=\frac{1}{\sqrt{\lambda }}.$$

Thus, we get the sequence of eigenvalues ${\lambda }_k={\theta }_k^2$, satisfying

$$\frac{(k-1)\Pi }{l}<{\theta }_k<\frac{(2k-1)\Pi }{2l},k=1,2,\cdots .$$

Related eigenfunctions are {s i n(θ _k x)}_k=1,2,⋯.

It is well known that an eigenvalue problem plays a very important role in the studying of all kinds of linear and nonlinear problems. Therefore, the research in present paper would be useful to the understanding of spectrum of nonlinear operator and related problems.

The sketch of the paper is as follows. We first establish the eigenvalue sequence in next section. Next, we consider the boundedness of eigenfunctions in section ‘Boundedness of eigenfunctions’. The simplicity and isolation of the first eigenvalue are considered in the section ‘Simplicity and isolation of the first eigenvalue’. In the section ‘Existence of the second eigenvalue’, we consider the existence of the second eigenvalue.

Results and discussion

Eigenvalue problem for the p-Laplacian

Weak solutions

Definition 2.1

A pair $(u,\lambda )\in X\times \mathbb{R}$ is a weak solution of (1.1) provided that

$${\int }_{\Omega }|\nabla u{|}^{p-2}\nabla u\nabla \mathit{\text{vdx}}=\lambda \left({\int }_{\Omega }|u{|}^{p-2}\mathit{\text{uvdx}}+{\int }_{\Gamma }|u{|}^{p-2}\mathit{\text{uvds}}\right),$$

(2.1)

for any v∈X as u=0 on σ. Where u is nontrivial, λ is an eigenvalue, and u is called an associated eigenfunction.

It follows from (2.1) that all eigenvalues λ are nonnegative (by choosing v=u). It shows that if Γ is of class C^1,γ, then eigenfunction of (2.1) belongs to $C^{1,\alpha }\left(\overline{\Omega }\right)$. Hence, ∇u exists on Γ, and the boundary conditions of the problem (1.1) make sense. The following lemma assures that if an eigenfunction u is smooth enough, then u solves the corresponding partial differential equation.

Lemma 2.2

Let (u,λ) be an eigenpair, i.e., a weak solution of (2.1) such that u∈W^2,p(Ω), then (u,λ) solves (1.1).

Proof

Let $(u,\lambda )\in W^{2,p}\left(\Omega \right)\times {\mathbb{R}}^{+}$ be an eigenpair of (2.1). We recall the first formula of Green [13], it follows from (2.1) that

$$\begin{array}{ll}{\int }_{\Omega }& \left(-{△}_{p}u\right)\mathit{\text{vdx}}+{\int }_{\Gamma }|\nabla u{|}^{p-2}\frac{\mathrm{\partial u}}{\mathrm{\partial n}}\mathit{\text{vds}}\\ =\lambda \left({\int }_{\Omega }|u{|}^{p-2}\mathit{\text{uvdx}}+{\int }_{\Gamma }|u{|}^{p-2}\mathit{\text{uvds}}\right)\end{array}$$

for any v∈X. Thus, taking any v in $C_0^{\infty }\left(\Omega \right)$ we have

$${\int }_{\Omega }\left({△}_{p}u+\lambda |u{|}^{p-2}u\right)\mathit{\text{vdx}}=0,$$

which implies −△ _p u=λ|u|^p−2u in Ω. Furthermore, since the range of the trace mapping X↪L^p(Γ) is continuous and compact (see [14]), and v=0 on σ, we have

$${\int }_{\Gamma }|\nabla u{|}^{p-2}\frac{\mathrm{\partial u}}{\mathrm{\partial n}}\mathit{\text{vds}}=\lambda {\int }_{\Gamma }|u{|}^{p-2}\mathit{\text{uvds}},\forall v\in L^p\left(\Gamma \right).$$

□

Therefore, $|\nabla u{|}^{p-2}\frac{\mathrm{\partial u}}{\mathrm{\partial n}}=\lambda |u{|}^{p-2}u$ on Γ.

Existence of L-S sequence for (1.1)

The existence of a sequence of eigenvalues can be proved by the Ljusternik-Schnirelman principle, we call this sequence as L-S sequence {λ _n }.

Let Ω be a bounded domain in ${\mathbb{R}}^N$ with C¹ boundary. We define the following functionals F and G on X

$$F\left(u\right)={\int }_{\Omega }a\left(x\right)\left|u\right(x){|}^p\mathit{\text{dx}}+{\int }_{\Gamma }b(s\left)\right|u\left(s\right){|}^p\mathit{\text{ds}},$$

(2.2)

$$G\left(u\right)={\int }_{\Omega }\left(|\nabla u(x){|}^p+|u\left(x\right){|}^p\right)\mathit{\text{dx}}+{\int }_{\Gamma }\beta \left(s\right)\left|u\right(s){|}^p\mathit{\text{ds}},$$

(2.3)

where a∈L^∞(Ω) and b,β∈L^∞(Γ) such that a,b,β>0. Consider the following eigenvalue problem

$$F^{\prime }\left(u\right)=\mu G^{\prime }\left(u\right),u\in S_G,\mu \in \mathbb{R},$$

where S _G is the level S _G ={u∈X:G(u)=1}.

For any positive integer n, denoted by ${\mathbb{A}}_n$ the class of all compact, symmetric subsets K of S _G such that F(u)>0 is on K and γ(K)≥n, where γ(K) denotes the genus of K, i.e., $\gamma \left(K\right):=inf\{k\in \mathbb{N}:\exists h:K\to {\mathbb{R}}^k\setminus \{0\}$ such that h is continuous and odd}.

Theorem 2.3

Let F(u), G(u) be defined in (2.2) and (2.3) with a(x)=b(x)=β(x)=1. Then there exists a nondecreasing sequence of nonnegative {λ _n } of (2.1) obtained by using the L-S principle such that ${\lambda }_n=\frac{1}{{\mu }_n}-1\to +\infty$ as n→+∞, where each μ _n is an eigenvalue of the corresponding equation $F^{\prime }\left(u\right)=\mu G^{\prime }\left(u\right)$ that satisfies ${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_k\ge \cdots >0,\underset{k\to +\infty }{lim}{\mu }_k=0$.

Proof

With a(x)=b(x)=β(x)=1 in (2.2) and (2.3), F(u) and G(u) become

$$\begin{array}{ll}F\left(u\right)& ={\int }_{\Omega }\left|u\right(x){|}^p\mathit{\text{dx}}+{\int }_{\Gamma }|u\left(s\right){|}^p\mathit{\text{ds}},G\left(u\right)\\ ={\int }_{\Omega }\left(|\nabla u(x){|}^p+|u\left(x\right){|}^p\right)\mathit{\text{dx}}+{\int }_{\Gamma }\left|u\right(s){|}^p\mathrm{ds.}\end{array}$$

□

Thus, $F^{\prime }\left(u\right)=\mu G^{\prime }\left(u\right)$ is equivalent to

$$\begin{array}{ll}{\int }_{\Omega }& |u{|}^{p-2}\mathit{\text{uvdx}}+{\int }_{\Gamma }|u{|}^{p-2}\mathit{\text{uvds}}\\ =\mu \left({\int }_{\Omega }\left(|\nabla u{|}^{p-2}\nabla u\nabla v+|u{|}^{p-2}\mathit{\text{uv}}\right)\mathit{\text{dx}}\right.\\ +\left.{\int }_{\Gamma }|u{|}^{p-2}\mathit{\text{uvds}}\right),\end{array}$$

for any v∈X, or

$$\begin{array}{ll}{\int }_{\Omega }|\nabla u{|}^{p-2}\nabla u\nabla \mathit{\text{vdx}}& =\left(\frac{1}{\mu }-1\right)\left({\int }_{\Omega }|u{|}^{p-2}\mathit{\text{uvdx}}\right.\\ +\left.{\int }_{\Gamma }|u{|}^{p-2}\mathit{\text{uvds}}\right),\forall v\in \mathrm{X.}\end{array}$$

Combining (2.1) and the existence of the L-S sequence principle, we obtain ${\lambda }_n=\frac{1}{{\mu }_n}-1\to +\infty$ as n→+∞.

Boundedness of eigenfunctions

Let Ω be a bounded domain in ${\mathbb{R}}^N$ with C¹ boundary and 1<p<+∞. We shall show the eigenfunctions are in L^∞(Ω), which is the boundedness for solutions of (1.1).

Theorem 3.1

Let (u,λ) be an eigensolution of the weak form (2.1), then u∈L^∞(Ω).

Proof

In this proof, we use the Moser iteration technique in [15]. We assume first that u≥0. We define v _M (x)= min{u(x),M} for M>0 and $\phi =v_M^{\mathit{\text{kp}}+1}$ for k>0, then $\nabla \phi =(\mathit{\text{kp}}+1)v_M^{\mathit{\text{kp}}}\nabla v_M$. It follows that $\phi \in X\cap L^{\infty }\left(\Omega \right)$ and v _M | _Γ = min{u| _Γ ,M}. Taking φ as a test function we have

$$\begin{array}{ll}(\mathit{\text{kp}}& +1\left){\int }_{\Omega }\right|\nabla u{|}^{p-2}\nabla u{v}_M^{\mathit{\text{kp}}}\nabla v_M\mathit{\text{dx}}\\ =\lambda \left({\int }_{\Omega }{\left|u\right|}^{p-2}u{v}_M^{\mathit{\text{kp}}+1}\mathit{\text{dx}}+{\int }_{\Gamma }{\left|u\right|}^{p-2}u{v}_M^{\mathit{\text{kp}}+1}\mathit{\text{ds}}\right),\end{array}$$

which implies that

$$\begin{array}{ll}\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}{\int }_{\Omega }|\nabla v_M^{k+1}{|}^p\mathit{\text{dx}}& \le \lambda \left({\int }_{\Omega }{\left|u\right|}^{(k+1)p}\mathit{\text{dx}}\right.\\ +\left.{\int }_{\Gamma }{\left|u\right|}^{(k+1)p}\mathit{\text{ds}}\right).\end{array}$$

□

Let M→∞; by Fatou’s lemma we obtain

$$\begin{array}{ll}\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}{\int }_{\Omega }|\nabla u^{k+1}{|}^p\mathit{\text{dx}}& \le \lambda \left({\int }_{\Omega }{\left|u\right|}^{(k+1)p}\mathit{\text{dx}}\right.\\ +\left.{\int }_{\Gamma }{\left|u\right|}^{(k+1)p}\mathit{\text{ds}}\right).\end{array}$$

That is,

$$\begin{array}{ll}\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}& {\int }_{\Omega }\left(|\nabla u^{k+1}{|}^p+|u^{k+1}{|}^p\right)\mathit{\text{dx}}\\ \le \left(\lambda +\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}\right){\int }_{\Omega }|u{|}^{(k+1)p}\mathit{\text{dx}}\\ +\lambda {\int }_{\Gamma }|u{|}^{(k+1)p}\mathit{\text{ds}},\end{array}$$

$$\begin{array}{ll}\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}\parallel u^{k+1}{\parallel }^p& \le \left(\lambda +\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}\right)\parallel u^{k+1}{\parallel }_{L^p\left(\Omega \right)}^p\\ +\lambda \parallel u^{k+1}{\parallel }_{L^p\left(\Gamma \right)}^p.\end{array}$$

(3.1)

When u=0 in σ, by the multiplicative inequality stated (see Chapter 1, Section 1.4.7, Corollary 2 in [11]) and the Moser iteration done in [12] of the form

$$\parallel u{\parallel }_{L^p\left(\mathrm{\partial \Omega }\right)}^p\le ϵ\parallel u{\parallel }^p+C\left(ϵ\right)\parallel u{\parallel }_{L^p\left(\Omega \right)}^p,ϵ>0,$$

we obtain

$$\parallel u^{k+1}{\parallel }_{L^p\left(\Gamma \right)}^p\le ϵ\parallel u^{k+1}{\parallel }^p+C\left(ϵ\right)\parallel u^{k+1}{\parallel }_{L^p\left(\Omega \right)}^p,ϵ>0.$$

(3.2)

Combining (3.1) and (3.2), it has

$$\begin{array}{ll}\left(\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}-\mathrm{\lambda ϵ}\right)\parallel u^{k+1}{\parallel }^p& \le \left(\lambda (1+C(ϵ\left)\right)+\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}\right)\\ \times \parallel u^{k+1}{\parallel }_{L^p\left(\Omega \right)}^p.\end{array}$$

Since ϵ→0, we may assume that $\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}-\mathrm{\lambda ϵ}>0$, then

$$\begin{array}{ll}\parallel u\parallel & \le {\left[\left(\lambda \left(1+C\left(ϵ\right)\right)+\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}\right)\frac{1}{\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}-\mathrm{\lambda ϵ}}\right]}^{\frac{1}{(k+1)p}}\\ \times \parallel u{\parallel }_{L^{(k+1)p}\left(\Omega \right)}.\end{array}$$

(3.3)

By Sobolev’s embedding function $X\hookrightarrow L^{p^{\ast }}\left(\Omega \right)$, where $p^{\ast }=\frac{\mathit{\text{Np}}}{N-p}$, if p<N and p^∗=2p, if p=N. Then there exists a constant c₁>0 such that

$$\parallel u^{k+1}{\parallel }_{L^{p^{\ast }}\left(\Omega \right)}\le c_1\parallel u^{k+1}\parallel ,$$

which is

$$\parallel u{\parallel }_{L^{(k+1)p^{\ast }}\left(\Omega \right)}\le c_1^{\frac{1}{k+1}}\parallel u\parallel .$$

(3.4)

By (3.3) and (3.4), for any k>0, we can find a constant c₂>0 such that

$${\left[\left(\lambda \left(1+C\left(ϵ\right)\right)+\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}\right)\frac{1}{\frac{\mathit{\text{kp}}+1}{{(k+1)}^p}-\mathrm{\lambda ϵ}}\right]}^{\frac{1}{(k+1)p}}<c_2^{\frac{1}{\sqrt{k+1}}},$$

which is

$${\parallel u\parallel }_{L^{(k+1)p^{\ast }}\left(\Omega \right)}\le c_1^{\frac{1}{k+1}}{c}_2^{\frac{1}{\sqrt{k+1}}}{\parallel u\parallel }_{L^{(k+1)p}\left(\Omega \right)}.$$

(3.5)

Choosing k₁ such that (k₁+1)p=p^∗, taking k=k₁ in (3.5), it has

$${\parallel u\parallel }_{L^{(k_1+1)p^{\ast }}\left(\Omega \right)}\le c_1^{\frac{1}{k_1+1}}{c}_2^{\frac{1}{\sqrt{k_1+1}}}{\parallel u\parallel }_{L^{p^{\ast }}\left(\Omega \right)}.$$

Next, we choose k₂ such that (k₂+1)p=(k₁+1)p^∗, then taking k=k₂ in (3.5), we have

$${\parallel u\parallel }_{L^{(k_2+1)p^{\ast }}\left(\Omega \right)}\le c_1^{\frac{1}{k_2+1}}{c}_2^{\frac{1}{\sqrt{k_2+1}}}{\parallel u\parallel }_{L^{(k_1+1)p^{\ast }}\left(\Omega \right)}.$$

Therefore,

$${\parallel u\parallel }_{L^{(k_n+1)p^{\ast }}\left(\Omega \right)}\le c_1^{\frac{1}{k_n+1}}{c}_2^{\frac{1}{\sqrt{k_n+1}}}{\parallel u\parallel }_{L^{(k_{n-1}+1)p^{\ast }}\left(\Omega \right)},$$

where the sequence {k _n } is chosen such that (k _n +1)p=(k_n−1+1)p^∗,k₀=0.

It is easy to see that $k_n+1={\left(\frac{p^{\ast }}{p}\right)}^n$, hence

$${\parallel u\parallel }_{L^{(k_n+1)p^{\ast }}\left(\Omega \right)}\le c_1^{\sum _{i=1}^n\frac{1}{k_i+1}}{c}_2^{\sum _{i=1}^n\frac{1}{\sqrt{k_i+1}}}{\parallel u\parallel }_{L^{p^{\ast }}\left(\Omega \right)}.$$

There exists C>0 such that

$$\parallel u{\parallel }_{L^{(k_n+1)p^{\ast }}\left(\Omega \right)}\le C\parallel u{\parallel }_{L^{p^{\ast }}\left(\Omega \right)},$$

for any n=1,2,⋯, with r _n =(k _n +1)p^∗→+∞ as n→+∞.

Next, we will prove u∈L^∞(Ω). Suppose u∉L^∞(Ω), then there exists ϵ₁>0 and a set A of positive measure in Ω such that $\left|u\right(x\left)\right|>C\parallel u{\parallel }_{L^{p^{\ast }}\left(\Omega \right)}+{ϵ}_1=K$, for all x∈A.

Hence

$$\begin{array}{ll}\underset{n\to +\infty }{liminf}\parallel u{\parallel }_{L^{r_n}\left(\Omega \right)}& \ge \underset{n\to +\infty }{liminf}{\left(\underset{A}{\int }{K}^{r_n}\right)}^{\frac{1}{r_n}}=\underset{n\to +\infty }{liminf}K|A{|}^{\frac{1}{r_n}}\\ =K>C\parallel u{\parallel }_{L^q\left(\Omega \right)},\end{array}$$

which contradicts what has been established above.

If u (as an eigenfunction of (2.1)) changes sign, we consider u⁺, and it is easy to know u⁺∈X. We define for each M>0,v _M (x)= min(u⁺(x),M). Taking again $\phi =v_M^{\mathit{\text{kp}}+1}$ as a test function in W^1,p(Ω), we obtain

$$\begin{array}{l}(\mathit{\text{kp}}+1){\int }_{\Omega }|\nabla u^{+}{|}^{p-2}\nabla u^{+}{v}_M^{\mathit{\text{kp}}}\nabla v_M\mathit{\text{dx}}\\ =\lambda \left({\int }_{\Omega }|u^{+}{|}^{p-2}{u}^{+}{v}_M^{\mathit{\text{kp}}+1}\mathit{\text{dx}}+{\int }_{\Gamma }|u^{+}{|}^{p-2}{u}^{+}{v}_M^{\mathit{\text{kp}}+1}\mathit{\text{ds}}\right).\end{array}$$

Proceeding the same way as above, we conclude that u⁺∈L^∞(Ω). Similarly we have u⁻∈L^∞(Ω). Therefore u=u⁺+u⁻ is in L^∞(Ω).

Simplicity and isolation of the first eigenvalue

In this section, we will study the characterization of the first eigenvalue of (1.1). In the succeeding text, we assume that Ω is a bounded domain in ${\mathbb{R}}^N$ with C^1,γ boundary, γ>0, and 1<p<+∞. By (2.1) we have ${\lambda }_1=\underset{u\in X\setminus \left\{0\right\}}{inf}\frac{{\int }_{\Omega }|\nabla u{|}^p\mathit{\text{dx}}}{{\int }_{\Omega }|u{|}^p\mathit{\text{dx}}+{\int }_{\Gamma }|u{|}^p\mathit{\text{ds}}}$.

Simplicity of the first eigenvalue

Proposition 4.1

If (u,λ) is an eigenpair of (2.1) with λ>λ₁, then u has to change sign in Ω.

Proof

If (u,λ) satisfies (2.1) for any v∈X, by choosing v≡1, we obtain

$${\int }_{\Omega }{\left|u\right|}^{p-2}\mathit{\text{udx}}+{\int }_{\Gamma }{\left|u\right|}^{p-2}\mathit{\text{uds}}=0.$$

□

Therefore, u has to change sign.

Theorem 4.2

The principal eigenvalue λ₁ is simple; i.e., if u,v are two eigenfunctions associated with λ₁, then there exists a constant k such that u=k v.

Proof

By proposition 4.1, we can assume that u,v are positive in Ω. We assume u,v are strictly positive in $\overline{\Omega }$, we take

$${\eta }_1=\frac{u^p-v^p}{u^{p-1}},{\eta }_2=\frac{v^p-u^p}{v^{p-1}}$$

as test functions in the weak form of (2.1) satisfied by u,v, respectively. We have

$$\begin{array}{ll}{\int }_{\Omega }& |\nabla u{|}^{p-2}\nabla u\nabla \left(\frac{u^p-v^p}{u^{p-1}}\right)\mathit{\text{dx}}\\ =\lambda \left({\int }_{\Omega }{\left|u\right|}^{p-2}u\left(\frac{u^p-v^p}{u^{p-1}}\right)\mathit{\text{dx}}+{\int }_{\Gamma }{\left|u\right|}^{p-2}u\right.\\ \times \left.\left(\frac{u^p-v^p}{u^{p-1}}\right)\mathit{\text{ds}}\right),\end{array}$$

(4.1)

$$\begin{array}{ll}{\int }_{\Omega }& |\nabla v{|}^{p-2}\nabla v\nabla \left(\frac{v^p-u^p}{v^{p-1}}\right)\mathit{\text{dx}}\\ =\lambda \left({\int }_{\Omega }{\left|v\right|}^{p-2}v\left(\frac{v^p-u^p}{v^{p-1}}\right)\mathit{\text{dx}}+{\int }_{\Gamma }{\left|v\right|}^{p-2}v\right.\\ \times \left.\left(\frac{v^p-u^p}{v^{p-1}}\right)\mathit{\text{ds}}\right).\end{array}$$

(4.2)

□

Combining (4.1) and (4.2) yields

$$\begin{array}{ll}0& ={\int }_{\Omega }|\nabla u{|}^{p-2}\nabla u\nabla \left(\frac{u^p-v^p}{u^{p-1}}\right)\mathit{\text{dx}}\\ +{\int }_{\Omega }|\nabla v{|}^{p-2}\nabla v\nabla \left(\frac{v^p-u^p}{v^{p-1}}\right)\mathrm{dx.}\end{array}$$

(4.3)

Using $\nabla \left(\frac{u^p-v^p}{u^{p-1}}\right)=\nabla u-p\frac{v^{p-1}}{u^{p-1}}\nabla v+(p-1)\frac{v^p}{u^p}\nabla u$, we obtain

$$\begin{array}{ll}{\int }_{\Omega }& |\nabla u{|}^{p-2}\nabla u\nabla \left(\frac{u^p-v^p}{u^{p-1}}\right)\mathit{\text{dx}}\\ ={\int }_{\Omega }|\nabla \mathit{\text{lnu}}{|}^p{u}^p-p{\int }_{\Omega }{v}^p|\nabla \mathit{\text{lnu}}{|}^{p-2}〈\nabla \mathit{\text{lnu}},\nabla \mathit{\text{lnv}}〉\\ +(p-1){\int }_{\Omega }|\nabla \mathit{\text{lnu}}{|}^p{v}^p\end{array}$$

(4.4)

and

$$\begin{array}{ll}{\int }_{\Omega }& |\nabla v{|}^{p-2}\nabla v\nabla \left(\frac{v^p-u^p}{v^{p-1}}\right)\mathit{\text{dx}}\\ ={\int }_{\Omega }|\nabla \mathit{\text{lnv}}{|}^p{v}^p-p{\int }_{\Omega }{u}^p|\nabla \mathit{\text{lnv}}{|}^{p-2}〈\nabla \mathit{\text{lnv}},\nabla \mathit{\text{lnu}}〉\\ +(p-1){\int }_{\Omega }|\nabla \mathit{\text{lnv}}{|}^p{u}^p.\end{array}$$

(4.5)

By (4.3), (4.4), and (4.5), we obtain

$$\begin{array}{ll}0& ={\int }_{\Omega }{u}^p\left(|\nabla \mathit{\text{lnu}}{|}^p-|\nabla \mathit{\text{lnv}}{|}^p-p|\nabla \mathit{\text{lnv}}{|}^{p-2}〈\nabla \mathit{\text{lnv}},\nabla \mathit{\text{lnu}}-\nabla \mathit{\text{lnv}}〉\right)\\ +{\int }_{\Omega }{v}^p\left(|\nabla \mathit{\text{lnv}}{|}^p-|\nabla \mathit{\text{lnu}}{|}^p-p|\nabla \mathit{\text{lnu}}{|}^{p-2}\right.\\ \times \left.〈\nabla \mathit{\text{lnu}},\nabla \mathit{\text{lnv}}-\nabla \mathit{\text{lnu}}〉\right).\end{array}$$

(4.6)

When p≥2 by reference [1], we have

$$\begin{array}{ll}|\nabla & \mathit{\text{lnu}}{|}^p-|\nabla \mathit{\text{lnv}}{|}^p-p|\nabla \mathit{\text{lnv}}{|}^{p-2}〈\nabla \mathit{\text{lnv}},\nabla \mathit{\text{lnu}}-\nabla \mathit{\text{lnv}}〉\\ \ge C\left(p\right)|\nabla \mathit{\text{lnv}}-\nabla \mathit{\text{lnu}}{|}^p,\\ |\nabla & \mathit{\text{lnv}}{|}^p-|\nabla \mathit{\text{lnu}}{|}^p-p|\nabla \mathit{\text{lnu}}{|}^{p-2}〈\nabla \mathit{\text{lnu}},\nabla \mathit{\text{lnv}}-\nabla \mathit{\text{lnu}}〉\\ \ge C\left(p\right)|\nabla \mathit{\text{lnu}}-\nabla \mathit{\text{lnv}}{|}^p.\end{array}$$

Therefore, (4.6) implies that

$$0\ge {\int }_{\Omega }C\left(p\right)|\nabla \mathit{\text{lnu}}-\nabla \mathit{\text{lnv}}{|}^p(u^p+v^p).$$

Hence,

$$0=|\nabla \mathit{\text{lnu}}-\nabla \mathit{\text{lnv}}|.$$

This also implies that u=k v, as we wanted to prove.

When p<2, we have

$$\begin{array}{ll}|\nabla & \mathit{\text{lnu}}{|}^p-|\nabla \mathit{\text{lnv}}{|}^p-p|\nabla \mathit{\text{lnv}}{|}^{p-2}〈\nabla \mathit{\text{lnv}},\nabla \mathit{\text{lnu}}-\nabla \mathit{\text{lnv}}〉\\ \ge C\left(p\right)\frac{|\nabla \mathit{\text{lnv}}-\nabla \mathit{\text{lnu}}{|}^p}{{\left(\right|\nabla \mathit{\text{lnu}}|+|\nabla \mathit{\text{lnv}}\left|\right)}^{2-p}},\\ |\nabla & \mathit{\text{lnv}}{|}^p-|\nabla \mathit{\text{lnu}}{|}^p-p|\nabla \mathit{\text{lnu}}{|}^{p-2}〈\nabla \mathit{\text{lnu}},\nabla \mathit{\text{lnv}}-\nabla \mathit{\text{lnu}}〉\\ \ge C\left(p\right)\frac{|\nabla \mathit{\text{lnu}}-\nabla \mathit{\text{lnv}}{|}^p}{{\left(\right|\nabla \mathit{\text{lnu}}|+|\nabla \mathit{\text{lnv}}\left|\right)}^{2-p}}.\end{array}$$

Arguing as above, we also conclude u=k v.

Theorem 4.3

Let u be an eigenfunction corresponding to λ≠λ₁, then u changes sign on Γ, that is, the sets {x∈Γ:u(x)>0} and {x∈Γ:u(x)<0} have positive measure.

Proof

Assume that u does not change sign in Ω, then we can assume that u>0 in Ω due to the Harnack inequality. Let u₁ be an eigenfunction with λ₁; making similar calculation as the ones performed in the proof of lemma 4.2, we arrive at

$$\begin{array}{ll}({\lambda }_1& -\lambda )\left({\int }_{\Omega }\left(u_1^p-u^p\right)\mathit{\text{dx}}+{\int }_{\Gamma }\left(u_1^p-u^p\right)\mathit{\text{ds}}\right)\\ =({\lambda }_1-\lambda ){\int }_{\Omega \cup \Gamma }\left(u_1^p-u^p\right)\mathit{\text{dx}}\\ \ge C{\int }_{\Omega }|\nabla \mathit{\text{ln}}{u}_1-\nabla \mathit{\text{lnu}}{|}^p\left(u_1^p+u^p\right)\mathrm{dx.}\end{array}$$

□

Hence, taking ku instead of u, for any k>0, we have

$${\int }_{\Omega \cup \Gamma }\left(u_1^p-k^p{u}^p\right)\mathit{\text{dx}}\le 0,$$

which is a contradiction if $k^p<\underset{\Omega \cup \Gamma }{\int }{u}_1^p\mathit{\text{dx}}/\underset{\Omega \cup \Gamma }{\int }{u}^p\mathit{\text{dx}}$. Therefore, u changes sign in Ω.

Suppose that u does changes sign on Γ, then we can assume u≤0 on Γ. Using u⁺ as a test function in (2.1), we conclude that

$${\int }_{\Omega }|\nabla u{|}^{p-2}\nabla u\nabla u^{+}\mathit{\text{dx}}=0.$$

Since u changes sign in Ω, the left hand side is strictly positive. This is a contradiction. Hence, u changes sign on Γ.

Isolation of the first eigenvalue

Given λ, an eigenvalue of (1.1) and u, an eigenfunction associated with λ, we define$Z\left(u\right)=\{x\in \overline{\Omega }:u(x)=0\},$N(u)= the number of components of $\overline{\Omega }\setminus Z\left(u\right),$N (λ) = s u p {N (u) :u is an eigenfunction associated with λ}.

We shall show N(λ) is finite.

Theorem 4.4

Let (u,λ) be a (weak) eigenpair of (1.1), λ≠λ₁, there exists a constant C such that |Γ⁺|≥C λ^−β and |Γ⁻|≥C λ^−β, where ${\Gamma }^{+}=\Gamma \cap \{u>0\},{\Gamma }^{-}=\Gamma \cap \{u<0\},\beta =(N-1)/(p-1)$ if 1<p<N and β=2 if p≥N.

Proof

If we let u⁻∈W^1,p(Ω) be a test function in (2.1), we obtain

$${\int }_{\Omega }|\nabla u^{-}{|}^p\mathit{\text{dx}}=\lambda \left({\int }_{\Omega }|u^{-}{|}^p\mathit{\text{dx}}+\underset{\Gamma \cap \{u<0\}}{\int }|u^{-}{|}^p\mathit{\text{ds}}\right),$$

(4.7)

that is,

$$\parallel u^{-}{\parallel }^p=(\lambda +1)\parallel u^{-}{\parallel }_{L^p\left(\Omega \right)}^p+\lambda \underset{\Gamma \cap \{u<0\}}{\int }|u^{-}{|}^p\mathrm{ds.}$$

□

When 1<p<N, we choose α=(N−1)/(N−p) and β=(N−1)/(p−1), by the Hölder inequality and the Sobolev embedding functions X↪L^αp(Γ) and X↪L^p(Ω), there exists constants C₁,C₂>0, such that

$$\parallel u^{-}{\parallel }^p\le (\lambda +1)C_1\parallel u^{-}{\parallel }^p+\lambda C_2\parallel u^{-}{\parallel }^p|{\Gamma }^{-}{|}^{1/\beta },$$

that is, |Γ⁻|≥C λ^−β, where $C={\left[\frac{1-(\lambda +1)C_1}{C_2}\right]}^{\beta }$.

When p≥N, we choose α=β=2 and by the embedding functions X↪L^2p(Γ), a similar argument works for u⁺ as above.

Theorem 4.5

The principal eigenvalue λ₁ of (1.1) is isolated. That is, there exists a>λ₁ such that λ₁ is the unique eigenvalue in [0,a].

Proof

We can prove this theorem as Theorem 5.16 of [3] by assuming

$${\Gamma }_n^{-}=\{x\in \Gamma :u_n(x)<0\},{\Gamma }_n^{+}=\{x\in \Gamma :u_n(x)>0\}.$$

□

Existence of the second eigenvalue

Proposition 5.1

For any eigenvalue λ of (2.1), we have

$${\lambda }_{N\left(\lambda \right)}\le \lambda ,$$

where N(λ) is the maximal number of nodal domains associated with λ (see Theorem 4.4), and λ_N(λ) is the N(λ)th eigenvalue taken from the L-S sequence of Theorem 2.3.

Proof

Let r=N(λ), then there is an eigenfunction u≠0 associated with λ such that r=N(u). Let ω₁,ω₂,⋯,ω _r be the r-components of $\overline{\Omega }\setminus Z\left(u\right)$. We define

$$v_i\left(x\right)=\left\{\begin{array}{ll}\frac{u\left(x\right)}{{[\underset{\Omega \cap {\omega }_i}{\int }|u{|}^p\mathit{\text{dx}}+\underset{\Gamma \cap {\omega }_i}{\int }|u{|}^p\mathit{\text{ds}}]}^{1/p}}& \text{if}x\in \overline{{\omega }_i},\\ 0& \text{if}x\in \overline{\Omega }\setminus \overline{{\omega }_i}.\end{array}\right.$$

□

Then by the Theorem C.3 in [3], we have v _i ∈X for i=1,2,⋯,r.

Let X _r denote the subspace of X which is spanned by {v₁,v₂,⋯,v _r }. For each v∈X _r , $v=\sum _{i=1}^r{\alpha }_i{v}_i$, we obtain

$$F\left(v\right)={\int }_{\Omega }|v{|}^p\mathit{\text{dx}}+{\int }_{\Gamma }|v{|}^p\mathit{\text{ds}}=\sum _{i=1}^r|{\alpha }_i{|}^{p}F\left(v_i\right)=\sum _{i=1}^r|{\alpha }_i{|}^p.$$

Thus, the map v↦F(v)^1/p is a norm on X _r . Hence, the compact set S _r is defined by

$$S_r=\left\{v\in X_r:F\left(v\right)=\frac{1}{\lambda +1}\right\},$$

which can be identified with the unit sphere of ${\mathbb{R}}^N$, and which is r. By choosing v=v _i as a test function, we obtain

$${\int }_{\Omega }|\nabla u{|}^{p-2}\nabla u\nabla v_i\mathit{\text{dx}}=\lambda \left({\int }_{\Omega }|u{|}^{p-2}u{v}_i\mathit{\text{dx}}+{\int }_{\Gamma }|u{|}^{p-2}u{v}_i\mathit{\text{ds}}\right).$$

Hence,

$$\underset{\Omega \cap {\omega }_i}{\int }|\nabla v_i{|}^p\mathit{\text{dx}}=\lambda \left(\underset{\Omega \cap {\omega }_i}{\int }|v_i{|}^p\mathit{\text{dx}}+\underset{\Gamma \cap {\omega }_i}{\int }|v_i{|}^p\mathit{\text{ds}}\right),$$

or

$$G\left(v_i\right)=(\lambda +1)F\left(v_i\right),i=1,2,\cdots ,\mathrm{r.}$$

Thus, for v∈S _r , we have

$$\begin{array}{ll}G\left(v\right)& =(\lambda +1)\sum _{i=1}^r\left|{\alpha }_i{|}^{p}F\right(v_i)=(\lambda +1\left)\sum _{i=1}^r\right|{\alpha }_i{|}^p\\ =(\lambda +1)F\left(v\right)=1.\end{array}$$

It implies $S_r\subset S_G$. Hence

$$\frac{1}{1+{\lambda }_r}={\mu }_r=\underset{H\in {\mathbb{A}}_r}{sup}\underset{v\in H}{inf}F\left(v\right)\ge infF\left(v\right)=\frac{1}{1+\lambda }.$$

Therefore λ _r ≤λ. This completes the proof.

Proposition 5.2

For any of the problems, λ₂= inf{λ:λ is an eigenvalue and λ>λ₁}.

Proof

The proof is similar to Theorem 5.19 in [3], we omit it here. □

Conclusions

There are four important conclusions that can really be drawn from this study: (1) there exists a nondecreasing sequence of nonnegative {λ _n } of (2.1); (2) there is boundedness of eigenfunctions; (3) the first eigenvalue is simple and isolated; and (4) there is an existence of a second eigenvalue.