Our analysis is based on a method used by Afrouzi and Brown [

2]. Consider, for fixed λ, the eigenvalue problem

$\left\{\begin{array}{cc}-{\Delta}_{p}u\left(x\right)-\lambda g\left(x\right){\left|u\left(x\right)\right|}^{p-2}u\left(x\right),=\mu {\left|u\left(x\right)\right|}^{p-2}u\left(x\right),\hfill & \mathsf{\text{in}}\phantom{\rule{2.77695pt}{0ex}}\Omega \mathsf{\text{,}}\hfill \\ \mathsf{\text{Ru}}={\left|\nabla u\left(x\right)\right|}^{p-2}\frac{\partial u}{\partial \nu}\left(x\right)+\alpha {\left|u\left(x\right)\right|}^{p-2}u\left(x\right)=0,\hfill & \mathsf{\text{on}}\phantom{\rule{2.77695pt}{0ex}}\partial \Omega \mathsf{\text{.}}\hfill \end{array}\right.$

(2.1)

We denote the lowest eigenvalue of (2.1) by

*μ*(

*α*, λ). Let

${S}_{\alpha ,\lambda}=\left\{\underset{\Omega}{\int}{\left|\nabla \varphi \right|}^{p}dx+\alpha \underset{\partial \Omega}{\int}{\left|\varphi \right|}^{p}d{S}_{x}-\lambda \underset{\Omega}{\int}g{\left|\varphi \right|}^{p}dx:\varphi \in {W}^{1,p}\left(\Omega \right),\underset{\Omega}{\int}{\left|\varphi \right|}^{p}=1\right\}$

When *α ≥* 0, it is clear that *S*_{α,λ}is bounded below. It is shown by variational arguments that *μ*(*α*, λ) = inf *S*_{α,λ}and that an eigenfunction corresponding to *μ*(*α*, λ) does not change sign on Ω [3]. Thus, clearly, λ is a principal eigenvalue of (1.1) if and only if *μ*(*α*, λ) = 0.

When *α* < 0, the boundedness below of *S*_{α,λ}is not obvious, but is a consequence of the following lemma.

**Lemma 2.1**.

*For every ε* > 0

*there exists a constant C*(

*ε*)

*such that*$\underset{\partial \Omega}{\int}{\left|\varphi \right|}^{p}d{S}_{x}\le \epsilon \underset{\Omega}{\int}{\left|\nabla \varphi \right|}^{p}dx+C\left(\epsilon \right)\underset{\Omega}{\int}{\left|\varphi \right|}^{p}dx$

*for all ϕ* ∈ *W*^{1,p}(Ω).

*Proof*. Suppose that the result does not hold. Then ε

_{0} > 0 and sequence {

*u*_{
n
}} ⊆

*W*^{1,p}(Ω) such that

*∫*_{Ω}|∇

*u*_{
n
}|

^{
p
}= 1 and

$\underset{\partial \Omega}{\int}{\left|{u}_{n}\right|}^{p}d{S}_{x}\ge {\epsilon}_{0}+n\underset{\Omega}{\int}{\left|{u}_{n}\right|}^{p}dx.$

(2.2)

Suppose first that {*∫*_{Ω} |*u*_{
n
}*|*^{
p
} *dx*} is unbounded. Let ${v}_{n}=\frac{{u}_{n}}{{\u2225{u}_{n}\u2225}_{{L}^{p}\left(\Omega \right)}}$. Clearly, {*υ*_{
n
}} is bounded in *W*^{1,p}(Ω), and so in *L*^{
p
}(*∂* Ω). But *∫*_{∂ Ω}|υ_{n}|^{
p
}*dS*_{
x
}≥ *n* *∫*_{Ω} |*υ*_{
n
}|^{
p
}*dx* = *n*, which is impossible.

Suppose now that {*∫*_{Ω} |*u*_{
n
}|^{
p
}*dx*} is bounded, then {*u*_{
n
}} is bounded in *W*^{1,p}and so has a subsequence, which we again denote by {*u*_{
n
}}, converging weakly to *u* in *W*^{1,p}. Since *W*^{1,p}is compactly embedded in *L*^{
p
}(*∂* Ω) and in *L*^{
p
}(Ω), it follows that {*u*_{
n
}} converges to some function *u* in *L*^{
p
}(*∂* Ω) and in *L*^{
p
}(Ω). Thus {*∫*_{
∂Ω
}|*u*_{
n
}|^{
p
}*dx*} is bounded, and so it follows from (2.2) that lim_{n→∞}*∫* Ω |*u*_{
n
}|^{
p
} *dx* = 0, i.e.,, {*u*_{
n
}} converges to zero in *L*^{
p
}(Ω). Hence {*u*_{
n
}} converges to zero in *L*^{
p
}(*∂* Ω), and this is impossible because (2.2).

Choosing $\epsilon <\frac{1}{\alpha}$, it is easy to deduce from the above result the *S*_{α,λ}is bounded below, and it follows exactly as in [3] that *μ*(*α*, λ) = inf *S*_{α,λ}and that an eigenfunction corresponding to *μ*(*α*, λ) does not change sign on Ω. Thus it is again λ is a principal eigenvalue of (1.1) if and only if *μ*(*α*, λ) = 0.

For fixed *ϕ* ∈ *W*^{1,p}(*Ω*), λ → *∫*_{Ω} |∇*ϕ*|^{
p
}*dx*+α *∫*_{∂ Ω}|*ϕ*|^{
p
}*dS*_{x}-λ *∫*_{Ω} *g|ϕ*|^{
p
}*dx* is an affine and so concave function. As the infimum of any collection of concave functions is concave, it follows that λ → *μ*(*α*, λ) is concave. Also, by considering test functions *ϕ*_{1}, *ϕ*_{2} ∈ *W*^{1,p}(Ω) such that *∫*_{Ω} *g |ϕ*_{1}|^{
p
}*dx* > 0 and *∫*_{Ω} *g|ϕ*_{2}|^{
p
}*dx* < 0, it is easy to see that *μ*(*α*, λ) → -∞ as λ → *±* ∞. Thus λ → *μ*(*α*, λ) is an increasing function until it attains its maximum, and is an decreasing function thereafter.

It is natural that the flux across the boundary should be outwards if there is a positive concentration at the boundary. This motivates the fact that the sign of *α* > 0. For a physical motivation of such conditions, see for example [4]. Suppose that 0 < *α* < ∞, i.e., we have the Robin boundary condition. Then, as can be seen from the the variational characterization of *μ*(*α*, λ) or -Δ_{
p
}has a positive principal eigenvalue, *μ*(*α*, 0) > 0 and so λ → *μ*(*α*, λ) must has exactly two zero. Thus in this case (1.1) exactly two principal eigenvalues, one positive and one negative.

Our results may be summarized in the following theorem.

**Theorem 2.2**. *If* 0 < *α* < ∞, *then (1.1) exactly two principal eigenvalues, one positive and one negative*.

However, for

*α* < 0 we have

*μ*(

*α*, 0) ≤ 0. For

*p* = 2, if

*u*_{0} is eigenfunction of (2.1) corresponding to principal eigenvalue

*μ*(

*α*, λ), then

$\frac{d\mu}{d\lambda}\left(\alpha ,\lambda \right)=-\frac{{\int}_{\Omega}g{u}_{0}^{2}dx}{{\int}_{\Omega}{u}_{0}^{2}dx}.$

(2.3)

Therefore, λ → *μ*(*α*, λ) is an increasing (decreasing) function, if we have $\frac{{\int}_{\Omega}g{u}_{0}^{2}dx}{{\int}_{\Omega}{u}_{0}^{2}dx}<0\left(>0\right)$ and at critical points we must have $\frac{{\int}_{\Omega}g{u}_{0}^{2}dx}{{\int}_{\Omega}{u}_{0}^{2}dx}=0$ (see, [2], Lemma 2]).

But, we cannot generalize it for

*p ≠* 2. Because, if

$v\left(\lambda \right)=\frac{d\mu}{d\lambda}$, then we have

$-\frac{d}{d\lambda}{\Delta}_{p}u\left(\lambda \right)=-\left(p-1\right)div\left(\nabla v{\left|\nabla u\right|}^{p-2}\right).$

So, we cannot get a similar result (2.3).

Now our analysis is based by Drabek and Schindler [

5]. We define the space

*V*_{
p
} as completion of

${W}^{1,p}\left(\Omega \right)\cap C\left(\stackrel{\u0304}{\Omega}\right)$ with respect to the norm

${{\u2225u\u2225}_{V}}_{p}={\left(\underset{\Omega}{\int}{\left|\nabla u\right|}^{p}dx+\underset{\partial \Omega}{\int}{\left|u\right|}^{p}ds\right)}^{\frac{1}{p}}.$

(2.4)

The spaces equivalent to *V*_{
p
} were introduced in [6]. In particular, *V*_{
p
} is a uniformly convex (and hence a reflexive) Banach space, *V*_{
p
} ↪ *L*^{
q
}(Ω) continuously for $1\le q\le \frac{Np}{N-1}$ and *V*_{
p
} ↪ *L*^{
q
}(Ω) compactly for $1\le q\le \frac{Np}{N-1}$ [6].

We say that

*u* **∈** *V*_{
p
} is a weak solution to (1.1) if for all

*ϕ* **∈** *V*_{
p
} we have

$\underset{\Omega}{\int}{\left|\nabla u\right|}^{p-2}\nabla u.\nabla \varphi dx+\alpha \underset{\partial \Omega}{\int}{\left|u\right|}^{p-2}u\varphi ds=\underset{\Omega}{\int}\lambda g\left(x\right){\left|u\right|}^{p-2}u\varphi dx.$

(2.5)

In fact there are domains Ω for which the embedding *V*_{
p
} ↪ *L*^{
p
} (Ω) is not injective. This is to the influence of the wildness of the boundary *∂* Ω. The domains for which the above embedding is injective are then called admissible. Ω is called admissible irregular domain for which *W*^{1,p}(Ω) is not subset *L*^{
q
}(Ω) for all *p* > *q*.

We assume that the domain Ω ⊂ ℝ

^{
N
}is bounded,

*N* > 1,

*α* > 0, and 1 <

*p* <

*N*. We apply variational for (1.1) with λ = 1. We introduce the

*C*^{1}-functionals

$l\left(u\right)=\underset{\Omega}{\int}{\left|\nabla u\right|}^{p}+\alpha \underset{\partial \Omega}{\int}{\left|u\right|}^{p}ds.$

(2.6)

and

$j\left(u\right)=\underset{\Omega}{\int}g\left(x\right){\left|u\right|}^{p}.$

(2.7)

If

*w* ∈

*V*_{
p
} be a global minimizer of

*l* subject to the constraint

*j*(

*w*) = 1, then the Lagrange multiplier method yields a λ ∈ ℝ such that

*l'*(

*u*) = λ

*j*'(

*u*), i.e.,

$p\underset{\Omega}{\int}{\left|\nabla w\right|}^{p-2}\nabla w.\nabla \varphi dx+p\alpha \underset{\partial \Omega}{\int}{\left|w\right|}^{p-2}w\varphi ds=\lambda p\underset{\Omega}{\int}g\left(x\right){\left|u\right|}^{p-2}u\varphi dx$

holds for any *ϕ* ∈ *V*_{
p
}. Then *w* is a weak solution (1.1). The existence of a minimizer follows from the fact that *l*(*u*) is bounded from below on the manifold *M =* {*u* ∈ *V*_{
p
} : *j*(*u*) = 1} and from Palais-Smale condition satisfied by the functional *l* on *M*.