# A note on some nonlinear principal eigenvalue problems

- Mohsen Boujary
^{1}Email author and - Ghasem Alizadeh Afrouzi
^{2}

**2012**:44

https://doi.org/10.1186/1687-2770-2012-44

© Boujary and Afrouzi; licensee Springer. 2012

**Received: **29 November 2011

**Accepted: **16 April 2012

**Published: **16 April 2012

## Abstract

We investigate the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) the boundary value problem

where Ω ⊆ ℝ^{
N
}is a bounded domain, 1 < *p* < ∞ and *α* is a real number.

**AMS Subject Classification:** 35J60; 35B30; 35B40.

## Keywords

*p*-Laplacianprincipal eigenvalue

## 1. Introduction

Mathematical models described by nonlinear partial differential equations have become more common recently. In particular, the *p*-Laplacian operator appears in subjects such as filtration problem, power-low materials, non-Newtonian fluids, reaction-diffusion problems, nonlinear elasticity, petroleum extraction, etc., see,[1]. The nonlinear boundary condition describes the flux through the boundary *∂* Ω which depends on the solution itself.

where Ω ⊆ ℝ^{
N
}is a bounded domain, 1 < *p* < ∞ and *α* is a real number. Attention has been confined mainly to the cases of Dirichlet and Neumann boundary conditions but we have the Robin boundary in (1.1).

We discuss about to exist principal eigenvalue for (1.1). In the case 0 < *α* < ∞, We shall show that there has exactly two principal eigenvalues, one positive and one negative.

## 2. Main result

*μ*(

*α*, λ). Let

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*

*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

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*.

*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

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]).

*p ≠*2. Because, if $v\left(\lambda \right)=\frac{d\mu}{d\lambda}$, then we have

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

*V*

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

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].

*u*

**∈**

*V*

_{ p }is a weak solution to (1.1) if for all

*ϕ*

**∈**

*V*

_{ p }we have

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*.

^{ 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

*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.,

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*.

## Declarations

## Authors’ Affiliations

## References

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