- Research
- Open Access
- Published:

# A note on some nonlinear principal eigenvalue problems

*Boundary Value Problems*
**volume 2012**, Article number: 44 (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.

## 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.

The purpose of this study is to discuss the existence of principal eigenvalues (i.e., eigenvalues corresponding to positive eigenfunctions) for the boundary value problem

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

Our analysis is based on a method used by Afrouzi and Brown [2]. Consider, for fixed λ, the eigenvalue problem

We denote the lowest eigenvalue of (2.1) by *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*.

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

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

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

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

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

and

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.,

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*.

## References

Diaz JI:

*Nonlinear Partial Differential Equatians and Free Boundaries.**Volume I*. Elliptic Equations, London; 1985.Afrouzi GA, Brown KJ: On principal eigenvalues for boundary value problems with indefinite weight and Robin boundary conditions.

*Proc Am Math Soc*1999, 127(1):125-130.Smoller J:

*Shock Waves and Reaction-Diffusion Equations*. Springer-Verlag, Berlin; 1983.Pao CV:

*Nonlinear Parabolic and Elliptic Equations*. Plenum Press, New York, London; 1992.Drabek P, Schindler I: Positive solutions for

*p*-Laplacian with Robin boundary conditions on irregular domains.*Appl Math Lett*2011, 24: 588-591.Maz'ja VG:

*Sobolev Spaces in: Springer Series in Soviet Mathematics (translated from the Russian by Shaposhnikova, TO).*Springer-Verlag, Berlin; 1985.

## Author information

### Authors and Affiliations

### Corresponding author

## Additional information

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

Boujary has presented the main purpose of the article and has used Afrouzi contribution due to reaching to conclusions. All authors read and approved the final manuscript.

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Boujary, M., Afrouzi, G.A. A note on some nonlinear principal eigenvalue problems.
*Bound Value Probl* **2012**, 44 (2012). https://doi.org/10.1186/1687-2770-2012-44

Received:

Accepted:

Published:

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

### Keywords

*p*-Laplacian- principal eigenvalue