Open Access

Multiplicity of positive solutions for eigenvalue problems of ( p , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq1_HTML.gif-equations

Boundary Value Problems20122012:152

DOI: 10.1186/1687-2770-2012-152

Received: 13 September 2012

Accepted: 7 December 2012

Published: 28 December 2012

Abstract

We consider a nonlinear parametric equation driven by the sum of a p-Laplacian ( p > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq2_HTML.gif) and a Laplacian (a ( p , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq1_HTML.gif-equation) with a Carathéodory reaction, which is strictly ( p 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq3_HTML.gif-sublinear near +∞. Using variational methods coupled with truncation and comparison techniques, we prove a bifurcation-type theorem for the nonlinear eigenvalue problem. So, we show that there is a critical parameter value λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq4_HTML.gif such that for λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq5_HTML.gif the problem has at least two positive solutions, if λ = λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq6_HTML.gif, then the problem has at least one positive solution and for λ ( 0 , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq7_HTML.gif, it has no positive solutions.

MSC: 35J25, 35J92.

Keywords

nonlinear regularity tangency principle p-Laplacian bifurcation-type theorem positive solutions

1 Introduction

Let Ω R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq8_HTML.gif be a bounded domain with a C 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq9_HTML.gif-boundary Ω. In this paper, we study the following nonlinear Dirichlet eigenvalue problem:
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equa_HTML.gif
Here, by Δ p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq10_HTML.gif we denote the p-Laplace differential operator defined by
Δ p u ( z ) = div ( u ( z ) p 2 u ( z ) ) u W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equb_HTML.gif

(with 2 < p < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq11_HTML.gif). In ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif, λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq13_HTML.gif is a parameter and f ( z , ζ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq14_HTML.gif is a Carathéodory function (i.e., for all ζ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq15_HTML.gif, the function z f ( z , ζ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq16_HTML.gif is measurable and for almost all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif, the function ζ f ( z , ζ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq18_HTML.gif is continuous), which exhibits strictly ( p 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq3_HTML.gif-sublinear growth in the ζ-variable near +∞. The aim of this paper is to determine the precise dependence of the set of positive solutions on the parameter λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq13_HTML.gif. So, we prove a bifurcation-type theorem, which establishes the existence of a critical parameter value λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq4_HTML.gif such that for all λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq5_HTML.gif, problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif has at least two nontrivial positive smooth solutions, for λ = λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq6_HTML.gif, problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif has at least one nontrivial positive smooth solution and for λ ( 0 , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq7_HTML.gif, problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif has no positive solution. Similar nonlinear eigenvalue problems with ( p 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq3_HTML.gif-sublinear reaction were studied by Maya and Shivaji [1] and Rabinowitz [2] for problems driven by the Laplacian and by Guo [3], Hu and Papageorgiou [4] and Perera [5] for problems driven by the p-Laplacian. However, none of the aforementioned works produces the precise dependence of the set of positive solutions on the parameter λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq13_HTML.gif (i.e., they do not prove a bifurcation-type theorem). We mention that in problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif the differential operator is not homogeneous in contrast to the case of the Laplacian and p-Laplacian. This fact is the source of difficulties in the study of problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif which lead to new tools and methods.

We point out that ( p , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq1_HTML.gif-equations (i.e., equations in which the differential operator is the sum of a p-Laplacian and a Laplacian) are important in quantum physics in the search for solitions. We refer to the work of Benci, D’Avenia-Fortunato and Pisani [6]. More recently, there have been some existence and multiplicity results for such problems; see Cingolani and Degiovanni [7], Sun [8]. Finally, we should mention the recent papers of Marano and Papageorgiou [9, 10]. In [9] the authors deal with parametric p-Laplacian equations in which the reaction exhibits competing nonlinearities (concave-convex nonlinearity). In [10], they study a nonparametric ( p , q ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq19_HTML.gif-equation with a reaction that has different behavior both at ±∞ and at 0 from those considered in the present paper, and so the geometry of the problem is different.

Out approach is variational based on the critical point theory, combined with suitable truncation and comparison techniques. In the next section, for the convenience of the reader, we briefly recall the main mathematical tools that we use in this paper.

2 Mathematical background

Let X be a Banach space and let X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq20_HTML.gif be its topological dual. By , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq21_HTML.gif we denote the duality brackets for the pair ( X , X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq22_HTML.gif. Let φ C 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq23_HTML.gif. A point x 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq24_HTML.gif is a critical point of φ if φ ( x 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq25_HTML.gif. A number c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq26_HTML.gif is a critical value of φ if there exists a critical point x 0 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq24_HTML.gif such that φ ( x 0 ) = c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq27_HTML.gif.

We say that φ C 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq23_HTML.gif satisfies the Palais-Smale condition if the following is true:

‘Every sequence { x n } n 1 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq28_HTML.gif, such that { φ ( x n ) } n 1 R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq29_HTML.gif is bounded and
φ ( x n ) 0 in  X , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equc_HTML.gif

admits a strongly convergent subsequence.’

This compactness-type condition is crucial in proving a deformation theorem which in turn leads to the minimax theory of certain critical values of φ C 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq23_HTML.gif (see, e.g., Gasinski and Papageorgiou [11]). A well-written discussion of this compactness condition and its role in critical point theory can be found in Mawhin and Willem [12]. One of the minimax theorems needed in the sequel is the well-known ‘mountain pass theorem’.

Theorem 2.1 If φ C 1 ( X ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq23_HTML.gif satisfies the Palais-Smale condition, x 0 , x 1 X https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq30_HTML.gif, x 1 x 0 > r > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq31_HTML.gif,
max { φ ( x 0 ) , φ ( x 1 ) } < inf { φ ( x ) : x x 0 = r } = η r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equd_HTML.gif
and
c = inf γ Γ max 0 t 1 φ ( γ ( t ) ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Eque_HTML.gif
where
Γ = { γ C ( [ 0 , 1 ] ; X ) : γ ( 0 ) = x 0 , γ ( 1 ) = x 1 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equf_HTML.gif

then c η r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq32_HTML.gif and c is a critical value of φ.

In the analysis of problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif, in addition to the Sobolev space W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq33_HTML.gif, we will also use the Banach space
C 0 1 ( Ω ¯ ) = { u C 1 ( Ω ¯ ) : u | Ω = 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equg_HTML.gif
This is an ordered Banach space with a positive cone:
C + = { u C 0 1 ( Ω ¯ ) : u ( z ) 0  for all  z Ω ¯ } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equh_HTML.gif
This cone has a nonempty interior given by
int C + = { u C + : u ( z ) > 0  for all  z Ω , u n ( z ) < 0  for all  z Ω } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equi_HTML.gif

where by n ( ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq34_HTML.gif we denote the outward unit normal on Ω.

Let f 0 : Ω × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq35_HTML.gif be a Carathéodory function with subcritical growth in ζ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq15_HTML.gif, i.e.,
| f 0 ( z , ζ ) | a 0 ( z ) + c 0 | ζ | r 1 for almost all  z Ω ,  all  ζ R , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equj_HTML.gif
with a 0 L ( Ω ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq36_HTML.gif, c 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq37_HTML.gif and 1 < r < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq38_HTML.gif, where
p = { N p N p if  p < N , + if  p N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equk_HTML.gif

(the critical Sobolev exponent).

We set
F 0 ( z , ζ ) = 0 ζ f 0 ( z , s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equl_HTML.gif
and consider the C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq39_HTML.gif-functional ψ 0 : W 0 1 , p ( Ω ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq40_HTML.gif defined by
ψ 0 ( u ) = 1 p u p p + 1 2 u 2 2 Ω F 0 ( z , u ( z ) ) d z u W 0 1 , p ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ1_HTML.gif
(2.1)

The next proposition is a special case of a more general result proved by Gasinski and Papageorgiou [13]. We mention that the first result of this type was proved by Brezis and Nirenberg [14].

Proposition 2.2 If ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq41_HTML.gif is defined by (2.1) and u 0 W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq42_HTML.gif is a local C 0 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq43_HTML.gif-minimizer of ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq41_HTML.gif, i.e., there exists ϱ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq44_HTML.gif such that
ψ 0 ( u 0 ) ψ 0 ( u 0 + h ) h C 0 1 ( Ω ¯ ) , h C 0 1 ( Ω ¯ ) ϱ 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equm_HTML.gif
then u 0 C 1 1 , β ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq45_HTML.gif for some β ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq46_HTML.gif and u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq47_HTML.gif is also a local W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq33_HTML.gif-minimizer of ψ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq41_HTML.gif, i.e., there exists ϱ 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq48_HTML.gif such that
ψ 0 ( u 0 ) ψ 0 ( u 0 + h ) h W 0 1 , p ( Ω ) , h ϱ 2 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equn_HTML.gif
Let g , h L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq49_HTML.gif. We say that g h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq50_HTML.gif if for all compact subsets K Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq51_HTML.gif, we can find ε = ε ( K ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq52_HTML.gif such that
g ( z ) + ε h ( z ) for almost all  z K . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equo_HTML.gif

Clearly, if g , h C ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq53_HTML.gif and g ( z ) < h ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq54_HTML.gif for all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif, then g h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq50_HTML.gif. A slight modification of the proof of Proposition 2.6 of Arcoya and Ruiz [15] in order to accommodate the presence of the extra linear term Δ u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq55_HTML.gif leads to the following strong comparison principle.

Proposition 2.3 If ξ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq56_HTML.gif, g , h L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq49_HTML.gif, g h https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq50_HTML.gif and u C 0 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq57_HTML.gif, v int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq58_HTML.gif are solutions of the problems
{ Δ p u ( z ) Δ u ( z ) + ξ | u ( z ) | p 2 u ( z ) = g ( z ) in Ω , Δ p v ( z ) Δ v ( z ) + ξ | v ( z ) | p 2 v ( z ) = h ( z ) in Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equp_HTML.gif

then v u int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq59_HTML.gif.

Let r ( 1 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq60_HTML.gif and let A r : W 0 1 , r ( Ω ) W 1 , r ( Ω ) = W 0 1 , r ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq61_HTML.gif (where 1 r + 1 r = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq62_HTML.gif) be a nonlinear map defined by
A r ( u ) , y = Ω u r 2 ( u , y ) R N d z u , y W 0 1 , r ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ2_HTML.gif
(2.2)

The next proposition can be found in Dinca, Jebelean and Mawhin [16] and Gasiński and Papageorgiou [11].

Proposition 2.4 If A r : W 0 1 , r ( Ω ) W 1 , r ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq63_HTML.gif (where 1 < r < + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq64_HTML.gif) is defined by (2.2), then A r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq65_HTML.gif is continuous, strictly monotone (hence maximal monotone too), bounded and of type ( S ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq66_HTML.gif, i.e., if u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq67_HTML.gif weakly in W 0 1 , r ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq68_HTML.gif and
lim sup n + A r ( u n ) , u n u 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equq_HTML.gif

then u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq67_HTML.gif in W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq33_HTML.gif.

If r = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq69_HTML.gif, then we write A 2 = A L ( H 0 1 ( Ω ) ; H 1 ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq70_HTML.gif.

In what follows, by λ ˆ 1 ( p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq71_HTML.gif we denote the first eigenvalue of the negative Dirichlet p-Laplacian ( Δ p , W 0 1 , p ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq72_HTML.gif. We know that λ ˆ 1 ( p ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq73_HTML.gif and it admits the following variational characterization:
λ ˆ 1 ( p ) = inf { u p p u p p : u W 0 1 , p ( Ω ) , u 0 } . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ3_HTML.gif
(2.3)
Finally, throughout this work, by https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq74_HTML.gif we denote the norm of the Sobolev space W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq33_HTML.gif. By virtue of the Poincaré inequality, we have
u = u p u W 0 1 , p ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equr_HTML.gif
The notation https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq74_HTML.gif will also be used to denote the norm of R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq75_HTML.gif. No confusion is possible since it will always be clear from the context which norm is used. For ζ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq15_HTML.gif, we set ζ ± = max { ± ζ , 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq76_HTML.gif. Then for u W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq77_HTML.gif, we define u ± ( ) = u ( ) ± https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq78_HTML.gif. We know that
u ± W 0 1 , p ( Ω ) , | u | = u + + u , u = u + u u W 0 1 , p ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equs_HTML.gif
If h : Ω × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq79_HTML.gif is superpositionally measurable (for example, a Carathéodory function), then we set
N h ( u ) ( ) = h ( , u ( ) ) u W 0 1 , p ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equt_HTML.gif

By | | N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq80_HTML.gif we denote the Lebesgue measure on R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq75_HTML.gif.

3 Positive solutions

The hypotheses on the reaction f are the following.

H: f : Ω × R R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq81_HTML.gif is a Carathéodory function such that f ( z , 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq82_HTML.gif for almost all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif, f ( z , ζ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq83_HTML.gif for almost all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif and all ζ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq84_HTML.gif and

(i) for every ϱ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq85_HTML.gif, there exists a ϱ L ( Ω ) + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq86_HTML.gif such that
f ( z , ζ ) a ϱ ( z ) for almost all  z Ω ,  all  ζ [ 0 , ϱ ] ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equu_HTML.gif

(ii) lim ζ + f ( z , ζ ) ζ p 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq87_HTML.gif uniformly for almost all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif;

(iii) lim ζ 0 + f ( z , ζ ) ζ p 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq88_HTML.gif uniformly for almost all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif;

(iv) for every ϱ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq85_HTML.gif, there exists ξ ϱ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq89_HTML.gif such that for almost all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif, the map ζ f ( z , ζ ) + ξ p ζ p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq90_HTML.gif is nondecreasing on [ 0 , ϱ ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq91_HTML.gif;

(v) if
F ( z , ζ ) = 0 ζ f ( z , s ) d s , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equv_HTML.gif
then there exists c R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq26_HTML.gif such that
F ( z , c ) > 0 for almost all  z Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equw_HTML.gif

Remark 3.1 Since we are looking for positive solutions and hypotheses H concern only the positive semiaxis R + = [ 0 , + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq92_HTML.gif, we may and will assume that f ( z , ζ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq93_HTML.gif for almost all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif and all ζ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq94_HTML.gif. Hypothesis H(ii) implies that for almost all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif, the map f ( z , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq95_HTML.gif is strictly ( p 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq3_HTML.gif-sublinear near +∞. Hypothesis H(iv) is much weaker than assuming the monotonicity of f ( z , ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq95_HTML.gif for almost all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif.

Example 3.2 The following functions satisfy hypotheses H (for the sake of simplicity, we drop the z-dependence):
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equx_HTML.gif

with 1 < q < p < τ < η https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq96_HTML.gif. Clearly f 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq97_HTML.gif is not monotone.

Let
Y = { λ > 0 : ( P ) λ  problem has a nontrivial positive solution } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equy_HTML.gif
and let S ( λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq98_HTML.gif be the set of solutions of ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif. We set
λ = inf Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equz_HTML.gif

(if Y = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq99_HTML.gif, then λ = + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq100_HTML.gif).

Proposition 3.3 If hypotheses H hold, then
S ( λ ) int C + and λ > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equaa_HTML.gif
Proof Clearly, the result is true if Y = https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq99_HTML.gif. So, suppose that Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq101_HTML.gif and let λ Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq102_HTML.gif. So, we can find u S ( λ ) W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq103_HTML.gif such that
{ Δ p u ( z ) Δ u ( z ) = λ f ( z , u ( z ) ) in  Ω , u | Ω = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equab_HTML.gif
From Ladyzhenskaya and Uraltseva [[17], p.286], we have that u L ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq104_HTML.gif. Then we can apply Theorem 1 of Lieberman [18] and have that u int C + { 0 } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq105_HTML.gif. Let ϱ = u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq106_HTML.gif and let ξ ϱ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq89_HTML.gif be as postulated by hypothesis H(iv). Then
Δ p u ( z ) Δ u ( z ) + ξ ϱ u ( z ) p 1 0 for almost all  z Ω , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equac_HTML.gif
so
Δ p u ( z ) + Δ u ( z ) ξ ϱ u ( z ) p 1 for almost all  z Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equad_HTML.gif
From the strong maximum principle of Pucci and Serrin [[19], p.34], we have that
u ( z ) > 0 z Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equae_HTML.gif

So, we can apply the boundary point theorem of Pucci and Serrin [[19], p.120] and have that u int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq107_HTML.gif. Therefore, S ( λ ) int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq108_HTML.gif.

By virtue of hypotheses H(ii) and (iii), we see that we can find c 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq109_HTML.gif such that
f ( z , ζ ) c 1 ζ p 1 for almost all  z Ω ,  all  ζ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ4_HTML.gif
(3.1)
Let λ 0 ( 0 , λ ˆ 1 ( p ) c 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq110_HTML.gif and ϑ ( 0 , λ 0 ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq111_HTML.gif. Suppose that ϑ Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq112_HTML.gif. Then from the first part of the proof, we know that we can find u ϑ S ( ϑ ) int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq113_HTML.gif. We have
A p ( u ϑ ) + A ( u ϑ ) = ϑ N f ( u ϑ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equaf_HTML.gif
so
u ϑ p p Ω ϑ f ( z , u ϑ ) u ϑ d z ϑ c 1 u ϑ p p λ 0 c 1 u ϑ p p < λ ˆ 1 ( p ) u ϑ p p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equag_HTML.gif

(see (3.1) and recall that ϑ λ 0 < λ ˆ 1 ( p ) c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq114_HTML.gif), which contradicts (2.3). Therefore, λ λ 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq115_HTML.gif. □

For λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq13_HTML.gif, let φ λ : W 0 1 , p ( Ω ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq116_HTML.gif be the energy functional for problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif defined by
φ λ ( u ) = 1 p u p p + 1 2 u 2 2 λ Ω F ( z , u ) d z u W 0 1 , p ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equah_HTML.gif

Evidently, φ λ C ( W 0 1 , p ( Ω ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq117_HTML.gif.

Proposition 3.4 If hypotheses H hold, then Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq101_HTML.gif.

Proof By virtue of hypotheses H(i) and (ii), for a given ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq118_HTML.gif, we can find c ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq119_HTML.gif such that
F ( z , ζ ) ε p ζ p + c ε for almost all  z Ω ,  all  ζ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ5_HTML.gif
(3.2)
Then for u W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq77_HTML.gif and λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq13_HTML.gif, we have
φ λ ( u ) = 1 p u p p + 1 2 u 2 2 λ Ω F ( z , u ) d z 1 p u p p λ ε p u + p p λ c ε | Ω | N = 1 p ( 1 λ ε λ ˆ 1 ( p ) ) u p λ c ε | Ω | N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ6_HTML.gif
(3.3)

(see (3.2) and (2.3)).

Let ε ( 0 , λ ˆ 1 ( p ) λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq120_HTML.gif. Then from (3.3) it follows that φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq121_HTML.gif is coercive. Also, exploiting the compactness of the embedding W 0 1 , p ( Ω ) L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq122_HTML.gif (by the Sobolev embedding theorem), we see that φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq121_HTML.gif is sequentially weakly lower semicontinuous. So, by the Weierstrass theorem, we can find u 0 W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq42_HTML.gif such that
φ λ ( u 0 ) = inf u W 0 1 , p ( Ω ) φ λ ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ7_HTML.gif
(3.4)
Consider the integral functional K : L p ( Ω ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq123_HTML.gif defined by
K ( u ) = Ω F ( z , u ( z ) ) d z u L p ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equai_HTML.gif
Hypothesis H(v) implies that K ( c ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq124_HTML.gif and since F ( z , ζ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq125_HTML.gif for almost all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif, all ζ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq94_HTML.gif, we may assume that c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq126_HTML.gif. Since W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq33_HTML.gif is dense in L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq127_HTML.gif and c > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq126_HTML.gif, we can find v ˆ W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq128_HTML.gif, v ˆ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq129_HTML.gif, such that K ( v ˆ ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq130_HTML.gif. Then for λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq13_HTML.gif large, we have
λ K ( v ˆ ) > 1 p v ˆ p p + 1 2 v ˆ 2 2 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equaj_HTML.gif
so
φ λ ( v ˆ ) < 0 for  λ > 0  large https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equak_HTML.gif
and thus
φ λ ( u 0 ) < 0 = φ λ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equal_HTML.gif
(see (3.4)), hence u 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq131_HTML.gif. From (3.4), we have
φ λ ( u 0 ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equam_HTML.gif
so
A p ( u 0 ) + A ( u 0 ) = λ N f ( u 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ8_HTML.gif
(3.5)
On (3.5), we act with u 0 W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq132_HTML.gif. Then
u 0 p p + u 0 2 2 = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equan_HTML.gif

hence u 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq133_HTML.gif, u 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq131_HTML.gif.

From (3.5), we have
{ Δ p u 0 ( z ) Δ u 0 ( z ) = λ f ( z , u 0 ( z ) ) in  Ω , u 0 | Ω = 0 , u 0 0 , u 0 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equao_HTML.gif

so u 0 S ( λ ) int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq134_HTML.gif (see Proposition 3.3).

So, for λ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq135_HTML.gif big, we have λ Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq136_HTML.gif and so Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq101_HTML.gif. □

Proposition 3.5 If hypotheses H hold and λ Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq102_HTML.gif, then [ λ , + ) Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq137_HTML.gif.

Proof Since by hypothesis λ Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq102_HTML.gif, we can find a solution u λ int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq138_HTML.gif of ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif (see Proposition 3.3). Let μ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq139_HTML.gif and consider the following truncation of the reaction in problem ( P ) μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq140_HTML.gif:
h μ ( z , ζ ) = { μ f ( z , u λ ( z ) ) if  ζ u λ ( z ) , μ f ( z , ζ ) if  u λ ( z ) < ζ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ9_HTML.gif
(3.6)
This is a Carathéodory function. Let
H μ ( z , ζ ) = 0 ζ h μ ( z , s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equap_HTML.gif
and consider the C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq39_HTML.gif-functional ψ μ : W 0 1 , p ( Ω ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq141_HTML.gif, defined by
ψ μ ( u ) = 1 p u p p + 1 2 u 2 2 Ω H μ ( z , u ( z ) ) d z u W 0 1 , p ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equaq_HTML.gif
As in the proof of Proposition 3.4, using hypotheses H(i) and (ii), we see that ψ μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq142_HTML.gif is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u μ W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq143_HTML.gif such that
ψ μ ( u μ ) = inf u W 0 1 , p ( Ω ) ψ μ ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equar_HTML.gif
so
ψ μ ( u μ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equas_HTML.gif
and thus
A p ( u μ ) + A ( u μ ) = N h μ ( u μ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ10_HTML.gif
(3.7)
On (3.7) we act with ( u λ u μ ) + W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq144_HTML.gif. Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equat_HTML.gif
(see (3.6) and use the facts that μ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq139_HTML.gif and f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq145_HTML.gif), so
{ u λ > u μ } ( u λ p 2 u λ u μ p 2 u μ , u λ u μ ) R N d z + ( u λ u μ ) + 2 2 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equau_HTML.gif
thus
| { u λ > u μ } | N = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equav_HTML.gif

and hence u λ u μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq146_HTML.gif.

Therefore, (3.7) becomes
A p ( u μ ) + A ( u μ ) = μ N f ( u μ ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equaw_HTML.gif
so
u μ S ( μ ) int C + , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equax_HTML.gif

hence μ Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq147_HTML.gif. This proves that [ λ , + ) Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq137_HTML.gif. □

Proposition 3.6 If hypotheses H hold, then for every λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq5_HTML.gif problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif has at least two positive solutions
u 0 , u ˆ int C + , u 0 u ˆ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equay_HTML.gif
Proof Note that Proposition 3.5 implies that ( λ , + ) Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq148_HTML.gif. Let λ < ϑ < λ < μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq149_HTML.gif. Then we can find u ϑ S ( ϑ ) int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq113_HTML.gif and u μ S ( μ ) int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq150_HTML.gif. We have
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ11_HTML.gif
(3.8)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ12_HTML.gif
(3.9)
(recall that f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq145_HTML.gif and ϑ < λ < μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq151_HTML.gif). As in the proof of Proposition 3.5, we can show that u ϑ u μ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq152_HTML.gif. We introduce the following truncation of the reaction in problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif:
g λ ( z , ζ ) = { λ f ( z , u ϑ ( z ) ) if  ζ < u ϑ ( z ) , λ f ( z , ζ ) if  u ϑ ( z ) ζ u μ ( z ) , λ f ( z , u μ ( z ) ) if  u μ ( z ) < ζ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ13_HTML.gif
(3.10)
This is a Carathéodory function. We set
G λ ( z , ζ ) = 0 ζ g λ ( z , s ) d s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equaz_HTML.gif
and consider the C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq39_HTML.gif-functional ψ ˆ λ : W 0 1 , p ( Ω ) R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq153_HTML.gif defined by
ψ ˆ λ ( u ) = 1 p u p p + 1 2 u 2 2 Ω G λ ( z , u ) d z u W 0 1 , p ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equba_HTML.gif
It is clear from (3.10) that ψ ˆ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq154_HTML.gif is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u 0 W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq42_HTML.gif such that
ψ ˆ λ ( u 0 ) = inf u W 0 1 , p ( Ω ) ψ ˆ λ ( u ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbb_HTML.gif
so
ψ ˆ λ ( u 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbc_HTML.gif
and thus
A p ( u 0 ) + A ( u 0 ) = N g λ ( u 0 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ14_HTML.gif
(3.11)
Acting on (3.11) with ( u ϑ u 0 ) + W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq155_HTML.gif and next with ( u 0 u μ ) + W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq156_HTML.gif (similarly as in the proof of Proposition 3.5), we get
u ϑ u 0 u μ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbd_HTML.gif
Hence, we have
u 0 [ u ϑ , u μ ] , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Eqube_HTML.gif

where [ u ϑ , u μ ] = { u W 0 1 , p ( Ω ) : u 0 ( z ) u ( z ) u μ ( z )  for almost all  z Ω } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq157_HTML.gif.

Then (3.11) becomes
A p ( u 0 ) + A ( u 0 ) = λ N f ( u 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbf_HTML.gif
(see (3.10)), so
u 0 S ( λ ) int C + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbg_HTML.gif
Let
a ( y ) = y p 2 y + y y R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbh_HTML.gif
Then a C 1 ( R N ; R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq158_HTML.gif (recall that p > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq2_HTML.gif) and
a ( y ) = y p 2 ( I + ( p 2 ) y y y 2 ) + I y R N , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbi_HTML.gif
so
( a ( y ) ξ , ξ ) R N ξ 2 y , ξ R N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbj_HTML.gif
Note that
div a ( u ) = Δ p u + Δ u u W 0 1 , p ( Ω ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbk_HTML.gif
So, we can apply the tangency principle of Pucci and Serrin [[19], p.35] and infer that
u ϑ ( z ) < u 0 ( z ) z Ω . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ15_HTML.gif
(3.12)
Let ϱ = u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq159_HTML.gif and let ξ ϱ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq89_HTML.gif be as postulated by hypothesis H(iv). Then
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbl_HTML.gif
(see hypothesis H(iv) and use the facts that λ > ϑ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq160_HTML.gif and f 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq145_HTML.gif), so
u 0 u ϑ int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ16_HTML.gif
(3.13)

(see (3.12) and Proposition 2.3).

In a similar fashion, we show that
u μ u 0 int C + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ17_HTML.gif
(3.14)
From (3.13) and (3.14), it follows that
u 0 int C 0 1 ( Ω ¯ ) [ u ϑ , u μ ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ18_HTML.gif
(3.15)
From (3.10), we see that
φ λ | [ u ϑ , u μ ] = ψ ˆ λ | [ u ϑ , u μ ] + ξ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbm_HTML.gif

for some ξ λ R https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq161_HTML.gif.

So, (3.15) implies that u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq47_HTML.gif is a local C 0 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq162_HTML.gif-minimizer of φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq121_HTML.gif. Invoking Proposition 2.3, we have that
u 0  is a local  W 0 1 , p ( Ω ) -minimizer of  φ λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ19_HTML.gif
(3.16)
Hypotheses H(i), (ii) and (iii) imply that for given ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq118_HTML.gif and r > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq163_HTML.gif, we can find c 2 = c 2 ( ε , r ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq164_HTML.gif such that
F ( z , ζ ) ε p ζ p + c 2 ζ r for almost all  z Ω ,  all  ζ 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ20_HTML.gif
(3.17)
Then for all u W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq77_HTML.gif, we have
φ λ ( u ) = 1 p u p p + 1 2 u 2 2 λ Ω F ( z , u ) d z 1 p u p p + 1 2 u 2 2 λ ε p u + p p λ c 2 u + r r 1 p ( 1 λ ε λ ˆ 1 ( p ) ) u p p + 1 2 u 2 2 λ c 3 u r 1 p ( 1 λ ε λ ˆ 1 ( p ) ) u p λ c 3 u r https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ21_HTML.gif
(3.18)

for some c 3 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq165_HTML.gif (see (3.17) and (2.3)).

Choose ε ( 0 , λ ˆ 1 ( p ) λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq166_HTML.gif. Then, from (3.18) and since r > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq163_HTML.gif, we infer that u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq47_HTML.gif is a local minimizer of φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq121_HTML.gif. Without any loss of generality, we may assume that φ λ ( 0 ) = 0 φ λ ( u 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq167_HTML.gif (the analysis is similar if the opposite inequality holds). By virtue of (3.16), as in Gasinski and Papageorgiou [20] (see the proof of Theorem 2.12), we can find 0 < ϱ < u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq168_HTML.gif such that
φ λ ( 0 ) = 0 φ λ ( u 0 ) < inf { φ λ ( u ) : u u 0 = ϱ } = η ϱ λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ22_HTML.gif
(3.19)
Recall that φ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq121_HTML.gif is coercive, hence it satisfies the Palais-Smale condition. This fact and (3.19) permit the use of the mountain pass theorem (see Theorem 2.1). So, we can find u ˆ W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq169_HTML.gif such that
η ϑ λ φ λ ( u ˆ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ23_HTML.gif
(3.20)
and
φ λ ( u ˆ ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ24_HTML.gif
(3.21)

From (3.20) and (3.19), we have that u ˆ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq170_HTML.gif, u ˆ = u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq171_HTML.gif. From (3.21), it follows that u ˆ S ( λ ) int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq172_HTML.gif. □

Next, we examine what happens at the critical parameter λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq173_HTML.gif.

Proposition 3.7 If hypotheses H hold, then λ Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq174_HTML.gif.

Proof Let { λ n } n 1 Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq175_HTML.gif be a sequence such that
λ < λ n n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbn_HTML.gif
and
λ n λ as  n + . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbo_HTML.gif
For every n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq176_HTML.gif, we can find u n int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq177_HTML.gif, such that
A p ( u n ) + A ( u n ) = λ n N f ( u n ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ25_HTML.gif
(3.22)
We claim that the sequence { u n } n 1 W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq178_HTML.gif is bounded. Arguing indirectly, suppose that the sequence { u n } n 1 W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq178_HTML.gif is unbounded. By passing to a suitable subsequence if necessary, we may assume that u n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq179_HTML.gif. Let
y n = u n u n n 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbp_HTML.gif
Then y n = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq180_HTML.gif and y n int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq181_HTML.gif for all n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq176_HTML.gif. From (3.22), we have
A p ( y n ) + 1 u n p 2 A ( y n ) = λ n N f ( u n ) u n p 1 n 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ26_HTML.gif
(3.23)
Recall that
f ( z , ζ ) c 1 ζ p 1 for almost all  z Ω ,  all  ζ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbq_HTML.gif
(see (3.1)), so the sequence { N f ( u n ) u n p 1 } n 1 L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq182_HTML.gif is bounded. This fact and hypothesis H(ii) imply that at least for a subsequence, we have
N f ( u n ) u n p 1 0 weakly in  L p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ27_HTML.gif
(3.24)
(see Gasinski and Papageorgiou [20]). Also, passing to a subsequence if necessary, we may assume that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ28_HTML.gif
(3.25)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ29_HTML.gif
(3.26)
On (3.23) we act with y n y W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq183_HTML.gif, pass to the limit as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq184_HTML.gif and use (3.24) and (3.26). Then
lim n + ( A p ( y n ) , y n y + 1 u n p 2 A ( y n ) , y n y ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbr_HTML.gif
so
lim n + A p ( y n ) , y n y 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbs_HTML.gif
Using Proposition 2.4, we have that
y n y in  W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbt_HTML.gif
and so
y = 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ30_HTML.gif
(3.27)
Passing to the limit as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq184_HTML.gif in (3.23) and using (3.24), (3.27) and the fact that p > 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq2_HTML.gif, we obtain
A p ( y ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbu_HTML.gif

so y = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq185_HTML.gif, which contradicts (3.27).

This proves that the sequence { u n } n 1 W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq178_HTML.gif is bounded. So, passing to a subsequence if necessary, we may assume that
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ31_HTML.gif
(3.28)
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ32_HTML.gif
(3.29)
On (3.22) we act with u n u W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq186_HTML.gif, pass to the limit as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq184_HTML.gif and use (3.28) and (3.29). Then
lim n + ( A p ( u n ) , u n u + A ( u n ) , u n u ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbv_HTML.gif
so
lim sup n + A p ( u n ) , u n u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbw_HTML.gif
(since A is monotone) and thus
u n u in  W 0 1 , p ( Ω ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ33_HTML.gif
(3.30)

(see Proposition 2.4).

Therefore, if in (3.22) we pass to the limit as n + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq187_HTML.gif and use (3.30), then
A p ( u ) + A ( u ) = λ N f ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbx_HTML.gif

and so u C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq188_HTML.gif is a solution of problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq189_HTML.gif.

We need to show that u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq190_HTML.gif. From (3.22), we have
{ Δ p u n ( z ) Δ u n ( z ) = λ n f ( z , u n ( z ) ) in  Ω u n | Ω = 0 n 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equby_HTML.gif
From Ladyzhenskaya and Uraltseva [[17], p.286], we know that we can find M 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq191_HTML.gif such that
u n M 1 n 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equbz_HTML.gif
Then applying Theorem 1 of Lieberman [18], we can find β ( 0 , 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq46_HTML.gif and M 2 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq192_HTML.gif such that
u n C 0 1 , β ( Ω ¯ ) and u n C 0 1 , β ( Ω ¯ ) M 2 n 1 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equca_HTML.gif
Recall that C 0 1 , β ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq193_HTML.gif is embedded compactly in C 0 1 ( Ω ¯ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq162_HTML.gif. So, by virtue of (3.28), we have
u n u in  C 0 1 ( Ω ¯ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equcb_HTML.gif
Suppose that u = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq194_HTML.gif. Then
u n 0 in  C 0 1 ( Ω ¯ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ34_HTML.gif
(3.31)
Hypothesis H(iii) implies that for a given ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq118_HTML.gif, we can find δ ( 0 , ε ] https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq195_HTML.gif such that
f ( z , ζ ) ε ζ p 1 for almost all  z Ω ,  all  ζ [ 0 , δ ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ35_HTML.gif
(3.32)
From (3.31), it follows that we can find n 0 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq196_HTML.gif such that
u n ( z ) [ 0 , δ ] z Ω ¯ ,  all  n n 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equ36_HTML.gif
(3.33)
Therefore, for almost all z Ω https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq17_HTML.gif and all n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq197_HTML.gif, we have
Δ p u n ( z ) Δ u n ( z ) = λ n f ( z , u n ( z ) ) λ n ε u n ( z ) p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equcc_HTML.gif
(see (3.32) and (3.33)), so
u n p p λ n ε u n p p λ n λ ˆ 1 ( p ) ε u n p p n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equcd_HTML.gif
(see (2.3)), thus
λ ˆ 1 ( p ) ε λ n n n 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equce_HTML.gif
and so
λ ˆ 1 ( p ) ε λ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equcf_HTML.gif

Let ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq198_HTML.gif to get a contradiction. This proves that u 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq190_HTML.gif and so u S ( λ ) int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq199_HTML.gif, hence λ Y https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq174_HTML.gif. □

The bifurcation-type theorem summarizes the situation for problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif.

Theorem 3.8 If hypotheses H hold, then there exists λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq4_HTML.gif such that

(a) for every λ > λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq5_HTML.gif problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif has at least two positive solutions:
u 0 , u ˆ int C + ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_Equcg_HTML.gif

(b) for λ = λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq6_HTML.gif problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif has at least one positive solution u int C + https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq200_HTML.gif;

(c) for λ ( 0 , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq7_HTML.gif problem ( P ) λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq12_HTML.gif has no positive solution.

Remark 3.9 As the referee pointed out, it is an interesting problem to produce an example in which, at the bifurcation point λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq201_HTML.gif, the equation has exactly one solution. We believe that the recent paper of Gasiński and Papageorgiou [21] on the existence and uniqueness of positive solutions will be helpful. Concerning the existence of nodal solutions for λ ( 0 , λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq202_HTML.gif, we mention the recent paper of Gasiński and Papageorgiou [22], which studies the ( p , 2 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2012-152/MediaObjects/13661_2012_Article_251_IEq1_HTML.gif-equations and produces nodal solutions for them.

Declarations

Acknowledgements

Dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.

The authors would like to express their gratitude to both knowledgeable referees for their corrections and remarks. This research has been partially supported by the Ministry of Science and Higher Education of Poland under Grants no. N201 542438 and N201 604640.

Authors’ Affiliations

(1)
Faculty of Mathematics and Computer Science, Institute of Computer Science, Jagiellonian University
(2)
Department of Mathematics, National Technical University

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