Positive solutions for parametric (p(z),q(z))-equations

Leszek Gasiński; Ireneusz Krech; Nikolaos S. Papageorgiou

doi:10.1515/math-2020-0074

Article Open Access

Positive solutions for parametric (p(z),q(z))-equations

Leszek Gasiński , Ireneusz Krech and Nikolaos S. Papageorgiou

Published/Copyright: October 7, 2020

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$Open Mathematics$

From the journal Open Mathematics Volume 18 Issue 1

Abstract

We consider a parametric elliptic equation driven by the anisotropic ( p , q ) -Laplacian. The reaction is superlinear. We prove a “bifurcation-type” theorem describing the change in the set of positive solutions as the parameter λ moves in ℝ + = ( 0 , + ∞ ) .

Keywords: anisotropic regularity; anisotropic maximum principle; positive solutions; minimal positive solution; superlinear reaction

MSC 2010: 35J20; 35J70

1 Introduction

Let Ω ⊆ ℝ N be a bounded domain with a C 2 -boundary ∂ Ω . We study the following parametric anisotropic ( p , q ) -equation:

− Δ p ( z ) u ( z ) − Δ q ( z ) u ( z ) = λ f ( z , u ( z ) ) in Ω , u | ∂ Ω = 0 , u > 0 , λ > 0 . ( P λ )

In this problem, the exponents p and q are Lipschitz continuous on Ω ¯ , that is, p , q ∈ C 0 , 1 ( Ω ¯ ) and 1 < q − = min Ω ¯ q ⩽ q + = max Ω ¯ q < p − = min Ω ¯ p ⩽ p + = max Ω ¯ p .

By Δ p ( z ) (respectively Δ q ( z ) ) we denote the p ( z ) -Laplacian (respectively the q ( z ) -Laplacian) defined by

Δ p ( z ) u = div ( | D u | p ( z ) − 2 D u ) ∀ u ∈ W 0 1 , p ( z ) ( Ω )

( respectively Δ q ( z ) u = div ( | D u | q ( z ) − 2 D u ) ∀ u ∈ W 0 1 , q ( z ) ( Ω ) ) .

In the reaction (right hand side of ( P λ ) ), f ( z , x ) is a Carathéodory function (that is, for all x ∈ ℝ , z ↦ f ( z , x ) is measurable and for almost all z ∈ Ω , x ↦ f ( z , x ) is continuous), which is ( p + − 1 ) -superlinear in the x-variable, but need not satisfy the Ambrosetti-Rabinowitz condition which is common in problems with superlinear reactions. Also, λ > 0 is a parameter. We are looking for positive solutions of ( P λ ) . More precisely, our aim is to determine the precise dependence on the parameter λ > 0 of the set of positive solutions. We prove a bifurcation-type result, which establishes the existence of a critical parameter value λ ∗ > 0 such that

for all λ ∈ ( 0 , λ ∗ ) problem ( P λ ) has at least two positive solutions;
for λ = λ ∗ problem ( P λ ) has at least one positive solution;
for all λ > λ ∗ there are no positive solutions for problem ( P λ ) .

Our work here extends those of Gasiński-Papageorgiou [1,2], who studied parametric equations driven by the isotropic p-Laplacian with a ( p − 1 ) -superlinear reaction. Nonlinear, nonparametric superlinear equations were also considered by Mugnai-Papageorgiou [3], Papageorgiou-Rădulescu [4], Papageorgiou-Scapellato [5] (isotropic problems) and Gasiński-Papageorgiou [6], Papageorgiou-Rădulescu-Repovš [7], Papageorgiou-Vetro [8] (anisotropic problems). They prove multiplicity results, producing also nodal (that is, sign changing) solutions. Also, we mention the relevant studies of Bahrouni-Rădulescu-Repovš [9] (existence of infinitely many solutions for anisotropic Dirichlet problems), Papageorgiou-Vetro-Vetro [10] (produce a continuous part of the spectrum for the Robin ( p , q ) -Laplacian), Vetro [11] (dealing with the asymptotic properties of the solutions of nonhomogeneous parametric isotropic equations), Vetro [12] (existence of a solution of an anisotropic Dirichlet problem), Vetro-Vetro [13] (a three-solution theorem for ( p , q ) -equations) and Vetro [14] (an infinity of solutions for isotropic ( p , q ) -equations).

Equations with variable exponents arise in many physical models. We refer to the book of Rǔžička [15] for such meaningful examples. The analysis of such problems requires the use of Lebesgue and Sobolev spaces with variable exponents. A comprehensive presentation of such spaces can be found in the book of Diening-Harjulehto-Hästö-Rǔžička [16] (see also the survey paper of Harjulehto-Hästö-Lê-Nuortio [17]). Various parametric boundary value problems with variable exponents can be found in the book of Rădulescu-Repovš [18]. Finally, we mention that we encounter ( p , q ) -equations (both isotropic and anisotropic), in many problems of mathematical physics. We refer to the studies of Bahrouni-Rădulescu-Repovš [19] (transonic flow problems), Benci-D’Avenia-Fortunato-Pisani [20] (quantum physics), Cherfils-Il’yasov [21] (reaction-diffusion systems) and Zhikov [22] (elasticity theory). We also mention the two informative survey papers by Marano-Mosconi [23] (isotropic problems) and Rădulescu [24] (isotropic and anisotropic problems).

2 Mathematical background – hypotheses

Let M ( Ω ) be the space of measurable functions u : Ω → ℝ . We identify two such functions that differ only on a set of zero Lebesgue measures. Also, let

E 1 = r ∈ C ( Ω ¯ ) : 1 < r − = min Ω ¯ r .

In the sequel given r ∈ C ( Ω ¯ ) , we define

r − = min Ω ¯ q and r + = max Ω ¯ q .

Given r ∈ E 1 , the variable exponent Lebesgue space L r ( z ) ( Ω ) is defined as follows:

L r ( z ) ( Ω ) = u ∈ M ( Ω ) : ∫ Ω | u | r ( z ) d z < + ∞ .

This space is equipped with the so-called “Luxemburg norm” defined by

∥ u ∥ r ( z ) = inf λ > 0 : ∫ Ω | u | λ r ( z ) d z ⩽ 1 .

Furnished with this norm, the space L r ( z ) ( Ω ) becomes a separable, reflexive (in fact, uniformly convex) Banach space. Let r ′ ∈ E 1 be defined by 1 r ( z ) + 1 r ′ ( z ) = 1 . We know that L r ( z ) ( Ω ) ∗ = L r ′ ( z ) ( Ω ) and we have the following Hölder-type inequality:

∫ Ω u h d z ⩽ 1 r − + 1 r − ′ ∥ u ∥ r ( z ) ∥ h ∥ r ′ ( z ) ∀ u ∈ L r ( z ) , h ∈ L r ′ ( z ) ( Ω ) .

If r 1 , r 2 ∈ E 1 and r 1 ( z ) ⩽ r 2 ( z ) for all z ∈ Ω ¯ , then the embedding L r 2 ( z ) ( Ω ) ⊆ L r 1 ( z ) ( Ω ) is continuous.

Using the variable exponent Lebesgue spaces, we can define in the usual way the variable exponent Sobolev spaces. So, if r ∈ E 1 , then we define

W 1 , r ( z ) ( Ω ) = { u ∈ L r ( z ) ( Ω ) : | D u | ∈ L r ( z ) ( Ω ) }

(where the gradient Du is understood in the weak sense). This space is equipped with the following norm:

∥ u ∥ 1 , r ( z ) = ∥ u ∥ r ( z ) + ∥ | D u | ∥ r ( z ) .

In the sequel for notational simplicity, we write ∥ D u ∥ r ( z ) = ∥ | D u | ∥ r ( z ) . Suppose that r ∈ E 1 is Lipschitz continuous (that is, r ∈ E 1 ∩ C 0 , 1 ( Ω ¯ ) ). Then we define

W 0 1 , r ( z ) ( Ω ) = C c ∞ ( Ω ) ¯ ∥ ⋅ ∥ 1 , r ( z ) .

Both W 1 , r ( z ) ( Ω ) and W 0 1 , r ( z ) ( Ω ) are separable, reflexive (in fact uniformly convex) Banach spaces.

For the space W 0 1 , r ( z ) ( Ω ) , the Poincaré inequality holds, namely

∥ u ∥ r ( z ) ⩽ c 0 ∥ D u ∥ r ( z ) ∀ u ∈ W 0 1 , r ( z ) ( Ω ) ,

for some c 0 > 0 . So, on W 0 1 , r ( z ) ( Ω ) (recall that r ∈ E 1 ∩ C 0 , 1 ( Ω ¯ ) ), we can consider the following equivalent norm:

∥ u ∥ 1 , r ( z ) = ∥ D u ∥ r ( z ) ∀ u ∈ W 0 1 , r ( z ) ( Ω ) .

For r ∈ E 1 , the critical Sobolev exponent corresponding to r is defined by

r ∗ ( z ) = N r ( z ) N − r ( z ) if r ( z ) < N , + ∞ if N ⩽ r ( z ) .

Suppose that r ∈ E 1 ∩ C 0 , 1 ( Ω ¯ ) , p ∈ E 1 , p + < N and 1 < p ( z ) ⩽ r ∗ ( z ) (respectively 1 < p ( z ) < r ∗ ( z ) ) for all z ∈ Ω ¯ . We have

W 0 1 , r ( z ) ( Ω ) ⊆ L p ( z ) ( Ω ) continuously

(respectively: compactly).

Useful in the analysis of these variable exponent spaces is the following modular function:

ϱ r ( u ) = ∫ Ω u r ( z ) d z ∀ u ∈ L r ( z ) ( Ω ) ,

with r ∈ E 1 . We write ϱ r ( D u ) = ϱ r ( | D u | ) .

There is a close relation between this modular function and the norm. We assume r ∈ E 1 .

Proposition 2.1

∥ u ∥ r ( z ) = λ ⇔ ϱ r u λ = 1 for all u ∈ L r ( z ) ( Ω ) , u ≠ 0 .
∥ u ∥ r ( z ) < 1 (resp. = 1 , > 1 ) ⇔ ϱ r ( u ) < 1 (resp. = 1 , > 1 ).
∥ u ∥ r ( z ) < 1 ⇒ ∥ u ∥ r ( z ) r + ⩽ ϱ r ( u ) ⩽ ∥ u ∥ r ( z ) r − .
∥ u ∥ r ( z ) > 1 ⇒ ∥ u ∥ r ( z ) r − ⩽ ϱ r ( u ) ⩽ ∥ u ∥ r ( z ) r + .
∥ u n ∥ r ( z ) → 0 ⇔ ϱ r ( u n ) → 0 .
∥ u n ∥ r ( z ) → + ∞ ⇔ ϱ r ( u n ) → + ∞ .

More details can be found in the book of Diening-Harjulehto-Hästö-Rǔžička [16].

Consider the map A r ( z ) : W 0 1 , r ( z ) ( Ω ) → W 0 1 , r ( z ) ( Ω ) ∗ = W − 1 , r ′ ( z ) ( Ω ) defined by

〈 A r ( z ) ( u ) , h 〉 = ∫ Ω | D u | r ( z ) − 2 ( D u , D h ) ℝ N d z ∀ u , h ∈ W 0 1 , p ( Ω ) .

This map has the following properties (see Gasiński-Papageorgiou [6, Proposition 2.5] and Rădulescu-Repovš [18, p. 40]).

Proposition 2.2

The map A r ( z ) : W 0 1 , r ( z ) ( Ω ) → W 0 1 , r ( z ) ( Ω ) ∗ is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone too) and of type ( S ) + , that is, “ u n → w u in W 0 1 , r ( z ) ( Ω ) and lim sup n → + ∞ 〈 A r ( z ) ( u n ) , u n − u 〉 ⩽ 0 , imply that u n → u in W 0 1 , r ( z ) ( Ω ) .”

In addition to the variable exponent spaces, we will also use the Banach space

C 0 1 ( Ω ¯ ) = { u ∈ C 1 ( Ω ¯ ) : u | ∂ Ω = 0 } .

This is an ordered Banach space with positive (order) cone

C + = { u ∈ C 0 1 ( Ω ¯ ) : u ( z ) ⩾ 0 for all z ∈ Ω ¯ } .

This cone has a nonempty interior given by

int C + = u ∈ C + : u > 0 , ∂ u ∂ n ∂ Ω < 0 ,

with n being the outward unit normal on ∂ Ω .

Given u , v ∈ W 1 , r ( z ) ( Ω ) with u ⩽ v , we define

[ u , v ] = { h ∈ W 0 1 , r ( z ) ( Ω ) : u ( z ) ⩽ h ( z ) ⩽ r ( z ) for a .a . z ∈ Ω } ,

[ u ) = { h ∈ W 0 1 , r ( z ) ( Ω ) : u ( z ) ⩽ h ( z ) for a .a . z ∈ Ω } .

If h 1 , h 2 : Ω → ℝ are measurable functions, then we write h 1 ≺ h 2 , if for every compact set K ⊆ Ω , we have 0 < c K ⩽ h 2 ( z ) − h 1 ( z ) for almost all z ∈ K . Evidently, if h 1 , h 2 ∈ C ( Ω ) and h 1 ( z ) < h 2 ( z ) for all z ∈ Ω , then h 1 ≺ h 2 .

A set S ⊆ W 0 1 , p ( z ) ( Ω ) is said to be “downward directed,” if for u 1 , u 2 ∈ S , we can find u ∈ S such that u ⩽ u 1 , u ⩽ u 2 .

By | ⋅ | N we denote the Lebesgue measure on ℝ N and by ∥ ⋅ ∥ ∗ the norm of W 0 1 , p ( z ) ( Ω ) ∗ .

In the sequel for notational economy, by ∥ ⋅ ∥ we denote the norm of the Sobolev space W 0 1 , p ( z ) ( Ω ) . Recall that

∥ u ∥ = ∥ D u ∥ p ( z ) ∀ u ∈ W 1 , p ( z ) ( Ω ) .

If X is a Banach space and φ ∈ C 1 ( X ) , then we set

K φ = { u ∈ X : φ ′ ( u ) = 0 }

(the critical set of φ ). We say that φ satisfies the “Cerami condition,” if the following property holds:

“Every sequence { u n } n ⩾ 1 ⊆ X such that { φ ( u n ) } n ⩾ 1 ⊆ ℝ is bounded and

( 1 + ∥ u n ∥ X ) φ ′ ( u n ) → 0 in X ∗ as n → + ∞ ,

admits a strongly convergent subsequence.”

Now we introduce the hypotheses on the data problem ( P λ ) .

H 0 : p , q ∈ E 1 ∩ C 0 , 1 ( Ω ¯ ) , q + < p − .

H 1 : f : Ω × ℝ → ℝ is a Carathéodory function such that f ( z , 0 ) = 0 for a.a. z ∈ Ω and

f ( z , x ) ⩽ a ( z ) ( 1 + x r − 1 ) for a.a. z ∈ Ω , all x ⩾ 0 , with a ∈ L ∞ ( Ω ) and p + < r < p ∗ ( z ) for all z ∈ Ω ¯ ;
if F ( z , x ) = ∫ 0 x f ( z , s ) d s , then
lim x → + ∞ F ( z , x ) x p + = + ∞ uniformly for a .a . z ∈ Ω ;
if σ ( z , x ) = f ( z , x ) x − p + F ( z , x ) , then there exists η ∈ L 1 ( Ω ) such that
σ ( z , x ) ⩽ σ ( z , y ) + η ( z ) for a .a . z ∈ Ω , all 0 ⩽ x ⩽ y ;
for every s > 0 , there exists m s > 0 such that
f ( z , x ) ⩾ m s > 0 for a .a . z ∈ Ω , all x ⩾ s ,
and
lim x → 0 + f ( z , x ) x q − − 1 = + ∞ uniformly for a .a . z ∈ Ω ;
for every ϱ > 0 , there exists ξ ^ ϱ > 0 such that for a.a. z ∈ Ω , the function x ↦ f ( z , x ) + ξ ^ ϱ x p ( z ) − 1 is nondecreasing on [ 0 , ϱ ] .

Remark 2.3

Since we want to find positive solutions and the aforementioned hypotheses concern the positive semiaxis ℝ + = [ 0 , + ∞ ) , without any loss of generality, we may assume that

(2.1) f ( z , x ) = 0 for a .a . z ∈ Ω , all x ⩽ 0 .

Hypotheses H 1 ( i i ) , ( i i i ) imply that f ( z , ⋅ ) is ( p + − 1 ) -superlinear. Usually in the literature, such problems are treated using the well-known Ambrosetti-Rabinowitz condition (see Ambrosetti-Rabinowitz [25]). Here instead we use the less restrictive condition H 1 ( i i i ) , which is an extension of a condition used by Li-Yang [26]. This quasimonotonicity condition on the function σ ( z , ⋅ ) is equivalent to saying that there exists M > 0 such that for a.a. z ∈ Ω , the quotient function x ↦ f ( z , x ) x p + − 1 is nondecreasing on [ M , + ∞ ) . This superlinearity condition incorporates in our framework superlinear nonlinearities with “slower” growth near + ∞ . For example, consider the following function:

f ( z , x ) = x τ ( z ) − 1 if 0 ⩽ x ⩽ 1 , x p + − 1 ln x + x μ ( z ) − 1 if 1 < x

(see (2.1)), with τ , μ ∈ E 1 and τ + < q − , μ ( z ) ⩽ p ( z ) for all z ∈ Ω ¯ . This function satisfies hypotheses H 1 , but fails to satisfy the Ambrosetti-Rabinowitz condition.

We introduce the following two sets:

ℒ = { λ > 0 : problem ( P λ ) admits a positive solution } ,

S λ = the set of positive solutions of ( P λ ) .

Also, we set

λ ∗ = sup ℒ .

3 Positive solutions

We start by showing that the set of admissible parameters ℒ is nonempty. Also, we determine the regularity properties of the elements in S λ .

Proposition 3.1

If hypotheses H 0 , H 1 ( i ) , ( i v ) hold, then ℒ ≠ ∅ and for every λ > 0 , S λ ⊆ int C + .

Proof

We consider the following auxiliary Dirichlet problem:

(3.1) − Δ p ( z ) u ( z ) − Δ q ( z ) u ( z ) = 1 in Ω , u | ∂ Ω = 0 .

The operator u ↦ A p ( z ) ( u ) + A q ( z ) ( u ) from W 0 1 , p ( z ) ( Ω ) into W 0 1 , p ( z ) ( Ω ) ∗ is continuous, strictly monotone (hence maximal monotone too) (see Proposition 2.2) and coercive. So, it is surjective (see Gasiński-Papageorgiou [27, Corollary 3.2.31, p. 319]). Hence, we can find u ¯ ∈ W 0 1 , p ( z ) ( Ω ) , u ¯ ⩾ 0 , u ¯ ≠ 0 such that

A p ( z ) ( u ¯ ) + A q ( z ) ( u ¯ ) = 1 in W 0 1 , p ( z ) ( Ω ) ∗ .

The strict monotonicity of the operator implies that this solution is unique. So, u ¯ is the unique positive solution of (3.1). Theorem 4.1 of Fan-Zhao [28] implies that u ¯ ∈ L ∞ ( Ω ) . Then from Fukagai-Narukawa [29, Lemma 3.3] (see also Tan-Fang [30, Corollary 3.1] and Lieberman [31] for the corresponding isotropic regularity theory), we have that u ¯ ∈ C 0 1 , α ( Ω ¯ ) = C 1 , α ( Ω ¯ ) ∩ C 0 1 ( Ω ¯ ) with α ∈ ( 0 , 1 ) . Hence, u ¯ ∈ C + \ { 0 } . From the anisotropic maximum principle (see Zhang [32]), we obtain that u ¯ ∈ int C + .

Let m = ∥ f ( ⋅ , u ¯ ( ⋅ ) ) ∥ ∞ (see hypothesis H 1 ( i ) ) and choose λ 0 > 0 such that λ 0 m ⩽ 1 . We have

(3.2) − Δ p ( z ) u ¯ − Δ q ( z ) u ¯ ⩾ λ f ( z , u ¯ ) in Ω ,

for all λ ∈ ( 0 , λ 0 ] . We introduce the Carathéodory function g ˆ ( z , x ) defined by

(3.3) g ˆ ( z , x ) = f ( z , x + ) if x ⩽ u ¯ ( z ) , f ( z , u ¯ ( z ) ) if u ¯ ( z ) < x .

We set

G ˆ ( z , x ) = ∫ 0 x g ˆ ( z , s ) d s

and for all λ ∈ ( 0 , λ 0 ] consider the C 1 -functional φ ˆ λ : W 0 1 , p ( z ) ( Ω ) → ℝ defined by

φ ˆ λ ( u ) = ∫ Ω 1 p ( z ) | D u | p ( z ) d z + ∫ Ω 1 q ( z ) | D u | q ( z ) d z − ∫ Ω λ G ˆ ( z , u ) d z ∀ u ∈ W 0 1 , p ( z ) ( Ω ) .

From (3.3) and Proposition 2.1, it is clear that φ ˆ λ is coercive. Also, the anisotropic Sobolev embedding theorem implies that φ ˆ λ is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find u λ ∈ W 0 1 , p ( z ) ( Ω ) such that

(3.4) φ ˆ λ ( u ˆ λ ) = min u ∈ W 0 1 , p ( z ) ( Ω ) φ ˆ λ ( u ) .

Hypothesis H 1 ( i v ) implies that given any θ > 0 , we can find δ = δ ( θ ) ∈ ( 0 , 1 ) such that

(3.5) F ( z , x ) ⩾ θ q − x q − for a .a . z ∈ Ω , all 0 ⩽ x ⩽ δ .

Let u ∈ int C + . Since u ¯ ∈ int C + , using Proposition 4.1.22 of Papageorgiou-Rădulescu-Repovš [33, p. 274], we can find t ∈ ( 0 , 1 ) small such that

(3.6) t u ( z ) ⩽ min{ u ¯ ( z ) , δ } for all z ∈ Ω ¯ .

From (3.3), (3.5) and (3.6) and since t ∈ ( 0 , 1 ) , we have

φ ˆ λ ( t u ) ⩽ t q − q − ( ϱ p ( D u ) + ϱ q ( D u ) − θ ∥ u ∥ q − q − ) .

Since θ > 0 is arbitrary, choosing θ > 0 big from the aforementioned inequality, we infer that

φ ˆ λ ( t u ) < 0 ,

φ ˆ λ ( u λ ) < 0 = φ ˆ λ ( 0 )

(see (3.4)) and thus u λ ≠ 0 .

From (3.4), we have

φ ˆ λ ′ ( u λ ) = 0 ,

(3.7) 〈 A p ( z ) ( u λ ) , h 〉 + 〈 A q ( z ) ( u λ ) , h 〉 = ∫ Ω λ g ˆ ( z , u λ ) h d z ∀ h ∈ W 0 1 , p ( z ) ( Ω ) .

We test (3.7) with h = − u λ − ∈ W 1 , p ( z ) ( Ω ) and obtain

ϱ p ( D u λ − ) + ϱ q ( D u λ − ) = 0 ,

so u λ ⩾ 0 , u λ ≠ 0 (see Proposition (2.1)).

Next in (3.7) we choose h = ( u λ − u ¯ ) + ∈ W 0 1 , p ( z ) ( Ω ) . We have

〈 A p ( z ) ( u λ ) , ( u λ − u ¯ ) + 〉 + 〈 A q ( z ) ( u λ ) , ( u λ − u ¯ ) + 〉 = ∫ Ω λ f ( z , u ¯ ) ( u λ − u ¯ ) + d z ⩽ 〈 A p ( z ) ( u ¯ ) , ( u λ − u ¯ ) + 〉 + 〈 A q ( z ) ( u ¯ ) , ( u λ − u ¯ ) + 〉

(see (3.3) and (3.2)), so

u λ ⩽ u ¯ .

So, we have proved that

(3.8) u λ ∈ [ 0 , u ¯ ] , u λ ≠ 0 .

From (3.8), (3.3) and (3.7), it follows that u λ is a positive solution of ( P λ ) . As before using the anisotropic regularity theorem (see Fan-Zhao [28], Fukagai-Narukawa [29]) and the anisotropic maximum principle (see Zhang [32]), we obtain that u λ ∈ int C + .

Therefore, we conclude that

( 0 , λ 0 ] ⊆ ℒ ≠ ∅

and

S λ ⊆ int C + λ > 0 . □

Next, we show that ℒ is connected.

Proposition 3.2

If hypotheses H 0 , H 1 ( i ) , ( i v ) hold, λ ∈ ℒ and 0 < μ < λ , then μ ∈ ℒ .

Proof

Since λ ∈ ℒ , we can find u λ ∈ S λ ⊆ int C + (see Proposition 3.1). We introduce the Carathéodory function g ˜ defined by

(3.9) g ˜ ( z , x ) = f ( z , x + ) if x ⩽ u λ ( z ) , f ( z , u λ ( z ) ) if u λ ( z ) < x .

We set

G ˜ ( z , x ) = ∫ 0 x g ˜ ( z , s ) d s

and consider the C 1 -functional φ ˜ μ : W 0 1 , p ( z ) ( Ω ) → ℝ defined by

φ ˜ μ ( u ) = ∫ Ω 1 p ( z ) | D u | p ( z ) d z + ∫ Ω 1 q ( z ) | D u | q ( z ) d z − ∫ Ω μ G ˜ ( z , u ) d z ∀ u ∈ W 0 1 , p ( z ) ( Ω ) .

On account of (3.9) and Proposition 2.1, φ ˜ μ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u μ ∈ W 0 1 , p ( z ) ( Ω ) such that

φ ˜ μ ( u μ ) = min u ∈ W 0 1 , p ( z ) ( Ω ) φ ˜ μ ( u ) < 0 = φ ˜ μ ( 0 )

(see the proof of Proposition 3.1), thus u μ ≠ 0 . We have

φ ˜ μ ′ ( u μ ) = 0

and from this as in the proof of Proposition 3.1, using (3.9), we obtain

u μ ∈ [ 0 , u λ ] , u μ ≠ 0 ,

u μ ∈ S μ ⊆ int C +

(see (3.9) and Proposition 3.1), hence μ ∈ ℒ .□

A byproduct of the above proof is the following monotonicity property of the solution multifunction λ ↦ S λ .

Corollary 3.3

If hypotheses H 0 , H 1 ( i ) , ( i v ) hold, λ ∈ ℒ , u λ ∈ S λ ⊆ int C + and 0 < μ < λ , then μ ∈ ℒ and we can find u μ ∈ S μ ⊆ int C + such that u μ ⩽ u λ .

We can improve this monotonicity property, if we bring in the picture hypothesis H 1 ( v ) .

Proposition 3.4

If hypotheses H 0 , H 1 ( i ) , ( i v ) , ( v ) hold, λ ∈ ℒ , u λ ∈ S λ ⊆ int C + and 0 < μ < λ , then μ ∈ ℒ and there exists u μ ∈ S μ ⊆ int C + such that

u λ − u μ ∈ int C + .

Proof

From Corollary 3.3, we already know that μ ∈ ℒ and that there exists u μ ∈ S μ ⊆ int C + such that u μ ⩽ u λ . Let ϱ = ∥ u λ ∥ ∞ and let ξ ˆ ϱ > 0 be as postulated by hypothesis H 1 ( v ) . We have

(3.10) − Δ p ( z ) u μ − Δ q ( z ) u μ + ξ ˆ ϱ u μ p ( z ) − 1 = μ f ( z , u μ ) + ξ ˆ ϱ u μ p ( z ) − 1 = λ f ( z , u μ ) + ξ ˆ ϱ u μ p ( z ) − 1 − ( λ − μ ) f ( z , u μ ) ⩽ λ f ( z , u λ ) + ξ ˆ ϱ u λ p ( z ) − 1 ( λ − μ ) f ( z , u μ ) ⩽ − Δ p ( z ) u λ − Δ q ( z ) u λ + ξ ˆ ϱ u λ p ( z ) − 1

(see hypothesis H 1 ( v ) ). Since u μ ∈ int C + , on account of hypothesis H 1 ( i v ) , we have that

0 ≺ ( λ − μ ) f ( ⋅ , u μ ( ⋅ ) ) .

Then from (3.10) and Proposition 2.4 of Papageorgiou-Rădulescu-Repovš [7], we conclude that u λ − u μ ∈ int C + .□

Next for every λ ∈ ℒ , we will produce a smallest (minimal) positive solution for problem ( P λ ) . To this end, we need some preparation.

Hypotheses H 1 ( i ) , ( i v ) imply that given β > 0 , we can find c 1 = c 1 ( β ) > 0 such that

(3.11) f ( z , x ) ⩾ β x q − − 1 − c 1 x r − 1 for a .a . z ∈ Ω , all x ⩾ 0 .

Motivated from this unilateral growth estimate for f ( z , ⋅ ) , we consider the following auxiliary Dirichlet problem:

− Δ p ( z ) u ( z ) − Δ q ( z ) u ( z ) = λ ( β u ( z ) q − − 1 − c 1 u ( z ) r − 1 ) in Ω , u | ∂ Ω = 0 , u > 0 , λ > 0 . ( Q λ )

Proposition 3.5

For every λ > 0 , we can choose β = β ( λ ) > 0 big such that ( Q λ ) has a unique positive solution u ˜ λ ∈ int C + .

Proof

First we show the existence of a positive solution. To this end, we consider the C 1 -functional σ λ : W 0 1 , p ( z ) ( Ω ) → ℝ defined by

σ λ ( u ) = ∫ Ω 1 p ( z ) | D u | p ( z ) d z + ∫ Ω 1 q ( z ) | D u | q ( z ) d z + λ c 1 r ∥ u + ∥ r r − λ β q − ∥ u + ∥ q − q − ∀ u ∈ W 0 1 , p ( z ) ( Ω ) .

Since q − ⩽ q ( z ) < p ( z ) ⩽ p + < r for all z ∈ Ω ¯ (see hypothesis H 0 ), we see that σ λ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u ˜ λ ∈ W 0 1 , p ( z ) ( Ω ) such that

(3.12) σ λ ( u ˜ λ ) = min u ∈ W 0 1 , p ( z ) ( Ω ) σ λ ( u ) .

Consider u ∈ C + with ∥ u ∥ ∞ ⩽ 1 . For t ∈ ( 0 , 1 ) , we have

σ λ ( t u ) ⩽ t p − p − ϱ p ( D u ) + t q − q − ( ϱ q ( D u ) − λ β ∥ u ∥ q − q − ) + λ c 1 r t r ∥ u ∥ r r ⩽ t q − q − ( ϱ p ( D u ) + ϱ q ( D u ) − λ β ∥ u ∥ q − q − ) + λ c 1 r t r ∥ u ∥ r r .

Recall that β > 0 is arbitrary. So, we choose β λ > ϱ p ( D u ) + ϱ q ( D u ) λ ∥ u ∥ q − q − and obtain

σ λ ( t u ) ⩽ λ c 2 t r − c 3 t q − ,

for some c 2 , c 3 > 0 . Since q − < r , choosing t ∈ ( 0 , 1 ) small, we have

σ λ ( t u ) < 0 ,

σ λ ( u ˜ λ ) < 0 = σ λ ( 0 )

(see (3.12)), hence u ˜ λ ≠ 0 .

From (3.12), we have

σ λ ′ ( u ˜ λ ) = 0 ,

(3.13) 〈 A p ( z ) ( u ˜ λ ) , h 〉 + 〈 A q ( z ) ( u ˜ λ ) , h 〉 = λ β λ ∫ Ω ( u ˜ λ + ) q − − 1 h d z − λ c 1 ∫ Ω ( u ˜ λ + ) r − 1 h d z ∀ h ∈ W 0 1 , p ( z ) ( Ω ) .

In (3.13), we choose h = − u ˜ − ∈ W 0 1 , p ( z ) ( Ω ) and obtain

ϱ p ( D u ˜ λ − ) + ϱ q ( D u ˜ λ − ) = 0 ,

so u ˜ λ ⩾ 0 , u ˜ λ ≠ 0 (see Proposition 2.1).

Then from (3.13) we infer that u ˜ λ is a positive solution of ( Q λ ) . Moreover, as before the anisotropic regularity theory and the anisotropic maximum principle imply

(3.14) u ˜ λ ∈ int C + .

Let v ˜ λ ∈ W 0 1 , p ( z ) ( Ω ) be another positive solution of ( Q λ ) . Again we have

(3.15) v ˜ λ ∈ int C + .

We consider the integral functional j : L 1 ( Ω ) → ℝ ¯ = ℝ ∪ { + ∞ } defined by

j ( u ) = ∫ Ω q − p ( z ) | D u 1 q − | p ( z ) d z + ∫ Ω q − q ( z ) | D u 1 q − | q ( z ) d z if u ⩾ 0 , u 1 q − ∈ W 0 1 , p ( z ) ( Ω ) , + ∞ , otherwise .

From Theorem 2.2 of Takáč-Giacomoni [34], we have that j is convex. Let

dom j = { u ∈ L 1 ( Ω ) : j ( u ) < + ∞ }

(the effective domain of j). From (3.14), (3.15) and Proposition 4.1.22 of Papageorgiou-Rădulescu-Repovš [33, p. 274], we have

u ˜ λ v ˜ λ ∈ L ∞ ( Ω ) , v ˜ λ u ˜ λ ∈ L ∞ ( Ω ) .

Let h ∈ C 0 1 ( Ω ¯ ) with | h | 1 q − ∈ W 0 1 , p ( z ) ( Ω ) . For t ∈ ( 0 , 1 ) small, we have

u ˜ λ q − + t h ∈ dom j and v ˜ λ q − + t h ∈ dom j .

Choose h = u ˜ λ q − − v ˜ λ q − . Evidently,

h ∈ C 0 1 ( Ω ¯ ) and | h | ⩽ u ˜ λ q − + v ˜ λ q − .

We have

| h | 1 q − ⩽ u ˜ λ + v ˜ λ ,

so | h | 1 q − ∈ W 0 1 , p ( z ) ( Ω ) .

Then on account of the convexity of j, it is Gâteaux differentiable at u ˜ λ q − and at v ˜ λ q − in the direction h = u ˜ λ q − − v ˜ λ q − . Moreover, we have (see also Takáč-Giacomoni [34])

j ′ ( u ˜ λ q − ) ( h ) = ∫ Ω − Δ p ( z ) u ˜ λ − Δ q ( z ) u ˜ λ u ˜ λ q − − 1 h d z ,

j ′ ( v ˜ λ q − ) ( h ) = ∫ Ω − Δ p ( z ) v ˜ λ − Δ q ( z ) v ˜ λ v ˜ λ q − − 1 h d z .

The convexity of j implies the monotonicity of j ′ . Hence,

0 ⩽ λ c 1 ∫ Ω ( u ˜ λ r − q − v ˜ λ r − q − ) ( v ˜ λ q − − u ˜ λ q − ) d z ⩽ 0

(recall that q − < r ), so

u ˜ λ = v ˜ λ ,

thus u ˜ λ ∈ int C + is the unique positive solution of ( Q λ ) .□

Using u ˜ λ ∈ int C + from Proposition 3.5, we can have a lower bound for the elements of S λ .

Proposition 3.6

If hypotheses H 0 , H 1 ( i ) , ( i v ) , ( v ) hold and λ ∈ ℒ , then u ˜ λ ⩽ u for all u ∈ S λ .

Proof

Let u ∈ S λ . We introduce the Carathéodory function k ( z , x ) defined by

(3.16) k ( z , x ) = β ( x + ) q − − 1 − c 1 ( x + ) r − 1 if x ⩽ u ( z ) , β u ( z ) q − − 1 − c 1 u ( z ) r − 1 if u ( z ) < x .

We set

K ( z , x ) = ∫ 0 x k ( z , s ) d s

and consider the C 1 -functional γ λ : W 0 1 , p ( z ) ( Ω ) → ℝ defined by

γ λ ( u ) = ∫ Ω 1 p ( z ) | D u | p ( z ) d z + ∫ Ω 1 q ( z ) | D u | q ( z ) d z − ∫ Ω λ K ( z , u ) d z ∀ u ∈ W 0 1 , p ( z ) ( Ω ) .

From Proposition 2.1 and (3.16) it is clear that γ λ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find u ¯ λ ∈ W 0 1 , p ( z ) ( Ω ) such that

(3.17) γ λ ( u ¯ λ ) = min u ∈ W 0 1 , p ( z ) ( Ω ) γ λ ( u ) .

As before (see the proof of Proposition 3.5), we have

γ λ ( u ¯ λ ) < 0 = γ λ ( 0 ) ,

so u ¯ λ ≠ 0 .

From (3.17), we have

γ λ ′ ( u ¯ λ ) = 0 ,

(3.18) 〈 A p ( z ) ( u ¯ λ ) , h 〉 + 〈 A q ( z ) ( u ¯ λ ) , h 〉 = λ ∫ Ω k ( z , u ¯ λ ) h d z ∀ h ∈ W 0 1 , p ( z ) ( Ω ) .

We test (3.18) with h = − u ¯ λ − ∈ W 0 1 , p ( z ) ( Ω ) and obtain

ϱ p ( D u ¯ λ − ) + ϱ q ( D u ¯ λ − ) = 0

(see (3.16)), so u ¯ λ ⩾ 0 , u ¯ λ ≠ 0 .

Next in (3.18) we choose h = ( u ¯ λ − u ) + ∈ W 0 1 , p ( z ) ( Ω ) . Then

〈 A p ( z ) ( u ¯ λ ) , ( u ¯ λ − u ) + 〉 + 〈 A q ( z ) ( u ¯ λ ) , ( u ¯ λ − u ) + 〉 = ∫ Ω λ ( β u q − − 1 − c 1 u r − 1 ) ( u ¯ λ − u ) + d z ⩽ ∫ Ω λ f ( z , u ) ( u ¯ λ − u ) + d z = 〈 A p ( z ) ( u ) , ( u ¯ λ − u ) + 〉 + 〈 A q ( z ) ( u ) , ( u ¯ λ − u ) + 〉

(see (3.16), (3.11) and use the fact that u ∈ S λ ), so u ¯ λ ⩽ u (see Proposition 2.2).

So, we have proved that

(3.19) u ¯ λ ∈ [0, u ] , u ¯ λ ≠ 0 .

From (3.19), (3.16) and (3.18), it follows that u ¯ λ is a positive solution of ( Q λ ) , hence u ¯ λ = u ˜ λ (see Proposition (3.6)). We conclude that u ˜ λ ⩽ u for all u ∈ S λ .□

Now we are ready to produce the minimal positive solution of problem ( P λ ) , λ ∈ ℒ .

Proposition 3.7

If hypotheses H 0 , H 1 ( i ) , ( i v ) , ( v ) hold and λ ∈ ℒ , then problem ( P λ ) admits a smallest positive solution u λ ∗ ∈ S λ ⊆ int C + (that is, u λ ∗ ⩽ u for all u ∈ S λ ).

Proof

From Papageorgiou-Rădulescu-Repovš [35] (proof of Proposition 7; see also Filippakis-Papageorgiou [36]), we know that S λ is downward directed. So, by Lemma 3.10 of Hu-Papageorgiou [37, p. 178], we can find a decreasing sequence { u n } n ⩾ 1 ⊆ S λ such that

(3.20) inf S λ = inf n ⩾ 1 u n

and

(3.21) u ˜ λ ⩽ u n ⩽ u 1 ∀ n ∈ ℕ

(see Proposition 3.6). We have

(3.22) 〈 A p ( z ) ( u n ) , h 〉 + 〈 A q ( z ) ( u n ) , h 〉 = ∫ Ω λ f ( z , u n ) h d z ∀ h ∈ W 0 1 , p ( z ) ( Ω ) , n ∈ ℕ .

In (3.22), we use h = u n ∈ W 0 1 , p ( z ) ( Ω ) . From (3.21), hypothesis H 1 ( i ) and Proposition 2.1, it follows that the sequence { u n } n ⩾ 1 ⊆ W 0 1 , p ( z ) ( Ω ) is bounded.

So, we may assume that

(3.23) u n → w u λ ∗ in W 0 1 , p ( z ) ( Ω ) and u n → u λ ∗ in L p ( z ) ( Ω ) .

We test (3.22) with h = u n − u λ ∗ ∈ W 0 1 , p ( z ) ( Ω ) , pass to the limit as n → + ∞ and use (3.23). We obtain

lim n → + ∞ ( 〈 A p ( z ) ( u n ) , u n − u λ ∗ 〉 + 〈 A q ( z ) ( u n ) , u n − u λ ∗ 〉 ) = 0 ,

limsup n → + ∞ ( 〈 A p ( z ) ( u n ) , u n − u λ ∗ 〉 + 〈 A q ( z ) ( u λ ∗ ) , u n − u λ ∗ 〉 ) ⩽ 0

(since A q ( z ) is monotone), thus

limsup n → + ∞ 〈 A p ( z ) ( u n ) , u n − u λ ∗ 〉 ⩽ 0

(see (3.23)) and hence

(3.24) u n → u λ ∗ in W 0 1 , p ( z ) ( Ω )

(see Proposition 2.2).

Then passing to the limit as n → + ∞ in (3.22) and using (3.24) and (3.21), we obtain

〈 A p ( z ) ( u λ ∗ ) , h 〉 + 〈 A q ( z ) ( u λ ∗ ) , h 〉 = ∫ Ω λ f ( z , u λ ∗ ) h d z ∀ h ∈ W 0 1 , p ( z ) ( Ω ) ,

u ˜ λ ⩽ u λ ∗

and hence

u λ ∗ ∈ S λ ⊆ int C + , u λ ∗ = inf S λ . □

We consider the map λ ↦ u λ ∗ from ℒ into C 0 1 ( Ω ¯ ) .

Proposition 3.8

If hypotheses H 0 , H 1 ( i ) , ( i v ) , ( v ) hold, then the map λ ↦ u λ ∗ from ℒ into C 0 1 ( Ω ¯ ) is

strictly increasing (that is, if 0 < μ < λ ∈ ℒ , then u λ ∗ − u μ ∗ ∈ int C + );
left continuous.

Proof

(a) Suppose that 0 < μ < λ ∈ ℒ . Let u λ ∗ ∈ S λ ⊆ int C + be the minimal solution of problem ( P λ ) (see Proposition 3.7). According to Proposition 3.4, we can find u μ ∈ S μ ⊆ int C + such that

u λ ∗ − u μ ∈ int C + ,

u λ ∗ − u μ ∗ ∈ int C +

and hence the map λ ↦ u λ ∗ is strictly increasing.

(b) Let λ n → λ − with λ ∈ ℒ . Let u n ∗ = u λ n ∗ ∈ int C + for all n ∈ ℕ . From part (a) and hypothesis H 1 ( i ) , we see that the sequence { u n ∗ } n ⩾ 1 ⊆ W 0 1 , p ( z ) ( Ω ) is bounded.

Then from the anisotropic regularity theory (see Fukagai-Narukawa [29] and Tan-Fang [30]), we can find α ∈ ( 0 , 1 ) and c 4 > 0 such that

u n ∗ ∈ C 0 1 , α ( Ω ¯ ) , ∥ u n ∗ ∥ C 0 1 , α ( Ω ¯ ) ⩽ c 4 ∀ n ∈ ℕ .

Exploiting the compactness of the embedding C 0 1 , α ( Ω ¯ ) ⊆ C 0 1 ( Ω ¯ ) , we have

(3.25) u n ∗ → u ˆ λ ∗ in C 0 1 ( Ω ¯ ) .

Evidently u ˆ λ ∗ ∈ S λ . If u ˆ λ ∗ ≠ u λ ∗ , then we can find z 0 ∈ Ω such that

u λ ∗ ( z 0 ) < u ˆ λ ∗ ( z 0 ) ,

u λ ∗ ( z 0 ) < u n ∗ ( z 0 ) ∀ n ⩾ n 0

(see (3.25)). This contradicts part (a). So, the map λ ↦ u λ ∗ is left continuous.□

So far, we only know that ℒ is nonempty and connected. We do not know if it is bounded or not. The next proposition shows that ℒ is bounded. In what follows, by φ λ : W 0 1 , p ( z ) ( Ω ) → ℝ we denote the energy (Euler) functional of problem ( P λ ) defined by

φ λ ( u ) = ∫ Ω 1 p ( z ) | D u | p ( z ) d z + ∫ Ω 1 q ( z ) | D u | q ( z ) d z − ∫ Ω λ F ( z , u ) d z ∀ u ∈ W 0 1 , p ( z ) ( Ω ) .

Proposition 3.9

If hypotheses H 0 , H 1 hold, then λ ∗ < + ∞ .

Proof

We argue by contradiction. So, suppose that λ ∗ = + ∞ (that is, ℒ = ( 0 , + ∞ ) ). Let { λ n } n ⩾ 1 ⊆ ℒ be such that λ n ↗ + ∞ . Then on account of Proposition 3.8 and hypothesis H 1 ( i i ) , we can find a nondecreasing sequence u n ∈ S λ n ⊆ int C + for n ∈ ℕ such that

(3.26) φ λ n ( u n ) ⩽ c 5 ∀ n ∈ ℕ ,

for some c 5 > 0 and

(3.27) φ λ n ′ ( u n ) = 0 ∀ n ∈ ℕ .

From (3.27), we have

(3.28) 〈 A p ( z ) ( u n ) , h 〉 + 〈 A q ( z ) ( u n ) , h 〉 = λ ∫ Ω f ( z , u n ) h d z ∀ h ∈ W 0 1 , p ( z ) ( Ω ) .

We test (3.28) with h = u n ∈ W 0 1 , p ( z ) ( Ω ) . Then

(3.29) − ϱ p ( D u n ) − ϱ q ( D u n ) + λ n ∫ Ω f ( z , u n ) u n d z = 0 ∀ n ∈ ℕ .

Also from (3.26), we have

∫ Ω 1 p ( z ) | D u n | p ( z ) d z + ∫ Ω 1 q ( z ) | D u n | q ( z ) d z − λ n ∫ Ω F ( z , u n ) d z ⩽ c 5 ∀ n ∈ ℕ ,

1 p + ( ϱ p ( D u n ) + ϱ q ( D u n ) ) − λ n ∫ Ω F ( z , u n ) d z ⩽ c 5 ∀ n ∈ ℕ ,

thus

(3.30) ϱ p ( D u n ) + ϱ q ( D u n ) − λ n ∫ Ω p + F ( z , u n ) d z ⩽ p + c 5 ∀ n ∈ ℕ .

Adding (3.29) and (3.30), we obtain

λ n ∫ Ω σ ( z , u n ) d z ⩽ p + c 5 ∀ n ∈ ℕ

(3.31) ∫ Ω σ ( z , u n ) d z ⩽ p + c 5 λ n ∀ n ∈ ℕ .

Suppose that the sequence { u n } n ⩾ 1 ⊆ W 0 1 , p ( z ) ( Ω ) is not bounded. We may assume that

(3.32) ∥ u n ∥ → + ∞ as n → + ∞ .

We set y n = u n ∥ u n ∥ for n ∈ ℕ . Then ∥ y n ∥ = 1 , y n ⩾ 0 for all n ∈ ℕ . We may assume that

(3.33) y n → w y in W 0 1 , p ( z ) ( Ω ) and y n → y in L p ( z ) ( Ω ) , y ⩾ 0 .

First suppose that y ≠ 0 . Let Ω ^ = { y > 0 } . Then | Ω ^ | N > 0 (see (3.33)) and u n ( z ) → + ∞ for almost all z ∈ Ω ^ . On account of hypothesis H 1 ( i i ) , we have

F ( z , u n ( z ) ) ∥ u n ∥ p + = F ( z , u n ( z ) ) u n ( z ) p + y n ( z ) p + → + ∞ for a .a . z ∈ Ω ^ .

Then by Fatou’s lemma, we have

(3.34) lim n → + ∞ ∫ Ω ^ F ( z , u n ) ∥ u n ∥ p + d z = + ∞ .

Hypotheses H 1 ( i ) , ( i i ) imply that we can find c 6 > 0 such that

(3.35) F ( z , x ) x p + ⩾ − c 6 for a .a . z ∈ Ω , all x ⩾ 0 .

We have

∫ Ω F ( z , u n ) ∥ u n ∥ p + d z = ∫ Ω ^ F ( z , u n ) ∥ u n ∥ p + d z + ∫ Ω \ Ω ^ F ( z , u n ) ∥ u n ∥ p + d z ⩾ ∫ Ω ^ F ( z , u n ) ∥ u n ∥ p + d z − c 7 ∀ n ∈ ℕ

for some c 7 > 0 (see (3.35)), so

(3.36) lim n → + ∞ ∫ Ω F ( z , u n ) ∥ u n ∥ p + d z = + ∞

(see (3.36)). From (3.29), we have

− ∫ Ω 1 ∥ u n ∥ p + − p ( z ) | D y n | p ( z ) − ∫ Ω 1 ∥ u n ∥ p + − q ( z ) | D y n | q ( z ) + λ n ∫ Ω f ( z , u n ) u n ∥ u n ∥ p + d z = 0 ∀ n ∈ ℕ ,

λ n ∫ Ω f ( z , u n ) u n ∥ u n ∥ p + d z ⩽ c 8 ∀ n ∈ ℕ ,

for some c 8 > 0 (see (3.32), recall that q + < p ( z ) ⩽ p + for all z ∈ Ω ¯ ), thus

λ n ∫ Ω p + F ( z , u n ) ∥ u n ∥ p + d z − λ n ∥ η ∥ 1 ⩽ c 8 ∀ n ∈ ℕ

(see hypothesis H 1 ( i v ) and recall that u n ⩾ 0 ), hence

(3.37) ∫ Ω p + F ( z , u n ) ∥ u n ∥ p + d z ⩽ c 8 λ n + ∥ η ∥ 1 ∀ n ∈ ℕ .

Comparing (3.36) and (3.37), we have a contradiction.

Next suppose that y = 0 . We consider the C 1 -functional φ λ ∗ : W 0 1 , p ( z ) ( Ω ) → ℝ defined by

φ λ ∗ ( u ) = 1 p + ϱ p ( D u ) − λ ∫ Ω F ( z , u ) d z ∀ u ∈ W 0 1 , p ( z ) ( Ω ) .

Evidently, we have

(3.38) φ λ ∗ ⩽ φ λ ∀ λ > 0 .

Let ϑ n ( t ) = φ λ n ∗ ( t u ∗ ) for all t ∈ [ 0 , 1 ] , all n ∈ ℕ . We can find t n ∈ [ 0 , 1 ] such that

ϑ n ( t n ) = max 0 ⩽ t ⩽ 1 ϑ n ( t ) .

Let β ⩾ 1 and set

v n ( z ) = ( 2 β ) 1 p ( z ) y n ( z ) ∀ n ∈ ℕ .

Clearly, we have

v n → 0 in L p ( z ) ( Ω )

(see (3.33) and recall that y = 0 ), so

(3.39) ∫ Ω F ( z , v n ) d z → 0 as n → + ∞ .

From (3.32), we see that we can find n 0 ∈ ℕ such that

( 2 β ) 1 p ( z ) 1 ∥ u n ∥ ⩽ 1 ∀ n ⩾ n 0 , z ∈ Ω ¯ .

It follows that

ϑ n ( t n ) ⩾ ϑ n ( 2 β ) 1 p ( z ) ∥ u n ∥ ∀ n ⩾ n 0 , z ∈ Ω ¯ ,

φ λ n ∗ ( t n u n ) ⩾ φ λ n ∗ ( 2 β ) 1 p ( z ) y n = φ λ n ∗ ( v n ) ∀ n ⩾ n 0 ,

thus

φ λ n ∗ ( t n u n ) ⩾ 2 β p + ϱ p ( D y n ) − ∫ Ω F ( z , v n ) d z ∀ n ⩾ n 0

and hence

(3.40) φ λ n ∗ ( t n u n ) ⩾ β p + ∀ n ⩾ n 1 ⩾ n 0

(see (3.39) and Proposition 2.1(a)).

Since β ⩾ 1 is arbitrary, from (3.40) we infer that

(3.41) φ λ n ∗ ( t n u n ) → + ∞ as n → + ∞ .

We have

0 ⩽ t n u n ⩽ u n ∀ n ∈ ℕ ,

σ ( z , t n u n ) ⩽ σ ( z , u n ) + η ( z ) for a .a . z ∈ Ω , all n ∈ ℕ

(see hypothesis H 1 ( i i i ) ), so

(3.42) ∫ Ω σ ( z , t n u n ) d z ⩽ ∫ Ω σ ( z , u n ) d z + ∥ η ∥ 1 ⩽ c 9 ∀ n ∈ ℕ

for some c 9 > 0 (see (3.31)). We know that

(3.43) φ λ n ∗ ( 0 ) = 0 and φ λ n ∗ ( u n ) ⩽ c 5 ∀ n ∈ ℕ

(see (3.26) and (3.38)). Then from (3.41) it follows that t n ∈ ( 0 , 1 ) for all n ⩾ n 2 . Therefore, we can say that

0 = t n d d t φ λ n ∗ ( t u n ) | t = t n ,

〈 ( φ λ n ∗ ) ′ ( t n u n ) , t n u n 〉 = 0

(by the chain rule), thus

ϱ p ( D ( t n u n ) ) + ϱ q ( D ( t n u n ) ) − λ n ∫ Ω f ( z , t n u n ) ( t n u n ) d z = 0 ∀ n ⩾ n 2

and hence

(3.44) p + φ λ n ∗ ( t n u n ) ⩽ c 9 ∀ n ⩾ n 2

(see (3.42)).

We compare (3.41) and (3.44) and have a contradiction. This proves that the sequence { u n } n ⩾ 1 ⊆ W 0 1 , p ( z ) ( Ω ) is bounded. Recall that

A p ( z ) ( u n ) + A q ( z ) ( u n ) = λ n N f ( u n ) in W 0 1 , p ( z ) ( Ω ) ∗ ∀ n ∈ ℕ ,

with N f ( u n ) ( ⋅ ) = f ( ⋅ , u n ( ⋅ ) ) (the Nemytskii map corresponding to f). From Proposition 2.2, it follows that

λ n ∥ N f ( u n ) ∥ ∗ ⩽ c 10 ∀ n ∈ ℕ

for some c 10 > 0 . Since u n ⩾ u 1 ∈ int C + , on account of hypothesis H 1 ( i v ) and since λ n → + ∞ , we have

λ n ∥ N f ( u n ) ∥ ∗ → + ∞ ,

a contradiction. This proves that λ ∗ < + ∞ .□

According to Proposition 3.9, we have

( 0 , λ ∗ ) ⊆ ℒ ⊆ ( 0 , λ ∗ ] .

Proposition 3.10

If hypotheses H 0 , H 1 hold and λ ∈ ( 0 , λ ∗ ) , then problem ( P λ ) has at least two positive solutions

u 0 , u ˆ ∈ int C + , u 0 ⩽ u ˆ , u 0 ≠ u ˆ .

Proof

Let λ , ϑ ∈ ( 0 , λ ∗ ) , λ < ϑ . We have λ , ϑ ∈ ℒ . We can find u ϑ ∈ S ϑ ⊆ int C + and u 0 ∈ S λ ⊆ int C + such that

u ϑ − u 0 ∈ int C +

(see Proposition 3.4). We introduce the Carathéodory function g ( z , x ) defined by

(3.45) g ( z , x ) = f ( z , u 0 ( z ) ) if x ⩽ u 0 ( z ) , f ( z , x ) if u 0 ( z ) < x .

We set

G ( z , x ) = ∫ 0 x g ( z , s ) d s

and consider the C 1 -functional ψ λ : W 1 , p ( z ) ( Ω ) → ℝ defined by

ψ λ ( u ) = ∫ Ω 1 p ( z ) | D u | p ( z ) d z + ∫ Ω 1 q ( z ) | D u | q ( z ) d z − λ ∫ Ω G ( z , u ) d z ∀ u ∈ W 1 , p ( z ) ( Ω ) .

Using (3.45) and the anisotropic regularity theory, we obtain

(3.46) K ψ λ ⊆ [ u 0 ) ∩ int C + .

We introduce the following truncation of g ( z , ⋅ )

(3.47) g ¯ ( z , x ) = g ( z , x ) if x ⩽ u 0 ( z ) , g ( z , u 0 ( z ) ) if u 0 ( z ) < x .

This is a Carathéodory function. We set

G ¯ ( z , x ) = ∫ 0 x g ¯ ( z , s ) d s

and consider the C 1 -functional ψ ˆ λ : W 1 , p ( z ) ( Ω ) → ℝ defined by

ψ ˆ λ ( u ) = ∫ Ω 1 p ( z ) | D u | p ( z ) d z + ∫ Ω 1 q ( z ) | D u | q ( z ) d z − λ ∫ Ω G ¯ ( z , u ) d z ∀ u ∈ W 1 , p ( z ) ( Ω ) .

For this functional, we have that

(3.48) K ψ ˆ λ ⊆ [ u 0 , u ϑ ] ∩ int C + .

We may assume that

(3.49) K ψ ˆ λ ∩ [ u 0 , u ϑ ] = { u 0 } .

Otherwise, on account of (3.46) and (3.45), we see that we already have a second positive smooth solution bigger than u 0 and so we are done.

The functional ψ ˆ λ is coercive (see Proposition 2.1 and (3.47)). Also, it is sequentially weakly lower semicontinuous. So, we can find u ˆ 0 ∈ W 0 1 , p ( z ) ( Ω ) such that

ψ ˆ λ ( u ˆ 0 ) = min u ∈ W 1 , p ( z ) ( Ω ) ψ ˆ λ ( u ) ,

so u ˆ 0 ∈ K ψ ˆ λ ⊆ [ u 0 , u ϑ ] ∩ int C + (see (3.33)).

Note that

ψ λ ′ | [ u 0 , u ϑ ] = ψ ˆ λ ′ | [ u 0 , u ϑ ]

(see (3.45) and (3.47)). So, it follows that u ˆ 0 = u 0 (see (3.49)).

Since u ϑ − u 0 ∈ int C + , we see that

u 0 is a local C 0 1 ( Ω ¯ ) -minimizer of ψ λ ,

(3.50) u 0 is a local W 0 1 , p ( z ) ( Ω ¯ ) -minimizer of ψ λ

(see Gasiński-Papageorgiou [6] and Tan-Fang [30]).

From (3.46), we see that we may assume that K ψ λ is finite (otherwise we already have infinity of positive smooth solutions bigger than u 0 and so we are done). Then on account of (3.50) and using Theorem 5.7.6 of Papageorgiou-Rădulescu-Repovš [33, p. 449], we can find ϱ ∈ ( 0 , 1 ) small such that

(3.51) ψ λ ( u 0 ) < inf { ψ λ ( u ) : ∥ u − u 0 ∥ = ϱ } = m λ .

Also, if u ∈ int C + , then from (3.45) and hypothesis H 1 ( i i ) we have that

(3.52) ψ λ ( t u ) → − ∞ as t → + ∞ .

Claim. ψ λ satisfies the Cerami condition.

Consider a sequence { u n } n ⩾ 1 ⊆ W 0 1 , p ( z ) ( Ω ) such that

(3.53) | ψ λ ( u n ) | ⩽ c 11 ∀ n ∈ ℕ ,

for some c 11 > 0 , so

(3.54) ( 1 + ∥ u n ∥ ) ψ λ ′ ( u n ) → 0 in W 0 1 , p ( z ) ( Ω ) ∗ as n → + ∞ .

From (3.54), we have

(3.55) 〈 A p ( z ) ( u n ) , h 〉 + 〈 A q ( z ) ( u n ) , h 〉 − λ ∫ Ω g ( z , u n ) h d z ⩽ ε n ∥ h ∥ 1 + ∥ u n ∥ ∀ h ∈ W 0 1 , p ( z ) ( Ω ) ,

with ε n → 0 + . In (3.55), we use h = − u n − ∈ W 0 1 , p ( z ) ( Ω ) and obtain

ϱ p ( D u n − ) + ϱ q ( D u n − ) ⩽ c 12 ∀ n ∈ ℕ ,

for some c 12 > 0 (see (3.45)), so

(3.56) the sequence { u n − } n ⩾ 1 ⊆ W 0 1 , p ( z ) ( Ω ) is bounded

(see Proposition 2.1).

Next in (3.55) we choose h = u n + ∈ W 1 , p ( z ) ( Ω ) . Then

− ϱ p ( D u n + ) − ϱ q ( D u n + ) + λ ∫ Ω g ( z , u n + ) u n + d z ⩽ ε n ∀ n ∈ ℕ ,

(3.57) − ϱ p ( D u n + ) − ϱ q ( D u n + ) + λ ∫ Ω f ( z , u n + ) u n + d z ⩽ c 13 ∀ n ∈ ℕ ,

for some c 13 > 0 .

From (3.53), (3.56) and (3.45), we have

(3.58) ϱ p ( D u n + ) + ϱ q ( D u n + ) − λ ∫ Ω p + F ( z , u n + ) d z ⩽ c 14 ∀ n ∈ ℕ ,

for some c 14 > 0 .

We add (3.57) and (3.58) and obtain

(3.59) λ ∫ Ω σ ( z , u n + ) d z ⩽ c 15 ∀ n ∈ ℕ ,

for some c 15 > 0 .

Using (3.59) and reasoning as in the proof of Proposition 3.9 (see the part of the proof after (3.31) up to (3.44)), we obtain that

(3.60) the sequence { u n + } ⊆ W 0 1 , p ( z ) ( Ω ) is bounded .

Then (3.50) and (3.60) imply that

the sequence { u n } ⊆ W 0 1 , p ( z ) ( Ω ) is bounded .

So, we may assume that

(3.61) u n → w u in W 0 1 , p ( z ) ( Ω ) and u n → u in L r ( Ω ) as n → + ∞ .

In (3.55), we test with h = u n − u ∈ W 0 1 , p ( z ) ( Ω ) and pass to the limit as n → + ∞ . As in the proof of Proposition 3.7, we obtain

u n → u in W 0 1 , p ( z ) ( Ω ) as n → + ∞

(see (3.24)), so ψ λ satisfies the Cerami condition. This proves the Claim.

Then (3.51), (3.52) and the Claim permit the use of the mountain pass theorem and find u ˆ ∈ W 0 1 , p ( z ) ( Ω ) such that

(3.62) u ˆ ∈ K ψ λ ⊆ [ u 0 ) ∩ int C + and m λ ⩽ ψ λ ( u ˆ )

(see (3.46) and (3.51)).

From (3.62), (3.51) and (3.45), we conclude that u ˆ ∈ int C + is a positive solution of ( P λ ) , u 0 ⩽ u ˆ , u 0 ≠ u ˆ .□

It remains to decide what happens with critical parameter value λ ∗ < + ∞ .

Proposition 3.11

If hypotheses H 0 , H 1 hold, then λ ∗ ∈ ℒ .

Proof

Let λ n ∈ ( 0 , λ ∗ ) , n ∈ ℕ be such that λ n ↗ λ ∗ . We can find u n ∈ S λ n ⊆ int C + nondecreasing such that

(3.63) φ λ n ( u n ) ⩽ c 16 ∀ n ∈ ℕ ,

for some c 16 > 0 , so

(3.64) φ λ n ′ ( u n ) = 0 ∀ n ∈ ℕ .

Using (3.63), (3.64) as in the proof of Proposition 3.9, first we obtain that the sequence { u n } n ⩾ 1 ⊆ W 0 1 , p ( z ) ( Ω ) is bounded and then via Proposition 2.2, at least for a subsequence, we have

(3.65) u n → u ∗ in W 0 1 , p ( z ) ( Ω ) .

From (3.64) and (3.65), in the limit as n → + ∞ , we obtain

〈 A p ( z ) ( u ∗ ) , h 〉 + 〈 A q ( z ) ( u ∗ ) , h 〉 = λ ∗ ∫ Ω f ( z , u ∗ ) h d z ∀ h ∈ W 0 1 , p ( z ) ( Ω ) ,

so u 1 ⩽ u ∗ . Therefore, u ∗ ∈ S λ ∗ ⊆ int C + and so λ ∗ ∈ ℒ .□

We conclude that

ℒ = ( 0 , λ ∗ ] .

So, summarizing our findings for problem ( P λ ) , we can state the following bifurcation-type theorem.

Theorem 3.12

If hypotheses H 0 , H 1 hold, then there exists λ ∗ > 0 such that

for all λ ∈ ( 0 , λ ∗ ) problem ( P λ ) has at least two positive solutions
u 0 , u ˆ ∈ int C + , u 0 ⩽ u ˆ , u 0 ≠ u ˆ ;
for λ = λ ∗ problem ( P λ ) has at least one positive solution
u ∗ ∈ int C + ;
for all λ > λ ∗ problem ( P λ ) has no positive solutions;
for all λ ∈ ( 0 , λ ∗ ] problem ( P λ ) has a smallest (minimal) positive solution u λ ∗ ∈ int C + and the map λ ↦ u λ ∗ from ℒ = ( 0 , λ ∗ ] into C 0 1 ( Ω ¯ ) is strictly increasing and left continuous.

Acknowledgements

The authors wish to thank the two knowledgeable referees for their corrections and remarks.

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Received: 2020-05-22

Revised: 2020-07-23

Accepted: 2020-08-17

Published Online: 2020-10-07

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Articles in the same Issue

https://doi.org/10.1515/math-2020-0074

Keywords for this article

anisotropic regularity; anisotropic maximum principle; positive solutions; minimal positive solution; superlinear reaction

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