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

Abstract We consider a parametric elliptic equation driven by the anisotropic ( p , q ) (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 \lambda moves in ℝ + = ( 0 , + ∞ ) {{\mathbb{R}}}_{+}=(0,+\infty ) .


Introduction
Let ⊆ Ω N be a bounded domain with a C 2 -boundary ∂Ω. We study the following parametric anisotropic By ( ) Δ p z (respectively ( ) Δ q z ) we denote the ( ) p z -Laplacian (respectively the ( ) q z -Laplacian) defined by In the reaction (right hand side of ( ) , is a Carathéodory function (that is, for all ∈ x , ↦ ( ) 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 λ 0, 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 .

Mathematical backgroundhypotheses
This space is equipped with the so-called "Luxemburg norm" defined by . We know that and we have the following Hölder-type inequality: (where the gradient Du is understood in the weak sense). This space is equipped with the following norm: In the sequel for notational simplicity, we write ∥ ∥ = ∥| |∥ ). Then we define ), we can consider the following equivalent norm: For ∈ r E 1 , the critical Sobolev exponent corresponding to r is defined by (respectively: compactly).
Useful in the analysis of these variable exponent spaces is the following modular function: There is a close relation between this modular function and the norm. We assume ∈ r E 1 .

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In addition to the variable exponent spaces, we will also use the Banach space This is an ordered Banach space with positive (order) cone This cone has a nonempty interior given by for a.a. Ω . 1 2 , we can find ∈ u S such that ⩽ u u 1 , ⩽ u u 2 . By |⋅| N we denote the Lebesgue measure on N and by ∥⋅∥ * the norm of ( ) In the sequel for notational economy, by ∥⋅∥ we denote the norm of the Sobolev space ( ) . We say that φ satisfies the "Cerami condition," if the following property holds: admits a strongly convergent subsequence." Now we introduce the hypotheses on the data problem ( ) uniformly for a.a. Ω; Ω ,a l l 0 ; 0f o ra . a . Ω ,a l l , uniformly for a.a. Ω; Since we want to find positive solutions and the aforementioned hypotheses concern the positive semiaxis = [ +∞) + 0, , without any loss of generality, we may assume that 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 iii 1 , 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 . This superlinearity condition incorporates in our framework superlinear nonlinearities with "slower" growth near + ∞. For example, consider the following function: This function satisfies hypotheses H 1 , but fails to satisfy the Ambrosetti-Rabinowitz condition.
We introduce the following two sets:

Positive solutions
We start by showing that the set of admissible parameters is nonempty. Also, we determine the regularity properties of the elements in S λ .
hold, then ≠ ∅ and for every > λ 0, ⊆ Proof. We consider the following auxiliary Dirichlet problem: 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 From the anisotropic maximum principle (see Zhang [32]), we obtain that ∈ for all ∈ ( ] λ λ 0, 0 . We introduce the Carathéodory function ( ) g z x , defined by We set From (3.3) and Proposition 2.1, it is clear that φ λ is coercive. Also, the anisotropic Sobolev embedding theorem implies that φ λ is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find
Hypotheses ( ) ( ) H i iv , 1 imply that given > β 0, we can find = ( ) > c c β 0 for a.a. Ω, all 0. Proof. First we show the existence of a positive solution. To this end, we consider the C 1 -functional , we see that σ λ is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find for some > c c , 0 . Then from (3.13) we infer that ũ λ is a positive solution of ( ) Q λ . Moreover, as before the anisotropic regularity theory and the anisotropic maximum principle imply be another positive solution of ( ) Q λ . Again we have We consider the integral functional 0, 1 small, we have We have Then on account of the convexity of j, it is Gâteaux differentiable at − ũ λ q and at − ṽ λ q in the direction The convexity of j implies the monotonicity of ′ j . Hence, Proof. Let ∈ u S λ . We introduce the Carathéodory function ( ) k z x , defined by if .
As before (see the proof of Proposition 3.5), we have be the minimal solution of problem ( ) P λ (see Proposition 3.7). According to Proposition 3.4, we can find ∈ ⊆ and hence the map ↦ * λ u λ is strictly increasing. , 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 is strictly increasing and left continuous.