Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 06, pp. 1–14. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2025.06 WEIGHTED (p, q)-EQUATIONS WITH GRADIENT DEPENDENT REACTION ZHAO JING, ZHENHAI LIU, NIKOLAOS S. PAPAGEORGIOU Abstract. We consider a weighted (p, q)-equation with a parametric reaction depending on the gradient. Using truncation and comparison techniques and the theory of nonlinear operators of monotone type, we show that for all small values of the parameter, the problem has a positive smooth solution. 1. Introduction Let Ω ⊆ RN be a bounded domain with a C2-boundary ∂Ω. In this article we study the parametric Dirichlet problem −∆a1 p u(z)−∆a2 q u(z) = f(z, u(z)) + λ|Du(z)|p−1 in Ω, u|∂Ω = 0, 1 < q < p < N, λ > 0, u > 0. (1.1) If a ∈ C0,1(Ω) with 0 < ĉ ≤ a(z) for all z ∈ Ω and s ∈ (1,∞), then by ∆a s we denote the weighted s-Laplace differential operator defined by ∆a su = div(a(z)|Du|s−2Du), for all u ∈W 1,s 0 (Ω). In problem (1.1) the equation is driven by the sum of two such operators with differ- ent exponents q < p and in general different weights (a weighted (p, q)-differential operator). So, the differential operator in (1.1) is not homogeneous. The reaction (right hand side) of (1.1) is gradient dependent. Combining varational tools from critical point theory, with the theory of nonlinear operators of monotone type, we show that for all small values of parameter λ > 0, problem (1.1) has a positive small solution. Nonlinear elliptic equations with gradient dependence, such as (1.1), were exam- ined by Candito-Gasiński-Papageorgiou [1], Faria-Miyagaki-Motreanu [4], Tanaka [16], Zeng-Papageorgiou [17]. All the aforementioned works deal with equations driven by autonomous differential operators and their method of proof is based on the fixed point theory. Our approach here is different and it is partly motivated by the work of Deuel-Hess [2] (see also Liu-Papageorgiou [10] on double phase equations). 2020 Mathematics Subject Classification. 35J60, 35J91. Key words and phrases. Weighted (p, q)-operator; principal eigenvalue; truncation; pseudo-monotone map; gradient dependence. ©2025. This work is licensed under a CC BY 4.0 license. Submitted February 25, 2024. Published January 13, 2025. 1 2 Z. JING, Z. LIU, N. PAPAGEORGIOU EJDE-2025/06 2. Mathematical background, hypotheses In the analysis of (1.1) the main spaces are the Sobolev space W 1,p 0 (Ω) and the Banach space C1 0 (Ω) = {u ∈ C1(Ω) : u|∂Ω = 0}. On account of the Poincaré inequality, on W 1,p 0 (Ω) we can use the following equivalent norm ∥u∥ = ∥Du∥p for all u ∈W 1,p 0 (Ω). The space C1 0 (Ω) is an ordered Banach space with positive (order) cone C+ = {u ∈ C1 0 (Ω) : u(z) ≥ 0 for all z ∈ Ω}. This cone has a nonempty interior given by intC+ = { u ∈ C+ : u(z) > 0 for all z ∈ Ω, ∂u ∂n |∂Ω < 0 } where ∂u ∂n = (Du,n)RN with n(·) being the outward unit normal on ∂Ω. Let a ∈ C0,1(Ω) with a(z) ≥ ĉ > 0 for all z ∈ Ω and 1 < s < ∞. We consider the following nonlinear eigenvalue problem −∆a su(z) = λ̂|u(z)|s−2u(z) in Ω, u|∂Ω = 0. This problem was examined by Liu-Papageorgiou [11] (see the Appendix in [11]), who established the existence of a smallest eigenvalue λ̂a1(s) > 0 which has the following variational characterization λ̂a1(s) = inf {ϱα,s(Du) ∥u∥ss : u ∈W 1,s 0 (Ω), u ̸= 0 } , (2.1) where ϱα,s(Du) = ∫ Ω a(z)|Du|s dz for all u ∈ W 1,s 0 (Ω). This eigenvalue is isolated in the spectrum and simple (that is, if û, v̂ ∈ W 1,s 0 (Ω) are two eigenfunctions corresponding to λ̂a1(s) > 0, then û = ϑv̂ for some ϑ ∈ R\{0}). So, the eigenspace corresponding to λ̂a1(s) is one-dimensional and its elements have fixed sign. In fact λ̂a1(s) is the only eigenvalue with eigenfunctions of constant sign. All the other eigenvalues have nodal (sign changing) eigenfunctions. The infimum in (2.1) is realized on the one dimensional eigenspace of λ̂a1(s). By û1 = û1(s) ∈W 1,s 0 (Ω), we denote the positive, Ls-normalized (that is, ∥û1∥s = 1) eigenfunction corresponding to λ̂a1(s). The nonlinear regularity theory and the nonlinear maximum principle (see Gasiński-Papageorgiou ([5], p.738), imply that û1 ∈ intC+. These properties of the principal eigenvalue λ̂a1(s) > 0 and of its eigenfunctions, lead to the following result (see Liu-Papageorgiou [11, Proposition 4.2])). Proposition 2.1. If ϑ ∈ L∞(Ω), ϑ(z) ≤ λ̂a1(s) for a.a. z ∈ Ω and ϑ ̸≡ λ̂α1 (s), then there exists, c0 > 0 such that c0∥u∥s ≤ ϱα,s(Du)− ∫ Ω ϑ(z)|u| dz for all u ∈W 1,p 0 (Ω). Suppose u : Ω → R is a measurable function. We define u+(z) = max{u(z), 0}, u−(z) = max{−u(z), 0} for all z ∈ Ω. We have u = u+ − u−, |u| = u+ + u− and if u ∈ W 1,s 0 (Ω), then u± ∈ W 1,s 0 (Ω). Also by | · |N we denote the Lebesgue measure on RN . If u, v : Ω → R are measurable functions and u(z) ≤ v(z) for a.a. z ∈ Ω, then we define [u, v] = {h ∈W 1,p 0 (Ω) : u(z) ≤ h(z) ≤ v(z) a.a. z ∈ Ω}. EJDE-2025/06 WEIGHTED (p, q)-EQUATIONS 3 If X is a Banach space and φ ∈ C1(X), then Kφ = {u ∈ X : φ′(u) = 0} (the critical set of φ). Suppose X is a reflexive Banach space and E : X → X∗ a bounded nonlinear map. We say that E(·) is “pseudomonotone”, if it has the following property: If un w−→ u in X, E(un) w−→ u∗ in X∗ and lim supn→∞⟨E(un), un− u⟩ ≤ 0, then u∗ = E(u) and ⟨E(un), un⟩ → ⟨E(un), u⟩. (see Gasiński-Papageorgiou [5, p.330]). Here by ⟨·, ·⟩ we denote the duality brackets for the pair (X∗, X). A maximal monotone operator is pseudomonotone. We say that E(·) is “strongly coercive”, if ⟨E(u), u⟩ ∥u∥X → +∞ as ∥u∥X → +∞. The next theorem reveals the importance of pseudomonotone maps. Theorem 2.2. If X is a reflexive Banach space and E : X → X∗ is a strongly coercive pseudomonotone map, then E(·) is surjective. Let Aa s : W 1,s 0 (Ω) → W−1,s′(Ω) = W 1,s 0 (Ω) ∗ ( 1s + 1 s′ = 1) be the nonlinear map defined by ⟨Aa s(u), h⟩ = ∫ Ω a(z)|Du|s−2(Du,Dh)RN dz for all u, h ∈W 1,s 0 (Ω). From Gasiński-Papageorgiou [6] (p.279), we have the following properties for this map. Proposition 2.3. Aa s : W 1,s 0 (Ω) → W−1,s′(Ω) is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (thus maximal monotone too) and of type (S)+, that is: if un w−→ u inW 1,s 0 (Ω) and lim supn→∞⟨Aa s(un), un−u⟩ ≤ 0, then un → u in W 1,s 0 (Ω). Our hypotheses on the data of problem (1.1), are the following: (H0) a1, a2 ∈ C0,1(Ω) and 0 < ĉ ≤ a1(z), a2(z) for all z ∈ Ω. (H1) f : Ω × R → R is a Carathéodory function such that f(z, 0) = 0 for a.a. z ∈ Ω and (i) for every ϱ > 0, there exists aϱ ∈ L∞(Ω) such that 0 ≤ f(z, x) ≤ aϱ(z) for a. a. z ∈ Ω and all 0 ≤ x ≤ ϱ; (ii) there exists a function ϑ̂ ∈ L∞(Ω) such that ϑ̂(z) ≤ λ̂a1 1 (p) for a.a. z ∈ Ω, ϑ̂ ̸≡ λ̂a1 1 (p) lim sup x→+∞ f(z, x) xp−1 ≤ ϑ̂(z) uniformly for a.a. z ∈ Ω; (iii) there exist τ ∈ (1, q) and δ > 0 such that cxτ−1 ≤ f(z, x) for a.a. z ∈ Ω, all 0 ≤ x ≤ δ, some c > 0. Remark 2.4. Since we are looking for positive solutions and the above hypotheses concern the positive semiaxis R+ = [0,∞) without any loss of generality we may assume that f(z, x) = 0 for a.a. z ∈ Ω, all x ≤ 0. 4 Z. JING, Z. LIU, N. PAPAGEORGIOU EJDE-2025/06 3. Auxiliary problems In this section we examine the following auxiliary parametric Dirichlet problem, with parameter µ > 0 −∆a1 p u(z)−∆a2 q u(z) = µcu(z)τ−1 in Ω, u|∂Ω = 0, u > 0, µ > 0, 1 < τ < q < p < N. (3.1) For this problem we have the following result. Proposition 3.1. If (H0) holds, then for every µ > 0 problem (3.1) has a unique positive solution such that: uµ ∈ intC+, {uµ}µ>0 is nondecreasing, and uµ → 0 in C1(Ω) as µ→ 0+. Proof. Consider the C1-functional ψµ :W 1,p 0 (Ω) → R defined by ψµ(u) = 1 p ϱa1,p(Du) + 1 q ϱa2,q(Du)− µc τ ∥u+∥ττ for all u ∈W 1,p 0 (Ω). Since τ < q < p, we see that ψµ(·) is coercive. Also, using the Sobolev embedding theorem, we see that ψµ(·) is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find uµ ∈W 1,p 0 (Ω) such that ψ′ µ(uµ) = min{ψµ(u) : u ∈W 1,p 0 (Ω)} < 0 = ψµ(0) (since τ < q < p) ⇒ uµ ̸= 0. (3.2) Then we have ⟨ψµ(uµ), h⟩ = 0 for all h ∈W 1,p 0 (Ω), ⇒ ⟨V (uµ), h⟩ = ∫ Ω µc(u+µ ) τ−1h dz for all h ∈W 1,p 0 (Ω). (3.3) Here V = Aa1 p + Aa2 q : W 1,p 0 (Ω) → W−1,p′ (Ω) and on account of Proposition 2.3 this map is bounded, continuous, strictly monotone (thus maximal monotone too) and of type (S)+. In (3.3) we use the test function h = −u−µ ∈W 1,p 0 (Ω) and obtain ϱa1,p(Du − µ ) ≤ 0 =⇒ ĉ∥Du−µ ∥pp ≤ 0 =⇒ uµ ≥ 0, uµ ̸= 0. From Ladyzhenskaya-Uraltseva [8, Theorem 7.1, p.286], we have uµ ∈ L∞(Ω). Then the nonlinear regularity theory of Lieberman [9], implies that uµ ∈ C+\{0}. Since ∆a1 p uµ + ∆a2 q uµ ≤ 0 in Ω, from the nonlinear maximum principle of Pucci-Serrin [15, pp.111, 120], we have that uµ ∈ int C+. Next we show the uniqueness of this positive solution. So, let v ∈ W 1,p 0 (Ω) be another positive solution of (Qµ). Again we have vµ ∈ int C+ We introduce the integral functional j : L1(Ω) → R̄ = R ∪ {+∞} defined by j(u) = { 1 pϱa1,p(Du 1/q) + 1 qϱa2,q(Du 1/q) if u ≥ 0, u1/q ∈W 1,p 0 (Ω), +∞ otherwise. From Diaz-Saa [3], see also Papageorgiou-Rădulescu [12] (proof of Proposition 3.5), we know that j(·) is convex. Let domj = {u ∈ L1(Ω) : j(u) < +∞} (the effective domain of j(·)). Let h = uqµ−vqµ ∈ C1 0 (Ω̄). Since uµ, vµ ∈ intC+, using Proposition 4.1.22, p.274, of Papageorgiou-Rădulescu-Repovš [14], we have that uµ vµ ∈ L∞(Ω) and vµ uµ ∈ L∞(Ω). EJDE-2025/06 WEIGHTED (p, q)-EQUATIONS 5 Then if t ∈ (0, 1) is small, we have uqµ + th ∈ dom j and vqµ + th ∈ dom j. Using the convexity of j(·), we can compute the directional derivatives of j(·) at uqµ and at vqµ in the direction h. A direct computation involving the nonlinear Green’s identity (see [14, pp.34-35]) gives j′(uqµ)(h) = 1 q ∫ Ω −∆a1 p uµ −∆a2 q uµ uq−1 µ h dz = 1 q ∫ Ω µc uq−τ µ h dz, j′(vqµ)(h) = 1 q ∫ Ω −∆a1 p vµ −∆a2 q vµ vq−1 µ h dz = 1 q ∫ Ω µc vq−τ µ h dz. The convexity of j(·) implies the monotonicity of the directional derivative j′(·). So, we have 0 ≤ ∫ Ω ( 1 uq−τ µ − 1 vq−τ µ )(uqµ − vqµ) dz ≤ 0, =⇒ uµ = vµ. This proves the uniqueness of the positive solution uµ ∈ intC+ of (Qµ) for all µ > 0. Next we show the monotonicity of the family {uµ}µ>0. So suppose that 0 < µ < η. We have −∆a1 p uη −∆a2 q uη ≥ ηcuτ−1 η ≥ µcuτ−1 η in Ω. (3.4) We introduce the Carathéodory function gµ(z, x) defined by gµ(z, x) = { µc(x+)z−1 if x ≤ uη(z) µcuη(z) τ−1 if uη(z) < x. (3.5) We set Gµ(z, x) = ∫ x 0 gµ(z, x)ds and consider the C1-functional σµ :W 1,p 0 (Ω) → R defined by σµ(u) = 1 p ϱa1,p(Du) + 1 q ϱa2,q(Du)− ∫ Ω Gµ(z, u) dz for all u ∈W 1,p 0 (Ω). From (3.5) it is clear that σµ(·) is coercive. Also it is sequentially weakly lower semicontinuous. So, we can find u∗µ ∈W 1,p 0 (Ω) such that σµ(u ∗ µ) = inf{σµ(u) : u ∈W 1,p 0 (Ω)}. (3.6) Let u ∈ C+\{0}. Since uµ ∈ intC+, we can find t ∈ (0, 1) small such that 0 ≤ tu ≤ uµ (see [14, p.274]). Then σµ(tu) = tp p ϱa1,p(Du) + tq q ϱa2,q(Du)− tτ τ µc∥u∥ττ . Since τ < q < p, choosing t ∈ (0, 1) even smaller if necessary, we have that σµ(tu) < 0 =⇒ σµ(u ∗ µ) < 0 = σµ(0) (see (3.6)) =⇒ u∗µ ̸= 0. From (3.6), we see that u∗µ ∈ Kσµ and so ⟨V (u∗µ), h⟩ = ∫ Ω gµ(z, u ∗ µ)h dz for all h ∈W 1,p 0 (Ω). (3.7) In (3.7). We choose the test function h = −(u∗µ) − ∈W 1,p 0 (Ω) and obtain ĉ∥D(u∗µ) − ∥pp≤ 0 (see hypotheses (H0)), 6 Z. JING, Z. LIU, N. PAPAGEORGIOU EJDE-2025/06 =⇒ u∗µ ≥ 0, u∗µ ̸= 0. Also, we test (3.7) with h = (u∗µ − uη) + ∈W 1,p 0 (Ω). We have ⟨V (u∗µ), (u ∗ µ − uη) +⟩ = ∫ Ω µcuτ−1 η (u∗µ − uη) + dz (see (3.5)) ≤ ⟨V (uη), (u ∗ µ − uη) +⟩ (see (3.4)), which implies u∗µ ≤ uη (since V (·) is strictly monotone). So, we have proved that u∗µ ∈ [0, uη], u ∗ µ ̸= 0, =⇒ u∗µ = uµ ≤ uη (see (3.6) and (3.7)). This proves the monotonicity of {uµ}µ>0. Standard Moser iteration, gives ∥uµ∥∞ ≤ c1∥u1∥ 1 p−1 ∞ for some c1 > 0, all µ ∈ (0, 1]. Then the nonlinear regularity theory of Lieberman [9], implies that there exist α ∈ (0, 1) and c2 > 0 such that uµ ∈ C1,α 0 (Ω) = C1,α(Ω) ∩ C1 0 (Ω), ∥uµ∥C1,α 0 (Ω) ≤ c2 for all µ ∈ (0, 1]. Recall that C1,α 0 (Ω) ↪→ C1 0 (Ω) compactly (Arzela-Ascoli theorem). Therefore, uµ → 0 in C1 0 (Ω) as µ→ 0+. □ Next we consider another auxiliary Dirichlet probem, −∆a1 p u(z)−∆a2 q u(z) = f(z, u(z)) + 1 in Ω, u|∂Ω = 0, u > 0. (3.8) Proposition 3.2. Under Assumptions (H0) and (H1), problem (3.6) has a smallest positive solution u ∈ intC+. Proof. We consider the nonlinear map E : W 1,p 0 (Ω) → W−1,p′ (Ω) = W 1,p 0 (Ω) ∗ defined by E(u) = V (u)−Nf (u +) for all u ∈W 1,p 0 (Ω), with Nf (v)(·) = f(·, v(·)) for all v ∈ W 1,p 0 (Ω) (the Nemytski map corresponding to f(z, x)). Note that on account of hypotheses H1(i), (ii) Nf (v) ∈ Lp′ (Ω) for all v ∈W 1,p 0 (Ω) and Lp′ (Ω) ↪→W−1,p′ (Ω) continuously, see [5, p. 141]. Evidently E(·) is bounded and continuous. Claim 1: E(·) is pseudomonotone. We consider a sequence {un}n∈N ⊆ W 1,p 0 (Ω) which satisfies un w−→ u in W 1,p 0 (Ω), E(un) w−→ u∗ in W−1,p′ 0 (Ω) =W 1,p 0 (Ω) ∗ , lim sup n→∞ ⟨E(un), un − u⟩ ≤ 0. (3.9) From (3.9) and since W 1,p 0 (Ω) ↪→ Lp(Ω) compactly, we have that un → u in Lp(Ω), as n→ ∞, =⇒ ⟨Nf (u + n ), un − u⟩ = ∫ Ω f(z, u+n )(un − u) dz → 0 as n→ ∞ (see (H1)(i), (H1)(ii)), EJDE-2025/06 WEIGHTED (p, q)-EQUATIONS 7 =⇒ lim sup n→∞ ⟨V (un), un − u⟩ ≤ 0(see (3.9)), =⇒ un → u in W 1,p 0 (Ω) (since V (·) is of type (S)+). Therefore u∗ = E(u) (since E(·) is continuous) and ⟨E(un), un⟩ → ⟨E(u), u⟩. So, E(·) is pseudomonotone and this proves Claim 1. Claim 2: E(·) is strongly coercive. Hypotheses (H1) (i), (ii) imply that given ε > 0, we can find cε > 0 such that 0 ≤ f(z, x) ≤ (ϑ̂(z) + ε)xp−1 + cε for a.a. z ∈ Ω and all x ≥ 0. (3.10) We have that ⟨E(u), u⟩ = ⟨V (u), u⟩ − ∫ Ω f(z, u+)u dz ≥ ϱa1,p(Du)− ∫ Ω ϑ̂(z)|u|p dz − ε∥u∥pp − cε|Ω|N (see (3.10)) ≥ [c0 − ε λ̂a1 1 (p) ]∥u∥p − cε|Ω|N (see Proposition 2.1 and (2.1)). Choosing ε ∈ (0, λ̂a1 1 (p)c0), we infer that ⟨E(u), u⟩ ≥ c3∥u∥p − c4 for some c3, c4 > 0 =⇒ E(·) is strongly coercive. This proves Claim 2. Claims 1 and 2, permit the use of Theorem 2.2. So, we have that the map E(·) is surjective. Hence we can find u ∈W 1,p 0 (Ω)\{0} such that E(u) = 1 in W−1,p′ (Ω) =W 1,p 0 (Ω) ∗ . With −u− ∈W 1,p 0 (Ω), we obtain ĉ∥Du−∥pp ≤ 0 =⇒ u ≥ 0, u ̸= 0. So u ∈W 1,p 0 (Ω) is a solution of (3.8) and as before the nonlinear regularity theory and the nonlinear maximum principle (see [9], [15]), imply that u ∈ intC+. We show that there is a smallest positive solution of (3.8). Let S+ be the set of positive solutions of (3.8). We have just seen that ∅ ≠ S+ ⊆ intC+. The set S+ is downward directed (that is, if u1, u2 ∈ S+, then we can find u ∈ S+ such that u ≤ u1, u ≤ u2; see [13, Proposition 7]). So, by Theorem 5.109, p.308, of Hu-Papageorgiou [7], we can find a decreasing sequence {un}n∈N ⊆ S+ such that inf S+ = inf n∈N un. We have ⟨V (un), h⟩ = ∫ Ω [f(z, un) + 1]h dz for all h ∈W 1,p 0 (Ω) and all n ∈ N, (3.11) 0 ≤ un ≤ u1. (3.12) If in (3.11), we choose h = un ∈ W 1,p 0 (Ω), then using (H0), (H1)(i) and (3.12), we obtain ĉ∥un∥p ≤ c5 for some c5 > 0, all n ∈ N, =⇒ un}n∈N ⊆W 1,p 0 (Ω) is bounded. 8 Z. JING, Z. LIU, N. PAPAGEORGIOU EJDE-2025/06 As before, the nonlinear regularity theory of Lieberman [9] implies that we can assume that un → u in C1 0 (Ω) as n→ ∞. If u = 0, then we can find n0 ∈ N such that 0 ≤ un(z) ≤ δ for all z ∈ Ω and all n ≥ n0, =⇒ cun(z) τ−1 ≤ f(z, ūn(z)) for a.a. x ∈ Ω and all n ≥ n0. (3.13) We introduce the Carathéodory function kn(z, x) defined by kn(z, x) = { c(x+)τ−1 if x ≤ un(z), n ≥ n0. cun(z) τ−1 if un(z) < x, n ≥ n0. (3.14) Now we consider the Dirichlet problem −∆p α1 u(z)−∆q α2 u(z) = kn(z, u(z)) in Ω u|∂Ω = 0, u > 0. (3.15) As before, using the Weierstrass-Tonelli theorem and since τ < q < p, we can find ũ1 ∈ intC+ solution of (3.15) and 0 ≤ ũ1 ≤ un (see (3.14), (3.13)). Hence ũ1 = u1 (see Proposition 3.1) =⇒ u1 ≤ un for all n ≥ n0, which contradicts our hypothesis that u = 0. So, u ̸= 0 and then u ∈ S+ ⊂ intC+, u = inf S+. □ 4. Positive solutions Let u ∈ intC+ be the minimal positive solution of (3.8) produced in Proposition 3.2. We can find λ0 > 0 such that λ|Du(z)| ≤ 1 for all z ∈ Ω, all 0 < λ ≤ λ0. We have −∆α1 p u−∆α2 q u = f(z, u) + 1 ≥ f(z, u) + λ|Du|p−1 (4.1) in Ω for all 0 < λ ≤ λ0. Also, using Proposition 3.1 and the fact that u ∈ intC+, we can find µ ∈ (0, 1] small such that 0 ≤ uµ ≤ min{δ, u(z)} for all z ∈ Ω. (4.2) So, we have −∆α1 p uµ −∆α2 1 uµ = µcuτ−1 µ ≤ cuτ−1 µ (since 0 < µ ≤ 1) ≤ f(z, uµ) + λ|Duµ| in Ω for all λ > 0 (4.3) (see (4.2)). Using that uµ ≤ u (see (4.2)), we introduce the truncation map τ0 : Lp(Ω) → Lp(Ω) defined by τ0(u)(z) =  uµ(z) if u(z) < uµ(z), u(z), if uµ(z) ≤ u(z) ≤ u(z), u(z) if u(z) < u(z). (4.4) Clearly τ0(·) is bounded and continuous. In fact we can say more. Note that if u ∈ W 1,p 0 (Ω), then τ0(u) ∈ W 1,p 0 (Ω) (see Papageorgiou- Rădulescu-Repovš [14, Proposition 1.4.5, p 23]). So, we can consider the map τ0 :W 1,p 0 (Ω) →W 1,p 0 (Ω). Proposition 4.1. τ0 :W 1,p 0 (Ω) →W 1,p 0 (Ω) is bounded and continuous. EJDE-2025/06 WEIGHTED (p, q)-EQUATIONS 9 Proof. We know that Dτ0(u)(z) =  Duµ(z) if u(z) < uµ(z), Du(z), if uµ(z) ≤ u(z) ≤ u(z) for all u ∈W 1,p 0 (Ω) Du(z) if u(z) < u(z). (4.5) (see [14, p.23]). From (4.5) it is clear that τ0 : W 1,p 0 (Ω) → W 1,p 0 (Ω) is bounded. We show that τ0(·) is also continuous. To this end let {un}n∈N such taht un → u in W 1,p 0 (Ω) as n→ ∞. So, we can assume that un(z) → u(z) in R, Dun(z) → Du(z) in RN for a.a. z ∈ Ω, |un(z)|, |Dun(z)| ≤ ĥ(z) for a.a. z ∈ Ω, all n ∈ N, with ĥ ∈ Lp(Ω). (4.6) Then from (4.6) and (3.13), (3.14) it follows that τ0(un)(z) → τ0(u)(z) in R, Dτ0(un)(z) → Dτ0(u)(z) in RN (4.7) for a.a. z ∈ Ω as n→ ∞. Moreover, from Gasinski-Papageorgiou[6, Problem 1.4, p.35]) we have that {|un − u|p}n∈N, {|D(un − u)|p}n∈N ⊆ L1(Ω), (4.8) are both unifirmly integrable. Then (4.7), (4.8) and Vitali’s theorem (see [7, The- orem 2.147, p.91]), imply that τ0(un) → τ0(u) in L p(Ω), Dτ0(un) → Dτ0(u) in L p(Ω,RN ) =⇒ τ0(un) → τ0(u) in W 1,p 0 (Ω). So, we have proved the continuity of τ0 :W 1,p 0 (Ω) →W 1,p 0 (Ω). □ The above result is due to Deuel-Hess[2, Lemma on p.53]. We have reproduced the proof, since in the proof of [2] there is a small gap, since they state that |τ0(un)(z)| ≤ |un(z)| for a.a. z ∈ Ω (see [2, p.54]), which is not true in general. For λ > 0, we introduce the Caratheodory function f̂λ(z, x, y) defined by f̂λ(z, x, y) = f(z, x) + λ|y|p−1 for all x ∈ Ω, all x ∈ R, and all y ∈ RN . Also let τ̂0 :W 1,p 0 (Ω) → Lp(Ω)× Lp(Ω,RN ) be defined by τ̂0(u) = (τ0(u), Dτ0(u)) for all u ∈W 1,p 0 (Ω). By Proposition 3.2, τ̂0(·) is bounded and continuous. Then we consider Kλ : W 1,p 0 (Ω) → Lp′ (Ω) ↪→W−1,p′ (Ω)∗ defined by Kλ(u) = (Nf̂λ ◦ τ̂0)(u) for all u ∈W 1,p 0 (Ω). Then Kλ(·) is bounded and continuous. Following Deuel-Hess [2], we introduce also the Caratheodory function b(z, x) =  −(uµ(z)− x)p−1 if x < uµ(z) 0, if uµ(z) ≤ x ≤ u(z) (x− u(z))p−1 if u(z) < x. (4.9) Let Nb : L p(Ω) → Lp′ (Ω) be the Nemytski map corresponding to b(z, x), that is, Nb(u)(·) = b(·, u(·)) for all u ∈ Lp(Ω). 10 Z. JING, Z. LIU, N. PAPAGEORGIOU EJDE-2025/06 Evidently Nb(·) is bounded and continuous. In what follows by (·, ·)pp′ , we denote the duality brackets for the dual pair (Lp′ (Ω), Lp(Ω)). We state the next proposition in a more general setting than the one we have here and so the estimate can be used in other more general situations. So, for the purposes of the next proposition, uµ and u are general functions in W 1,p 0 (Ω)∩L∞(Ω) not necessarily positive (as the case here), which satisfy uµ ≤ u. Proposition 4.2. (Nb(u), u)pp′ = ∫ Ω b(z, u)u dz ≥ c6∥u∥pp − c7 for some c6, c7 > 0 all u ∈W 1,p 0 (Ω). Proof. We have (Nb(u), u)pp′ = ∫ {u>u} (u− u)p−1u dz − ∫ {u u} we have |u|p−1 = |(u− u) + u|p−1 ≤ [(u− u) + |u|]p−1 ≤ ĉ1[(u− u)p−1 + |u|p−1] for some ĉ1 > 0. This implies 1 ĉ1 |u|p−1 − |u|p−1 ≤ (u− u)p−1. If u(z) > 0, then on {u > u} we have (u− u)p−1u ≥ 1 ĉ1 |u|p − |u|p−1|u| (since |u(z)| = u(z) > 0). (4.10) If u(z) < 0, then on {u > u} we have u < u < 0 ⇒ |u| < |u|. It follows that( |u| |u| )p−1 − ( |u| |u| − 1 )p−1 ≥ ĉ2 > 0, =⇒ |u|p−1 |u|p−1 − (u− u)p−1 |u|p−1 ≥ ĉ2 > 0 (since (|u| − |u|)p−1 = (u− u)p−1 on {u < u < 0}) =⇒ |u|p−1 − (u− u)p−1 ≥ ĉ2|u|p−1 =⇒ ĉ2|u|p − |u|p−1|u| ≤ (u− u)p−1u on {u < u < 0}. (4.11) From (4.10) and (4.11), we see that∫ {u>u} (u− u)p−1u dz ≥ ∫ {u>u} [ĉ3|u|p − |u|p−1|u|]dz for some ĉ3 > 0. (4.12) Next we estimate the set {u < uµ}. If u(z) < 0, then on {u < uµ} we have −(uµ − u)p−1u = (uµ − u)p−1|u| = |uµ − u|p−1|u| ≥ ĉ4|u|p − ĉ5|uµ|p−1|u| for some ĉ4, ĉ5 > 0. (4.13) If u(z) > 0, then uµ > u > 0 on {u < uµ} and so −(uµ − u)p−1u ≥ up − ĉ7u p−1 µ u with ĉ7 > 0. (4.14) EJDE-2025/06 WEIGHTED (p, q)-EQUATIONS 11 From (4.13) and (4.14), it follows that∫ {u 0. Using (4.12) and (4.15), we see that∫ Ω b(z, u) dz ≥ ∫ {u>u} [ĉ3|u|p − |u|p−1|u|]dz + ∫ {u 0 ≥ ĉ10∥u∥pp − ĉ12∥u∥ − ĉ13 for some ĉ12, ĉ13 > 0. (4.16) Using Young’s inequality with ε ∈ (0, ρĉ10), from (4.16) we obtain (Nb(u), u)pp′ ≥ c6∥u∥pρ − c7 for some c6, c7 > 0. □ We introduce the map Gλ :W 1,p 0 (Ω) →W−1,p′ (Ω)∗ defined by Gλ(u) = V (u) + θNb(u) − Kλ(u) for all u ∈ W 1,p 0 (Ω), all 0 < λ ≤ λ0, θ > 0. Evidently Gλ(·) is bounded and continuous. Proposition 4.3. If (H0), (H1), 0 < λ ≤ λ0 and θ > 0 hold, then Gλ(·) is pseudomonotone. Proof. We consider a sequence {un}n∈N ⊆W 1,p 0 (Ω) such that un w→ u in W 1,p 0 (Ω), Gλ(un) w→ u∗ in W−1,p′ (Ω) =W 1,p 0 (Ω)∗ lim sup n→∞ ⟨Gλ(un), un − u⟩ ≤ 0. (4.17) From this and the fact thatW 1,p 0 (Ω) ↪→ Lp(Ω) compactly, we have un → u in Lp(Ω) as n→ ∞. Then using Hölder’s inequality, we obtain∫ Ω b(z, un)(un − u)dz → 0 as n→ ∞,∫ Ω f̂λ(z, τ0(un), Dτ0(un))(un − u)dz → 0 as n→ ∞. From (4.17) it follows that lim sup n→∞ ⟨V (un), un − u⟩ ≤ 0 =⇒ un → u in W 1,p 0 (Ω) (since V (·) is an (S)+-map). Therefore, u∗ = Gλ(u) and ⟨Gλ(un), un⟩ → ⟨Gλ(u), u⟩. This proves that Gλ(·) is a pseudomonotone map. □ Proposition 4.4. If (H0), (H1) hold and 0 < λ ≤ λ0, then for θ > 0 large the map Gλ(·) is strongly coercive. 12 Z. JING, Z. LIU, N. PAPAGEORGIOU EJDE-2025/06 Proof. For every u ∈W 1,p 0 (Ω), we have |⟨Kλ(u), u⟩| = |(Kλ(u), u)pp′ | = ∣∣ ∫ Ω [f(z, τ0(u)) + λ|Dτ0(u)|p−1]u dz ∣∣ ≤ ∫ Ω [f(z, τ0(u)) + λ0|Dτ0(u)|p−1]|u| dz ≤ c8[∥u∥p + λ0∥Dτ0(u)∥p−1 p ∥u∥p] for some c8 > 0 (use hölder’s inequality) ≤ c9[∥u∥+ λ0(∥u∥p−1 + 1)∥u∥p] for some c9 > 0 (recall that W 1,p 0 (Ω) ↪→ Lp(Ω) continuously and see (4.5)) ≤ c10[∥u∥+ λ0∥u∥p−1∥u∥p] for some c10 > 0 ≤ c11[∥u∥+ λ0(ε∥u∥p + 1 ε ∥u∥pp)] for some c11 > 0 (use Young’s inequality with ε > 0). (4.18) From Proposition 4.2 we have θ⟨Nb(u), u⟩ = θ(Nb(u), u)pp′ ≥ θc6∥u∥pp − θc7 for all u ∈W 1,p 0 (Ω). (4.19) Using (4.18), (4.19), and hypotheses(H0), we have ⟨Gλ(u), u⟩ ≥ [ĉ− λ0c11ε]∥u∥ρ + [θc6 − λ0c11 ε ]∥u∥pp − θc7. (4.20) First we choose ε ∈ (0, ĉ λ0c11 ). So, ĉ − λc11ε > 0. Then using this choice of ε > 0, we choose θ > λ0c11 εc6 . From (4.20), we see that ⟨Gλ(u), u⟩ ≥ c12∥u∥ρ − θc7 for some c12 > 0 and all u ∈W 1,p 0 (Ω) implies that Gλ(·) is strongly coercive. □ We can now state and prove the existence theorem for problem (1.1) Theorem 4.5. If (H0), (H1) hold and 0 < λ ≤ λ0, then problem (1.1) has a positive solution ûλ ∈ intC+. Proof. Proposition 4.3 and 4.4 permit the use of Theorem 2.2. So Gλ(·) is surjective and we can find ûλ ∈W 1,p 0 (Ω) such that Gλ(ûλ) = 0 in W−1,p′ (Ω)∗ =W 1,p 0 (Ω)∗ =⇒ ⟨V (ûλ), h⟩+ θ ∫ Ω b(z, ûλ)hdz = ∫ Ω [f(z, τ0(u)) + λ|Dτ0(u)|p−1]hdz for all h ∈W 1,p 0 (Ω). (4.21) In (4.21) first we used the test function h = (ûλ − u)+ ∈W 1,p 0 (Ω). Then ⟨V (ûλ), (ûλ − u)+⟩+ θ ∫ Ω (ûλ − u)p−1(ûλ − u)+ dz = ∫ Ω [f(z, u) + λ|Du|p−1](ûλ − u)+dz (see (4.9), (4.4), (4.5)) ≤ ⟨V (u), (ûλ − u)+⟩ (see (4.1)) EJDE-2025/06 WEIGHTED (p, q)-EQUATIONS 13 which implies ⟨V (ûλ)− V (u), (ûλ − u)+⟩ ≤ −θ ∫ Ω (ûλ − u)p−1(ûλ − u)+ dz ≤ 0, which in turn implies ûλ ≤ u. Next, we test (4.21) with h = (uµ − ûλ) + ∈W 1,p 0 (Ω). We have ⟨V (ûλ), (uµ − u)+⟩ − θ ∫ Ω (uµ − u)p−1(uµ − ûλ) + dz = ∫ Ω [f(z, uµ) + λ|Duµ|p−1](uµ − ûλ) +dz (see (4.9), (4.4), (4.5) ) ≥ ⟨V (uµ), (uµ − ûλ) +⟩ (see (4.3)) which implies ⟨V (V (uµ)− ûλ), (uµ − ûλ) +⟩ ≤ −θ ∫ Ω (uµ)− ûλ) p−1(uµ)− ûλ) + dz ≤ 0, which in turn implies uµ ≤ ûλ. So, we have proved that ûλ ∈ [uµ, u]. (4.22) From (4.22), (4.9), (4.4), (4.5), and (4.21), it follows that −∆p α1 ûλ −∆1 α2 ûλ = f(z, ûλ) + λ|Dûλ|p−1 in Ω implies ûλ ∈ inf C+ (see [9], (4.22) and recall that uµ ∈ intC+). □ Acknowledgments. Z. Jing was supported by the Guangxi First-class Discipline Statistics Construction Project Fund. Z. Liu was supported by the NNSF of China Grant No. 12071413, and by the European Union’s Horizon 2020 Research and Inno- vation Programme under the Marie Sklodowska-Curie grant agreement No. 823731 CONMECH. N. Papageorgiou was supported by the grand “Nonlinear Differential Systems in Applied Sciences” of the Romanian Ministry of Research, Innovation and Digitization within PNRR-III-C9-2022-I8 (Grant No. 22). References [1] P. Candito, L. Gasiński, N. S. Papageorgiou; Nonlinear nonhomogeneous Robin problems with convection, Ann. Acad. Sci. Fenn. Math., 44 (2019), 755-767. [2] J. Deuel, P. Hess; A criterion for the existence of solutions of nonlinear elliptic boundary value problems, Proc. Royal Soc. Edinburgh 74A (1974/75),49-54. [3] J. J. Diaz, J. E. Saa; Existence of unicité de solutions positives pour certaines equations elliptiques quasineaires, CRAS Paris, L305 (1967),521-524. [4] L. F. O. Faria, O. H. Miyagaki, D. Motreanu; Comparison and positive solutions for problems with the (p, q)-Laplacian and a convection term, Proc. Edinburgh Math. Soc. 57 (2014),667- 698. [5] L. Gasiński, N. S. Papageorgiou; Nonlinear Analysis, Series in Mathematical Analysis and Applications, 9. Chapman & Hall/CRC, Boca Raton, FL, 2006. [6] L. Gasiński, N. S. Papageorgiou; Exercises in Analysis. Part 2, Nonlinear Analysis. Problem Books in Mathematics. Springer, Cham, 2016. viii+1062 pp. [7] S. Hu, N. S. Papageorgiou; Research Topics in Analysis. Volume I: Grounding Theory, Birkhäuser, Cham, 2022. [8] O. A. Ladyzhenskaya, N. N. Uraltseva; Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis Academic Press, New York-London 1968 xviii+495 pp. [9] G. Lieberman; The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations, Comm. Partial Differential Equations, 16 (1991), 311-361. 14 Z. JING, Z. LIU, N. PAPAGEORGIOU EJDE-2025/06 [10] Z. H. Liu, N. S. Papageorgiou; A double phase equation with convection, Electr. Jour. Qual. Theory Diff. Equ., 2021:91 (2021). [11] Z. H. Liu, N. S. Papageorgiou; A weighted (p, 2)-equations with double resonance, Electr. Jour. Diff. Equ.,2023:30(2023). [12] N. S. Papageorgiou, V. D. Rădulescu; Coercive and noncoercive nonlinear Neumann problems with indefinite potential, Forum Math., 28(2016),545-571. [13] N. S. Papageorgiou, V. D. Rădulescu, D. Repovš; Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential, Discr. Cont. Dyn. Systems -A, 37(2017),2589-2618. [14] N. S. Papageorgiou, V. D. Rădulescu, D. Repovš; Nonlinear Analysis-Theory and Methods, Springer Nature, Swizerland AG, 2019. [15] P. Pucci, J. Serrin; The Maximum Principle, Birkhäuser, Basel, 2007. [16] M. Tanaka; Existence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient, Boundary Value Problems, 2013:173 (2013) [17] S. Zeng, N. S. Papageorgiou; Positive solutions for (p, q)-equations with convection and sign- changing reaction, Adv. Nonlin. Anal., 11 (2022), 40-57. Zhao Jing School of Mathematics and Quantitative economics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, China Email address: jingzhao100@126.com Zhenhai Liu Guangxi Colleges and Universities Key Laboratory of Optimization Control and Engi- neering Calculation, Guangxi Minzu University, Nanning, Guangxi, 530006, China Center for Applied Mathematics of Guangxi, Yulin Normal University, Yulin 537000, China Email address: zhhliu@hotmail.com Nikolaos S. Papageorgiou Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece. Department of Mathematics, University of Craiova, 200585 Craiova, Romania Email address: npapg@math.ntua.gr 1. Introduction 2. Mathematical background, hypotheses 3. Auxiliary problems 4. Positive solutions Acknowledgments References