Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 41, pp. 1–24. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.40 POSITIVE SOLUTION FOR NONLINEAR ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS LUIZ F. O. FARIA, MARCELO MONTENEGRO Abstract. We show the existence of a positive solution for the Schrödinger quasilinear equation with variable exponents above the critical regime. For that matter, we show an embedding into an Orlicz space of functions modeled over radially symmetric domains. Then we use a Galerkin method combined with a fixed-point argument to obtain a solution. 1. Introduction Our aim is to find positive radially symmetric solutions u(r), r = |x|, for the equation −∆pu+ up−1 = λa(r)uq(r)−1 + uθ(r)−1 in D, (1.1) where the exponents θ(r) and q(r) are functions satisfying 1 < q(r) < p < N, (1.2) θ(r) = pN N − p + h(r), (1.3) λ > 0 is a parameter and a(r), h(r), θ(r), q(r) are positive radially symmetric con- tinuous functions, and D ⊆ RN , N ≥ 2, is an open symmetric set centered at the origin (that is, if x ∈ D, then |x| ∈ D). The domain D may be a ball, an annu- lus, RN or RN minus a ball. Here 1 < p < N and ∆p = div(|∇u|p−2∇u) is the so-called p-Laplacian. Equation (1.1) is sometimes called quasilinear Schrödinger equation. Including the case p = 2, it was studied in many papers see for instance [1, 2, 4, 7, 8, 9, 10, 11, 12, 13, 17, 18, 19, 21, 23, 24, 28, 31, 33, 42, 45, 44]. Because θ(r) > pN/(N − p) we say that (1.1) is in the supercritical range in the sense of Sobolev embedding, see Remark 1.5 below. Throughout this article h is a function with the following properties: h : [0,∞) → [0,∞) is continuous, and h(0) = 0; (1.4) there are constants β > 2 and c > 0 such that h(r) ≤ c| log r|−β for r near 0; (1.5) there is a constant c > 0 such that h(r) ≤ c |1− r| for r close to 1. (1.6) 2020 Mathematics Subject Classification. 65N30, 35B09, 35J60, 46E30. Key words and phrases. Positive solution; nonlinear elliptic equation; variable exponents. ©2024. This work is licensed under a CC BY 4.0 license. Submitted March 14, 2024. Published July 30, 2024. 1 2 L. F. O. FARIA, M. MONTENEGRO EJDE-2024/41 We denote by W 1,p r (D) the closed subspace of W 1,p(D) composed by radially sym- metric functions on D, i.e., W 1,p r (D) = {u ∈W 1,p(D) : u = u(r), r = |x|}, (1.7) endowed with the standard norm ∥u∥W 1,p(D) = (∫ D (|∇u|p + |u|p)dx )1/p . (1.8) Theorem 1.1. Let θ(r) = p∗ + h(r), p∗ = pN/(N − p), and h satisfy (1.4)–(1.6). Then sup {∫ RN |u(x)|θ(r)dx : u ∈W 1,p r (RN ), ∥u∥W 1,p(RN ) = 1 } <∞. (1.9) By the conditions on h(r) and the decay of u(r), the integrals are well defined and nonsingular. We denote byW 1,p 0,r (D) the closed subspace ofW 1,p 0 (D) composed by radially symmetric functions on D, i.e., W 1,p 0,r (D) = {u ∈W 1,p 0 (D) : u = u(r), r = |x|}, (1.10) endowed with the norm (1.8). The continuity of the embedding into the Orlicz space reads as follows, see [20, 39] or Section 2 below. Corollary 1.2. Let θ(r) = p∗+h(r) with h ∈ L∞ + (RN ) satisfying (1.4)–(1.6). Then the following embedding is continuous W 1,p r (RN ) ↪→ Lθ(r)(RN ). (1.11) In what follows, we consider an equation more general than (1.1). Let f : RN ×R → R be a continuous function, radially symmetric in the first variable and satisfying the growth condition 0 ≤ f(r, t)t ≤ b1t θ(r) for every r ∈ R and t ≥ 0, (1.12) where b1 > 0 is a constant, θ(r) = p∗+h(r), with h ∈ L∞ + (RN ) satisfying (1.4)–(1.5) (see also (2.3) below), and p∗ = pN/(N − p). Wyhen D is a bounded domain, we deal with the problem −∆pu+ up−1 = λa(r)uq(r)−1 + f(r, u) in D u > 0 in D u = 0 on ∂D. (1.13) Note that D encompasses balls and an annulus centered at the origin, in which we have the following result. Theorem 1.3. If λ > 0 is a constant, q and a are radially symmetric continu- ous functions, such that 1 < q− ≤ q(r) ≤ q+ < p, q−, q+ ∈ R, a ∈ L p p−q(r) (RN ), a(r) > 0 in RN , r = |x|, f : RN × R → R is a continuous function satisfying (1.12). Then there exists λ∗ > 0 such that for every λ ∈ (0, λ∗) problem (1.13) possesses at least one positive radially symmetric solution uλ ∈W 1,p 0 (D). Further- more, ∥uλ∥W 1,p 0 (D) → 0 as λ→ 0+. EJDE-2024/41 ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS 3 Let BR ⊆ RN be the open ball with radius R centered at the origin. (For R = 0 we have B0 = ∅.) We study the problem −∆pu+ up−1 = λa(r)uq(r)−1 + f(r, u) in RN\BR u > 0 in RN\BR u = 0 on ∂(RN\BR). (1.14) Note that RN\BR admits the whole space RN and exterior domains such as RN\BR. The next result applies to exterior domains RN\BR and the whole space RN . Theorem 1.4. If λ > 0 is a constant, q and a are radially symmetric continuous functions, such that 1 < q− ≤ q(r) ≤ q+ < p, q−, q+ ∈ R, a ∈ L p p−q(r) (RN ), a(r) > 0 in RN , r = |x|, and f : RN × R → R is a continuous function satisfying (1.12). Then there exists λ∗ > 0 such that for every λ ∈ (0, λ∗) problem (1.14) possesses at least one positive radially symmetric solution uλ ∈ W 1,p 0 (RN\BR). Furthermore, ∥uλ∥W 1,p 0 (RN\BR) → 0 as λ→ 0+. Remark 1.5. The Sobolev embedding with θ(r) ≤ pN/(N − p) can be found in [15]. In such cases we say that f in (1.12) or equations (1.13)–(1.14) are subcritical if θ(r) < pN/(N − p) and critical when θ(r) = pN/(N − p). The equations studied in this article have the supercritical term f related to θ(r) > pN/(N − p). About this matter we prove the boundedness of the Sobolev constant in Theorem 1.1 and the embedding stated as Corollary 1.2. Equation (1.13) and Theorem 1.3 are associated with the existence of a positive solution on domains D which can be balls or an annulus. Other equations in the supercritical range were treated in [7, 23] when D is a ball. Problems on D being an annulus and f with critical growth are studied in [11, 12], non-radial solutions were studied in [16], and uniqueness questions in [17, 18, 42, 45]. With respect to (1.14), when D is an exterior domain RN\BR or the whole RN we prove Theorem 1.4. The equation with subcritical nonlinearity was treated in the seminal article [13] on a domain like RN\BR. The critical case was addressed in [2] and on exterior domains without symmetry in [33]. The p-Laplacian equation on an exterior domain was studied in [21] by ODE methods, and with a supercritical f in [28] by means of variational methods. Equations with Neumann condition on the inner boundary ∂BR were investigated in [1, 24, 31, 44] and symmetry issues were analyzed in [36]. The problem in RN was studied in [8] for a p-Laplacian equation with critical f , see also [3, 14]. The supercritical case was treated in [10, 19, 9]. The equations considered in this article also correspond to the concave-convex problems of [4]. The outline of this article is as follows. Section 2 presents some preliminary re- sults about extension of Sobolev functions, Brouwer theorem, comparison principles and function spaces with variable exponents. Sometimes in the paper we pass from the domain D to the whole RN in order to stress that the estimates are independent of D and since we are keeping in mind the use of Corollary 1.2. In Section 3 we prove of Theorem 1.1 and Corollary 1.2 by means of integral estimates which are nonsingular and well defined because of the conditions on h(r) and the polynomial decay of u(r). In Section 4 we present the Lipschitz approximate functions of f that will be useful for solving the approximated equation in Section 5 with the aid of a Galerkin type scheme. In doing so we use a Schauder basis. In the limit the solution of those approximate equations tend to a solution of the original equation 4 L. F. O. FARIA, M. MONTENEGRO EJDE-2024/41 (1.13) on a ball. We apply the Palais principle to show that the solution is radi- ally symmetric. We prove Theorems 1.3 and 1.4 in Sections 6 and 7, respectively. Obtaining a positive solution in RN is done by means of a diagonal argument that allows us to let R approach ∞. 2. Preliminaries From now on, when a function defined in D is radially symmetric, for conve- nience, we will use the same notation to represent the function on x or r = |x|. 2.1. Extension. Let u ∈W 1,p 0 (D). In what follows, we denote by ũ the canonical extension of u by 0 outside D, that is, ũ(x) = { u(x) if x ∈ D, 0 if x ∈ RN \D. (2.1) It is well known that u ∈W 1,p 0 (D) implies ũ ∈W 1,p(RN ) (see e.g. [15, Proposition 9.18]). 2.2. Brouwer Theorem. The following lemma is a generalization of the classical Brouwer fixed point theorem found in Kesavan [32], the proof is performed in [6]. We denote ⟨·, ·⟩1/2 the Euclidean inner product with its induced norm and | · |m is any other norm. Lemma 2.1. Let F : (Rm, | · |m) → (Rm, | · |m) be a continuous function such that ⟨F (ξ), ξ⟩ ≥ 0 for every ξ ∈ Rm with |ξ|m = ϑ for some ϑ > 0, and | · |m is any norm. Then, there exists z0 in the closed ball B m ϑ (0) := {z ∈ Rm; |z|m ≤ ϑ} such that F (z0) = 0. 2.3. Comparison principle. Assume that D is a bounded domain in RN with C2 boundary ∂D. Next we present a subtle adaptations of the results achieved in [26, Theorems 3 and 5] for a non-autonomous function. We present a couple of comparison principles for a subsolution and for a super- solution of the problem −∆pu+ |u|p−2u = g(x, u) in D u = 0 on ∂D, (2.2) where g : R → R is a continuous function. We say that u1 ∈W 1,p(D) is a subsolution of (2.2) if u1 ≤ 0 on ∂D and∫ D (|∇u1|p−2∇u1∇φ+ |u1|p−2u1φ)dx ≤ ∫ D g(x, u1)φdx for all φ ∈ W 1,p 0 (D) with φ ≥ 0 in D provided the integral ∫ D g(x, u1)φdx exists. We say that u2 ∈ W 1,p(D) is a supersolution of (2.2) if the reversed inequalities are satisfied with u2 in place of u1 for all φ ∈W 1,p 0 (D) with φ ≥ 0 in D. The next comparison results are particular cases of the ones achieved in [27, Theorems 3 and 5]. Proposition 2.2. Let g : D×R → R be a continuous function such that g(x, t)/tp−1 is decreasing for t > 0. Assume that u1 and u2 are a positive subsolution and a positive supersolution of problem (2.2), respectively. If u2(x) > u1(x) = 0 for all x ∈ ∂D, ui ∈ C1,α(D) with some α ∈ (0, 1), ∆pui ∈ L∞(D), for i, j = 1, 2, then u2 ≥ u1 in D. EJDE-2024/41 ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS 5 Whenever u1 and u2 satisfy the homogeneous Dirichlet boundary condition we can state the following result. Proposition 2.3. Let g : R → R be a continuous function such that g(x, t)/tp−1 is decreasing for t > 0. Assume that u1, u2 ∈ C1,α 0 (D), with some α ∈ (0, 1), are a positive subsolution and a positive supersolution of problem (2.2), respectively. If ∆pui ∈ L∞(D), for i, j = 1, 2, u1/u2 ∈ L∞(D) and u2/u1 ∈ L∞(D), then u2 ≥ u1 in D. 2.4. Function spaces with variable exponents. In the sequel we display some results of the Lebesgue spaces with variable exponents (see [20, 25, 39] for more details). Let D be an open domain in RN , 1 < p < N and p∗ = pN/(N−p). Define L∞ + (D) = {y : y ∈ L∞(D), inf x∈D y(x) > 1}. (2.3) For each y ∈ L∞ + (D), we define y− = y−(D) = inf x∈D y(x), y+ = y+(D) = sup x∈D y(x). For y ∈ L∞ + (D), the space Ly(x) = { u : is real measurable, ∫ D |u(x)|y(x)dx ≤ ∞ } (2.4) is equipped with a Banach norm ∥u∥y(x) = inf { σ > 0 : ∫ D |u(x) σ |y(x)dx ≤ 1 } . And Ly(x) is called Orlicz space (see also Musielak-Orlicz spaces in [20, 39]) and ∥ · ∥y(x) is the Luxemburg norm. Proposition 2.4. Let u ∈ Ly(x)(D) and ∥u∥y(x) = λ. If λ ≥ 1, then λy− ≤ ∫ D |u(x)|y(x)dx ≤ λy+ . If λ ≤ 1, then λy+ ≤ ∫ D |u(x)|y(x)dx ≤ λy− . Proposition 2.5. The conjugate space of Ly(x)(D) is Lyo(x)(D), where 1/y(x) + 1/yo(x) = 1. Furthermore, for u ∈ Ly(x)(D), v ∈ Lyo(x)(D), we have the inequality∣∣ ∫ D u(x)v(x)dx ∣∣ ≤ 2∥u∥y(x)∥v∥yo(x). Proposition 2.6. Let D be an open bounded domain in RN with the cone property. If y ∈ L∞ + (D) satisfies y(x) ≤ y+ < p∗ for all x ∈ D, then the following embedding is compact W 1,p(D) ↪→ Ly(x)(D). 6 L. F. O. FARIA, M. MONTENEGRO EJDE-2024/41 3. Proof of Theorem 1.1 Let u : RN → R be a radial function and v : [0,∞) → R such that u(x) = v(r) for all x ∈ RN . For convenience, we will keep the same notation u for both cases, i.e. u(x) = v(r) = u(r). It is well-known that (see e.g. [29])∫ RN u(x)dx = ωN ∫ ∞ 0 u(r)rN−1dr, where ωN = 2πN/2 Γ(N/2) . Thus, if u ∈W 1,p 0,r (RN ) we can write ∥u∥W 1,p(RN ) = ( ωN ∫ ∞ 0 (|Du′|p + |u|p)rN−1dr )1/2 . (3.1) Let C∞ 0 (RN ) be the space of infinitely differentiable functions with compact sup- port. We denote by C∞ 0,r(RN ) the subspace of C∞ 0 (RN ) of radially symmetric functions. Lemma 3.1. Let u ∈W 1,p 0,r (RN ). Then |u(r)| ≤ Cmin { 1 r(N−p)/p , 1 r(N−1)/p } ∥u∥W 1,p(RN ). (3.2) Proof. If ϕ ∈ C∞ 0,r(RN ), then |ϕ(r)| ≤ ∣∣ ∫ ∞ r ϕ′(s)ds ∣∣ ≤ ∫ ∞ r ∣∣ϕ′(s)s(N−1)/p 1 s(N−1)/p ds ∣∣ ≤ (∫ ∞ r |ϕ′(s)|psN−1ds )1/p(∫ ∞ r 1 s N−1 p−1 ds ) p−1 p ≤ (∫ ∞ r (|ϕ′(s)|p + |ϕ(s)|p)sN−1ds )1/p(∫ ∞ r 1 s N−1 p−1 ds ) p−1 p ≤ 1 ω 1/p N ( p− 1 N − p ) p−1 p 1 r(N−p)/p ∥ϕ∥W 1,p(RN ). (3.3) On the other hand, (ϕ(r))p = −p ∫ ∞ r ϕ′(s)ϕ(s)p−1ds ≤ p ∫ ∞ r |ϕ′(s)||ϕ(s)|p−1sN−1 1 sN−1 ds ≤ p rN−1 ∫ ∞ r ( |ϕ′(s)|p p + |ϕ(s)|p p/(p− 1) )sN−1ds ≤ 1 ωN max{1, p− 1} 1 rN−1 ∥ϕ∥p W 1,p(RN ) . Thus, we arrive at |ϕ(r)| ≤ C 1 r(N−1)/p ∥ϕ∥W 1,p(RN ). (3.4) Since C∞ 0,r(RN ) is dense in W 1,p 0,r (RN ), we obtain (3.2). □ EJDE-2024/41 ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS 7 Proof of Theorem 1.1. Let u ∈ W 1,p 0,r (RN ) with ∥u∥W 1,p(RN ) = 1. Let ρ ∈ (0, 1) to be chosen later on, we can write 1 wp ∫ RN |u(r)|p ∗+h(r)dx = ∫ ρ 0 |u(r)|p ∗+h(r)rN−1dr︸ ︷︷ ︸ I + ∫ 1 ρ |u(r)|p ∗+h(r)rN−1dr︸ ︷︷ ︸ J + ∫ ∞ 1 |u(r)|p ∗+h(r)rN−1dr︸ ︷︷ ︸ K . We estimate I, J,K in 3 steps. In the first two steps we use ideas similar to those in [23, Theorem 2.1]. Step 1. Estimate of I. I = ∫ ρ 0 |u(r)|p ∗+h(r)rN−1dr ≤ ∫ ρ 0 |u(r)|p ∗ (|u(r)|h(r) − 1)rN−1dr + ∫ 1 0 |u(r)|p ∗ rN−1dr ≤ ∫ ρ 0 |u(r)|p ∗ (|u(r)|h(r) − 1)rN−1dr + CN,p, where CN,p is the constant in the Sobolev constant embedding W 1,p(RN ) ↪→ Lp∗ (RN ). By Lemma (3.1), we have∫ ρ 0 |u(r)|p ∗ (|u(r)|h(r) − 1)rN−1dr ≤ C ∫ ρ 0 1 r(N−p)p∗/p ( 1 r(N−p)h(r)/p − 1 ) rN−1dr = C ∫ ρ 0 1 r ( exp[ −(N − p)h(r) p log r]− 1 ) dr ≤ Cd N − p p ∫ ρ 0 h(r) | log r| r dr <∞, whence the above inequality follows from (1.5). Step 2. Estimate of J . By (1.6) and Lemma (3.1), we obtain J = ∫ 1 ρ |u(r)|p ∗+h(r)rN−1dr ≤ C ∫ 1 ρ 1 r (N−p) p (p∗+h(r)) rN−1dr = C ∫ 1 ρ 1 r1+ N−p p h(r) dr = −C ∫ 0 1−ρ 1 (1− s)1+ N−p p h(1−s) ds ≤ C ∫ 1−ρ 0 1 (1− s)1+ N−p p c s ds = C ∫ 1−ρ 0 1 exp (1 + N−p p c s ) log(1− s) ds <∞. 8 L. F. O. FARIA, M. MONTENEGRO EJDE-2024/41 Step 3. Estimate of K. The third integral K can be bounded with the aid of (1.4) and Lemma 3.1. Indeed K = ∫ ∞ 1 |u(r)|p ∗+h(r)rN−1dr ≤ C ∫ ∞ 1 r− N−1 p (p∗+h(r))+(N−1)dr ≤ C ∫ ∞ 1 r− N−1 p p∗+(N−1)dr = C ∫ ∞ 1 r−p(N−1)/(N−p)dr <∞. (3.5) From the previous steps, we infer (1.9). The proof of Theorem 1.1 is complete. □ Proof of Corollary 1.2. Notice that θ(r) = p∗ + h(r) belongs to L∞ + (RN ) since θ(r) ≥ p∗ > 1. Define the space Lp∗+h(r) = { u : RN → R : real measurable, ∫ RN |u(r)|p ∗+h(r)dx ≤ ∞ } equipped with the norm ∥u∥p∗+h(r) = inf { σ > 0 : ∫ RN |u(r) σ |p ∗+h(r)dx ≤ 1 } . Taking that u ∈ W 1,p r (RN ) with ∥u∥W 1,p r (RN ) = 1. Theorem 1.1 yields that there exists C such that ∫ RN |u(r)|p ∗+h(r)dx ≤ C <∞, where C does not depend on u. By Proposition 2.4, we obtain ∥u∥Lp∗+h(r) ≤ C0 = max{Cp−, Cp+}. Thus, ∥u∥Lp∗+h(r) ≤ C0∥u∥W 1,p r (RN ), for every u ∈W 1,p r (RN ). □ 4. Approximate functions The continuous function f satisfying (1.12) can be approximated by Lipschitz functions fk : RN × R → R defined by fk(r, s) =  −k[G(r,−k − 1 k )−G(r,−k)], if s ≤ −k −k[G(r, s− 1 k )−G(r, s)], if − k ≤ s ≤ − 1 k k2s[G(r,− 2 k )−G(r,− 1 k )], if − 1 k ≤ s ≤ 0 k2s[G(r, 2k )−G(r, 1k )], if 0 ≤ s ≤ 1 k k[G(r, s+ 1 k )−G(r, s)], if 1 k ≤ s ≤ k k[G(r, k + 1 k )−G(r, k)], if s ≥ k, (4.1) where G(r, s) = ∫ s 0 f(r, ζ)dζ, Gs = f and G(r, 0)=0. The following approximation result was proved in [41] and it uses the explicit expression of the sequence (4.1). EJDE-2024/41 ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS 9 Lemma 4.1. Let f : RN × R → R be a continuous function such that sf(r, s) ≥ 0 for every s ∈ R. Then there exists a sequence fk : RN × R → R of continuous functions satisfying (i) sfk(r, s) ≥ 0 for every s ∈ R; (ii) for each k ∈ N there is a continuous function ck(r) such that |fk(r, ξ)− fk(r, η)| ≤ ck(r)|ξ − η| for every ξ, η ∈ R; (iii) fk converges uniformly to f in bounded sets. The following result gives the growth behavior to the sequence of functions ck(·), and the proof can be found in [7, Proposition 5]. Lemma 4.2. One can choose the sequence of Lipschitz constants ck(·), defined in Lemma 4.1, satisfying the following estimates ck(r) ≤ Ck sup t { |f(r, t)|; t ∈ [−k − 1 k , k + 1 k ] } , ∀r ∈ [0, R], (4.2) where the constant C does not depend on neither r nor k. A similar version of the next lemma was presented in [7, Lemma 2]. However, since here we can deal with unbounded domains, a slight different statement is needed and deserves a proof. Lemma 4.3. Let f : RN × R → R be a continuous function satisfying (1.12) for every s ∈ R. Then the sequence (fk) of Lemma 4.1 satisfies (i) for all k ∈ N, 0 ≤ sfk(r, s) ≤ K1|s|θ(r) for every |s| ≥ 1/k; (ii) for all k ∈ N, 0 ≤ sfk(r, s) ≤ K1 1 kp−−1 |s| for every |s| ≤ 1/k, where C1 is a positive constant independent of k. Proof. In this proof the constant b1 is the one of (1.12). According to the definition (2.3), p− = inf r≥0 θ(r) and p+ = sup r≥0 θ(r). Step 1. Suppose that −k ≤ s ≤ −1/k. By the mean value theorem, there exists η ∈ (s− 1 k , s) such that fk(r, s) = −k[G(r, s− 1 k )−G(r, s)] = −kGs(r, η)(s− 1 k − s) = f(r, η), sfk(r, s) = sf(r, η). Since s− 1 k < η < s < 0 and f(r, η) < 0, we have sf(r, η) ≤ ηf(r, η). Therefore, 0 ≤ sfk(r, s) ≤ ηf(r, η) ≤ b1|η|θ(r) ≤ b1|s− 1 k |θ(r) ≤ b1(|s|+ 1 k )θ(r) ≤ b1(2|s|)θ(r) ≤ b12 p+ |s|θ(r). 10 L. F. O. FARIA, M. MONTENEGRO EJDE-2024/41 Step 2. Assume 1 k ≤ s ≤ k. By the mean value theorem, there exists η ∈ (s, s+ 1 k ) such that fk(r, s) = k[G(r, s+ 1 k )−G(r, s)] = kGs(r, η)(s+ 1 k − s) = f(r, η), sfk(r, s) = sf(r, η). Since 0 < s < η < s+ 1 k and f(r, η) > 0, we have sf(r, η) ≤ ηf(r, η). Therefore, 0 ≤ sfk(r, s) ≤ ηf(r, η) ≤ b1|η|θ(r) ≤ b1|s+ 1 k |θ(r) ≤ b12 p+ |s|θ(r). Step 3. Suppose that |s| ≥ k, then fk(r, s) = { −k[G(r,−k − 1 k )−G(r,−k)], if s ≤ −k k[G(r, k + 1 k )−G(r, k)], if s ≥ k. (4.3) If s ≤ −k, again by the mean value theorem, there exists η ∈ (−k − 1 k ,−k) such that fk(r, s) = k[G(r,−k − 1 k )−G(r,−k)] = −kGs(r, η)(−k − 1 k − (−k)) = f(r, η), sfk(r, s) = sf(r, η). Since −k − 1 k < η < −k < 0 and k < |η| < k + 1 k , we conclude that 0 ≤ sfk(r, s) = s η ηf(r, η) ≤ |s| |η| b1|η|θ(r) = b1|s||η|θ(r)−1 ≤ b1|s|(k + 1 k )θ(r)−1 ≤ b1|s|(|s|+ 1 k )θ(r)−1 ≤ b1|s|(2|s|)θ(r)−1 ≤ b12 p+ |s|θ(r). (4.4) If s ≥ k, by the mean value theorem, there exists η ∈ (k, k + 1 k ) such that fk(r, s) = k[G(r, k + 1 k )−G(r, k)] = kGs(r, η)(k + 1 k − k) = f(r, η). By using similar computations as for (4.4) one has 0 ≤ sfk(r, s) = sf(r, η) = s η ηf(r, η) ≤ |s| |η| b1|η|θ(r) ≤ b12 p+ |s|θ(r). Step 4. Assume − 1 k ≤ s ≤ 1 k . Then fk(r, s) = { k2s[G(r,−2/)−G(r,−1/k)], if − 1/k ≤ s ≤ 0 k2s[G(r, 2/k)−G(r, 1/k)], if 0 ≤ s ≤ 1/k. If −1/k ≤ s ≤ 0, by the mean value theorem, there exists η ∈ (−2/k,−1/k) such that fk(r, s) = k2s[G(r,−2 k )−G(r,−1 k )] = k2sGs(r, η)(− 2 k − (−1 k )) = −ksf(r, η). EJDE-2024/41 ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS 11 Therefore 0 ≤ sfk(r, s) = −ks2f(r, η) = −k s 2 η ηf(r, η) ≤ k s2 |η| ηf(r, η) ≤ b1k|s|2|η|θ(r)−1 ≤ b1k|s|2( 2 k )θ(r)−1 ≤ b1 2θ(r)−1 kp−−2 |s|2 ≤ b12 p+ 1 kp−−1 |s|. (4.5) If 0 ≤ s ≤ 1/k, by the mean value theorem, there exists η ∈ (1/k, 2/k) such that fk(r, s) = k2s[G(r, 2 k )−G(r, 1 k )] = k2sGs(r, η)( 2 k − 1 k ) = ksf(r, η). Using similar computations as for (4.5) one obtains 0 ≤ sfk(r, s) = ks2f(r, η) = k s2 |η| ηf(r, η) ≤ b12 p+ 1 kp−−1 |s|. The proof of the lemma follows by taking K1 = b12 p+ , where b1 is given in (1.12). □ 5. Approximate equation Let D ⊆ RN be an open, bounded and symmetric set centered at the origin. We say that u ∈W 1,p 0 (D) is a solution of (1.13) if u(x) > 0 in D and∫ D |∇u|p−2∇u∇ϕdx+ ∫ D |u|p−2uϕdx = λ ∫ D a(x)|u|q(r)−2uϕdx+ ∫ D f(r, u)ϕdx, for all ϕ ∈W 1,p 0 (D). We will employ the following auxiliary problem in our reasoning later. −∆pu+ |u|p−2u = λa(x)|u|q(r)−2u+ fn(r, u) + φ n in D u > 0 in D u(x) = 0 on ∂D, (5.1) with n > 0 a integer number, φ > 0 is a fixed function such that φ ∈ L∞(RN ) ∩ Lp′ (RN ) and fn is given by Lemmas 4.1 and 4.3. Lemma 5.1. There exists λ∗ > 0 and n∗ ∈ N such that (5.1) has a solution un ∈ C1 0 (D) such that ∂un/∂ν < 0 on ∂D for every λ ∈ (0, λ∗) and n ≥ n∗. Furthermore, ∥un∥W 1,p(D) ≤ ϑ, where ϑ does not depend on n. Proof. Let B = {w1, w2, . . . , wn, . . . } be a Schauder basis (see [15] for details) for the Banach space (W 1,p 0,r (D), ∥ · ∥W 1,p(D)). For each positive integer m, let Wm = [w1, w2, . . . , wm] 12 L. F. O. FARIA, M. MONTENEGRO EJDE-2024/41 be the m-dimensional subspace of W 1,p 0,r (D) generated by {w1, w2, . . . , wm} with norm induced from W 1,p 0,r (D). Let ξ = (ξ1, . . . , ξm) ∈ Rm, notice that |ξ|m := ∥∥ m∑ j=1 ξjwj ∥∥ W 1,p(D) (5.2) defines a norm in Rm (see [5] for the details). Using the above notation, we can identify the spaces (Wm, ∥ · ∥W 1,p(D)) and (Rm, | · |m) by the isometric linear transformation u = m∑ j=1 ξjwj ∈Wm 7→ ξ = (ξ1, . . . , ξm) ∈ Rm. (5.3) Define the function F : Rm → Rm such that F (ξ) = (F1(ξ), F2(ξ), . . . , Fm(ξ)), where ξ = (ξ1, ξ2, . . . , ξm) ∈ Rm, Fj(ξ) = ∫ D |∇u|p−2∇u∇wjdx+ ∫ D |u|p−2uwjdx− λ ∫ D a(x)(u+) q(r)−1wjdx − ∫ D fn(r, u+)wjdx− 1 n ∫ D φwjdx, j = 1, 2, . . . ,m, and u = ∑m i=1 ξiwi ∈Wm. Therefore, ⟨F (ξ), ξ⟩ = ∫ D |∇u|pdx+ ∫ D |u|pdx− λ ∫ D a(x)(u+) q(r)dx − ∫ D fn(r, u+)u+dx− 1 n ∫ D φudx, (5.4) where u+ = max{u, 0}, u− = u+ − u. For a given u ∈Wm we define D+ n = {x ∈ D : |u(r)| ≥ 1 n }, D− n = {x ∈ D : |u(r)| < 1 n }. Thus, we can write (5.4) as ⟨F (ξ), ξ⟩ = ⟨F (ξ), ξ⟩P + ⟨F (ξ), ξ⟩N , where ⟨F (ξ), ξ⟩P = ∫ D+ n |∇u|pdx+ ∫ D+ n |u|pdx− λ ∫ D+ n a(x)(u+) q(r)dx − ∫ D+ n fn(r, u+)u+dx− 1 n ∫ D+ n φudx and ⟨F (ξ), ξ⟩N = ∫ D− n |∇u|pdx+ ∫ D− n |u|pdx− λ ∫ D− n a(x)(u+) q(r)dx − ∫ D− n fn(r, u+)u+dx− 1 n ∫ D− n φudx. Now we consider |ξ|m = ∥u∥W 1,p 0 (D) = ϑ (5.5) for some 0 < ϑ ≤ 1 to be chose later. Step 1. Since the embedding W 1,p(RN ) ⊂ EJDE-2024/41 ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS 13 Lτ (RN ) is continuous for all p ≤ τ ≤ p∗ (see [15, Corollary 9.11]), we have∫ D+ n |a(r)|(u+)q(r)dx ≤ C∥a∥Lp/(p−q(x))(RN )∥ũ∥ q− W 1,p(RN ) ≤ C1 2 ∥ũ∥q− W 1,p(RN ) . (5.6) By Lemma 4.3 and Corollary 1.2, we obtain∫ D+ n fn(r, u+)u+dx ≤ K1 ∫ D+ n |u+|θ(r)dx ≤ C2∥ũ∥p− W 1,p(RN ) . (5.7) Since φ ∈ Lp′ (RN ), we have∫ D+ n φudx ≤ ∥φ∥Lp′ (RN )∥ũ∥Lp(RN ) ≤ C3 2 ∥ũ∥W 1,p(RN ). (5.8) It follows from (5.6), (5.7) and (5.8) that ⟨F (ξ), ξ⟩P ≥ ∫ D+ n |∇u|pdx+ ∫ D+ n |u|pdx− λ C1 2 ∥ũ∥q− W 1,p(RN ) − C2∥ũ∥p− W 1,p(RN ) − C3 2n ∥ũ∥W 1,p(RN ). (5.9) Step 2. In a similarly way, we obtain∫ D− n |a(x)|(u+)qdx ≤ C1 2 ∥ũ∥q− W 1,p(RN ) . (5.10) By Lemma 4.3, we obtain∫ D− n fn(r, u+)u+dx ≤ K1 1 np−−1 ∫ D− n |u+|dx ≤ K1 1 np−−1 (∫ D− n dx ) 1 p′ (∫ RN |ũ|pdx )1/p ≤ K1|D|1/p ′ 1 np−−1 ∥ũ∥W 1,p(RN ). (5.11) Also we have ∫ D− n φudx ≤ C3 2 ∥ũ∥W 1,p(RN ). (5.12) It follows from (5.10), (5.11) and (5.12) that ⟨F (ξ), ξ⟩N ≥ ∫ D− n |∇u|pdx+ ∫ D− n |u|pdx− λ C1 2 ∥ũ∥q− W 1,p(RN ) −K1|D|1/p ′ 1 np−−1 ∥ũ∥W 1,p(RN ) − C3 2n ∥ũ∥W 1,p(RN ). (5.13) Using that ∥u∥pW 1,p(D) = ∥ũ∥p W 1,p(RN ) = ∥∇ũ∥p LN (RN ) + ∥ũ∥p LN (RN ) , inequalities (5.9) and (5.13) imply ⟨F (ξ), ξ⟩ ≥ ∥ũ∥p W 1,p(RN ) − λC1∥ũ∥q−W 1,p(RN ) − C2∥ũ∥p− W 1,p(RN ) − ( K1|D|1/p ′ 1 np−−1 + C3 n ) ∥ũ∥W 1,p(RN ). (5.14) Since |ξ|m = ∥ũ∥W 1,p(RN ) = ϑ, we have ⟨F (ξ), ξ⟩ ≥ ϑp − λC1ϑ q− − C2ϑ p− − (K1|D|1/p′ np−−1 + C3 n ) ϑ. 14 L. F. O. FARIA, M. MONTENEGRO EJDE-2024/41 If ϑ is such that ϑ ≤ 1 (2C2) 1 p−−p , then ϑp − C2ϑ p− ≥ ϑp 2 . Thus, choosing ϑ := min { 1 (2C2) 1 p−−p , 1 } , (5.15) we obtain ⟨F (ξ), ξ⟩ ≥ ϑp 2 − λC1ϑ q− − (K1|D|1/p′ np−−1 + C3 n ) ϑ. We define ς := ϑp 2 − λC1ϑ q− . If we choose λ∗ := ϑp−q− 4C1 > 0, then ς > ϑp 4 for all 0 < λ < λ∗. We choose n∗ ∈ N such that(K1|D|1/p′ np−−1 + C3 n ) ϑ < ς 2 for every n ≥ n∗. Notice that n∗ depends on the domain D. Since ξ ∈ Rm is such that |ξ|m = ϑ, then for λ < λ∗ and n ≥ n∗ we obtain ⟨F (ξ), ξ⟩ ≥ ς 2 > 0. (5.16) Since fn is a Lipschitz function (for each n ∈ N), it easy to see that F : Rm → Rm is a continuous function. Thus, for each λ < λ∗ and n > n∗ fixed, Lemma 2.1 ensure the existence of y ∈ Rm with |y|m ≤ ϑ and such that F (y) = 0. In other words, there exists um ∈Wm satisfying ∥um∥W 1,p(D) ≤ ϑ, (5.17) and such that∫ D |∇um|p−2∇um∇wdx+ ∫ D |um|p−2umw dx = λ ∫ D a(x)(um+) q(x)−1wdx+ ∫ D fn(r, um+)wdx+ 1 n ∫ D φw dx, (5.18) for all w ∈Wm. Remark 5.2. It is important to mention that ϑ, given in (5.15), does not depend on the domain D, m nor n. For this matter, Corollary 1.2 plays an important role. SinceWm ⊂W 1,p 0,r (D) for all m ∈ N, and ϑ does not depend on m, then (um)m∈N is a bounded sequence in W 1,p 0 (D). Therefore, for some subsequence, there exists u ∈W 1,p 0 (D) such that um ⇀ u weakly in W 1,p 0,r (D), (5.19) um → u in Ls(D), p ≤ s < p∗, (5.20) um → u, a.e. in , D. (5.21) Thus, ∥u∥W 1,p(D) ≤ lim inf m→∞ ∥um∥W 1,p(D) ≤ ϑ. (5.22) EJDE-2024/41 ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS 15 Now we claim that um → u in W 1,p 0,r (D). (5.23) Indeed, using that B = {w1, w2, . . . , wn, . . . } is a Schauder basis of W 1,p 0,r (D), for every u ∈ W 1,p 0,r (D) there exists a unique sequence (αn)n≥1 in R such that u =∑∞ j=1 αjwj , so that ψm := m∑ j=1 αjwj → u in W 1,p 0,r (D) as m→ ∞. (5.24) Using w = (um − ψm) ∈Wm as test function in (5.18), we obtain∫ D |∇um|p−2∇um∇(um − ψm)dx+ ∫ D |um|p−2um(um − ψm)dx = λ ∫ D a(x)(um+) q(x)−1(um − ψm)dx+ ∫ D fn(r, um+)(um − ψm)dx + 1 n ∫ D φ(um − ψm)dx. (5.25) It is easy to see that∫ D ( |um|p−1 + |λa(x)(um+) q(x)−1|+ 1 n |φ| ) |um − ψm|dx ≤ ( ∥un∥p−1 Lp(D) + λ∥a∥Lp/(p−q(x))(D)∥un∥ q−−1 Lp(D) + 1 n ∥φ∥Lp′ (D) ) ∥um − ψm∥Lp(D). (5.26) By Lemma 4.3, one has∫ D [fn(r, um+)] θ(r) θ(r)−1 dx = ∫ D+ n [fn(r, um+)] θ(r) θ(r)−1 dx+ ∫ D− n [fn(r, um+)] θ(r) θ(r)−1 dx,∫ D+ n [fn(r, um+)] θ(r) θ(r)−1 dx ≤ (max{K1, 1})p ∗ ∫ D+ n |um+|θ(r)dx ≤ (max{K1, 1})p ∗ ∫ D |um+|θ(r)dx ≤ K2∥um∥W 1,p(D)∫ D− n [fn(r, um+)] θ(r) θ(r)−1 dx ≤ (max{K1, 1})p ∗ 1 np∗−2 ∫ D− n |um+|θ(r)dx ≤ (max{K1, 1})p ∗ |D|. Since ∥um∥W 1,p(D) ≤ ϑ, by the estimates above, we obtain∫ D [fn(r, um+)] θ(r) θ(r)−1 dx ≤ C, (5.27) where C does not depend on m. Hence fn(r, um+) is bounded in L θ(r) θ(r)−1 (D). Applying Corollary 1.2, (5.24) and (5.27) we conclude that lim m→∞ ∫ D fn(r, um+)(um − ψm)dx = 0. (5.28) 16 L. F. O. FARIA, M. MONTENEGRO EJDE-2024/41 Notice that∫ D fn(r, um+)(um − ψm)dx ≤ C̃∥um − ψm∥θ(r) ≤ ˜̃C∥um − ψm∥W 1,p(D) → 0 as m→ ∞. (5.29) By (5.17) and (5.24), we obtain lim m→∞ ∫ D |∇um|p−2∇um∇(u− ψm)dx = 0. (5.30) By (5.25), (5.26), (5.28) and (5.30), we obtain lim m→∞ ∫ D |∇um|p−2∇um∇(um − u)dx = 0. (5.31) It is sufficient to apply the (S+)− property of −∆p (see, e.g., [38, Proposition 3.5.]) to obtain (5.23). For every m ≥ k we obtain∫ D |∇um|p−2∇um∇wkdx+ ∫ D |um|p−2umwkdx = λ ∫ D a(x)(um+) q(x)−1wkdx+ ∫ D fn(r, um+)wkdx+ 1 n ∫ D φwkdx, (5.32) for all wk ∈Wk. It follows from (5.23) that∫ D |∇u|p−2∇u∇wkdx+ ∫ D |u|p−2uwkdx = λ ∫ D a(x)(u+) q(x)−1wkdx+ ∫ D fn(r, u+)wkdx+ 1 n ∫ D φwkdx, (5.33) for all wk ∈Wk. Since [Wk]k∈N is dense in W 1,p 0,r (D) we conclude that∫ D |∇u|p−2∇u∇wdx+ ∫ D |u|p−2uw dx = λ ∫ D a(x)(u+) q(x)−1wdx+ ∫ D fn(r, u+)w dx+ 1 n ∫ D φw dx, (5.34) for all w ∈W 1,p 0,r (D). Before concluding, we will check that u satisfy (5.34) for all w ∈ W 1,p 0 (D). Indeed, we will use a symmetric critical principle of Palais [40] in Banach spaces developed in [22]. Let O(N) be the subgroup of isometries g :W 1,p 0 (D) →W 1,p 0 (D) corresponding to all rotations, that is, O(N) is the orthogonal group of dimensionN . The subspace of W 1,p 0 (D) consisting of radially symmetric functions, O(N)-invariant, is given by W 1,p 0,r (D) = {u ∈W 1,p 0 (D) : g(u) = u, for all g ∈ O(N)}, see (1.10). Let u ∈W 1,p 0,r (D) satisfying (5.34). Define Φ(u) ∈W 1,p 0 (D)∗ (the dual space) by (Φ(u), w) = ∫ D |∇u|p−2∇u∇wdx+ ∫ D |u|p−2uw dx − λ ∫ D a(x)(u+) q(x)−1wdx− ∫ D fn(r, u+)wdx− 1 n ∫ D φw dx. (5.35) EJDE-2024/41 ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS 17 By (5.34), we have that (Φ(u), w) = 0, w ∈W 1,p 0,r (D), and Φ(u) is invariant under the action of O(N). By [22], we can infer that Φ(u) ≡ 0. In other words, u satisfy (5.34) for all w ∈W 1,p 0 (D), i.e.∫ D |∇u|p−2∇u∇wdx+ ∫ D |u|p−2uw dx = λ ∫ D a(x)(u+) q(x)−1wdx+ ∫ D fn(r, u+)wdx+ 1 n ∫ D φw dx, (5.36) for all w ∈W 1,p 0 (D). Furthermore, u ≥ 0 in D. In fact, since u− ∈ W 1,p 0 (D) then from (5.36) we obtain −∥u−∥pW 1,p(D) = ∫ D |∇u|p−2∇u∇u−dx+ ∫ D |u|p−2uu−dx = λ ∫ D a(x)(u+) q(x)−1u−dx+ ∫ D fn(r, u+)u−dx+ 1 n ∫ D φu−dx ≥ 0. Then u− ≡ 0 a.e. in D, whence u ≥ 0 a.e. in D. Moreover, u ̸≡ 0 is valid due to φ n > 0 in D. Applying the strong maximum principle [43] we obtain u > 0 in D and ∂u/∂ν < 0 on ∂D. By Lemma 4.1 and [34, Theorem 7.1] we conclude that u ∈ L∞(D). Thus, [35, Theorem 1] ensure the regularity up to the boundary u ∈ C1,β(D), for some β ∈ (0, 1). Therefore, we conclude that proof of the lemma taking un = u. □ 6. Proof of Theorem 1.3 For the proof of Theorem 1.3 we need the following result. Lemma 6.1. For each constant b > 0, the problem −∆pu+ up−1 = buq(r)−1 in D u > 0 in D u = 0 on ∂D, (6.1) where D is a bounded domain in RN with C2 boundary ∂D, admits a solution u0 ∈ C1,β(D) such that ∂u0/∂ν < 0 on ∂D. Proof. This result is more or less standard, but we will sketch the proof. Given a constant b > 0, we define the functional I :W 1,p 0 (D) → R by I(u) = 1 p ∫ D |∇u|pdx+ 1 p ∫ D |u|pdx− b ∫ D 1 q(r) (u+)q(r)dx for all u ∈W 1,p 0 (D), where u+ = max{0, u}. Notice that I is of class C1. Using the Sobolev embedding, Proposition 2.4 and Proposition 2.6, we have the estimate I(u) ≥ 1 p ∥u∥pW 1,p(D) − c(∥u∥q+W 1,p(D) + ∥u∥q−W 1,p(D)) for all u ∈W 1,p 0 (D), with a constant c > 0. Since p > q+ ≥ q− > 1, I is bounded from below and coercive. Considering that the first two terms in the expression of I are convex and continuous on W 1,p 0 (D) and the embedding of W 1,p 0 (D) into Lq(r)(D) is compact, 18 L. F. O. FARIA, M. MONTENEGRO EJDE-2024/41 we infer that I is sequentially weakly lower semi-continuous. Therefore, there exists u0 ∈W 1,p 0 (D) such that I(u0) = inf u∈W 1,p 0 (D) I(u) (see, e.g., [37, Theorems 1.1, 1.2]). Hence u0 is a critical point of I that reads as∫ D |∇u0|p−2∇u0∇vdx+ ∫ D |u0|p−2u0vdx = b ∫ D (u+0 ) q(r)−1vdx (6.2) for all v ∈W 1,p 0 (D). Since the variable exponent q(r) is subcritical, we obtain with standard bootstrap arguments that u0 ∈ C1,β(D). It remains to justify that u0 > 0. Inserting v = −u−0 = −max{0,−u0} in (6.2) leads to u−0 = 0, so u0 ≥ 0 in D. We observe that the condition 1 < q− ≤ q+ < p ensures that I(tu) < 0 provided u ̸= 0 and t > 0 is sufficiently small, which implies that u0 ̸= 0. Finally, we can verify that the strong maximum principle applies in the case of equation (6.2). We conclude that u0 > 0 in D, so u0 is a (weak) solution of problem (6.1). Applying the Hopf boundary point lemma we obtain ∂u0/∂ν < 0 on ∂D. □ Proof of Theorem 1.3. First we show that (1.13) has a positive solution. For each n ∈ N we know, by Lemma 5.1, that equation (5.1) has a (weak) solution un ∈ W 1,p 0,r (D) ∩ C1,β(D), thus∫ D |∇un|p−2∇un∇wdx+ ∫ D |un|p−2unw dx = λ ∫ D a(x)(un) q(x)−1wdx+ ∫ D fn(r, un)wdx+ 1 n ∫ D φw dx, (6.3) for all w ∈W 1,p 0 (D). By (5.22) we have ∥un∥W 1,p(D) ≤ ϑ ≤ 1, ∀n ∈ N, (6.4) and ϑ does not depend on n (indeed, see Remark 5.2). Thus, for a subsequence again relabeled as (un), there exists u ∈W 1,p 0 (D) such that un ⇀ u weakly in W 1,p 0 (D) as n→ ∞. (6.5) Since un → u a.e. in D, by the uniform convergence of Lemma 4.1 (ii) we have fn(·, un(·)) → f(·, u(·)) a.e. in D. (6.6) By Lemma 4.3, one has∫ D [fn(r, un)] θ(r) θ(r)−1 dx = ∫ D+ n [fn(r, un)] θ(r) θ(r)−1 dx+ ∫ D− n [fn(r, un)] θ(r) θ(r)−1 dx,∫ D+ n [fn(r, un)] θ(r) θ(r)−1 dx ≤ (max{K1, 1})p ∗ ∫ D+ n |un|θ(r)dx ≤ (max{K1, 1})p ∗ ∫ D |un|θ(r)dx ≤ K2∥un∥p ∗ W 1,p(D),∫ D− n [fn(r, un)] θ(r) θ(r)−1 dx ≤ (max{K1, 1})p ∗ 1 np∗−2 ∫ D− n |un|θ(r)dx ≤ (max{K1, 1})p ∗ |D|. EJDE-2024/41 ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS 19 Since ∥un∥W 1,p(D) ≤ ϑ, by the estimates before, we obtain∫ D [fn(r, un)] θ(r) θ(r)−1 dx ≤ C, where C does not depend on n. Hence fn(r, un) is bounded in L θ(r) θ(r)−1 (D) and by similar arguments like [30, Theorem 13.44] leads to fn(r, un)⇀ f(r, u) weakly in L θ(r) θ(r)−1 (D). (6.7) By (6.5), (6.7) and Proposition 2.6, we can pass to the limit in (6.3) to obtain∫ D |∇u|p−2∇u∇wdx+ ∫ D |un|p−2uw dx = λ ∫ D a(x)(u)q(x)−1wdx+ ∫ D f(r, u)wdx, (6.8) for all w ∈W 1,p 0 (D). Thus, u is a solution of (1.13). We need to prove that the limit function u does not vanish. For this matter, fix a positive constant λ ∈ (0, λ∗) such that λ∗ = ϑ2−q− 4C1 (6.9) was given in Lemma 5.1. Since a > 0 is a continuous function, define aτ = inf D a(r). Then, according to Lemma 6.1, there exists a positive solution uλ,R of −∆pu+ up−1 = λaτu q(r)−1 in D u > 0 in D u = 0 on ∂D. Let un be a positive solution of problem (5.1) obtained by Lemma 5.1. We observe that uλ/un, un/uλ ∈ L∞(D) because uλ,R and un are positive functions belonging to C1,β 0 (D) and satisfying ∂un/∂ν < 0, ∂uλ,R/∂ν < 0 on ∂D. Hence we are able to apply Proposition 2.3 with u1 = uλ,R, u2 = un, g(r, t) = λaτ t q(r). Notice that λtq(r) + fn(r, t) + ϕ n ≥ λaτ t q(r) = g(r, t). Hence, u1 and u2 are a positive subsolution and a positive supersolution of problem (6.1), respectively. In this way, by Proposition 2.3 we see that un ≥ uλ,R > 0 in D for every n ≥ 1. Therefore, in the limit as n → ∞ we obtain that u ≥ uλ,R a.e. in D. Thus, by letting to the limit, we conclude that u is a positive solution of problem (1.13). The solution we just found is being written as uλ with explicit dependence on λ. We will deduce that ∥uλ∥W 1,p(D) → 0 as λ → 0. Fix the pair (λ, uλ), where λ ∈ (0, λ∗) and uλ is the corresponding solution of problem (1.13),given by the previous steps. Using w = uλ as a test function in (6.8) and recalling (2.1), we obtain 20 L. F. O. FARIA, M. MONTENEGRO EJDE-2024/41 ∫ RN |∇ũλ|pdx+ ∫ RN ũpλdx = ∫ D |∇uλ|pdx+ ∫ D upλdx = λ ∫ D a(r)u q(r) λ dx+ ∫ D f(r, uλ)uλdx = λ ∫ RN a(r)ũ q(r) λ dx+ ∫ RN f(r, ũλ)ũλdx ≤ λC1∥ũλ∥q−W 1,p(RN ) + C2∥ũλ∥p− W 1,p(RN ) , (6.10) where C1, C2 are given in (5.6), (5.7), respectively. Since ũλ ̸= 0, from (6.10), we have the estimate ∥ũλ∥p−q− W 1,p(RN ) (1− C2∥ũλ∥p−−2 W 1,p(RN ) ) ≤ λC1. (6.11) Combining (5.22), (6.4), we obtain ∥ũλ∥p−−2 W 1,p(RN ) ≤ 1 2C2 . Thus, ∥uλ∥W 1,p(D) = ∥ũλ∥W 1,p(RN ) ≤ (2λC1) 1/(p−q−). (6.12) We conclude that ∥uλ∥W 1,p(RN ) → 0 as λ → 0. The proof of the theorem is complete. □ 7. Proof of Theorem 1.4 Let R > 0. In what follows, we denote Bn = Bn(0) the open ball centered at the origin and of radius n ∈ N, for some n > R. The space W 1,p(Bn) is equipped with the norm ∥u∥2,n = (∫ Bn ( |∇u|p + |u|p ) dx )1/p . Proof of Theorem 1.4. Applying Theorem 1.3 with D = Bn\BR (and n > R), we obtain a positive solution un ∈W 1,p 0 (Bn) ∩ C1,β(Bn) of the problem −∆pu+ up−1 = λa(r)uq(r)−1 + f(r, u) in Bn u > 0 in Bn u = 0 on ∂Bn, (7.1) Again, (5.22) and (6.12) show the uniform boundedness of the sequence (un)n>R in W 1,p 0 (Bn), that is, defining ϑ̃ = min { ϑ, (2λC1) 1/(p−q−) } we obtain ∥un∥2,n ≤ ϑ̃ for all n ∈ N. (7.2) Fix m ∈ N. If n ≥ m > R, by (7.2) we have ∥un∥2,m ≤ ∥un∥2,n ≤ ϑ̃. (7.3) Therefore, for a subsequence if necessary, there exists u ∈W 1,p(Bm) such that un ⇀ u in W 1,p(Bm), un → u for a.e. x ∈ Bm, un → u in Lp(Bm), EJDE-2024/41 ELLIPTIC EQUATIONS ON SYMMETRIC DOMAINS 21 un ⇀ u in Lθ(r)/(θ(r)−1)(Bm), un ⇀ u in Lq(r)−1(Bm), as n→ ∞. Recalling that un > 0 in Bm, by the above convergences, we infer that u is a nonnegative solution of the problem −∆pu+ up−1 = λa(r)uq(r)−1 + f(r, u) in Bm, u ≥ 0texton∂Bm. By a diagonal argument we obtain a subsequence of (ũn) and a function u ∈ W 1,p(RN ) such that ũn ⇀ u in W 1,p(RN ), ũn → u for a.e. x ∈ RN , ũn ⇀ u in Lθ(r)/(θ(r)−1)(RN ), ũn ⇀ u in Lq(r)−1(RN ) as n→ ∞. (7.4) Indeed, fix φ ∈ C∞ 0 (RN ), let m ∈ N such that supp(φ) ⊂ Bm. For n large enough, we have ∫ RN |∇ũn|p−2∇ũn∇φdx = ∫ Bm |∇ũn|p−2∇ũn∇φdx → ∫ Bm |∇u|p−2∇u∇φdx = ∫ RN |∇u|p−2∇u∇φdx,∫ RN |ũn|p−2ũnφdx = ∫ Bm |ũn|p−2ũnφdx → ∫ Bm |u|p−2uφdx = ∫ RN |u|p−2uφdx,∫ RN a(x)(ũn) q(x)−1φdx = ∫ Bm a(x)(ũn) q(x)−1φdx → ∫ Bm a(x)uq(x)−1φdx = ∫ RN a(x)uq(x)−1φdx,∫ RN f(x, ũn)φdx = ∫ Bm f(x, ũn)φdx → ∫ Bm f(x, u)φdx = ∫ RN f(x, u)φdx. Since C∞ 0 (RN ) is dense in W 1,p(RN ), these convergence properties ensure that u is a weak solution of problem (1.14). The next step is to show that the limit function u does not vanish in RN . Fix λ ∈ (0, λ∗), with λ∗ satisfying (6.9). Lemma 6.1 provides a solution uλ,m of the problem −∆pu+ up−1 = λaτu q(r)−1 in Bm u > 0 in Bm u = 0 on ∂Bm. Since λa(r)tq(r)−1 + f(r, t) ≥ λtq(r)−1 for all x ∈ RN and t > 0, we can apply Proposition 2.2 to the functions uλ,m and ũn with n ≥ m, in place of u1 = uλ,m 22 L. F. O. FARIA, M. MONTENEGRO EJDE-2024/41 and u2 = ũn, respectively, which renders ũn ≥ uλ,m in Bm for every n ≥ m. 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Faria Universidade Federal de Juiz de Fora, ICE, Departamento de Matemática, Campus Uni- versitário, Rua José Lourenço Kelmer, s/n, Juiz de Fora, MG, CEP 36036-900, Brazil Email address: luiz.faria@ufjf.edu.br Marcelo Montenegro Universidade Estadual de Campinas, IMECC, Departamento de Matemática, Rua Sérgio Buarque de Holanda, 651 Campinas, SP, CEP 13083-859, Brazil Email address: msm@ime.unicamp.br 1. Introduction 2. Preliminaries 2.1. Extension 2.2. Brouwer Theorem 2.3. Comparison principle 2.4. Function spaces with variable exponents 3. Proof of Theorem ?? 4. Approximate functions 5. Approximate equation 6. Proof of Theorem ?? 7. Proof of Theorem ?? Acknowledgments References