Special Issue in honor of Alan C. Lazer Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 327–344. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu GENERALIZED QUASILINEAR EQUATIONS WITH CRITICAL GROWTH AND NONLINEAR BOUNDARY CONDITIONS LILIANE DE ALMEIDA MAIA, JOSÉ CARLOS OLIVEIRA JUNIOR, RICARDO RUVIARO Abstract. We study the quasilinear problem − div(h2(u)∇u) + h(u)h′(u)|∇u|2 + u = −λ|u|q−2u+ |u|2·2 ∗−2u in Ω, ∂u ∂η = µg(x, u) on ∂Ω, where Ω ⊂ R3 is a bounded domain with regular boundary ∂Ω, λ, µ > 0, 1 < q < 4, 2 ·2∗ = 12, ∂ ∂η is the outer normal derivative and g has a subcritical growth in the sense of the trace Sobolev embedding. We prove a regularity result for all weak solutions for a modified, and introducing a new type of constraint, we obtain a multiplicity of solutions, including the existence of a ground state. 1. Introduction We study the quasilinear Schrödinger equation i∂tψ = −∆ψ + V (x)ψ − η(|ψ|2)ψ − κ[∆ρ(|ψ|2)]ρ′(|ψ|2)ψ, (1.1) where ψ:R × RN → C, V :RN → R is a given potential, N ≥ 1, κ is a positive constant and ρ, η:R+ → R are suitable functions. This equation arises in various branches of mathematical physics, see for example [28]. When κ 6= 0, (1.1) models phenomena in plasma physics and fluid mechanics [15, 16, 18, 21], laser theory [2, 29], and in condensed matter theory [24]. The case ρ(s) = s occurs in theory of superfluids (see [15, 16, 19] and the references in [17]), whereas ρ(s) = (1 + s)1/2 appears in the self-channeling of a high-power ultra short laser in matter (see [3, 4]). Looking for standing wave solutions for (1.1), one takes ψ(t, x) := exp(−iEt)u(x) with E ∈ R and u:RN → R a function, which leads to consider the elliptic equation −∆u+ V (x)u− κ∆(ρ(u2))ρ′(u2)u = g(u), in Ω ⊆ RN , (1.2) where we have replaced V (x)− E by V (x) and g(u) = η(u2)u. 2020 Mathematics Subject Classification. 35J25, 35J62, 35B33. Key words and phrases. Quasilinear equations; variational methods; concave nonlinearities; critical exponent; ground state solution. ©2021 This work is licensed under a CC BY 4.0 license. Published June 27, 2022. 327 328 L. MAIA, J. C. OLIVEIRA JUNIOR, R. RUVIARO EJDE/SI/01 In this article, we are interested in the quasilinear problem −div(h2(u)∇u) + h(u)h′(u)|∇u|2 + u = −λ|u|q−2u+ |u|2·2 ∗−2u in Ω, ∂u ∂η = µg(x, u) on ∂Ω, (1.3) where Ω ⊂ R3 is a bounded domain with regular boundary ∂Ω, λ, µ > 0, 1 < q < 4, 2 · 2∗ = 12, and ∂ ∂η is the outer normal derivative. Note that if we take h2(s) = 1 + 1 2 ( d ds ρ(s2) )2 , then equation (1.3) becomes (1.2), see [31]. We consider nonlinearities h, satisfying the following: (A1) h ∈ C2(R, (0,+∞)) is even, non-decreasing in [0,+∞) and h∞ := lim t→∞ h(t) t ∈ (0,+∞). (1.4) (A2) It holds that β := sup t∈R th′(t) h(t) ≤ 1. (1.5) (A3) The mapping t 7→ h′(t)h(t)/t is non-increasing for t > 0. Remark 1.1. Hypotheses (A1) and (A3) together imply that, for all t > 0, t2h′′(t) h(t) + t2[h′(t)]2 h2(t) ≤ th′(t) h(t) . Since h is an even function, we have that h′ is an odd function and h′′ is an even function. Therefore, the above inequality still holds for t ≤ 0. We refer the reader to [26] and references therein, for a review of the semilinear case, i.e., problem (1.2) when κ = 0, in bounded domains Ω ⊂ RN . Whether Ω = RN and again κ = 0, there are [22] and its references. The literature on the subcritical case of problem (1.2) with κ 6= 0 is extensive for Ω = RN (see [8, 20, 23, 25]), as well as a bounded domain Ω ⊂ RN (see [7, 10]). Furthermore, recent results concerning the case of the critical power in RN , g(u) = up for p = 2 · 2∗ = 4N/(N − 2) are found in publications such as Deng et al. [9]. In their introduction they present a complete review for this class of problems. We highlight the seminal papers [8, 19] in which the particular case ρ(s) = s, that is, h(s) = (1 + 2s2)1/2, was cleverly studied. Since the energy functional associated to the problem is not well defined in the whole Sobolev space, the authors considered the change of variables u = f(v), where f is defined by f ′(t) := 1√ 1 + 2f2(t) in [0,+∞) f(t) := −f(−t) in (−∞, 0], (1.6) and for some adequate growth for function g, they applied variational methods to establish the existence of a nontrivial solution for (1.2). We point out that this change of variables has become a powerful tool for solving problem (1.2) when ρ(s) = s. For more details, see [1, 23, 30] and references therein. Note that problem (1.2) in a bounded domain Ω is also relevant, for example, in physical models that describe electrons on lattices and applications to nanotubes [14]. Semilinear and quasilinear problems of this type in bounded domains, on EJDE-2021/SI/01 QUASILINEAR EQUATIONS WITH CRITICAL GROWTH 329 either Dirichlet or Neumann boundary conditions, appear in [6, 26, 27] and its references. To tackle problem (1.3), we use a new type of constraint for the energy functional related to a modified problem. Alternative to the usual method of Nehari (for example, [20]), we define, in Section 3, the constraint based on the change of variable that we will apply. One of the advantages of this definition is that we can consider values of q in the interval (1, 4) that may not be considered when applying the usual Nehari manifold as a constraint. In exchange, we restrict the approach to three dimensions because of technical issues related to the Sobolev embeddings, as explained in Remark 3.6. The lack of compactness issues, which naturally appear due to the critical exponent, are circumvent by proving that, for µ sufficiently large, there exists a (PS) sequence in the range ( 0, 4−N/2(Sh∞)N/2 N ) where compactness holds (see Proposition 2.6 below). Here, S is the best constant to the Sobolev embedding D1,2(RN ) ↪→ L2∗(RN ) and h∞ > 0 is defined in (1.4). Next we give some examples of functions that appear in physics models satisfy conditions (A1)–(A3). Lemma 1.2. The following functions h : R→ (0,+∞) satisfy (A1)–(A3). (a) h(t) = √ 1 + 2t2; (b) h(t) = √ 1 + t2 2(1+t2) + t2; (c) h(t) = √ 1 + 3t2 1+t2 + ln(1 + et2); (d) h(t) = √ 1− e−t2 2 + 1 2 ln(1 + et2). In this article, we will use either the notations 2 · 2∗ = 4N/(N − 2) and 2 · 2∗ = 4(N − 1)/(N − 2), or respectively, 12 and 8 in dimension N = 3. We assume that the function g : ∂Ω×R→ R, satisfies the following hypotheses: (A4) g ∈ C1,θ(∂Ω× R,R) for some θ ∈ (0, 1); (A5) Let G(x, s) = ∫ s 0 g(x, t)dt. There exists a constant σ satisfying 1 2·2∗ < σ ≤ 1 4 such that σg(x, s)s ≥ G(x, s) > 0 for all s 6= 0 and almost every x ∈ ∂Ω; (A6) lims→0 g(x,s) s3 = 0 and lim|s|→+∞ |g(x,s)| |s|p−1 = g∞(x) uniformly for x ∈ ∂Ω, for some g∞ ∈ L∞(∂Ω), and 4 ≤ p < 2 · 2∗; (A7) The function defined by s 7→ g(x, s)/s3 for s ∈ (−∞, 0) ∪ (0,+∞) is non- decreasing for almost every x ∈ ∂Ω; (A8) There exist c1, c2 > 0 such that |g′(x, s)| ≤ c1|s|p−2 + c2. Remark 1.3. We note that hypothesis (A5) includes the 3-asymptotically linear case, that is, it may occurs that lim |s|→+∞ |g(x, s)| |s|3 = g∞(x) uniformly on x ∈ ∂Ω. If we consider the functions g(x, s) = g∞(x)|s|p−2s or g(x, s) = g∞(x) s5 1+s2 , where g∞ ∈ L∞(∂Ω) such that 0 < g0 ≤ g∞(x) ≤ g∞ almost everywhere x ∈ ∂Ω and 4 ≤ p < 2 · 2∗, then g satisfies all conditions (g1)− (g5). 330 L. MAIA, J. C. OLIVEIRA JUNIOR, R. RUVIARO EJDE/SI/01 The first difficulty in directly applying variational methods to solve problem (1.3) is that the energy functional associated with this problem may not be well defined in the whole space H1(Ω). Precisely, the functional Tλ,µ : H1(Ω) → R associated with equation (1.3), given by Tλ,µ(u) = 1 2 ∫ Ω (h2(u)|∇u|2 + u2)dx+ λ q ∫ Ω |u|qdx − 1 2 · 2∗ ∫ Ω |u|2·2 ∗ dx− µ ∫ ∂Ω G(x, u)dσx, (1.7) for u ∈ H1(Ω), where dσx is the measure on the boundary, is not well defined, because the term ∫ Ω h2(u)|∇u|2dx is not finite for all u ∈ H1(Ω) and for all h that we are considering. Indeed, without loss of generality, assume B2(0) ⊂ Ω and let h(t) = √ 1 + 2t2 (item a) from Lemma 1.2), and φ ∈ C∞0 (Ω, [0, 1]) be such that φ ≡ 1 in B1(0) = {x ∈ Ω; |x| < 1} and φ ≡ 0 in Ω \ B2(0) = {x ∈ Ω; |x| ≥ 2}. Now taking u(x) = |x|−1 4 φ(x) for x 6= 0, it is easy to see that u ∈ H1(Ω), however∫ Ω h2(u)|∇u|2dx ≥ 2 ∫ Ω u2|∇u|2dx = +∞. To overcome this difficulty, the main idea is to take the primitive H(s) := ∫ s 0 h(t)dt, and consider the change of variable w = H(u), then look for critical point of the functional Iλ,µ : H1(Ω)→ R defined by Iλ,µ(u) := Tλ, µ(H−1(w)) for w ∈ H1(Ω). It can be proved that w ∈ H1(Ω) is a critical point of Iλ,µ if, and only if, u = H−1(w) is a weak solution of problem (1.3). We list below the main properties of the change of variable which will be used throughout this work. Lemma 1.4. The function H−1 : R→ R satisfies the following properties: (1) H−1 ∈ C1(R,R); (2) 0 < d dt ( H−1(t) ) = 1 h(H−1(t)) ≤ 1 h(0) for all t ∈ R; (3) |H−1(t)| ≤ |t| h(0) for all t ∈ R; (4) H−1(t) t → 1 h(0) as t→ 0; (5) 1 ≤ H−1(t)h(H−1(t)) t ≤ 2 for all t 6= 0. (6) ∣∣∣ t h(t) ∣∣∣ ≤ 1 h∞ for all t ∈ R; (7) H−1(t)√ t is non-decreasing in (0,+∞) and |H−1(t)| ≤ (2/h∞)1/2 √ |t| for all t ∈ R; (8) H−1(t)√ t → √ 2 h∞ as t→ +∞; (9) |H−1(t)| ≥ H−1(1) √ |t| for all |t| ≥ 1; (10) 1 2 (H−1(t))2 ≤ H−1(t)(H−1)′(t)t ≤ (H−1(t))2 for all t ∈ R. Proof. Properties (1)–(9) can be found in [12, Lemma 2.1]. Property (10) follows from property (5) and the fact that h is even, H is odd and so H−1 as well. � After the change of variable u = H−1(w) in (1.7), we obtain Iλ,µ(w) = 1 2 ∫ Ω (|∇w|2 + |H−1(w)|2)dx+ λ q ∫ Ω |H−1(w)|qdx EJDE-2021/SI/01 QUASILINEAR EQUATIONS WITH CRITICAL GROWTH 331 − 1 2 · 2∗ ∫ Ω |H−1(w)|2·2 ∗ dx− µ ∫ ∂Ω G(x,H−1(w))dσx, and the functional Iλ,µ is associated with the problem −∆w +H−1(w)(H−1)′(w) = p(w), in Ω, ∂w ∂η = µg(x,H−1(w))(H−1)′(w), on ∂Ω (1.8) for w ∈ H1(Ω), where p(w) = −λ|H−1(w)|q−2H−1(w)(H−1)′(w) + |H−1(w)|2·2 ∗−2H−1(w)(H−1)′(w). To show that Iλ,µ is well defined and belongs to C1(H1(Ω),R), we use that, for every ε > 0, by conditions (g1)− (g3), there exists Cε := C(ε, q, σ) > 0 such that |g(x, s)| ≤ ε|s|3 + Cε|s|p−1 and |G(x, s)| ≤ ε|s|4 + Cε|s|p (1.9) for all s ∈ R and x ∈ ∂Ω. Here, we may choose 4 ≤ p = 1/σ < 2 · 2∗ (see Lemma 2.1 below). Then, it is enough to use (1.9), properties (1) and (7) and the Sobolev embeddings to conclude that Iλ,µ is continuous and is well defined in H1(Ω). The C1 regularity of Iλ,µ follows from Lemma 1.4, the properties of the functions H−1 and (H−1)′. In this article, let ‖ · ‖ denote the norm u 7→ √∫ Ω (|∇u|2 + u2)dx in H1(Ω) and | · |r denote the usual norm in the Lebesgue space Lr(Ω) for r ≥ 1. The main contributions of this article are the following. Theorem 1.5. Under assumptions (A1)–(A8), for λ > 0, there exists µλ > 0 such that, for every µ ≥ µλ, one of the following cases occurs: 1. Problem (1.8) has two solutions, one of which is nonnegative and ground state solution and the other is non-positive; 2. Problem (1.8) has two solutions, one of which is non-positive and ground state solution and the other is nonnegative. Corollary 1.6. Let uλ,µ ≥ 0 and vλ,µ ≤ 0 be the solutions given in Theorem 1.5. It holds that Iλ,µ(uλ,µ) → 0 and Iλ,µ(vλ,µ) → 0 as µ → +∞ uniformly on λ in a bounded set. Theorem 1.7. Under assumptions (A1)–(A8), every weak solution w ∈ H1(Ω) for problem (1.8) is a classical solution in the sense that w ∈ C2,γ(Ω), for some γ ∈ (0, 1), and w satisfies pointwisely equation (1.8). 2. A compactness result The next lemma is a direct consequence of hypothesis (A6) and Remark 1.3. Lemma 2.1. Let p ≤ τ < 2 · 2∗. For all ε > 0, there exists a positive constant Cε > 0 such that |g(x, s)| ≤ ε|s|3 + Cε|s|τ−1, |G(x, s)| ≤ ε|s|4 + Cε|s|τ for all s ∈ R and x ∈ ∂Ω. 332 L. MAIA, J. C. OLIVEIRA JUNIOR, R. RUVIARO EJDE/SI/01 In what follows, we show that H−1 has an appropriate behavior at the origin and at infinity in order to use a general theorem due to Brezis-Lieb, involving the change of variableH−1, result that will be essential to demonstrate the Palais-Smale condition ((PS) condition) for functional Iλ,µ. Lemma 2.2. For each ε > 0, there exists Ĉε = Ĉ(ε) > 0 such that |H−1(t)| ≤ ε|t|1/2 + Ĉε|t|1/2 for all t ∈ R. Moreover, (Ĉε) is uniformly bounded for ε in a bounded set. Proof. Let ε > 0 be any positive real number. By property (3), we have |H −1(t)| |t|1/2 → 0 as t→ 0 and then there exists δ = δ(ε) > 0 such that |H−1(t)| ≤ ε|t|1/2 for all |t| < δ. (2.1) Property (8) ensures that there exists γ = γ(ε) > 0 such that |H−1(t)| ≤ ( ε+ √ 2 h∞ ) |t|1/2 for all |t| > γ. (2.2) Since from property (7) we have |H−1(t)| ≤ (2/h∞)1/2|t|1/2 for all δ ≤ |t| ≤ γ, it follows from (2.1) and (2.2) that, for all t ∈ R, |H−1(t)| ≤ ε|t|1/2 + (ε+ (2/h∞)1/2)|t|1/2 + (2/h∞)1/2|t|1/2, that is, |H−1(t)| ≤ ε|t|1/2 + Ĉε|t|1/2 for all t ∈ R, where Ĉε = ε + 2(2/h∞)1/2 > 0. Clearly (Ĉε) is uniformly bounded for ε in a bounded set, and the lemma follows. � Lemma 2.3. Let j : R → R be defined as j(t) = |H−1(t)|2·2∗ . Given ε > 0, there exist two nonnegative continuous functions ϕε, ψε : R → R such that, for all a, b ∈ R, it holds |j(a+ b)− j(b)| ≤ εϕε(a) + ψε(b). Proof. We apply the same arguments as in [11, Lemma 3.2], replacing f for H−1, using property (6),the Mean Value Theorem, Young inequality, and Lemma 2.2. Then, for all ε > 0 and a, b ∈ R, we obtain the existence of some constants C,Bε > 0 such that |j(a+ b)− j(b)| ≤ εϕε(a) + ψε(b), where Ĉε = ε + 2(2/h∞)1/2 > 0 is given in Lemma 2.2 and ϕε(a) = Cε2·2∗−2(1 + Ĉε)|a|2 ∗ and ψε(b) = (ε2·2∗−2 + Ĉε + Bε)|b|2 ∗ are the two nonnegative continuous functions required. The lemma is proved. � Lemma 2.4. Given ε > 0, let (wn) ⊂ H1(Ω) be a sequence that converges weakly to w in H1(Ω) and let j, ϕε, ψε : R→ R be as in Lemma 2.3. Then (i) j(w) ∈ L1(Ω); (ii) ∫ Ω ϕε(wn − w)dx ≤ C < +∞ for some constant C > 0, which does not depend on 0 < ε < 1 and n ∈ N; (iii) ∫ Ω ψε(w)dx < +∞ for all ε > 0. EJDE-2021/SI/01 QUASILINEAR EQUATIONS WITH CRITICAL GROWTH 333 Proof. Items (i) and (iii) follow directly from Sobolev embedding. To prove (ii), we have from Lemma 2.2 that Ĉε = ε+2(2/h∞)1/2 < 1+2(2/h∞)1/2 for all 0 < ε < 1, whence ϕε(wn − w) = Cε2·2∗−2(1 + Ĉε)|wn − w|2 ∗ ≤ C(2 + 2(2/h∞)1/2)|wn − w|2 ∗ . Thus, item (ii) is proved since (wn) is also bounded in L2∗(Ω) by hypothesis. � Proposition 2.5. Let (wn) ⊂ H1(Ω) be a sequence that converges weakly to w in H1(Ω). Then∫ Ω ∣∣∣|H−1(wn − w)|2·2 ∗ − |H−1(wn)|2·2 ∗ + |H−1(w)|2·2 ∗ ∣∣∣ dx→ 0 as n→ +∞. In particular, |H−1(wn − w)|2·2 ∗ 2·2∗ + |H−1(w)|2·2 ∗ 2·2∗ = |H−1(wn)|2·2 ∗ 2·2∗ + on(1), with on(1)→ 0 as n→ +∞. The proof of the above propsition is a direct consequence of Lemma 2.4 with the general Brezis-Lieb Lemma (see Theorem 2 in [5]). 2.1. (PS) condition in the correct range. In the sequel, we will show that Iλ,µ satisfies (PS) condition in a particular range for bounded sequences. Proposition 2.6. Let (wn) ⊂ H1(Ω) be a bounded (PS)c sequence for the func- tional Iλ,µ. If c < 4−N/2(Sh∞)N/2 N , then (wn) possesses a strongly convergent subse- quence. Proof. Let (wn) be a bounded (PS)c sequence for functional Iλ,µ. So, up to a subsequence, we may suppose that wn ⇀ w in H1(Ω), wn → w in L2(Ω), wn → w in Lq(Ω), wn(x)→ w(x) a.e. in Ω. (2.3) By the Sobolev compact embeddings, we obtain I ′λ,µ(w)v = ∫ Ω ∇w∇vdx+ ∫ Ω H−1(w)(H−1)′(w)vdx + λ ∫ Ω |H−1(w)|q−2H−1(w)(H−1)′(w)vdx − ∫ Ω |H−1(w)|2·2 ∗−2H−1(w)(H−1)′(w)vdx − µ ∫ ∂Ω g(x,H−1(w))(H−1)′(w)vdσx = 0 (2.4) for all v ∈ H1(Ω). Thus, from (A5), Iλ,µ(w) = Iλ,µ(w)− σI ′λ,µ(w)H−1(w)h(H−1(w)) ≥ ∫ Ω (1 2 − σ(1 + β) ) |∇w|2dx+ (1 2 − σ ) ∫ Ω |H−1(w)|2dx + λ (1 q − σ ) ∫ Ω |H−1(w)|qdx+ ( σ − 1 2 · 2∗ ) ∫ Ω |H−1(w)|2·2 ∗ dx + ∫ ∂Ω ( σg(x,H−1(w))H−1(w)−G(x,H−1(w)) ) dσx ≥ 0. (2.5) 334 L. MAIA, J. C. OLIVEIRA JUNIOR, R. RUVIARO EJDE/SI/01 Let us denote vn := wn − u and prove that vn → 0 in H1(Ω). From (2.3), we have vn → 0 in L2(Ω) and Lq(Ω). So, Iλ,µ(w) + Iλ,µ(vn) = 1 2 |∇w|22 + 1 2 |H−1(w)|22 + λ q |H−1(w)|qq − µ 2 · 2∗ |H−1(w)|2·2 ∗ 2·2∗ − µ ∫ ∂Ω G(x,H−1(w))dx+ 1 2 |∇vn|22 + 1 2 |H−1(vn)|22 + λ q |H−1(vn)|qq − 1 2 · 2∗ |H−1(vn)|2·2 ∗ 2·2∗ − µ ∫ ∂Ω G(x,H−1(vn))dσx = 1 2 |wn|22 + 1 2 |H−1(wn)|22 + λ q |H−1(wn)|qq − 1 2 · 2∗ ( |H−1(w)|2·2 ∗ 2·2∗ + |H−1(vn)|2·2 ∗ 2·2∗ ) − µ ∫ ∂Ω G(x,H−1(wn))dσx + on(1), where we used (2.3) and Lemma 2.1 to ensure the following convergences:∫ ∂Ω G(x,H−1(vn))dσx = on(1), |H−1(vn)|q = on(1), |H−1(vn)|2 = on(1), |H−1(wn)|q = |H−1(w)|q + on(1),∫ ∂Ω G(x,H−1(wn))dσx = ∫ ∂Ω G(x,H−1(w))dσx + on(1). Therefore, by Proposition 2.5 and (2.5), it holds Iλ,µ(vn) ≤ 1 2 |∇wn|22 + 1 2 |H−1(wn)|22 + λ q |H−1(wn)|qq − 1 2 · 2∗ |H−1(wn)|2·2 ∗ 2·2∗ − µ ∫ ∂Ω G(x,H−1(wn))dσx + on(1) = Iλ,µ(wn) + on(1) = c. (2.6) Now, applying Proposition 2.5 and definition of (PS)c sequence one more time, we obtain from convergences in (2.3) and from (2.4) that on(1) = I ′λ,µ(wn)wn − 2 ∫ Ω ∇wn∇wdx+ 2|∇w|22 − I ′λ,µ(w)w = |∇wn|22 − 2 ∫ Ω ∇wn∇wdx+ |∇w|22 + on(1) − ∫ Ω |H−1(wn)|2·2 ∗−2H−1(wn)(H−1)′(wn)wndx + ∫ Ω |H−1(w)|2·2 ∗−2H−1(w)(H−1)′(w)wdx − µ ∫ ∂Ω g(x,H−1(wn))(H−1)′(wn)wndσx + µ ∫ ∂Ω g(x,H−1(w))(H−1)′(w)wdσx = |∇vn|22 −An + on(1), (2.7) EJDE-2021/SI/01 QUASILINEAR EQUATIONS WITH CRITICAL GROWTH 335 where An := ∫ Ω |H−1(wn)|2·2 ∗−2H−1(wn)(H−1)′(wn)wndσx − ∫ Ω |H−1(w)|2·2 ∗−2H−1(w)(H−1)′(w)wdσx. It follows from Proposition 2.5 and property (5) that we can also prove the equality An = ∫ Ω |H−1(vn)|2·2 ∗−2H−1(vn)(H−1)′(vn)vndx+ on(1). Thus, (2.7) yields on(1) = |∇vn|22 − ∫ Ω |H−1(vn)|2·2 ∗−2H−1(vn)(H−1)′(vn)vndx. Since both sequence (|∇vn|22) and ( ∫ Ω |H−1(vn)|2·2∗−2H−1(vn)(H−1)′(vn)vndx ) are bounded, let us suppose that |∇vn|22 → d and ∫ Ω |H−1(vn)|2·2 ∗−2H−1(vn)(H−1)′(vn)vndx→ d as n→ +∞. By property (7), |H−1(vn)|22·2∗ ≤ (2/h∞)|vn|2∗ , and from Sobolev embedding and property (10),(∫ Ω |H−1(vn)|2·2 ∗−2H−1(vn)(H−1)′(vn)vndx )2/2∗ ≤ (∫ Ω |H−1(vn)|2·2 ∗ dx )2/2∗ = |H−1(vn)|42·2∗ ≤ (2/h∞)2|vn|22∗ ≤ 4 Sh∞ |∇vn|22. Then, taking n→ +∞, one obtains Sh∞d 2/2∗ ≤ 4d. Now, suppose by contradiction that d 6= 0. This implies 4−N/2(Sh∞) N 2 ≤ d. On the other hand, since Iλ,µ(vn) = 1 2 |∇vn| 2 2 − 1 2·2∗ |H −1(vn)|2·2∗2·2∗ + on(1), it follows from property (10) that 1 2 |∇vn|22 − 1 2∗ ∫ Ω |H−1(vn)|2·2 ∗−2H−1(vn)(H−1)′(vn)vndx ≤ 1 2 |∇vn|22 − 1 2 · 2∗ |H−1(vn)|2·2 ∗ 2·2∗ = Iλ,µ(vn) + on(1), which, passing to a subsequence if necessary, by (2.6), produces d (1 2 − 1 2∗ ) ≤ c. Hence, 4−N/2(Sh∞)N/2 N ≤ d (1 2 − 1 2∗ ) ≤ c < 4−N/2(Sh∞)N/2 N , what is clearly an absurd. Necessarily, d = 0 and, then, wn → w strongly in H1(Ω), as we wished to prove. � 336 L. MAIA, J. C. OLIVEIRA JUNIOR, R. RUVIARO EJDE/SI/01 3. Existence of two solutions Consider I±λ : H1(Ω)→ R the C1-functional defined by I±λ,µ(w) = 1 2 ∫ Ω (|∇w|2 + |H−1(w)±|2)dx+ λ q ∫ Ω (H−1(w)±)qdx − 1 2 · 2∗ ∫ Ω |H−1(w)±|2·2 ∗ (w)dx− µ ∫ ∂Ω G(x,H−1(w)±)dσx, where u+ = max{u, 0} and u− = max{−u, 0}. Suppose that w ∈ H1(Ω) satisfies (I±λ,µ)′(w) = 0. Since H−1(s) has the same sign of s, we have 0 = (I±λ,µ)′(w)H−1(w∓)h(H−1(w∓)) = ∫ Ω [( 1 + H−1(w∓)h′(H−1(w∓)) h(H−1(w∓)) ) |∇w∓|2 + |H−1(w∓)|2) ] dx, that is, w∓ = 0. This shows that every critical point of I+ λ,µ is non-negative and every critical point of I−λ,µ is non-positive. Therefore, they both are critical points of Iλ,µ as well. To find solutions, we will consider a type of Nehari set defined by N± = { w ∈ H1(Ω) \ {0} : (I±λ,µ)′(w)H−1(w)h(H−1(w)) = 0 } . Every nontrivial critical point of I±λ,µ is contained in N±. For simplicity, we prove all results taking in account the functional Iλ,µ instead of I+ λ,µ and I−λ,µ because all the calculations are exactly the same in the three cases: Iλ,µ, I+ λ,µ and I−λ,µ. We mean that, in the sequel, finding a critical point of Iλ,µ, we prove simultaneously that also I+ λ,µ and I−λ,µ possess critical points. Henceforth, N = {w ∈ H1(Ω) \ {0}; I ′λ,µ(w)H−1(w)h(H−1(w)) = 0} and I ′λ,µ(w)H−1(w)h(H−1(w)) = ∫ Ω ( 1 + H−1(w)h′(H−1(w)) h(H−1(w)) ) |∇w|2dx+ ∫ Ω |H−1(w)|2dx + λ ∫ Ω |H−1(w)|qdx− ∫ Ω |H−1(w)|2·2 ∗ dx + µ ∫ ∂Ω g(x,H−1(w))H−1(w)dσx. (3.1) Lemma 3.1. If w ∈ H1(Ω) \ {0}, with w ≥ 0, there exists tw = tλ,µ(w) > 0 such that tww ∈ N . In particular, N 6= ∅. Proof. Consider the continuous function ξ(t) := I ′λ,µ(tw)H−1(tw)h(H−1(tw)), t > 0. From (3.1) we have ξ(t) ≥ t2 [ ∫ Ω |∇w|2dx− 1 t2 ∫ Ω |H−1(tw)|2·2 ∗ dx− µ t2 ∫ ∂Ω g(x,H−1(tw))H−1(tw)dσx ] . Property (4) ensures that 1 t2 ∫ Ω |H−1(tw)|2·2∗dx → 0 as t → 0+ and hypothesis (A6) guarantees that µ t2 ∫ ∂Ω g(x,H−1(tw))H−1(tw)dσx → 0 as t→ 0+. Therefore, ξ(t) > 0 for t > 0 small enough. (3.2) EJDE-2021/SI/01 QUASILINEAR EQUATIONS WITH CRITICAL GROWTH 337 On the other hand, from (A5) and (A2), ξ(t) = t2 [ ∫ Ω ( 1 + H−1(tw)h′(H−1(tw)) h(H−1(tw)) ) |∇w|2dx + 1 t2 ∫ Ω |H−1(tw)|2dx+ λ t2−q/2 ∫ Ω |H−1(tw)|q tq/2 dx − t2 ∗−2 ∫ Ω |H−1(tw)|2·2∗ t2∗ dx− µ t2 ∫ ∂Ω g(x,H−1(tw))H−1(tw)dσx ] ≤ t2 [ 2 ∫ Ω |∇w|2dx+ 1 t2 ∫ Ω |H−1(tw)|2dx + λ t2−q/2 ∫ Ω |H−1(tw)|q tq/2 dx− t2 ∗−2 ∫ Ω |H−1(tw)|2·2∗ t2∗ dx ] . (3.3) By property (8), we obtain the following three convergences: λ t2−q/2 ∫ Ω |H−1(tw)|q tq/2 dx→ 0,∫ Ω |H−1(tw)|2·2∗ t2∗ dx→ (2/h∞)2∗ ∫ Ω |w|2 ∗ dx > 0, 1 t2 ∫ Ω |H−1(tw)|2dx→ 0 as t→ +∞ since q < 4. These convergences applied in (3.3) yield ξ(t) < 0 (3.4) for values of t > 0 large enough. Since ξ is a continuous function, from (3.2) and (3.4), there exists at least one tw > 0 such that ξ(tw) = 0, that is, tww ∈ N , and the lemma is proved. � Remark 3.2. In the case of I−λ,µ in the previous lemma, we consider w ≤ 0 instead of w ≥ 0. Lemma 3.3. The set N is a C1 manifold. Proof. Define Jλ,µ(w) := I ′λ,µ(w)h(H−1(w))H−1(w) and let w ∈ N . A direct calculation gives us d ds ( 1 + H−1(s)h′(H−1(s)) h(H−1(s)) ) = 1 h2(H−1(s)) ( h′(H−1(s) +H−1(s)h′′(H−1(s)))− H−1(s)(h′(H−1(s)))2 h(H−1(s)) ) , and then, by (A8), J ′λ,µ(w)h(H−1(w))H−1(w) = ∫ Ω H−1(w)h′(H−1(w)) h(H−1(w)) |∇w|2dx+ ∫ Ω |H−1(w)|2h′′(H−1(w)) h(H−1(w)) |∇w|2dx − ∫ Ω |H−1(w)|2(h′(H−1(w)))2 h2(H−1(w)) |∇w|2dx + 2 ∫ Ω ( 1 + H−1(w)h′(H−1(w)) h(H−1(w)) )2 |∇w|2dx+ 2 ∫ Ω |H−1(w)|2dx 338 L. MAIA, J. C. OLIVEIRA JUNIOR, R. RUVIARO EJDE/SI/01 + λq ∫ Ω |H−1(w)|qdx− 2 · 2∗ ∫ Ω |H−1(w)|2·2 ∗ dx − µ ∫ ∂Ω (g′(x,H−1(w))H−1(w)2 + g(x,H−1(w))H−1(w))dσx = 2 ∫ Ω ( 1 + H−1(w)h′(H−1(w)) h(H−1(w)) ) |∇w|2dx + 2 ∫ Ω H−1(w)h′(H−1(w)) h(H−1(w)) |∇w|2dx+ ∫ Ω (H−1(w)h′(H−1(w)) h(H−1(w)) + |H−1(w)|2h′′(H−1(w)) h(H−1(w)) + [H−1(w)]2(h′(H−1(w)))2 h2(H−1(w)) ) |∇w|2dx + 2 ∫ Ω |H−1(w)|2dx+ λq ∫ Ω |H−1(w)|qdx− 2 · 2∗ ∫ Ω |H−1(w)|2·2 ∗ dx − µ ∫ ∂Ω (g′(x,H−1(w))H−1(w)2 + g(x,H−1(w))H−1(w))dσx. Applying firstly hypothesis (A3) (see Remark 1.1) and after using (A2), we obtain J ′λ,µ(w)h(H−1(w))H−1(w) ≤ 4 ∫ Ω ( 1 + H−1(w)h′(H−1(w)) h(H−1(w)) ) |∇w|2dx+ 2 ∫ Ω |H−1(w)|2dx + λq ∫ Ω |H−1(w)|qdx− 2 · 2∗ ∫ Ω |H−1(w)|2·2 ∗ dx − µ ∫ ∂Ω (g′(x,H−1(w))H−1(w)2 + g(x,H−1(w))H−1(w))dσx. Since w ∈ N , it follows that 4 ∫ Ω ( 1 + H−1(w)h′(H−1(w)) h(H−1(w)) ) |∇w|2dx = −4 ∫ Ω |H−1(w)|2dx− 4λ ∫ Ω |H−1(w)|qdx +4 ∫ Ω |H−1(w)|2·2 ∗ dx, + 4µ ∫ ∂Ω g(x,H−1(w))H−1(w)dσx and, once q − 4 < 0, one obtains from assumption (A7) that J ′λ,µ(w)h(H−1(w))H−1(w) ≤ −2 ∫ Ω |H−1(w)|2dx+ λ(q − 4) ∫ Ω |H−1(w)|qdx+ (4− 2 · 2∗) ∫ Ω |H−1(w)|2·2 ∗ dx − µ ∫ ∂Ω (g′(x,H−1(w))H−1(w)2 − 3g(x,H−1(w))H−1(w))dσx < (4− 2 · 2∗) ∫ Ω |H−1(w)|2·2 ∗ dx < 0. Hence, J ′λ(w) 6= 0 for all w ∈ N , what proves thatN is a C1 manifold and completes the proof. � Lemma 3.4. Let (wn) be a sequence such that wn ∈ N and Iλ,µ(wn) → c, as n→ +∞. Then (wn) is bounded. EJDE-2021/SI/01 QUASILINEAR EQUATIONS WITH CRITICAL GROWTH 339 Proof. Firstly, we claim that the sequence (H−1(wn)) ⊂ H1(Ω) is bounded. Indeed, consider the sequence (ϕn) defined by ϕn = H−1(wn)h(H−1(wn)), observe that by (5), we have |ϕn|22 ≤ 4|wn|22 for all n ≥ 1. Since by property (2), d dt ( H−1(t) ) h(H−1(t)) = 1 for all t ∈ R, we obtain ∇ϕn = d dt [H−1(t)h(H−1(t))] ∣∣∣ t=wn ∇wn = ( 1 + H−1(wn)h′(H−1(wn)) h(H−1(wn)) ) ∇wn. Therefore, |∇ϕn| = ( 1 + H−1(wn)h′(H−1(wn)) h(H−1(wn)) ) |∇wn| ≤ (1 + β)|∇wn|, where we used (1.5), choosing t = H−1(wn). Thus, ϕn ∈ H1(Ω) with ‖ϕn‖ ≤ C‖wn‖ for some C > 0. Recalling that (wn) ⊂ N , i.e., I ′λ,µ(wn)ϕn = 0, we have c+ on(1) ≥ Iλ,µ(wn)− σI ′λ,µ(wn)ϕn ≥ ∫ Ω (1 2 − σ(1 + β) ) |∇wn|2dx+ (1 2 − σ ) ∫ Ω |H−1(wn)|2dx + λ (1 q − σ ) ∫ Ω |H−1(wn)|qdx+ ( σ − 1 2 · 2∗ ) ∫ Ω |H−1(wn)|2·2 ∗ dx + ∫ ∂Ω ( σg(x,H−1(wn))H−1(wn)−G(x,H−1(wn)) ) dσx, (3.5) where (1.5) was used. By hypothesis (A5), it follows that∫ Ω (1 2 − σ(1 + β) ) |∇wn|2dx+ (1 2 − σ ) ∫ Ω |H−1(wn)|2dx ≤ c+ on(1). (3.6) Suppose by contradiction that, up to a subsequence, ‖wn‖ → +∞ as n→ +∞ and consider vn := wn ‖wn‖ . Since ‖vn‖ = 1, by the Sobolev embedding, vn → v strongly in L2(Ω). From (3.6) and hypothesis (A2), we have∫ Ω |∇vn|2dx ≤ on(1). Since 1 = ‖vn‖2 = ∫ Ω |∇vn|2dx+ ∫ Ω v2 ndx, one has ∫ Ω v2dx = 1 and therefore v 6= 0. Dividing (3.6) by ‖wn‖, and using (A5) and (A2), one obtains on(1) ≥ ∫ Ω |H−1(wn)|2 ‖wn‖ dx = ∫ Ω ( H−1(vn‖wn‖) |vn|1/2‖wn‖1/2 )2 |vn|dx. By property (8) and noting that v 6= 0 and ‖wn‖ → +∞ as n → +∞ in a subset Ω0 of Ω of positive measure, we obtain 0 ≥ lim inf n→+∞ ∫ Ω ( H−1(vn‖wn‖) |vn|1/2‖wn‖1/2 )2 |vn|dx ≥ ∫ Ω0 2 h∞ |v|dx > 0. This contradiction shows that (wn) is bounded in H1(Ω). � Let us define mλ,µ = inf N Iλ,µ(w). (3.7) The next result will provide a positive lower bound for the function defined by Ψ(w) = |∇w|22 + |H−1(w)|22 for w ∈ N , and consequently a positive lower bound 340 L. MAIA, J. C. OLIVEIRA JUNIOR, R. RUVIARO EJDE/SI/01 for mλ,µ. Its proof depends on the essential role played by the linear term u in problem (1.3), which is responsible for the term |H−1(w)|22 in the definition of the functional Iλ,µ. Lemma 3.5. There exists a positive constant c0 > 0, which does not depend on λ but does on µ, such that |∇w|22 + |H−1(w)|22 ≥ c0 for all w ∈ N . Furthermore, it holds that mλ,µ > c1 > 0 for some c1 > 0. Proof. We have∫ Ω ( 1 + H−1(w)h′(H−1(w)) h(H−1(w)) ) |∇w|2dx+ ∫ Ω |H−1(w)|2dx+ λ ∫ Ω |H−1(w)|qdx = ∫ Ω |H−1(w)|2·2 ∗ dx+ µ ∫ ∂Ω g(x,H−1(w))H−1(w)dσx that yields (since sh′(s) ≥ 0 for all s ∈ R) Ψ(w) ≤ ∫ Ω |H−1(w)|2·2 ∗ dx+ µ ∫ ∂Ω g(x,H−1(w))H−1(w)dσx. (3.8) From Lemma 2.1 with p = 2 · 2∗, for all ε > 0, there is a positive constant Cε > 0 such that Ψ(w) ≤ |H−1(w)|2·2 ∗ 2·2∗ + µε ∫ ∂Ω |H−1(w)|4dσx + µCε ∫ ∂Ω |H−1(w)|2·2∗dσx. (3.9) By the trace Sobolev embeddings H1(Ω) ↪→ L4(∂Ω), for 2 ≤ 4 ≤ 2∗ = 4, and it follows from property (2) that∫ ∂Ω |H−1(w)|4dσx ≤ C (∫ Ω (|∇H−1(w)|2 + |H−1(w)|2)dx )2 = C [ ∫ Ω ( 1 h2(H−1(w)) |∇w|2 + |H−1(w)|2 ) dx ]2 ≤ C (∫ Ω (|∇w|2 + |H−1(w)|2)dx )2 = CΨ(w)2. (3.10) Finally, the trace Sobolev embeddings one more time, now applied to (H−1(w))2, together with property (H2), produce∫ ∂Ω |H−1(w)|2·2∗dσx ≤ C (∫ Ω (|∇(H−1(w))2|2 + |H−1(w)|4)dx )2∗/2 ≤ C (∫ Ω 4|H−1(w)|2 h2(H−1(w)) |∇w|2 + |H−1(w)|4)dx )2∗/2 ≤ C (∫ Ω (|∇w|2 + |H−1(w)|4)dx )2∗/2 ≤ CΨ(w)2∗/2 + C (∫ Ω |H−1(w)|4)dx )2∗/2 ≤ CΨ(w)2∗/2 + CΨ(w)2∗ , (3.11) where, in the last inequality, we used the same calculations as in (3.10), and applied the Sobolev embedding H1(Ω) ↪→ L4(Ω) for 2 ≤ 4 ≤ 2∗ = 6. The same arguments EJDE-2021/SI/01 QUASILINEAR EQUATIONS WITH CRITICAL GROWTH 341 also show that ∫ Ω |H−1(w)|2·2 ∗ dx ≤ CΨ(w)2∗/2 + CΨ(w)2∗ . (3.12) Now, using (3.10), (3.11) and (3.12) in (3.9), it follows that Ψ(w) ≤ C ( Ψ(w)2∗/2 + Ψ(w)2∗ + Ψ(w)2 + Ψ(w)2∗/2 + Ψ(w)2∗ ) . Since Ψ(w) > 0 and 2∗/2, 2∗/2, 2∗ and 2∗ are bigger than 1, necessarily, there exists a positive constant c0 > 0 such that Ψ(w) ≥ c0 > 0 and we prove the first part of this result. The second part may be obtained following the calculation in (3.5) of Lemma 3.4 and by using the first part of this lemma. � Remark 3.6. Here is the point that we highlight the reason for having fixed the dimension of the Euclidean space in N = 3. In the previous result, we need to relate the term |H−1(w)|44 with the gradient norm |∇w|22, for w ∈ H1(Ω), to get the positive lower bound for Ψ(w). However, since Ω is a bounded domain in RN and the space H1(Ω) contains functions that are not zero on ∂Ω, every embedding theorem brings up the norm of w in L2(Ω), which does not compare with the gradient norm. Thus, we use the trace Sobolev embedding to deal with ∫ ∂Ω |H−1(w)|4dσx and then we need that 2 ≤ 4 ≤ 2 · 2∗, what implies N = 3. Lemma 3.7. Let (wn) be a (PS)c sequence for Iλ,µ|N restrict to the set N . Then I ′λ,µ(wn)→ 0 as n→ +∞ in the dual space ( H1(Ω) )∗ . Proof. Let ϕn = H−1(wn)h(H−1(wn)) as in (3.5). We claim that the sequence( ∫ Ω |H−1(wn)|2·2∗dx ) does not converge to zero as n → +∞. Otherwise, once we have I ′λ,µ(wn)ϕn = 0 and the growth of g is subcritical, by Hölder inequality and since |Ω| <∞, one obtains∫ Ω ( 1 + H−1(wn)h′(H−1(wn)) h(H−1(wn)) ) |∇wn|2dx+ on(1) = ∫ Ω |H−1(wn)|2·2 ∗ dx = on(1). Therefore, by Lemma 3.5, 0 < c0 ≤ Ψ(wn) = ∫ Ω |∇wn|2dx+ on(1) ≤ ∫ Ω ( 1 + H−1(wn)h′(H−1(wn)) h(H−1(wn)) ) |∇wn|2dx = on(1), which is a contradiction. Hence, there exists C > 0 such that∫ Ω |H−1(wn)|2·2 ∗ dx ≥ C > 0. This and Lemma 3.3 imply that the sequence (J ′λ, µ(wn)ϕn) does not converge to zero as n→ +∞. The next arguments are standard and the lemma follows. � Despite being a minimizing sequence in N for functional Iλ,µ, it may not be a sequence that converges weakly to a solution of problem (1.8). In the next result, we will show the existence of an appropriate minimizing sequence for our purpose. 342 L. MAIA, J. C. OLIVEIRA JUNIOR, R. RUVIARO EJDE/SI/01 Proposition 3.8. Let mλ,µ as in (3.7). There exists a bounded (PS)mλ,µ sequence (wn) ⊂ N for functional Iλ,µ. Proof. Note that the functional Jλ,µ(w) := I ′λ,µ(w)H−1(w)h(H−1(w)) belongs to the space C(H1(Ω),R), whence N is a complete metric subspace. By Lemma 3.5, Iλ,µ is bounded from below on N and Iλ,µ is a C1-functional, hence we may apply Ekeland’s Principle to ensure the existence of a (PS)mλ,µ sequence (wn) ⊂ N for functional Iλ,µ|N . Finally, by Lemma 3.4, it is bounded and, by Lemma 3.7, it is a (PS)mλ,µ sequence in the whole space H1(Ω). � Lemma 3.9. There exists µ∗ > 0 such that, for all µ ∈ [µ∗,+∞) and for λ in a bounded set, it holds that mλ,µ < 4−N/2(Sh∞)N/2/N . Proof. Let w ∈ H1(Ω) \ {0}, w ≥ 0 (in the case of I−λ,µ, we choose w ≤ 0), and consider tλ,µ > 0 given by Lemma 3.1, which shows that tλ,µw ∈ N . Then, from hypothesis (A2), we have 2t2λ,µ ∫ Ω |∇w|2dx+ ∫ Ω |H−1(tλ, µw)|2dx+ λ ∫ Ω |H−1(tλ,µw)|qdx ≥ ∫ Ω |H−1(tλ,µw)|2·2 ∗ dx+ µ ∫ ∂Ω g(x,H−1(tλ,µw))H−1(tλ,µw)dσx, (3.13) which implies from property (3), and by assumption (A5) that C(t2λ,µ + t q/2 λ,µ) ≥ ∫ Ω |H−1(tλ,µw)|2·2 ∗ dx, for some C > 0 which does not depend on µ and λ in a bounded set. We claim that (tλ,µ)µ≥1 is bounded as µ→ +∞. Otherwise, it follows that C ( 1 + 1 t 2−q/2 λ,µ ) ≥ ∫ Ω 1 t2λ,µ |H−1(tλ,µw)|2·2 ∗ dx = ∫ Ω |H−1(tλ,µw)|4 t2λ,µ |H−1(tλ,µw)|2·2 ∗−4dx. But this is an absurd in view of properties (H8)− (H9) and q/2 < 2. So, let t0 ≥ 0 be such that tλ,µ → t0 as µ→ +∞ uniformly on λ in a bounded set. This implies the following boundedness 2t2λ,µ ∫ Ω |∇w|2dx+ ∫ Ω |H−1(tλ,µw)|2dx+ λ ∫ Ω |H−1(tλ,µw)|qdx ≤ C for all µ > 0 and for some constant C > 0, whence by (3.13) µ ∫ ∂Ω g(x,H−1(tλ,µw))H−1(tλ,µw)dσx ≤ C, for all µ > 0 and λ in a bounded set. By assumptions (A4) and (A5), this implies, necessarily, that t0 = 0. Therefore mλ,µ ≤ Iλ,µ(tλ,µw) ≤ t2λ,µ 2 |∇w|22 + 1 2 |H−1(tλ, µw)|22 + λ q |H−1(tλ,µw)|qq → 0 (3.14) as µ → +∞ and λ is in a bounded set. The lemma follows choosing µ sufficiently large. � EJDE-2021/SI/01 QUASILINEAR EQUATIONS WITH CRITICAL GROWTH 343 Proof of Theorem 1.5. By Lemma 3.9, we take µ > 0 sufficiently large such that mλ,µ < 4−N/2(Sh∞)N/2 N and from Proposition 3.8, there is a (PS)mλ,µ sequence (wn) for functional Iλ,µ, which converges strongly to wλ,µ ∈ H1(Ω) in view of Lemma 2.6. By Lemma 3.5, Iλ,µ(w) = limn→+∞ Iλ,µ(wn) ≥ mλ,µ > 0 and, consequently, wλ,µ 6= 0. Since I ′λ, µ(wλ, µ) = 0, then wλ,µ ∈ N is a nontrivial ground state solution of problem (1.8). � Remark 3.10. We observe that any ground state solution wλ, µ obtained in Theo- rem 1.5 as a minimum on this new natural constraint N is always a signed solution since w+ λ,µ and w−λ,µ belong to N . Proof of Corollary 1.6. This is a direct consequence of the (3.14) and the fact that both solutions are ground state (for the respective functional). � Proof of Theorem 1.7. It is an application of [13, Theorem 6.31] and of the remark there subsequent to the theorem. � Acknowledgment. The second author would like to thank the warm hospitality of the Department of Mathematics at University of Brasilia, where part of this research was developed. Acknowledgments. This research was partially supported by FAPDF, CAPES, CNPq grant 309866/2020-0, and grant 316386/2021-9. References [1] J. F. L. Aires, M. A. S. Souto; Existence of solutions for a quasilinear Schrödinger equation with vanishing potentials, J. Math. Anal. Appl., 416 (2014), 924–946. [2] A. V. Borovskii, A. L. Galkin; Dynamical modulation of an ultrashort high-intensity laser pulse in matter, JETP 77, 4 (1993), 562–573. [3] A. de Bouard, N. Hayashi, J. C. Saut; Global existence ofsmall solutions to a relativistic nonlinear Schrödinger equation, Comm. Math. Phys., 189 (1997), 73–105. [4] A. de Bouard, N. Hayashi, J. C. Saut; Scattering problem and asymptotics for a relativistic nonlinear Schrödinger equation, Nonlinearity, 12 (1999), 1415–1425. [5] H. Brezis, E. H. Lieb; A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc., 88, no. 3, (1983), 486–490. [6] J. Chabrowski, J. Yang; On the Neumann problem with combined nonlinearities, Ann. Polon. Math. 85 (2005), 239–250. [7] W. Cintra, E. Medeiros, U. Severo; On positive solutions for a class of quasilinear elliptic equations, Z. Angew. Math. Phys. 70 (2019), no. 3, Paper No. 79, 17 pp. [8] M. Colin, L. Jeanjean; Solutions for a quasilinear Schrödinger equation: a dual approach, Nonlinear Analysis, 56 (2004), 213–226. [9] Y. Deng, W. Huang, S. Zhang; Ground state solutions for quasilinear Schrödinger equa- tions with critical growth and lower power subcritical perturbation, Adv. Nonlinear Stud., 19 (2019), no. 1, 219–237. [10] G. M. Figueiredo, J. R. S. Júnior, A. Suárez; Structure of the set of positive solutions of a non-linear Schrödinger equation, Israel Journal of Mathematics, 227 (2018), 485–505. [11] G. M. Figueiredo, R. Ruviaro, J.C. Oliveira Junior; Quasilinear Equations Involving Critical Exponent and Concave Nonlinearity at the Origin, Milan Journal of Mathematics, 88 (2020), 295–314. [12] M. Furtado, E. D. Silva, M. L. Silva; Existence of solution for a generalized quasilinear elliptic problem. J. Math. Phys., 58 (2017), no. 3, 031503, 14 pp. [13] D. Gilbarg, N. S. Trundiger; Elliptic Partial Differential Equation of Second Order, Grundlehren, 2nd, edn., vol. 224. Springer, Berlin (1983). [14] B. Hartmann, W. J. Zakrzewski; Electrons on hexagonal lattices and applications to nan- otubes, Phys. Rev. B, 68 (2003), 184302. 344 L. MAIA, J. C. OLIVEIRA JUNIOR, R. RUVIARO EJDE/SI/01 [15] S. Kurihura; Large-amplitude quasi-solitons in superfluid films, J. Phys. Soc. Japan, 50 (1981), 3262–3267. [16] E. W. Laedke, K. H. Spatschek, L. Stenflo; Evolution theorem for a class of perturbed envelope soliton solutions, J. Math. Phys., 24 (1983), 2764–2769. [17] H. Lange, M. Poppenberg, H. Teismann; Nash–Moser methods for the solution of quasilinear Schrödinger equations, Comm. Partial Differential Equations, 24 (1999), 1399–1418. [18] A. G. Litvak, A. M. Sergeev; One dimensional collapse of plasma waves, JETP Lett., 27 (1978), 517–520. [19] J. Liu, Y. Wang, Z. Wang; Soliton solutions for quasilinear Schrödinger equations II, J. Differential Equations, 187 (2003), 473–493. [20] J. Liu, Y. Wang, Z. Q. Wang; Solutions for quasilinear Schrödinger equation via Nehari method, Commun. PDE., 29 (2004), 879–901. [21] S. Lojasiewicz Jr., E. Zehnder; An inverse function theorem in Fréchet spaces, J. Funct. Anal., 33 (1979), 165–174. [22] L. A. Maia, J. C. Oliveira Junior, R. Ruviaro; A non-periodic and asymptotically linear indefinite variational problem in RN , Indiana University Mathematics Journal, 66 (2017), no. 1, 31–54. [23] L. A. Maia, J. C. Oliveira Junior, R. Ruviaro; A quasi-linear Schrödinger equation with indefinite potential, Complex Var. Elliptic Equ., 61 (2016), no, 4, 574–586. [24] V. G. Makhankov, V. K. Fedyanin; Non-linear effects in quasi-one-dimensional models of condensed matter theory, Phys. Rep. 104, (1984), 1–86. [25] J. M. do Ó, U. Severo; Quasilinear Schrödinger equations involving concave and convex nonlinearities, Comm. on pure and app. anal., 8 (2009), 621–644. [26] F. O. de Paiva, A. E. Presoto; Semilinear elliptic problems with asymmetric nonlinearities, J. Math. Anal. Appl., 409 (2014), 254–262. [27] F. O. V. de Paiva, E. Massa; Multiple solutions for some elliptic equations with a nonlinearity concave at the origin, Nonlinear Anal., 66 (2007), 2940–2946. [28] M. Poppenberg, K. Schmitt, Z. Q. Wang; On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations, 14 (2002), 329–344. [29] B. Ritchie; Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. 50E, 2 (1994), 687–689. [30] D. Ruiz, G. Siciliano; Existence of ground states for a nonlinear Schrödinger equation, Non- linearity, 23 (2010), 1221–1233. [31] Y. Shen, Y. Wang; Soliton solutions for generalized quasilinear Schrödinger equations. Non- linear Analysis, 80 (2013) 194-201. Liliane de A. Maia Universidade de Braśılia, Departamento de Matemática, 70.910-900, Braśılia, DF, Brazil Email address: lilimaia@unb.br José Carlos Oliveira Junior Universidade Federal do Tocantins, Departamento de Matemática, 77.824-838, Araguáına, TO, Brazil Email address: jc.oliveira@uft.edu.br Ricardo Ruviaro Universidade de Braśılia, Departamento de Matemática, 70.910-900, Braśılia, DF, Brazil Email address: ruviaro@unb.br 1. Introduction 2. A compactness result 2.1. (PS) condition in the correct range 3. Existence of two solutions Acknowledgment Acknowledgments References