Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 99, pp. 1–14. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ASYMPTOTICALLY LINEAR AND SUPERLINEAR ELLIPTIC EQUATIONS WITH GRADIENT TERMS YUANHONG WEI, JIAN TIAN Abstract. In this article we establish the existence of solutions for elliptic problem involving a gradient term. To handle the so-called non-variational problem, we use a variational methods. We assume that the nonlinear term satisfies an asymptotically linear growth condition or a superlinear growth condition. We show the existence of at least one positive solution and one negative solution. 1. Introduction This article concerns the existence of solutions for nonlinear elliptic equations with a gradient term, −∆u = f(x, u,∇u) in Ω, u = 0 on ∂Ω, (1.1) where Ω ⊂ Rn, n ≥ 1, is bounded, smooth and open with the boundary ∂Ω, f : Rn × R× Rn → R is continuous. There is considerable attention on the existence of solution for nonlinear elliptic problems without the gradient term by using various variational methods. If f de- pends on the gradient of the solution, the problem is non-variational where the well developed critical point theory does not work. There have been quite a few works focusing on this kind of problems by using the topological degree theory; see for ex- ample, Amann-Crandall [1], Brezis-Turner [4], Pohožaev [9], Xavier [14], Yan [15]. Some innovative ideas were proposed by De Figueiredo-Girardi-Matzeu [5], based on the application of variational methods for the problem with the fixed gradient term, as well as the iterative method. The existence of solution was established while f satisfies the classical condition by Ambrosetti-Rabinowitz [2]: (AR) there exist ν > 2 and t0 > 0 such that 0 < νF (x, s, ξ) ≤ sf(x, s, ξ), x ∈ Ω, t ≥ t0, ξ ∈ Rn, where F (x, s, ξ) = ∫ s 0 f(x, t, ξ)dt. The main purpose of this article is to establish the existence of solution for (1.1) under the asymptotically linear growth condition or the superlinear growth con- dition. To handle the so-called non-variational problem, we follow the framework developed by De Figueiredo-Girardi-Matzeu [5]. 2010 Mathematics Subject Classification. 35B09, 35J20, 35A01. Key words and phrases. Positive solution; nonlinearity; gradient term; iterative method; Mountain pass theorem. c©2020 Texas State University. Submitted March 9, 2020. Published September 23, 2020. 1 2 Y. WEI, J. TIAN EJDE-2020/99 It is well-known that the role of (AR) is to ensure the boundedness of the Palais- Smale sequence of the Euler-Lagrange functional. However, the asymptotically linear growth condition eliminates (AR) condition, thereby bringing some new ob- stacles to the argument. Besides, the nonlinearity of asymptotically linear type will compete with the spectra of the linear operator, which requests us to develop a new and different argument from the one for the superlinear case. There are some works related to asymptotically linear problems, such as Jeanjean-Tanaka [8], Stuart-Zhou [10] for second order elliptic equation, Wei-Su [13] for non-local elliptic equation, and Wei [12] for fourth-order elliptic equation etc. For more applications of this problem, we refer to Girardi-Matzeu [7] for periodic solutions of Hamiltonian system and Dong-Wei [6] for radial solutions of elliptic equation etc. For the asymptotically linear case, we assume that the nonlinearity f satisfies the following assumptions. (H1) f(x, 0, ξ) = 0 for all x ∈ Ω, ξ ∈ Rn. (H2) The following holds uniformly for x ∈ Ω, ξ ∈ Rn: 0 ≤ lim inf s→0 f(x, s, ξ) s ≤ lim sup s→0 f(x, s, ξ) s < λ1 < lim inf |s|→+∞ f(x, s, ξ) s ≤ lim sup |s|→+∞ f(x, s, ξ) s < +∞, where λ1 is the first eigenvalue of −∆ with the Dirichlet boundary condi- tion. (H3) There exists M > 0 such that for any x ∈ Ω, s ∈ R, ξ ∈ Rn, it holds∣∣f(x, s, ξ) s ∣∣ ≤M. (H4) f satisfies the local Lipschitz conditions: there exist constants L and K such that |f(x, s1, ξ)− f(x, s2, ξ)| ≤ L|s1 − s2| for any x ∈ Ω, |s1| ≤ ρ1, |s2| ≤ ρ1, |ξ| ≤ ρ2, and |f(x, s, ξ1)− f(x, s, ξ2)| ≤ K|ξ1 − ξ2| for any x ∈ Ω, |s| ≤ ρ1, |ξ1| ≤ ρ2 and |ξ2| ≤ ρ2, where ρ1, ρ2 are positive constants to be determined. Moreover, the Lipschitz constants L and K satisfy L+ √ λ1K < λ1. The following theorem concerns the asymptotically linear case. Theorem 1.1. Under hypotheses (H1)–(H4), equation (1.1) possesses at least one positive solution and one negative solution. Remark 1.2. Consider f(x, s, ξ) = h(s)(1 + τg(ξ)), where τ is a constant satisfying |τ | < 1/2, g ∈ C1(Rn), |g(ξ)| < 1, and h(s) =  λ1(2s+ 3 2Λ), s < −Λ; λ1 2 s, |s| ≤ Λ; λ1(2s− 3 2Λ), s > Λ. EJDE-2020/99 ELLIPTIC EQUATIONS WITH GRADIENT TERMS 3 It is apparent that h is continuous. Then (H1)–(H4) are satisfied for τ small enough and Λ large enough. In addition, we study the superlinear problem under the following hypotheses which are weaker than (AR). (H5) lims→0 f(x, s, ξ)/s = 0 uniformly for x ∈ Ω, ξ ∈ Rn. (H6) For every l > 0 there exists C1 > 0 such that F (x, s, ξ) ≥ ls2 − C1, x ∈ Ω, s ∈ R, ξ ∈ Rn, where F (x, s, ξ) = ∫ s 0 f(x, t, ξ)dt. (H7) There exist constants c0 > 0 and q ∈ (1, 2∗ − 1) such that for any x ∈ Ω, s ∈ R, ξ ∈ Rn, it holds |f(x, s, ξ)| ≤ c0(1 + |s|q), where 2∗ = { 2n n−2 , n > 2; +∞, n ≤ 2. (H8) f(x,s,ξ) |s| is increasing with respect to s in (−∞, 0) and (0,+∞). Theorem 1.3. Under hypotheses (H4)–(H8), equation (1.1) possesses at least one positive solution and one negative solution. Remark 1.4. Consider the superlinear case f(x, s, ξ) = εh(x)|s|αsg(ξ), where ε > 0, α ∈ (0, 2∗ − 2), h ∈ C(Ω), and g ∈ C1(Rn) ∩ L∞(Rn). Suppose that there exists a constant b such that 0 < b ≤ h(x) and 0 < b ≤ g(ξ). Then, for ε small enough, all assumptions of Theorem 1.3 are satisfied. There exists ε0 > 0, such that for all 0 < ε < ε0, problem (1.1) has at least one positive solution and one negative solution. This article is mainly motivated by De Figueiredo-Girardi-Matzeu [5], while both main results and approaches are different from the existing ones. On the one hand, unlike the assumptions in the above reference, the condition (AR) is not imposed. This means even in the superlinear case, the assumptions of this paper are slightly weaker. The asymptotically linear problem is also studied, which can be seen as an asymptotically linear version of [5]. On the other hand, we try to consider the superlinear problem in a different variational framework, including the Nehari manifold technique. Our arguments are based on some methods of nonlinear analysis. Mountain pass theorem, iterative technique and contraction mapping theorem are essential to the proofs of main results. This article is organized as follows. In Section 2, we introduce some preliminaries and an auxiliary problem. The existence of solution for the auxiliary problem of the asymptotically linear case is established in Section 3 by means of Mountain pass theorem. Some uniform estimates are obtained to describe the property of the solution. In Section 4, we study the superlinear auxiliary problem. The Nehari manifold is defined, which transfers the nontrivial solution to the extreme point of Euler-Lagrange functional on the constraint manifold. The proofs of main results are given in Section 5, by the fixed point theorem and the iterative method. 4 Y. WEI, J. TIAN EJDE-2020/99 2. Preliminaries and auxiliary problem For any v ∈ C1 0 (Ω), consider the auxiliary problem −∆u = f(x, u,∇v) in Ω, u = 0 on ∂Ω. (2.1) The Euler-Lagrange functional of (2.1) is Jv(u) = 1 2 ∫ Ω |∇u|2dx− ∫ Ω F (x, u,∇v)dx, u ∈ H1 0 (Ω). It is well known that the norm ‖u‖ = (∫ Ω |∇u|2dx )1/2 is an equivalent norm in H1 0 (Ω). Denote Φv(u) = ∫ Ω F (x, u,∇v)dx, then Jv(u) = 1 2 ‖u‖2 − Φv(u). Since (H3) of Theorem 1.1 or (H7) of Theorem 1.3 holds, we know that Jv is C1, and Φ′v is completely continuous. The weak solution of (2.1) is equivalent to the critical point of Jv. Let λ1 be the first eigenvalue of −∆ with the Dirichlet boundary condition and the corresponding eigenfunctions of λ1 is denoted by ϕ1. It is well known that λ1 > 0 is simple and ϕ1 is positive. Set u+ = max{u, 0}, u− = min{u, 0}. For any v ∈ C1 0 (Ω), consider the problem −∆u = f±(x, u,∇v) in Ω, u = 0 on ∂Ω, (2.2) where f+(x, s, ξ) = { f(x, s, ξ), s ≥ 0, 0, s < 0, f−(x, s, ξ) = { 0, s > 0, f(x, s, ξ), s ≤ 0. We define the corresponding functional J±v : H1 0 (Ω)→ R, by J±v (u) = 1 2 ‖u‖2 − ∫ Ω F±(x, u,∇v)dx, where F±(x, u, v) = ∫ u 0 f±(x, s, v)ds. Denote Φ±v (u) = ∫ Ω F±(x, u,∇v)dx and thus J±v (u) = 1 2 ‖u‖2 − Φ±v (u). EJDE-2020/99 ELLIPTIC EQUATIONS WITH GRADIENT TERMS 5 Obviously, J±v ∈ C1(H1 0 (Ω),R). If u is a critical point of J+ v (J−v ), then u is a weak solution of (2.2). By the weak maximum principle it follows that u ≥ 0 (≤ 0) a.e. in Ω. Thus u is also a solution of problem (2.1). Hence, the nontrivial critical point of J+ v (J−v ) is actually a positive (negative) solution of (2.1). Throughout this paper, denote by ‖ · ‖p the Lp norm in Ω. 3. Asymptotically linear case In this section, we study (2.1) under asymptotically linear conditions. We first show that the functional J±v has the mountain pass geometry. Lemma 3.1. Under the assumptions (H1)–(H3), J±v is unbounded from below. Proof. Since (H1) holds, from (H3) it is apparent that∣∣F (x, s, ξ) s2 ∣∣ ≤ M 2 for x ∈ Ω, s ∈ R, ξ ∈ Rn. Then (H2) implies that there exist ε > 0 and Cε > 0 such that F±(x, s, ξ) ≥ 1 2 (λ1 + ε)|s±|2 − Cε, x ∈ Ω, s ∈ R, ξ ∈ Rn. (3.1) From (3.1) it follows that J±v (±tϕ1) ≤ 1 2 ‖tϕ1‖2 − 1 2 (λ1 + ε) ∫ Ω t2ϕ2 1dx+ ∫ Ω Cεdx ≤ t2 2 ‖ϕ1‖2 − t2 2 (λ1 + ε)‖ϕ1‖22 + Cε|Ω| ≤ 1 2 (1− λ1 + ε λ1 )t2‖ϕ1‖2 + Cε|Ω|, (3.2) where |Ω| denotes the Lebesgue measure of Ω. Then lim t→+∞ J±v (±tϕ1) = −∞, which completes the proof. � Remark 3.2. Obviously, there exists γ > 0, independent of v, such that J±v (±sϕ1) ≤ 0, for all s ≥ γ. Lemma 3.3. Assume that (H1)–(H3) hold. Then there exist r,R > 0 such that J±v (u) ≥ R, if ‖u‖ = r. Proof. From (H1)-(H3), we can find ε0 > 0 and C0 > 0, such that F±(x, s, ξ) ≤ 1 2 (λ1 − ε0)|s|2 + C0|s|2 ∗ . (3.3) Combining (3.3) with Poincaré inequality as well as Sobolev embedding, we have J±v (u) ≥ 1 2 ‖u‖2 − λ1 − ε0 2 ∫ Ω |u|2dx− C0 ∫ Ω |u|2 ∗ dx ≥ ( 1 2 − λ1 − ε0 2λ1 )‖u‖2 − CsC0‖u‖2 ∗ , (3.4) where Cs is the Sobolev constant. Choosing ‖u‖ = r > 0 small enough, it follows that J±v (u) ≥ R > 0. � 6 Y. WEI, J. TIAN EJDE-2020/99 Lemma 3.4. Suppose that (H2) and (H3) hold. Then every Palais-Smale sequence of J±v has a convergent subsequence in H1 0 (Ω). Proof. Since Ω is bounded and (H2) and (H3) hold, it suffices to show that every (PS) sequence {un} is bounded in H1 0 (Ω). We only need to prove the case of J+ v , because the case of J−v is similar. Assume that {un} ⊂ H1 0 (Ω) is a (PS) sequence of J+ v , i.e., J+ v (un)→ c, (J+ v )′(un)→ 0 as n→ +∞. (3.5) From (H2) and (H3) we know that |f+(x, s, ξ)s| ≤ C(1 + |s|2). Then (3.5) implies that for all ϕ ∈ H1 0 (Ω),∫ Ω ( ∇un · ∇ϕ− f+(x, un,∇v)ϕ ) dx→ 0. (3.6) Setting ϕ = un and using Hölder inequality we have ‖un‖2 = ∫ Ω f+(x, un,∇v)undx+ 〈(J+ v )′(un), un〉 ≤ ∫ Ω f+(x, un,∇v)undx+ o(1)‖un‖ ≤ C|Ω|+ C‖un‖22 + o(1)‖un‖. (3.7) We claim that ‖un‖2 is bounded. Assume, by contradiction, that passing to a subsequence, it holds ‖un‖22 → +∞, as n→ +∞. Set ωn = un ‖un‖2 and thus ‖ωn‖2 = 1. From (3.7) we know that ‖ωn‖2 ≤ o(1) + C + o(1) ‖un‖2 · ‖un‖ ‖un‖2 ≤ o(1) + C + o(1)‖ωn‖, which implies that ‖ωn‖ is bounded. Hence, there exists ω ∈ H1 0 (Ω), ‖ω‖2 = 1, such that ωn ⇀ ω in H1 0 (Ω), ωn → ω in L2(Ω), ωn(x)→ ω(x) a.e. in Ω. From (3.6) it follows that∫ Ω ∇ωn · ∇ϕ− ∫ Ω f+(x, un,∇v) ‖un‖2 ϕdx = o(1), ϕ ∈ H1 0 (Ω). (3.8) Taking ϕ = ω−n , we have ‖ω−n ‖ = o(1), which implies ω−(x) = 0 a.e. in Ω and thus ω(x) ≥ 0. If ω(x) = 0, from (H3) it follows that |f+(x, un,∇v)| ‖un‖2 = |f +(x, un,∇v) un |ωn ≤Mωn → 0. Then we have lim n→+∞ f+(x, un,∇v) ‖un‖2 = 0. EJDE-2020/99 ELLIPTIC EQUATIONS WITH GRADIENT TERMS 7 If ω(x) > 0, it follows un = ωn‖un‖2 → +∞. Then (H2) implies that there exists δ > 0, such that lim inf n→+∞ f+(x, un,∇v) ‖un‖2 = lim inf n→+∞ f+(x, un,∇v) un ωn ≥ (λ1 + δ)ω. Hence, from the above two cases we derive that lim inf n→+∞ f+(x, un,∇v) ‖un‖2 ≥ (λ1 + δ)ω (3.9) for all x ∈ Ω. Taking ϕ = ϕ1 in (3.8), we obtain that λ1 ∫ Ω ωϕ1dx = ∫ Ω ∇ω · ∇ϕ1dx = lim n→+∞ ∫ Ω ∇ωn · ∇ϕ1dx = lim n→+∞ ∫ Ω f+(x, un,∇v) ‖un‖2 ϕ1dx. (3.10) Since ϕ1 > 0, it is known from Fatou’s Lemma that∫ Ω lim inf n→+∞ f+(x, un,∇v) ‖un‖2 ϕ1dx ≤ lim n→+∞ ∫ Ω f+(x, un,∇v) ‖un‖2 ϕ1dx. (3.11) Then, from (3.9), (3.10) and (3.11) we obtain λ1 ∫ Ω ωϕ1dx = lim n→+∞ ∫ Ω f+(x, un,∇v) ‖un‖2 ϕ1dx ≥ ∫ Ω lim inf n→+∞ f+(x, un,∇v) ‖un‖2 ϕ1dx ≥ (λ1 + δ) ∫ Ω ωϕ1dx, which implies that ω ≡ 0. However, this fact contradicts with ‖ωn‖ = 1 and hence ‖un‖2 is bounded. Therefore, from (3.7) we know that {un} is bounded in H1 0 (Ω). � Lemma 3.5. Let (H1)–(H3) hold. Then, for any v ∈ C1 0 (Ω), problem (2.1) has at least one positive weak solution u+ v ∈ H1 0 (Ω) and one negative weak solution u−v ∈ H1 0 (Ω). Proof. We define Ψ± = {ψ ∈ C([0, 1], H1 0 (Ω)) : ψ(0) = 0, ψ(1) = ±γϕ1}, where γ is given by Remark 3.2. Let c±v = inf ψ∈Ψ± max s∈[0,1] J±v (ψ(s)). (3.12) Since Lemma 3.1, Lemma 3.3 and Lemma 3.4 hold, Mountain pass theorem implies that c+v (c−v ) is a critical value of J+ v (J−v ). Namely, (J±v )′(u±v ) = 0, J±v (u±v ) = inf ψ∈Ψ± max s∈[0,1] J±v (ψ(s)), which completes the proof. � Now, we establish some uniform estimates for solutions u±v of (2.1) obtained by Lemma 3.5. 8 Y. WEI, J. TIAN EJDE-2020/99 Lemma 3.6. Let v ∈ C1 0 (Ω), and (H2) and (H3) hold. Then there exists a positive constant c0, independent of v, such that ‖u±v ‖ ≥ c0 for all solutions u±v of (2.1) obtained by Lemma 3.5. Proof. Since u±v is a solution of (2.1), we know∫ Ω |∇u±v |2dx = ∫ Ω f±(x, u±v ,∇v)u±v dx. From (H2) and (H3), we know there exist positive constants ε and cε such that |f±(x, s±, ξ)| ≤ (λ1 − ε)|s±|+ cε|s±|2 ∗−1, for any x ∈ Ω, s ∈ R, ξ ∈ Rn. Hence, ∫ Ω |∇u±v |2dx ≤ (λ1 − ε) ∫ Ω |u±v |2dx+ cε ∫ Ω |u±v |2 ∗ dx. By Poincaré inequality and Sobolev inequality, we obtain (1− λ1 − ε λ1 )‖u±v ‖2 ≤ cε‖u±v ‖2 ∗ 2∗ ≤ cεc2 ∗ 0 ‖u±v ‖2 ∗ , which implies the conclusion. � Lemma 3.7. Let (H1)–(H3) hold. Then there exists a positive constant ρ, inde- pendent of v, such that ‖u±v ‖ ≤ ρ for all solutions u±v obtained by Lemma 3.5. Proof. We only give the proof for the case of J+ v , the case of J−v is similar. We suppose, by contradiction, there exist subsequences {vj} and {uvj}, such that {vj} ⊂ C1 0 (Ω), {uvj} ⊂ H1 0 (Ω) and (J+ vj )′(uvj ) = 0, ‖uvj‖ → +∞ as j → +∞. Then for all ϕ ∈ H1 0 (Ω), it holds∫ Ω ( ∇uvj · ∇ϕ− f+(x, uvj ,∇vj)ϕ ) dx = 0. (3.13) We set ωj = uvj ‖uvj ‖ and thus ‖ωj‖ = 1. Hence, there exists ω ∈ H1 0 (Ω), ‖ω‖ = 1 such that ωj ⇀ ω in H1 0 (Ω), ωj → ω in L2(Ω), ωj(x)→ ω(x) a.e. in Ω. From (3.13) it follows∫ Ω ( ∇ωj · ∇ϕ− f+(x, uvj ,∇vj) ‖uvj‖ ϕ ) dx = 0. (3.14) Taking ϕ = ω−j we know ‖ω−j ‖ = 0, which implies ω(x) ≥ 0. If ω(x) = 0, from (H3) it follows that |f+(x, uvj ,∇vj)| ‖uvj‖ = ∣∣f+(x, uvj ,∇vj) uvj ∣∣ωj ≤Mωj → 0. EJDE-2020/99 ELLIPTIC EQUATIONS WITH GRADIENT TERMS 9 Then we have lim j→+∞ f+(x, uvj ,∇vj) ‖uvj‖ = 0. If ω(x) > 0, it follows that uvj = ωj‖uvj‖ → +∞. Then (H2) implies that there exists δ > 0, such that lim inf j→+∞ f+(x, uvj ,∇vj) ‖uvj‖ = lim inf j→+∞ f+(x, uvj ,∇vj) uvj ωj ≥ (λ1 + δ)ω. Hence, from the above two cases, lim inf j→+∞ f+(x, uvj ,∇vj) ‖uvj‖ ≥ (λ1 + δ)ω (3.15) for all x ∈ Ω. Taking ϕ = ϕ1 in (3.14), since ϕ1 > 0, ω ≥ 0, from (3.15) and Fatou’s Lemma we derive λ1 ∫ Ω ωϕ1dx = ∫ Ω ∇ω · ∇ϕ1dx = lim j→+∞ ∫ Ω ∇ωj · ∇ϕ1dx = lim j→+∞ ∫ Ω f+(x, uvj ,∇vj) ‖uvj‖ ϕ1dx ≥ ∫ Ω lim inf j→+∞ f+(x, uvj ,∇vj) ‖uvj‖ ϕ1dx ≥ (λ1 + δ) ∫ Ω ωϕ1dx. Hence, ω ≡ 0, which contradicts with ‖ω‖ = 1. The proof is complete. � Lemma 3.8. Assume that (H1)–(H3) hold. Then there exist positive constants ρ1 and ρ2, independent of v, such that max x∈Ω |u±v (x)| ≤ ρ1, max x∈Ω |∇u±v (x)| ≤ ρ2. Proof. Since f is continuous in all variables and v ∈ C1 0 (Ω), using the regularity theory we know that u±v is C2, see Brezis [3]. Hence, Sobolev embedding theorem and Lemma 3.7 imply the conclusion. � 4. Nehari manifold for superlinear case This section is devoted to the existence of critical point of Jv for superlinear case. The critical points will be obtained by means of constrained minimization. For fixed v ∈ C1 0 (Ω), define Nehari manifold Nv := {u ∈ H1 0 (Ω) \ {0} : J ′v(u)u = 0}. Lemma 4.1. Under assumptions (H5), (H7), (H8), there exists a positive constant c0, independent of v, such that ‖u‖ ≥ c0 for all solutions u ∈ Nv. Proof. Since u ∈ Nv, we know∫ Ω |∇u|2dx = ∫ Ω f(x, u,∇v)udx. 10 Y. WEI, J. TIAN EJDE-2020/99 From (H5) and (H7), for any ε > 0, there exists cε > 0 such that |f(x, s, ξ)| ≤ ε|s|+ cε|s|q, x ∈ Ω, s ∈ R, ξ ∈ Rn. Then ∫ Ω |∇u|2dx ≤ ε ∫ Ω |u|2dx+ cε ∫ Ω |u|q+1dx. Hence, by Poincaré inequality and Sobolev inequality we obtain (1− ε λ1 )‖u‖2 ≤ cε‖u‖q+1 q+1 ≤ cεc q+1 0 ‖u‖q+1, which implies the conclusion. � Lemma 4.2. Assume that (H5) and (H7) hold. Then Φ′v(u) = o(‖u‖), Φv(u) = o(‖u‖2) as u→ 0 in H1 0 (Ω). Proof. (H5) and (H7) imply that for any given ε > 0, there exists a positive cε such that F (x, s, ξ) ≤ ε|s|2 + cε|s|q+1, x ∈ Ω, s ∈ R, ξ ∈ Rn. (4.1) Then, by Hölder inequality and Sobolev inequality, it is standard to prove the lemma. � To prove the main result, we will apply the following lemma, which can be found in Szulkin and Weth [11, Theorem 12]. Lemma 4.3. Let E be a Hilbert space and J(u) = 1 2‖u‖ − Φ(u), where (i) Φ′(u) = o(‖u‖) as u→ 0; (ii) s 7→ Φ′(su)u/s is strictly increasing for all u 6= 0 and s > 0; (iii) Φ(su)/s2 → +∞ uniformly for u on weakly compact subset of E \ {0} as s→ +∞; (iv) Φ′ is completely continuous. Then equation J ′(u) = 0 has a ground state solution. Lemma 4.4. Let (H5)–(H8) hold. Then, for any v ∈ C1 0 (Ω), problem (2.1) has a ground state solution uv ∈ H1 0 (Ω). Proof. It suffices to check (i)–(iv) of Lemma 4.3. Actually, Lemma 4.2 and (H8) imply (i) and (ii), respectively. For (iii), let W be a weakly compact subset of H1 0 (Ω) \ {0} and {un} ⊂W . Passing to a subsequence, it holds un ⇀ u ∈ H1 0 (Ω) \ {0}. Then un(x)→ u(x) a.e. in Ω. Hence, the set Ω∗ := {x ∈ Ω : u(x) 6= 0} is a subset of Ω with positive measure. Taking sn → +∞, we know that for x ∈ Ω∗, |snun(x)| → +∞, as n→ +∞. Then Fatou’s Lemma yields Φv(snun) s2 n = ∫ Ω F ( x, snun(x),∇v(x) ) (snun)2 u2 ndx→ +∞. Finally, since Ω is bounded and (H7) holds, from the compact embedding we know (iv) holds. � EJDE-2020/99 ELLIPTIC EQUATIONS WITH GRADIENT TERMS 11 Remark 4.5. In the above lemma, a ground state solution is found, which is a critical point of functional Jv. From the proof of the above lemma, it can be seen that if Jv is replaced by J±v , then a ground state solution u±v can also be obtained. Remark 4.6. It is easy to check the following minimax characterization (see Szulkin and Weth [11]): cv := inf u∈Nv Jv(u) = inf u∈H1 0 (Ω)\{0} max s>0 Jv(su) = inf u∈H1 0 (Ω),‖u‖=1 max s>0 Jv(su). Lemma 4.7. Assume that (H5)–(H8) hold. Then, there exists a positive constant d, such that cv ≤ d uniformly for v ∈ C1 0 (Ω). Proof. Because of the minimax characterization in Remark 4.6, it suffices to show that there exists φ ∈ H1 0 (Ω) \ {0}, such that max s>0 Jv(sφ) ≤ d, uniformly for v ∈ C1 0 (Ω). From (H6), for every l > 0 there exists C1 > 0 such that F (x, s, ξ) ≥ ls2 − C1, x ∈ Ω, s ∈ R, ξ ∈ Rn. Fix φ ∈ H1 0 (Ω) with ‖φ‖ = 1. From the above we obtain Jv(sφ) = s2 2 ∫ Ω |∇φ|2dx− ∫ Ω F (t, sφ,∇v)dx (4.2) ≤ s2 2 ‖φ‖ 2 − ∫ Ω ls2φ2dx+ ∫ Ω C1dx (4.3) ≤ s2 ( 1 2 − l ∫ Ω φ2dx ) + C1|Ω|. (4.4) Setting l = 1∫ Ω φ2dx , it follows that Jv(sφ) ≤ −1 2 s2 + C1|Ω| ≤ C1|Ω| uniformly for v ∈ C1 0 (Ω). � Lemma 4.8. There exists a positive constant ρ, independent of v, such that for every ground state solution uv given in Lemma 4.4, ‖uv‖ ≤ ρ. Proof. By contradiction, suppose that there exist subsequences {vj} ⊂ C1 0 (Ω) and {uvj} ⊂ H1 0 (Ω), such that uvj ∈ Nvj , J(uvj ) = inf u∈Nvj J(u), ‖uvj‖ → +∞ as j → +∞. Set ωj = uvj/‖uvj‖ and thus ‖ωj‖ = 1. Then, there exists ω ∈ H1 0 (Ω) such that ωj ⇀ ω in H1 0 (Ω), ωj → ω in L2(Ω), ωj(x)→ ω(x) a.e. in Ω. We claim that ω(x) ≡ 0 a.e. in Ω. Denote Ω∗ = {x ∈ Ω, ω(x) 6= 0}. If Ω∗ 6= ∅, then for x ∈ Ω∗, |uvj (x)| → +∞ as j → +∞. By (H6) we have lim j→+∞ F ( x, uvj (x),∇vj(x) )( uvj (x) )2 ( ωj(x) )2 = +∞. (4.5) 12 Y. WEI, J. TIAN EJDE-2020/99 Then Fatou’s Lemma implies∫ Ω lim j→+∞ F ( x, uvj (x),∇vj(x) )( uvj (x) )2 ( ωj(x) )2 dx ≤ lim inf j→+∞ 1 ‖uvj‖2 ∫ Ω F ( x, uvj (x),∇vj(x) ) dx = lim j→+∞ 1 ‖uvj‖2 (1 2 ‖uvj‖2 − Jvj (uvj ) ) . (4.6) From the property of Nehari manifold we know that Jvj (uvj ) = max s>0 Jvj (suvj ). Then Lemma 4.2 implies Jvj (uvj ) ≥ 0. Hence, from (4.6) we obtain∫ Ω lim j→+∞ F ( x, uvj (x),∇vj(x) )( uvj (x) )2 ( ωj(x) )2 dx ≤ 1 2 , which contradicts with (4.5). Therefore, Ω∗ has zero measure and ω(t) ≡ 0 a.e. in Ω. Since Φvj is weakly continuous, from Lemma 4.7 we obtain d ≥ Jvj (uvj ) ≥ Jvj (sωj) ≥ 1 2 s2 − Φvj (sωj)→ 1 2 s2, which is a contradiction, for s large enough. � Lemma 4.9. Assume that (H5)–(H8) hold. Then there exist positive constants ρ1 and ρ2, independent of v, such that max x∈Ω |uv(x)| ≤ ρ1, max x∈Ω |∇uv(x)| ≤ ρ2 for all solutions uv obtained in Lemma 4.4. The proof of the above lemma is as same as the proof of Lemma 3.8. Remark 4.10. Actually, a similar result can also be established for problem (2.2). We can find a critical point u±v for functional J±v and positive constants ρ1 and ρ2, independent of v, such that max x∈Ω |u±v (x)| ≤ ρ1, max x∈Ω |∇u±v (x)| ≤ ρ2. 5. Proofs of main results In this section, we prove our main results by the iterative argument, which was established by De Figueiredo, Girardi and Matzeu [5]. Define the map T± : H1 0 (Ω)→ H1 0 (Ω), T±v 7→ u±v , with domain D(T±) = C1 0 (Ω) ⊂ H1 0 (Ω). Here u±v is the solution of (2.1) given by Lemma 3.5 for the asymptotically linear case and Remark 4.5 for the superlin- ear case, respectively. For any v ∈ C1 0 (Ω), the map is well-defined, and actually, T±(C1 0 (Ω)) ⊂ C1 0 (Ω) because of the regularity theory. Moreover, denote Bρ := {u ∈ H1 0 (Ω), ‖u‖ ≤ ρ}, where ρ > 0 is the uniform bound in Lemma 3.7 for the asymptotically linear case and Lemma 4.8 for the superlinear case, respectively. Then, T±(C1 0 (Ω)) ⊂ Bρ. EJDE-2020/99 ELLIPTIC EQUATIONS WITH GRADIENT TERMS 13 Hence, T±(C1 0 (Ω)) ⊂ Bρ ∩ C1 0 (Ω). Recall that a point x is a fixed point of map T , if and only if x ∈ T (x). Choosing u±0 ∈ Bρ∩C1 0 (Ω), we construct a sequence {u±n } ⊂ Bρ∩C1 0 (Ω) as solutions of −∆u±n = f±(x, u±n ,∇u±n−1) in Ω, u±n = 0 on ∂Ω, (5.1) obtained in Lemma 3.5 for asymptotically linear case and in Lemma 4.4 for super- linear case, respectively. Proof of Theorems 1.1 and 1.3. By (5.1) for n and for n+ 1, we have∫ Ω ∇u±n · (∇u±n+1 −∇u±n )dx = ∫ Ω f±(x, u±n ,∇u±n−1)(u±n+1 − u±n )dx,∫ Ω ∇u±n+1 · (∇u ± n+1 −∇u±n )dx = ∫ Ω f±(x, u±n+1,∇u±n )(u±n+1 − u±n )dx. According to Lemma 3.8 for the asymptotically linear case and Remark 4.10 for the superlinear case, we know that max x∈Ω |u±v (x)| ≤ ρ1, max x∈Ω |∇u±v (x)| ≤ ρ2. Combining (H4) with Poincaré inequality as well as Cauchy-Schwarz inequality, it follows that ‖u±n+1 − u±n ‖2 = ∫ Ω ( f±(x, u±n+1,∇u±n )− f±(x, u±n ,∇u±n−1) ) (u±n+1 − u±n )dx = ∫ Ω ( f±(x, u±n+1,∇u±n )− f±(x, u±n ,∇u±n ) ) (u±n+1 − u±n )dx + ∫ Ω ( f±(x, u±n ,∇u±n )− f±(x, u±n ,∇u±n−1) ) (u±n+1 − u±n )dx ≤ L ∫ Ω |u±n+1 − u±n |2dx+K ∫ Ω |∇u±n −∇u±n−1‖u ± n+1 − u±n |dx ≤ L λ1 ‖u±n+1 − u±n ‖2 + K√ λ1 ‖u±n − u±n−1‖ · ‖u ± n+1 − u±n ‖. Hence, ‖u±n+1 − u±n ‖ ≤ K √ λ1 λ1 − L ‖u±n − u±n−1‖. From L + √ λ1K < λ1 we know {u±n } ⊂ H1 0 (Ω) is a Cauchy sequence, and thus there exists u±∗ ∈ H1 0 (Ω) such that u±∗ ∈ T±(u±∗ ). Finally, from Lemma 3.6 for the asymptotically linear case and Lemma 4.1 for the superlinear case we know that ‖u±∗ ‖ ≥ c0, which means that u±∗ is a nontrivial solution. � Acknowledgements. This work was supported by the National Natural Science Foundation of China No. 11871242, and by the Natural Science Foundation of Jilin Province No. 20200201248JC. 14 Y. WEI, J. 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Su; On a class of non-local elliptic equations with asymptotically linear term, Discrete Contin. Dyn. Syst., 38 No. 12 (2018), 6287-6304. [14] J. Xavier; Some existence theorems for equations of the form −∆u = f(x, u,Du), Nonlinear Anal., 15 No. 1 (1990), 59-67. [15] Z. Yan; A note on the solvability in W 2,p(Ω) for the equation −∆u = f(x, u,Du), Nonlinear Anal., 24 No. 9 (1995), 1413-1416. Yuanhong Wei (corresponding author) School of Mathematics, Jilin University, 130012, Changchun, China Email address: weiyuanhong@jlu.edu.cn Jian Tian School of Mathematics, Jilin University, 130012, Changchun, China Email address: tianjian@jlu.edu.cn 1. Introduction 2. Preliminaries and auxiliary problem 3. Asymptotically linear case 4. Nehari manifold for superlinear case 5. Proofs of main results Acknowledgements References