Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 56, pp. 1–17. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu POSITIVE SOLUTIONS FOR ASYMPTOTICALLY 3-LINEAR QUASILINEAR SCHRÖDINGER EQUATIONS GUOFA LI, BITAO CHENG, YISHENG HUANG Abstract. In this article, we study the quasilinear Schrödinger equation −∆u+ V (x)u− κ 2 [∆(1 + u2)1/2] u (1 + u2)1/2 = h(u), x ∈ RN , where N ≥ 3, κ > 0 is a parameter, V : RN → R is a given potential. The nonlinearity h ∈ C(R,R) is asymptotically 3-linear at infinity. We obtain the nonexistence of a least energy solution and the existence of a positive solution, via the Pohožaev manifold and a linking theorem. Our results improve recent results in [4, 22]. 1. Introduction and statement of main results In this article, we study the quasilinear Schrödinger equation −∆u+ V (x)u− κ 2 [∆(1 + u2)1/2] u (1 + u2)1/2 = h(u), x ∈ RN , (1.1) where N ≥ 3, κ > 0 is a parameter, V : RN → R is a given potential and h is a real function. Solutions of (1.1) are related to standing waves for the following quasilinear Schrödinger equation izt = −∆z +W (x)z − a(x)η(|z|2)z − κ[∆(ϕ(|z|2)ϕ′(|z|2)]z, x ∈ RN , (1.2) where z : R × RN → C, W : RN → R is a given potential and a, ϕ, η : R → R are real functions. Note that (1.2) is a generalized nonlinear Schrödinger equation, which has been derived as mathematical models of several physical phenomena corresponding to various types of the nonlinear terms ϕ and η, see [3, 8, 9, 12, 24]. Substituting z(t, x) = exp(−iEt)u(x) into (1.2), we obtain the equation −∆u+ V (x)u− κ[∆(ϕ(|u|2))ϕ′(|u|2)u = a(x)η(|u|2)u, x ∈ RN , (1.3) where V (x) := W (x) − E is the new potential function. Setting h(t) := η(t2)t, then if ϕ(t) = √ 1 + t, a(x) = 1, Equation (1.3) turns into (1.1) and if ϕ(t) = t and κ = 1, Equation (1.3) becomes the quasilinear problem −∆u+ V (x)u−∆(u2)u = a(x)h(u), x ∈ RN . (1.4) Since the behavior of h at infinity plays an important role in searching the weak solutions of (1.3), many authors have studied (1.3) with particular forms of ϕ via 2010 Mathematics Subject Classification. 35J20, 35J62. Key words and phrases. Quasilinear Schrödinger equations; asymptotically 3-linear; Pohožaev identity; linking theorem; positive solution. c©2020 Texas State University. Submitted June 16, 2019. Published June 4, 2020. 1 2 G. LI, B. CHENG, Y. HUANG EJDE-2020/56 variational methods under various conditions on the nonlinearity h; e.g. h hass superlinear growth [5, 6, 15, 17, 18, 23, 25] or has asymptotically linear growth [16] at infinity. Moreover, the existence of positive solutions for (1.4) was obtained in [4, 22] for asymptotically 2-linear growth of the nonlinear term h(t) at infinity, where h(t) was assumed to satisfy (H1) h ∈ C1(R+,R+) and limt→0+ h(t) t = 0; (H2) limt→∞ h(t) t2 = 1; (H3) If Q(t) = 1 4h(t)t−H(t), H(t) = ∫ t 0 h(s)ds, and thus a constant D ≥ 1 exists such that 0 < Q(s) ≤ DQ(t) for 0 < s ≤ t, and limt→∞Q(t) = +∞. We note that if h(t) is positive for t > 0, then from (H1) and (H3) it follows that H(t) > ct4 for some positive constant c and large t > 0, which means that (H2) does not occur when h(t) satisfies assumptions (H1) and (H3). The purpose of this article is to investigate the existence of positive solutions to (1.1) for the nonlinear term h(t) satisfying the modified assumptions (H1)-(H3). More precisely, we suppose that h satisfies the following assumptions (H1’) h ∈ C1(R+,R+) and limt→0+ h(t) t = 0; (H2’) limt→+∞ h(t) t3 = 1; (H3’) Q(t) = 1 4h(t)t−H(t) > 0 for all t > 0, where H(t) = ∫ t 0 h(s)ds. Also, we assume that the following conditions on the potential function V (x) (H4) V ∈ C2(RN ,R); (H5) lim|x|→+∞ V (x) = V∞ < 1, 1/2 < V∞ < V (x) for all x ∈ RN ; (H6) 〈∇V (x), x〉 ≤ 0 for all x ∈ RN with the strict inequality holding on a subset of positive Lebesgue measure of RN ; (H7) NV (x) + 〈∇V (x), x〉 ≥ NV∞ for all x ∈ RN ; (H8) xHV (x)x N + 〈∇V (x), x〉 ≤ 0 for all x ∈ RN , where HV is the Hessian matrix of the function V (x). We want to point out that similar method can be applied to (1.4). Now, we study the quasilinear Equation (1.1). The associated energy functional of the Euler- Lagrange equation (1.1) is I(u) = 1 2 ∫ RN [ 1 + κu2 2(1 + u2) ] |∇u|2dx+ 1 2 ∫ RN V (x)|u|2dx− ∫ RN H(u)dx. When V (x) ≡ V∞, we are led to the limiting problem of (1.1), −∆u+ V∞u− κ 2 [∆(1 + u2)1/2] u (1 + u2)1/2 = h(u), x ∈ RN . (1.5) The associated energy functional of (1.5) is I∞(u) = 1 2 ∫ RN [ 1 + κu2 2(1 + u2) ] |∇u|2dx+ 1 2 ∫ RN V∞|u|2dx− ∫ RN H(u)dx. Making a change of variable, i.e. using the dual approach (cf. [7]), we can reduce the quasilinear Schrödinger equation into a semilinear equation like the case of κ = 0. Let v = G(u) = ∫ u 0 g(t)dt (1.6) EJDE-2020/56 QUASILINEAR SCHRÖDINGER EQUATIONS 3 with G satisfying (G−1(t))′ = 1 g(G−1(t)) = 1√ 1 + κ(G−1(t))2 2(1+(G−1(t))2) for t ∈ [0,+∞) and G−1(t) = −G−1(−t) for t ∈ (−∞, 0]. Then, (1.1) and (1.5) will be reduced to the semilinear equation −∆v + V (x) G−1(v) g(G−1(v)) = h(G−1(v)) g(G−1(v)) , x ∈ RN , (1.7) and −∆v + V∞ G−1(v) g(G−1(v)) = h(G−1(v)) g(G−1(v)) , x ∈ RN . (1.8) Clearly, weak solutions of (1.7) and (1.8) correspond to critical points of the energy functional J(v) = 1 2 ∫ RN |∇v|2dx+ 1 2 ∫ RN V (x)|G−1(v)|2dx− ∫ RN H(G−1(v))dx, (1.9) and J∞(v) = 1 2 ∫ RN |∇v|2dx+ 1 2 ∫ RN V∞|G−1(v)|2dx− ∫ RN H(G−1(v))dx. (1.10) Moreover, for ψ ∈ H1(RN ), the derivative of J in the direction ψ at v is 〈J ′(v), ψ〉 = ∫ RN ∇v∇ψdx+ ∫ RN V (x) G−1(v) g(G−1(v)) ψdx− ∫ RN h(G−1(v)) g(G−1(v)) ψdx. If v ∈ H1(RN ) is a weak solution of (1.7), then v satisfies the Pohožaev identity γ(v) = 0, where γ(v) = N − 2 2 ∫ RN |∇v|2dx+ N 2 ∫ RN V (x)|G−1(v)|2dx + 1 2 ∫ RN 〈∇V (x), x〉|G−1(v)|2dx−N ∫ RN H(G−1(v))dx. Furthermore, we defined the Pohožaev manifold associated with (1.7) by P := {v ∈ H1(RN )\{0} : γ(v) = 0}. Motivated by [11], we will employ the minimization methods restricted to the Pohožaev manifold to obtain the existence of positive solutions for (1.1). Now, we state our first result. Theorem 1.1. Assume that (H1’)–(H3’), (H4)–(H8) hold. Then P is a natural constraint of (1.1), i.e. any critical point of J |P is a critical point of J in H1(RN ). Moreover, p = infv∈P J(v) is not a critical level for the function J . We define Γ∞ = {ξ ∈ C([0, 1], H1(RN )) : ξ(0) = 0 6= ξ(1), J∞(ξ(1)) < 0}, as well as the mountain pass min-max level c∞ = inf ξ∈Γ∞ max t∈[0,1] J∞(ξ(t)), where J∞ is defined by (1.10). We will use the linking theorem and the barycenter function is restricted to the Pohožaev manifold to obtain the existence of weak solution for (1.1). Here is our second result. 4 G. LI, B. CHENG, Y. HUANG EJDE-2020/56 Theorem 1.2. Assume that (H1’)–(H3’), (H4)–(H8) and the following conditions hold: (1) h ∈ C1(R) ∩ Lip(R+,R+); (2) ‖V (x)− V∞‖∞ is sufficiently small; (3) the least energy level c∞ of J∞ is an isolated radial critical level or equation (1.5) admits a unique positive solution which is radially symmetric about some point. Then (1.1) admits a positive solution whose energy is above c∞. There are functions satisfying (H1’)–(H3’), for example h(t) = t5 1+t2 . There are also functions satisfying (H4)–(H8), for example V (x) = c1 + c2 1+|x|2 , where 1/2 < c1 < 1, c2 > 0. In fact, since N ≥ 3, we know that 〈∇V (x), x〉 = − 2c2|x|2 (1 + |x|2)2 < 0, NV (x) + 〈∇V (x), x〉 = Nc1 + Nc2 + c2|x|2(N − 2) (1 + |x|2)2 ≥ Nc1 = NV∞, xHV (x)x N + 〈∇V (x), x〉 = 4c2(|x|4 − |x|2) N(1 + |x|2)3 − 2c2|x|2 (1 + |x|2)2 = 2c2|x|2 (1 + |x|2)3 [( 2 N − 1 ) |x|2 − ( 2 N + 1 )] ≤ 0. In this article: ‖u‖q (1 ≤ q ≤ ∞) denotes the standard norm in Lq(RN ). 〈·, ·〉 denotes the duality pairing between a Banach space and its dual space. → and ⇀ denote strong convergence and weak convergence in the related function space, respectively. on(1) denotes the quantities tending to 0 as n → ∞. C,C0, C1, . . . denote positive constants. BR(0) denotes a ball centered at the origin with radius R > 0. 2. Preliminaries We shall work in the space H1(RN ) with the norm ‖u‖2 = ∫ RN (|∇u|2 + V (x)u2)dx, because of (H4) and (H5), this norm is equivalent to the standard H1(RN ) norm. If u is a solution of (1.1), then for all ϕ ∈ H1(RN ) we have 〈I ′(u), ϕ〉 = ∫ RN [g2(u)∇u∇ϕ+ g(u)g′(u)|∇u|2ϕ]dx+ ∫ RN V (x)uϕdx − ∫ RN h(u)ϕdx = 0, u ∈ H1(RN ), (2.1) where g(t) = √ 1 + κt2 2(1+t2) . On the one hand, if we choose ϕ = ψ g(u) in (2.1), combining (1.6) and (1.9), we obtain 〈J ′(v), ψ〉 = ∫ RN ∇v∇ψdx+ ∫ RN V (x) G−1(v) g(G−1(v)) ψdx − ∫ RN h(G−1(v)) g(G−1(v)) ψdx = 0. (2.2) EJDE-2020/56 QUASILINEAR SCHRÖDINGER EQUATIONS 5 On the other hand, let ψ = g(u)ϕ in (2.2), we obtain (2.1). Thus (2.1) is equivalent to (2.2). Hence, u is a weak solution of (1.1) if and only if v is a critical point of the functional J . Note that a function v ∈ H1(RN ) is a least energy solution if and only if v is a solution of (1.5) and J∞(v) = m∞, where m∞ = inf { J∞(v) : v ∈ H1(RN )\{0} is a solution of (1.5) } . To see the smoothness of J , we need the following lemma. Lemma 2.1. The functions g(t) and G(t) = ∫ t 0 g(s)ds satisfy the following prop- erties: (1) G(t) and G−1(t) are odd functions. (2) 1 ≤ g(t) ≤ √ 1 + κ 2 . (3) 0 ≤ t g(t)g ′(t) ≤ √ 2(2+κ)−2√ 2(2+κ)+2 for all t ≥ 0. (4) √ 2 2+κ |t| ≤ |G −1(t)| ≤ |t| for all t ∈ R. Proof. From the definition of G(t) we can prove (1) and (2). (3) Setting Z(t) = t g(t)g ′(t), direct computations show that Z(t) = κt2 (1 + t2)[2 + (2 + κ)t2] = Φ(t2). Then 0 ≤ Z(t) for κ > 0. Moreover, Φ(r) attains its maximum at r0 = √ 2 2+κ and Zmax(t) = Z(t)| t2= √ 2 2+κ = √ 2(2 + κ)− 2√ 2(2 + κ) + 2 . Then, (3) holds. (4) Since g(t) is nondecreasing for t ≥ 0, by the differential mean value theorem, we know that t = g(0)t ≤ G(t) = ∫ t 0 g(s)ds = g(ξ)t ≤ g(t)t ≤ g(∞)t = √ 1 + κ 2 t, ξ ∈ [0, t]. Then, √ 2/(2 + κ)t ≤ G−1(t) ≤ t. When t < 0, it deduce from the oddness of G−1(t) that t ≤ G−1(t) ≤ √ 2 2+κ t. Thus the proof is complete. � Now, by Lemma 2.1, J is well defined and is of C1 if h(t) satisfies the conditions (H1’)–(H3’). Next, we show another property of the change of variable G which will play important roles in proving our results. Lemma 2.2. For t > 0, it holds 1 2 G−1(t)g(G−1(t)) ≤ t ≤ G−1(t)g(G−1(t)). Proof. Let η(s) = G(s)− 1 2sg(s), then by Lemma 2.1(3), we have η′(s) = g(s)− 1 2 g(s)− 1 2 sg′(s) = 1 2 g(s) [ 1− sg′(s) g(s) ] ≥ 0. Thus η(s) ≥ η(0), let s = G−1(t), we have 1 2G −1(t)g(G−1(t)) ≤ t for t > 0. Moreover, we set θ(t) = G−1(t)g(G−1(t))− t. 6 G. LI, B. CHENG, Y. HUANG EJDE-2020/56 Direct computation shows that θ′(t) = 1 g(G−1(t)) g(G−1(t)) +G−1(t)g′(G−1(t))− 1 = G−1(t) (√ 1 + κ(G−1(t))2 2(1 + (G−1(t))2) )′ = G−1(t) κG−1(t) 2[1 + κ(G−1(t))2 2(1+(G−1(t))2) ][1 + (G−1(t))2]2 ≥ 0, ∀κ > 0, t > 0. Then θ(t) ≥ θ(0), which implies that t ≤ G−1(t)g(G−1(t)) for t > 0. � 3. Pohožaev manifold In this section, we will show the nonexistence of solution for (1.1). First, for the Pohožaev manifold P, we have the following properties. Lemma 3.1. The functional γ : H1(RN ) → R and the Pohožaev manifold P satisfy: (1) {v ≡ 0} is an isolated point of γ−1({0}); (2) P is a closed set; (3) P is a C1 manifold; (4) there exists σ > 0 such that ‖v‖ > σ for all v ∈ P. Proof. (1) By (H1’) and (H2’), we can deduce that for any ε > 0 and 4 ≤ q ≤ 2∗, there is Cε such that |H(s)| ≤ ε 2 |s|2 + Cε q |s|q, (3.1) and |h(s)| ≤ ε|s|+Cε|s|q−1 for all s ∈ R. Thanks to (H5), (H7) and Lemma 2.1(2), (4), if we choose ε = V∞/2, then γ(v) = N − 2 2 ∫ RN |∇v|2dx+ N 2 ∫ RN V (x)|G−1(v)|2dx + 1 2 ∫ RN 〈∇V (x), x〉|G−1(v)|2dx−N ∫ RN H(G−1(v))dx ≥ N − 2 2 ∫ RN |∇v|2dx+ N 2 ∫ RN V∞|G−1(v)|2dx −N ∫ RN [ε 2 |G−1(v)|2 + Cε q |G−1(v)|q ] dx ≥ N − 2 2 ∫ RN |∇v|2dx+ N 2 ∫ RN V∞|G−1(v)|2dx− N 2 ∫ RN V∞ 2 |G−1(v)|2dx − NCε q ∫ RN |G−1(v)|qdx ≥ N − 2 2 ∫ RN |∇v|2dx+ N 2(2 + κ) ∫ RN V∞|v|2dx− NCε q ∫ RN |v|qdx ≥ min {N − 2 2 , N 2(2 + κ) } C‖v‖2 − NCε q ‖v‖q. EJDE-2020/56 QUASILINEAR SCHRÖDINGER EQUATIONS 7 Let ‖v‖ = ρ > 0 be small enough such that min { N−2 2 , N 2(2+κ) } Cρ2 > 2NCε ρq q , we obtain γ(v) ≥ min {N − 2 2 , N 2(2 + κ) } Cρ2 − NCε q ρq > 1 2 {N − 2 2 , N 2(2 + κ) } Cρ2 > 0. (2) The functional γ(v) is a C1 functional, thus P ∪ {0} = γ−1(0) is a closed subset. Moreover, {v ≡ 0} is an isolated point in γ−1({0}) and the assertion follows. (3) Since v ∈ P, we have (N − 2) ∫ RN |∇v|2dx+N ∫ RN V (x)|G−1(v)|2dx + ∫ RN 〈∇V (x), x〉|G−1(v)|2dx = 2N ∫ RN H(G−1(v))dx. (3.2) Combining (H3’,) (H7), and Lemma 2.2, we obtain 〈γ′(v), v〉 = (N − 2) ∫ RN |∇v|2dx+N ∫ RN V (x) G−1(v) g(G−1(v)) v dx + ∫ RN 〈∇V (x), x〉 G−1(v) g(G−1(v)) v dx−N ∫ RN h(G−1(v)) g(G−1(v)) v dx = ∫ RN (NV (x) + 〈∇V (x), x〉) [ G−1(v) g(G−1(v)) v − |G−1(v)|2 ] dx +N ∫ RN [ 2H(G−1(v))− h(G−1(v)) g(G−1(v)) v ] dx ≤ ∫ RN (NV (x) + 〈∇V (x), x〉) [ G−1(v) g(G−1(v)) g(G−1(v))G−1(v)− |G−1(v)|2 ] dx +N ∫ RN [ 2H(G−1(v))− h(G−1(v)) g(G−1(v)) 1 2 g(G−1(v))G−1(v) ] dx < 0. This shows that P is a C1 manifold. (4) Since 0 is an isolated point in γ−1({0}), there must be a ball ‖v‖ ≤ σ which doesn’t intersect P and the assertion is proved. � Next, we obtain relations between the Pohožaev manifold P associated with (1.7) and the Pohožaev manifold P∞ associated with limiting problem (1.8). Recall that P∞ := {v ∈ H1(RN ) \ {0} : γ∞(v) = 0}, where γ∞(v) = N − 2 2 ∫ RN |∇v|2dx+ N 2 ∫ RN V∞|G−1(v)|2dx −N ∫ RN H(G−1(v))dx. (3.3) Next, we obtain the Lemmas 3.2–3.6 and 3.8 which will be used for proving Theorem 1.1. Their proofs can be found in [11] and [14]. 8 G. LI, B. CHENG, Y. HUANG EJDE-2020/56 Lemma 3.2. Assume that ∫ RN [H(G−1(v)) − 1 2V∞|G −1(v)|2]dx > 0, then there exist unique t1 > 0 and t2 > 0 such that v ( · t1 ) ∈ P and v ( · t2 ) ∈ P∞. If v ∈ P, then there exists 0 < tv < 1 such that v ( · tv ) ∈ P∞. If w ∈ P∞, then there exists tw > 1 such that w ( · tw ) ∈ P. Lemma 3.3. Assume that Ω = {v ∈ H1(RN ) \ {0} : ∫ RN [H(G−1(v))− 1 2 V∞|G−1(v)|2]dx > 0} . Then the function t1 : Ω → R+ is given by v 7→ t1(v) such that v ( · t1(v) ) ∈ P is continuous. Lemma 3.4. Assume that v ∈ P∞, then for all y ∈ RN . Then v(· − y) ∈ P∞. Moreover, there exists ty > 1 such that v ( · − y ty ) ∈ P and lim |y|→∞ ty = 1. Lemma 3.5. It holds supy∈RN ty := t̄ < +∞ and t̄ > 1. Lemma 3.6. There exists a real number σ̂ > 0 such that infv∈P ‖∇v‖2 ≥ σ̂. Lemma 3.7. If v ∈ H1(RN ) satisfies ∫ RN [H(G−1(v))− 1 2V∞|G −1(v)|2]dx > 0 and tv > 0 are such that v ( · tv ) ∈ P∞, then J∞ ( v ( x tv )) = tN−2 v N ∫ RN |∇v|2dx. Proof. If v ( · tv ) ∈ P∞, by (3.3), we know that N − 2 2 tN−2 v ∫ RN |∇v|2dx+ N 2 tNv ∫ RN V∞|G−1(v)|2dx = NtNv ∫ RN H(G−1(v))dx. Then J∞ ( v ( x tv )) = tN−2 v 2 ∫ RN |∇v|2dx+ tNv 2 ∫ RN V∞|G−1(v)|2dx− tNv ∫ RN H(G−1(v))dx = (1 2 − N − 2 2N ) tN−2 v ∫ RN |∇v|2dx = tN−2 v N ∫ RN |∇v|2dx. (3.4) � Lemma 3.8. It holds p = infv∈P J(v) > 0 and p = c∞. Proof of Theorem 1.1. Arguing by contradiction, we suppose that there is v ∈ H1(RN ) such that J(v) = p and J ′(v) = 0. Then v ∈ P. By Lemma 3.2, there is 0 < tv < 1 such that v ( · tv ) ∈ P∞. From (3.4) and (H6) we obtain p = J(v) = 1 2 ∫ RN |∇v|2dx+ 1 2 ∫ RN V (x)|G−1(v)|2dx− ∫ RN H(G−1(v))dx = ( 1 2 − N − 2 2N ) ∫ RN |∇v|2dx− 1 2N ∫ RN 〈∇V (x), x〉|G−1(v)|2dx EJDE-2020/56 QUASILINEAR SCHRÖDINGER EQUATIONS 9 = 1 N ∫ RN |∇v|2dx− 1 2N ∫ RN 〈∇V (x), x〉|G−1(v)|2dx > tN−2 v N ∫ RN |∇v|2dx = J∞ ( v ( · tv )) > c∞, which contradicts Lemma 3.8. Moreover, by [22, Lemma 2.3], any critical point of J |P is a critical point of J in H1(RN ), then p is not a critical level for the function J . � 4. Existence of a positive solution In this section, we will show the existence of a positive solution for (1.1). Sim- ilarly to what done in [11], first, we prove the existence of a positive solution for limiting problem (1.8) by a global compactness lemma. Second, we prove the exis- tence of positive for (1.1) using barycenter constrains and a version of the Linking Theorem. Lemma 4.1. (1) There exist ρ, a > 0 such that J(v) ≥ a, ‖v‖ = ρ. (2) There exists e ∈ H1(RN ) with ‖e‖ > ρ such that J(e) < 0. Proof. (1) By (3.1), (H5), Lemma 2.1 (4) and Sobolev embedding, select ε = V∞ 2+κ , we know that J(v) ≥ 1 2 ∫ RN |∇v|2dx+ 1 2 + κ ∫ RN V∞v 2dx− ε 2 ∫ RN v2dx− C q ∫ RN |v|qdx = 1 2 ∫ RN |∇v|2dx+ 1 2(2 + κ) ∫ RN V∞v 2dx− C q ∫ RN |v|qdx ≥ C 2(2 + κ) ‖v‖2 − C1 q ‖v‖q. Thereby, choosing ‖v‖ = ρ is small enough, we have J(v) ≥ C 2(2+κ)ρ 2 − C1 q ρ q > 0. (2) Let w ∈ H1(RN ) be a least energy solution of (1.8), motivated by [10, Lemma 2.2], we define a continuous path α : [0,+∞)→ H1(RN ) by setting α(t)(x) = w(xt ), if t > 0 and α(0) = 0. Then J∞(0) = 0 and J∞(α(t)) = 1 2 ∫ RN ∣∣∇w( x t ) ∣∣2dx+ 1 2 ∫ RN V∞ ∣∣G−1 ( w( x t ) )∣∣2dx − ∫ RN H ( G−1 ( w( x t ) )) dx = 1 2 tN−2 ∫ RN |∇w(x)|2dx+ 1 2 tN ∫ RN V∞|G−1(w(x))|2dx − tN ∫ RN H(G−1(w(x)))dx. Taking the derivative, we have d dt J∞(α(t)) = N − 2 2 tN−3 ∫ RN |∇w|2dx+ N 2 tN−1 ∫ RN V∞|G−1(w)|2dx −NtN−1 ∫ RN H(G−1(w))dx. 10 G. LI, B. CHENG, Y. HUANG EJDE-2020/56 Since w is a solution of (1.8), it satisfies the Pohožaev identity N − 2 2 ∫ RN |∇w|2dx+ N 2 ∫ RN V∞|G−1(w)|2dx = N ∫ RN H(G−1(w))dx. Therefore, d dt J∞(α(t)) = N − 2 2 tN−3(1− t2) ∫ RN |∇w|2dx. Since N ≥ 3, the map t 7→ J∞(α(t)) achieves the maximum value at t = 1. Choosing L > 0 is sufficiently large, we have J∞(α(L)) < 0. Taking ζ(t) = α(tL), we have ζ ∈ Γ∞. If ζy(t) = w ( ·−y tL ) , by (V2) and Lebesgue Dominated Convergence Theorem, we know J(ζy(1)) = 1 2 ∫ RN ∣∣∇w(x− y L )∣∣2dx+ 1 2 ∫ RN V (x) ∣∣G−1 ( w (x− y L ))∣∣2dx − ∫ RN H ( G−1 ( w (x− y L ))) dx = 1 2 ∫ RN ∣∣∇w( x L ) ∣∣2dx+ 1 2 ∫ RN V (x+ y)|G−1 ( w( x L ) ) |2dx − ∫ RN H ( G−1 ( w( x L ) )) dx = J∞(ζy(1)) + 1 2 ∫ RN (V (x+ y)− V∞)|G−1(ζy(1))|2dx < 0, for |y| large. Choosing e = ζy(1), we complete the proof. � From Lemma 4.1, the min-max mountain pass level for the function J is c = inf ξ∈Γ max t∈[0,1] J(ξ(t)), where Γ = {ξ ∈ C([0, 1], H1(RN )) : ξ(0) = 0 6= ξ(1), J(ξ(1)) < 0}. Then there is a Cerami sequence {vn} for the functional J at level c such that J(vn)→ c and ‖J ′(vn)‖(1 + ‖vn‖)→ 0. Now, we state the following Lemma, the proof is similar to the proof of [11, Lemmas 4.1 and 4.2], we omit it. Lemma 4.2. It holds c = c∞ = p. Lemma 4.3. For all ξ ∈ Γ, there is s ∈ (0, 1) such that ξ(s) ∈ P. Proof. Since γ(v) = N − 2 2 ∫ RN |∇v|2dx+ N 2 ∫ RN V (x)|G−1(v)|2dx + 1 2 ∫ RN 〈∇V (x), x〉|G−1(v)|2dx−N ∫ RN H(G−1(v))dx = NJ(v)− ∫ RN |∇v|2dx+ 1 2 ∫ RN 〈∇V (x), x〉|G−1(v)|2dx, EJDE-2020/56 QUASILINEAR SCHRÖDINGER EQUATIONS 11 by (H6), we have γ(v) < NJ(v), for every v ∈ H1(RN ). If ξ ∈ Γ, we have γ(ξ(0)) = 0 and γ(ξ(1)) < NJ(ξ(1)) < 0. Now, there is s ∈ (0, 1) such that γ(ξ(s)) = 0 for ‖ξ(s)‖ > ρ. Then ξ(s) ∈ P. � Lemma 4.4. If {vn} ⊂ H1(RN ) is a (Ce)c sequence with c > 0, then {vn} is bounded. Proof. Since G−1(vn)g(G−1(vn)) ∈ H1(RN ), we have J(vn) = 1 2 ∫ RN |∇vn|2dx+ 1 2 ∫ RN V (x)|G−1(vn)|2dx− ∫ RN H(G−1(vn))dx = c+ on(1), and 〈J ′(vn), G−1(vn)g(G−1(vn))〉 = ∫ RN [ 1 + G−1(vn) g(G−1(vn)) g′(G−1(vn)) ] |∇vn|2dx + ∫ RN V (x)|G−1(vn)|2dx− ∫ RN h(G−1(vn))G−1(vn)dx = on(1). Then, by (H3’) and Lemma 2.1 (3), we obtain c+ on(1) = J(vn)− 1 4 〈J ′(vn), G−1(vn)g(G−1(vn))〉 = 1 4 ∫ RN [ 1− G−1(vn) g(G−1(vn)) g′(G−1(vn)) ] |∇vn|2dx+ 1 4 ∫ RN V (x)|G−1(vn)|2dx − ∫ RN [ H(G−1(vn))− 1 4 h(G−1(vn))G−1(vn) ] dx ≥ 1 4 [ 1− √ 2(2 + κ)− 2√ 2(2 + κ) + 2 ] ∫ RN |∇vn|2dx+ 1 2(2 + κ) ∫ RN V (x)|vn|2dx ≥ min { 4√ 2(2 + κ) + 2 , 1 2(2 + κ) } ‖vn‖2, hence, {vn} is bounded in H1(RN ). � Lemma 4.5 (Splitting). Let {vn} ⊂ H1(RN ) be a bounded sequence such that J(vn)→ c > 0 and (1 + ‖vn‖)‖J ′(vn)‖ → 0. Replacing {vn} by a subsequence, if necessary, there exists a solution v̄ of (1.1), a number k ∈ N ⋃ {0}, k functions v1, v2, . . . , vk and k sequence of points {yjn} ∈ RN , 1 ≤ j ≤ k, satisfying (1) vn → v̄ in H1(RN ) or (2) vj are nontrivial solutions of (1.8); (3) |yjn| → ∞ and |yjn − yin| → ∞, i 6= j; (4) vn − ∑k i=1 v i(x− yin)→ v̄; (5) J(vn)→ J(v̄) + ∑k i=1 J∞(vi). Proof. The proof is a version of concentration compactness of Lions in [13, 19], one can mimic the proof of [21, Theorem 8.4]. � 12 G. LI, B. CHENG, Y. HUANG EJDE-2020/56 Corollary 4.6. If J(vn) → c∞ and ‖J ′(vn)‖(1 + ‖vn‖) → 0, then either {vn} is relatively compact or the splitting lemma 4.5 holds with k = 1 and v̄ = 0. Let c] := inf{c > c∞ : c is a radial critical value of J∞} . Then we have the following lemma. Lemma 4.7. Assume that c∞ is an isolated radial critical level for J∞. Then c] > c∞ and J satisfies condition (Ce) at level d ∈ (c∞,min{c], 2c∞}). Assume now that the limiting problem (1.8) admits a unique positive radial solution. Then J satisfies condition (Ce) at level d ∈ (c∞, 2c∞). The proof of the above lemma is analogous to the proof of [11, Lemma 5.9], we omit it. Lemma 4.8. Let J(vj)→ d > 0 and {vj} ⊂ P, then {vj} is bounded in H1(RN ). Proof. Since {vj} ⊂ P, by (H6) and (3.2), we obtain d+ 1 ≥ J(vj) = 1 2 ∫ RN |∇vj |2dx+ 1 2 ∫ RN V (x)|G−1(vj)|2dx− ∫ RN H(G−1(vj))dx = 1 N ∫ RN |∇vj |2dx− 1 2N ∫ RN 〈∇V (x), x〉|G−1(vj)|2dx ≥ 1 N ∫ RN |∇vj |2dx. Then, ‖∇vj‖2 is bounded. By Sobolev inequality, the sequence ‖vj‖2∗ is also bounded. Setting ε = V∞/2, combining this with (H1’) and (H2’), (H5), Lemma 2.1 (4), we have d+ 1 = J(vj) = 1 2 ∫ RN |∇vj |2dx+ 1 2 ∫ RN V (x)|G−1(vj)|2dx− ∫ RN H(G−1(vj))dx ≥ 1 2 ∫ RN |∇vj |2dx+ V∞ 2 ∫ RN |G−1(vj)|2dx− ε 2 ∫ RN |G−1(vj)|2dx − Cε q ∫ RN |G−1(vj)|2 ∗ dx ≥ 1 2 ∫ RN |∇vj |2dx+ V∞ 2 + κ ∫ RN |vj |2dx− ε 2 ∫ RN |vj |2dx− Cε q ∫ RN |vj |2 ∗ dx = 1 2 ∫ RN |∇vj |2dx+ V∞ 2(2 + κ) ∫ RN |vj |2dx− Cε q ∫ RN |vj |2 ∗ dx. If ‖vj‖2 →∞, we obtain a contradiction. � Next, we introduce the barycenter function, see [1, 20], which is crucial for proving the existence of a solution for (1.1). Definition 4.9. The barycenter function of a function u ∈ H1(RN )\{0} is defined by µ(u)(x) := 1 |B1| ∫ B1(x) |u(y)|dy. EJDE-2020/56 QUASILINEAR SCHRÖDINGER EQUATIONS 13 It follows that µ(u) ∈ L∞(RN ) ∩ C(RN ). Subsequently, take û(x) := [ µ(u)(x)− 1 2 maxµ(u) ]+ , we know that û ∈ C0(RN ). Now we define the barycenter of u by β(u) = 1 ‖û‖1 ∫ xû(x)dx. Since µ(u) has compact support, by definition, β(u) is well defined. β satisfies the following properties (1) β is a continuous function in H1(RN )\{0}. (2) If u is radially symmetric, then β(u) = 0. (3) Given y ∈ RN and setting uy(x) := u(x− y), we have β(uy) = β(u) + y. Lemma 4.10. Let {un}, {vn} ⊂ H1(RN ) be such that ‖un−vn‖ → 0 and J ′(vn)→ 0 as n→∞. Then J ′(un)→ 0 as n→∞. Proof. For each ϕ ∈ H1(RN ), we have 〈J ′(un)− J ′(vn), ϕ〉 = ∫ RN ∇(un − vn)∇ϕdx+ ∫ RN V (x) [ G−1(un) g(G−1(un)) − G−1(vn) g(G−1(vn)) ] ϕdx − ∫ RN [h(G−1(un)) g(G−1(un)) − h(G−1(vn)) g(G−1(vn)) ] ϕdx. From ‖un − vn‖ → 0, we have∫ RN ∇(un − vn)∇ϕdx ≤ (∫ RN |∇(un − vn)|2dx )1/2(∫ RN |∇ϕ|2dx )1/2 → 0, as n → ∞. Since the function G−1(s) g(G−1(s)) is continuous for s, by (H5), when un is sufficiently close to vn, we can conclude that∫ RN V (x) [ G−1(un) g(G−1(un)) − G−1(vn) g(G−1(vn)) ] ϕdx→ 0. Moreover, by the assumption h ∈ Lip(R+,R+), we deduce from Lemma 2.1 (2), (4) that ∣∣∣h(G−1(un)) g(G−1(un)) − h(G−1(vn)) g(G−1(vn)) ∣∣∣ = |h(G−1(un))g(G−1(vn))− h(G−1(vn))g(G−1(un))| g(G−1(un))g(G−1(vn)) ≤ |h(G−1(un))− h(G−1(vn))|g(G−1(vn)) g(G−1(un))g(G−1(vn)) + |h(G−1(vn))||g(G−1(vn))− g(G−1(un))| g(G−1(un))g(G−1(vn)) ≤ √ 1 + κ 2 C|G−1(un)−G−1(vn)| + C|G−1(vn)− 0||g′(vn + θ1(un − vn))||G−1(vn)−G−1(un)| ≤ √ 1 + κ 2 C|(G−1(vn + θ2(un − vn)))′||un − vn| 14 G. LI, B. CHENG, Y. HUANG EJDE-2020/56 + C|vn|C̃|(G−1(vn + θ2(un − vn)))′||un − vn| = (√ 1 + κ 2 C + CC̃|vn| ) 1 g(G−1(vn + θ2(un − vn))) |un − vn| ≤ (√ 1 + κ 2 C + CC̃|vn| ) |un − vn|, where g′(t) = κt√ 1 + κt2 2(1+t2) (1 + t2)2 ≤ C̃, θ1, θ2 ∈ (0, 1). By the Hölder’s inequality,∫ RN |(un − vn)ϕ|dx ≤ (∫ RN |un − vn|2dx )1/2(∫ RN |ϕ|2dx )1/2 → 0 and ∫ RN |vn||(un − vn)ϕ|dx ≤ (∫ RN |un − vn|2dx )1/2(∫ RN |vnϕ|2dx )1/2 → 0 as n→∞. Therefore,∫ RN ∣∣∣[h(G−1(un)) g(G−1(un)) − h(G−1(vn)) g(G−1(vn)) ] ϕ ∣∣∣dx→ 0 as n→∞. Then J ′(un)→ 0 as n→∞. � Now, we define b := inf{J(v) : v ∈ P, β(v) = 0}. From Lemma 4.10, similarly to the proof of [22, Lemma 4.11], we have the following result. Lemma 4.11. b > c∞. Let us consider a positive, radially symmetric, ground state solution w ∈ H1(RN ) to the autonomous problem at infinity. We define the operator Π : RN → P by Π[y](x) = w (x− y θy ) , where θy projects w(·− y) onto P. Π is continuous as θy is unique and θy(w(·− y)) is a continuous function of w(·−y). The following lemma describes some properties of Π, its proof can be founded in [10], [11]. Lemma 4.12. It holds that β(Π[y](x)) = y and J(Π[y])→ c∞, |y| → ∞. Lemma 4.13. Assume that (H9) ‖V∞ − V ‖∞ < 2(min{c],2c∞}−c∞) θ̄N‖w‖22 , θ̄ = supy∈RN θy. Then J(Π[y]) < min{c], 2c∞}. Proof. Since J∞ is translation invariant, the maximum of θ 7→ J](w(·/θ)) is attained at θ = 1 and θy > 1. It follows from (H9) and Lemma 2.1 (4) that J(Π[y]) = J∞(Π[y]) + J(Π[y])− J∞(Π[y]) = J∞(Π[y]) + 1 2 ∫ RN (V (x)− V∞)|G−1(Π[y])|2dx EJDE-2020/56 QUASILINEAR SCHRÖDINGER EQUATIONS 15 < c∞ + min{c], 2c∞} − c∞ θ̄N‖w‖22 ∫ RN |Π[y]|2dx = c∞ + min{c], 2c∞} − c∞ θ̄N‖w‖22 ∫ RN ∣∣w(x− y θy )∣∣2dx = c∞ + min{c], 2c∞} − c∞ θ̄N‖w‖22 θNy ‖w‖22 ≤ min{c], 2c∞}. � Remark 4.14. Replacing (H9) with ‖V∞ − V ‖∞ < 2c∞ θ̄N‖w‖22 yields J(Π[y]) < 2c∞. We recall a version of the Linking Theorem with Cerami condition by [2, Theorem 2.3], which we state here for the sake of completeness. Definition 4.15. Let S be a closed subset of a Banach space X and Q be a submanifold of X with relative boundary ∂Q. We say that S and ∂Q link if the following facts hold (1) S ∩ ∂Q = ∅; (2) for any f ∈ C0(X,X) with f |∂Q = id, then f(Q) ∩ S 6= ∅. Moreover, if S and Q are as above and B is a subset of C0(X,X), then S and ∂Q link with respect to B if (1) and (2) hold for any f ∈ B. Lemma 4.16. Suppose that J ∈ C1(X,R) is a functional satisfying (Ce)c con- dition. Consider a closed subset S ⊂ X and a submanifold Q ⊂ X with relative boundary ∂Q are such that (1) S and ∂Q link; (2) α = infv∈S J(v) > supv∈∂Q J(v) = α0; (3) supv∈Q J(v) < +∞. If B = {f ∈ C0(X,X) : f |∂Q = id}, then τ = inff∈B supv∈Q J(f(v)) ≥ α is a critical value of J . Proof of Theorem 1.2. By Lemmas 4.11 and 4.12, we have b > c∞ and J(Π[y]) → c∞, |y| → ∞, there is ρ̄ > 0 such that c∞ < max |y|=ρ̄ J(Π[y]) < b. (4.1) To apply the Linking Theorem 4.16, we take Q := Π ( Bρ̄(0) ) , S := {v ∈ H1(RN ) : v ∈ P, β(v) = 0}, and we show that ∂Q and S link with respect to H = {f ∈ C(Q,P) : f |∂Q = id}. Since β(Π[y](x)) = y from Lemma 4.12, we have that ∂Q ∩ S = ∅, as if v ∈ S, then β(v) = 0, and if v ∈ ∂Q, v = Π[y] for some y ∈ RN with |y| = ρ̄ and then β(v) = y 6= 0. Now we show that f(Q) ∩ S 6= ∅ for any f ∈ H. Given f ∈ H, let T : Bρ̄(0)→ RN is defined by T (y) = β◦f ◦Π[y]. Then the function T is continuous. Moreover, for |y| = ρ̄, we have that Π[y] ∈ ∂Q, thus f ◦ Π[y] = Π[y] as f |∂Q = id, and hence T (y) = y by Lemma 4.12. By Brouwer Fixed Point Theorem there is 16 G. LI, B. CHENG, Y. HUANG EJDE-2020/56 ỹ ∈ Bρ̄(0) with T (ỹ) = 0, which implies that f(Π[ỹ]) ∈ S. Then f(Q) ∩ S 6= ∅ and S and ∂Q link. Now, from (4.1), we may write b = inf S J > max ∂Q J. Let us define k = inf f∈H max v∈Q J(f(v)). Then k ≥ b. In fact, if f ∈ H, there exists w ∈ S with w = f(u) for some u ∈ Π ( Bρ̄(0) ) . Therefore, max v∈Q J(f(v)) ≥ J(f(u)) = J(w) ≥ inf v∈S J(v) = b, and hence k ≥ b, which implies that k > c∞. Furthermore, if f = id, by Lemma 4.13, we have k = inf f∈H max v∈Q J(f(v)) < max v∈Q J(v) < min{c], 2c∞}. Then k ∈ (c∞,min{c], 2c∞}) and it deduces from Lemma 4.7 that the (Ce)c con- dition at level k is satisfied. Then, by the linking theorem, k is a critical level of J . � Remark 4.17. Theorems 1.1 and 1.2 hold for (1.4) with a(x) = 1 under assump- tions (H1’)–(H3’) and (H4)–(H9). Acknowledgments. 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Cui; Existence of infinitely solutions for a modified nonlinear Schrödinger equation via dual approach. Electron. J. Differential Equations, 2018 no. 147 (2018), 1-15. Guofa Li College of Mathematics and Statistics, Qujing Normal University, Qujing 655011, China Email address: liguofa2013@163.com Bitao Cheng College of Mathematics and Statistics, Qujing Normal University, Qujing 655011, China Email address: chengbitao2006@126.com Yisheng Huang (corresponding author) Department of Mathematics, Soochow University, Suzhou 215006, China Email address: yishengh@suda.edu.cn 1. Introduction and statement of main results 2. Preliminaries 3. Pohožaev manifold 4. Existence of a positive solution Acknowledgments References