On the asymptotically cubic generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation

In this paper, we consider the non-existence and existence of solutions for a generalized quasilinear Schrödinger equation with a Kirchhoff-type perturbation. When the non-linearity h(u) shows critical or supercritical growth at infinity, the non-existence result for a quasilinear Schrödinger equation is proved via the Pohožaev identity. If h(u) shows asymptotically cubic growth at infinity, the existence of positive radial solutions for the quasilinear Schrödinger equation is obtained when b is large or equal to 0 and b is equal to 0 by the variational methods. Moreover, some properties are established as the parameter b tends to be 0.

1 Introduction

The Schrödinger equation [1] is of paramount importance in physics, and there are many modifications in literature, for example, the Chen–Lee–Liu equation [2] and stochastic Schrödinger equation [3]. However, the generalized quasilinear Schrödinger equation with a Kirchhoff-type perturbation was rarely studied in literature, which can be written as

$\left(1+b{\int }_{{\mathbb{R}}^3}{g}^2\left(u\right)|\nabla u{|}^{2}dx\right)\left[-\mathrm{d}\mathrm{i}\mathrm{v}\left(g^2\left(u\right)\nabla u\right)+g\left(u\right)g^{\prime }\left(u\right)|\nabla u{|}^2\right]+V\left(x\right)u=h\left(u\right),$(1.1)

where $x\in {\mathbb{R}}^3,b\ge 0,V:{\mathbb{R}}^3\to \mathbb{R}$ and $h:\mathbb{R}\to \mathbb{R}$ are continuous functions, $g\in C^1(\mathbb{R},{\mathbb{R}}^{+})$ satisfies (g₁), g is even, g′(t) ≤ 0, $g(0)=1,\underset{t\to +\infty }{\lim }g(t)=l,l\in (\mathrm{0,1})$, and ∀ t ≥ 0.

When b = 0, Eq. 1.1 is reduced to the following quasilinear Schrödinger equation:

$-\mathrm{d}\mathrm{i}\mathrm{v}\left(g^2\left(u\right)\nabla u\right)+g\left(u\right)g^{\prime }\left(u\right)|\nabla u{|}^2+V\left(x\right)u=h\left(u\right),x\in {\mathbb{R}}^3.$(1.2)

According to [4], let $g(u)=\sqrt{1+2{({\phi }^{\prime }(|u{|}^2))}^2{u}^2}$, then, Eq. 1.2 is transformed into

$\begin{array}{c}\hfill -△u-\left[△\left(\phi \left(|u{|}^2\right)\right)\right]{\phi }^{\prime }\left(|u{|}^2\right)u+V\left(x\right)u=h\left(u\right),x\in {\mathbb{R}}^3.\end{array}$(1.3)

It is well-known that the classical case is φ(s) = s or $\phi (s)=\sqrt{1+s}$ [5–12].

For Eq. 1.1, another interesting question is b > 0. When g(t) = 1 for all $t\in \mathbb{R}$, it is reduced to the following classical Kirchhoff equation:

$-\left(1+b{\int }_{{\mathbb{R}}^3}|\nabla u{|}^{2}dx\right)\mathrm{\Delta }u+V\left(x\right)u=h\left(u\right),x\in {\mathbb{R}}^3.$(1.4)

It is well-known that Eq. 1.4 is related to the stationary analog of the following Kirchhoff-type equation:

$u_{tt}+\left(1+b{\int }_{{\mathbb{R}}^3}|\nabla u{|}^{2}dx\right)\mathrm{\Delta }u+V\left(x\right)u=h\left(u\right),x\in {\mathbb{R}}^3,$(1.5)

which was proposed by Kirchhoff as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings [13,14]. More physical background can be found in [15] and the references therein. Based on the aforementioned analysis, it is necessary to study Eq. 1.1.

1.1 Related works and main results

At first, let us briefly review the predecessors’ pioneering works about the problem [16–20]. However, to the best of our knowledge, there are no works involving Eq. 1.1 when the non-linearity h(u) is asymptotically cubic at infinity. More information about the asymptotically cubic problems is given in [21,22] and the references therein. The main goal of the present paper is to investigate this problem. Precisely, we suppose that

(V₁) $V(x)=V(|x|),0<V_0\le V(x)\le V_{\infty }≔\underset{|x|\to +\infty }{\lim }V(x)<\infty$;

(V₂) $V\in C^1({\mathbb{R}}^3,\mathbb{R})$ and $⟨\nabla V(x),x⟩\le 0,\forall x\in {\mathbb{R}}^3$;

(h₁) $h\in C(\mathbb{R},\mathbb{R})$, h(t) = 0, ∀ t ≤ 0, and $\underset{t\to 0}{\lim }\frac{h(t)}{t}=0$;

(h₂) $\underset{|t|\to +\infty }{\lim }\frac{|h(t)|}{|t{|}^3}=\gamma ,\gamma >b{l}^4{\lambda }_1$, where

${\lambda }_1≔\inf \left\{{\left({\int }_{{\mathbb{R}}^3}|\nabla w{|}^{2}dx\right)}^2:w\in \mathcal{H},{\int }_{{\mathbb{R}}^3}|w{|}^{4}dx=1\right\}$

and $\mathcal{H}$ is defined in Section 2;

(h₃) $\frac{1}{4}h(t)t\ge H(t)$ for all t > 0, where $H(t)={\int }_0^{t}h(s)ds$.

Remark 1.1: For example, $h(t)=\frac{\gamma t^5}{1+t^2}$. By direct calculations, we have

$H\left(t\right)=\frac{\gamma t^4}{4}-\frac{\gamma t^2}{2}+\frac{\gamma }{2}\ln \left(1+t^2\right).$

It is easy to observe that h satisfies the assumption (h₁) − (h₃).

The first result involves non-existence for the Kirchhoff-type perturbation problem.

Theorem 1.1: Assume that (g₁) holds with $\frac{1}{3}\le l\le 1$ and ⟨∇V(x), x⟩≥ 0. For any b > 0, Eq. 1.1 has no non-trivial solutions with h(u) = |u|^p−2u, p ≥ 6.

The next result describes the existence for generalized quasilinear Schrödinger equations with the Kirchhoff term.

Theorem 1.2: Assume that (V₁), (V₂), (g₁), (h₁), and (h₂) are satisfied. Then, Eq. 1.1 has a positive radial solution.

The third result shows the existence for generalized quasilinear Schrödinger equations without the Kirchhoff term.

Theorem 1.3: Assume that (V₁), (V₂), (g₁), and (h₁) − (h₃) are satisfied. Then, Eq. 1.2 has a positive radial solution.

Compared with Theorem 1.2, without the Kirchhoff term ${\int }_{{\mathbb{R}}^3}{g}^2(u)|\nabla u{|}^{2}dx$, we find that we need to add the condition (h₃). Until now, we have not been able to remove it. A natural question is that what happens if Kirchhoff-type perturbation occurs, that is, when b → 0, can we build a relationship between Theorem 1.2 and 1.3? In this regard, we state the following.

Theorem 1.4: Assume that (V₁), (V₂), (g₁), and (h₁) − (h₃) hold and $\{u_{b_n}\}\subset \mathcal{H}$ are the positive radial solutions obtained in Theorem 1.2 for each $n\in \mathbb{N}$. Then, $u_{b_n}\to u_0$ in $\mathcal{H}$ as b_n → 0, n → ∞, where u₀ is a positive radial solution for Eq. 1.2.

1.2 Our contributions and methods

We should mention that our results are new since we focus on the asymptotically cubic case. Compared with [16,19,20], we know that in Theorem 1.1, our non-linear term in the autonomy problem Eq. 1.1 is supercritical, so we invoke the Pohožaev-type identity. As for Theorem 1.2, the problem is asymptotically 3-linear at infinity (i.e., h(t) ∼ t³), so it is different from [16]. We take full advantage of the condition h₂, and this is our paper’s highlight. We borrow the idea from [16], but we require more elaborate estimates (see Lemma 3.2–3.4) to prove Theorem 1.3. It is worth pointing out that in Theorem 1.3, it seems that the condition (h₃) is fussy, but our pursuit is not to relax the condition. Our condition (h₃) is different from ([16], h₅), and we adopt the idea from [23], Lemma 2.2 to obtain mountain pass geometry (see Lemma 3.5). Finally, we study the behavior of the positive radial solutions as b → 0. Since we do not know whether u₀ is unique, we cannot draw the conclusion that u₀ is obtained in Theorem 1.2.

1.3 Organization

This paper is organized as follows. Section 2 provides some preliminaries, and Section 3 is divided into three parts, which will prove Theorems 1.1–1.3, respectively. The proof of Theorem 1.4 is given in Section 3. Throughout this paper, the following notations are used:

• ‖u‖_p (1 < p ≤ ∞) is the norm in $L^p({\mathbb{R}}^3)$;

• → and ⇀ denote strong and weak convergence, respectively;

• ⟨⋅, ⋅⟩ denotes the duality pairing between a Banach space and its dual space;

• o_n(1) denotes o_n(1) → 0 as n → ∞.

2 Preliminary results

Since the condition (V₁), we use the work space

$\mathcal{H}≔\left\{u\in H^1\left({\mathbb{R}}^3\right):u\left(x\right)=u\left(|x|\right)\right\},$

equipped with the norm

$\Vert u{\Vert }_{\mathcal{H}}^2={\int }_{{\mathbb{R}}^3}\left(|\nabla u{|}^2+V\left(x\right)u^2\right)dx.$(2.1)

According to [16], the energy functional associated with Eq. 1.1 is

$\begin{array}{cc}\hfill I_b\left(u\right)=& \frac{1}{2}{\int }_{{\mathbb{R}}^3}{g}^2\left(u\right)|\nabla u{|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^3}V\left(x\right)|u{|}^{2}dx+\frac{b}{4}{\left({\int }_{{\mathbb{R}}^3}{g}^2\left(u\right)|\nabla u{|}^{2}dx\right)}^2\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^3}H\left(u\right)dx,\hfill \end{array}$

where $H(t)={\int }_0^{t}h(s)ds$. We require the change of variable [24–27]

$v=G\left(u\right)={\int }_0^{u}g\left(t\right)dt,$(2.2)

and I(u) can be reduced to

$\begin{array}{cc}\hfill J_b\left(v\right)& =\frac{1}{2}{\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^3}V\left(x\right)|G^{-1}\left(v\right){|}^{2}dx+\frac{b}{4}{\left({\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx\right)}^2\hfill \\ \hfill & -{\int }_{{\mathbb{R}}^3}H\left(G^{-1}\left(v\right)\right)dx,\hfill \end{array}$(2.3)

where G⁻¹(v) is the inverse of G(u).

Clearly, we have the following lemma (see [16]).

Lemma 2.1: Assume that (V₁) holds. If $v\in \mathcal{H}$ is a critical point of J_b, then u = G⁻¹(v) is a weak solution of Eq. 1.1.

3 Proof of the main results

3.1 Proof of Theorem 1.1

By a standard argument in [28], we can obtain the following Pohožaev type.

Lemma 3.1: If $v\in \mathcal{H}$ is a weak solution of Eq. 1.1 with h(t) = |t|^p−2t, p ≥ 6, then v satisfies

$\begin{array}{cc}\hfill \frac{1}{2}{\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx& +\frac{3}{2}{\int }_{{\mathbb{R}}^3}V\left(x\right)|G^{-1}\left(v\right){|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^3}⟨\nabla V\left(x\right),x⟩|G^{-1}\left(v\right){|}^{2}dx\hfill \\ \hfill & +\frac{b}{2}{\left({\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx\right)}^2=\frac{3}{p}{\int }_{{\mathbb{R}}^3}|G^{-1}\left(v\right){|}^{p}dx.\hfill \end{array}$

Based on the identity, we can provide the proof of Theorem 1.1. Indeed, v satisfies

$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx+{\int }_{{\mathbb{R}}^3}V\left(x\right)\frac{G^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}vdx+b{\left({\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx\right)}^2\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^3}\frac{|G^{-1}\left(v\right){|}^{p-2}{G}^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}vdx.\hfill \end{array}$

Since $\frac{1}{3}\le l\le 1$, using (5) of Lemma 2.1 in [29], jointly with ⟨∇V(x), x⟩≥ 0, we can obtain 0 = u = G⁻¹(v).

3.2 Proof of Theorem 1.2

This section provides the proof of Theorem 1.2. Clearly, as mentioned previously, we are devoted to studying the functional J_b [Eq. 2.3]. Since our case is asymptotically cubic, it is hard to prove the boundedness of the PS-sequences of J_b. Hence, we use [30], Theorem 1.1 to find a special bounded PS-sequence of J_b,μ, where

$\begin{array}{cc}\hfill J_{b,\mu }\left(v\right)≔& \frac{1}{2}{\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^3}V\left(x\right)|G^{-1}\left(v\right){|}^{2}dx+\frac{b}{4}{\left({\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx\right)}^2\hfill \\ \hfill & -\mu {\int }_{{\mathbb{R}}^3}H\left(G^{-1}\left(v\right)\right)dx,\hfill \end{array}$

μ ∈ [1, 2]. We have the following lemma.

Lemma 3.2: Assume that (h₁)–(h₂) are satisfied, then

(i) for μ ∈ [1, 2], there exists $v\in \mathcal{H}\backslash \{0\}$ such that J_b,μ(v) < 0.

(ii) there exists ρ, α > 0 such that J_b,μ(v) ≥ α and $\Vert v{\Vert }_{\mathcal{H}}=\rho$.

Proof. (i) It is well-known that λ₁ > 0 is attained [ ([31]; Section 1.7)]. In other words, $\varphi \in \mathcal{H}$ satisfied ${\int }_{{\mathbb{R}}^3}|\varphi {|}^{4}dx=1$ and ϕ > 0 such that

${\lambda }_1={\left({\int }_{{\mathbb{R}}^3}|\nabla \varphi {|}^{2}dx\right)}^2.$

In view of (h₂), $1<\frac{1}{l^2}$, and 1 ≤ μ ≤ 2, jointly with (3) and (4) of Lemma 2.1 in [29], we have

$\begin{array}{cc}\hfill & \underset{t\to +\infty }{\lim }\frac{J_{b,\mu }\left(t\varphi \right)}{t^4}\hfill \\ \hfill & <\underset{t\to +\infty }{\lim }\left[\frac{1}{l^2{t}^2}\Vert \varphi {\Vert }_{\mathcal{H}}^2+\frac{b}{4}{\left({\int }_{{\mathbb{R}}^3}|\nabla \varphi {|}^{2}dx\right)}^2-\mu {\int }_{{\mathbb{R}}^3}\frac{H\left(G^{-1}\left(t\varphi \right)\right)}{|G^{-1}\left(t\varphi \right){|}^4}\frac{|G^{-1}\left(t\varphi \right){|}^4}{|t\varphi {|}^4}|\varphi {|}^{4}dx\right]\hfill \\ \hfill & \le \frac{b}{4}{\left({\int }_{{\mathbb{R}}^3}|\nabla \varphi {|}^{2}dx\right)}^2-\frac{b{\lambda }_1}{4}{\int }_{{\mathbb{R}}^3}|\varphi {|}^{4}dx\hfill \\ \hfill & =0.\hfill \end{array}$

Hence, when t is large, let v ≔ tϕ, and we obtain the results.

(ii) Let $\epsilon \in \left(0,\frac{l^2{V}_0}{2\mu }\right)$, then we obtain

$J_{b,\mu }\left(v\right)\ge \frac{1}{2}{\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^3}\left(V_0-\frac{\mu \epsilon }{l^2}\right)|v{|}^{2}dx-\frac{C_{\epsilon }\mu }{q{l}^q}{\int }_{{\mathbb{R}}^3}|v{|}^{q}dx.$(3.1)

Hence, we can choose $\Vert v{\Vert }_{\mathcal{H}}=\rho >0$ small enough such that J_b,μ(v) > 0.

Define

$\begin{array}{cc}\hfill A\left(v\right)≔& \frac{1}{2}{\int }_{{\mathbb{R}}^3}\left[|\nabla v{|}^2+V\left(x\right)|G^{-1}\left(v\right){|}^2\right]dx+\frac{b}{4}{\left({\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx\right)}^2,\hfill \\ \hfill B\left(v\right)≔& {\int }_{{\mathbb{R}}^3}H\left(G^{-1}\left(v\right)\right)dx.\hfill \end{array}$

It is deduced from (V₁) and (3) of Lemma 2.1 in [29] that

$A\left(v\right)>\frac{1}{2}\Vert v{\Vert }_{\mathcal{H}}^2\to +\infty ,\text{as}\Vert v{\Vert }_{\mathcal{H}}\to \infty ,\forall v\in \mathcal{H}.$

Moreover, from (h₁), it can be observed that $B(v)={\int }_{{\mathbb{R}}^3}H(G^{-1}(v))dx\ge 0,\forall v\in \mathcal{H}$.

Using [30], Theorem 1.1 or [16], Theorem 4.1, it shows that for a.e. μ ∈ [1, 2], there is a bounded ${(PS)}_{c_{\mu }}$ sequence $\{v_n\}\subset \mathcal{H}$, where c_μ is the mountain pass level.

Lemma 3.3: Up to a subsequence, v_n → v_μ in $\mathcal{H}$.

Proof: Since $\{v_n\}\subset \mathcal{H}$ is bounded, up to a subsequence, there exists $v_{\mu }\in \mathcal{H}$ such that v_n ⇀ v_μ, in $\mathcal{H}$, v_n → v_μ, in $L^p({\mathbb{R}}^3)$ for 2 < p < 6, and v_n(x) → v_μ(x) a.e. $x\in {\mathbb{R}}^3$. Obviously, $J_{b,\mu }^{\prime }(v_{\mu })=0$. Then,

$\begin{array}{c}{o}_n\left(1\right)=⟨J_{b,\mu }^{\prime }\left(v_n\right)-J_{b,\mu }^{\prime }\left(v_{\mu }\right),v_n-v_{\mu }⟩\hfill \\ ={\int }_{{\mathbb{R}}^3}|\nabla \left(v_n-v_{\mu }\right){|}^{2}dx+{\int }_{{\mathbb{R}}^3}V\left(x\right)\left[\frac{G^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}-\frac{G^{-1}\left(v_{\mu }\right)}{g\left(G^{-1}\left(v_{\mu }\right)\right)}\right]\\ \left(v_n-v_{\mu }\right)dx+b\left[{\int }_{{\mathbb{R}}^3}|\nabla v_n{|}^{2}dx{\int }_{{\mathbb{R}}^3}\nabla v_n\nabla \left(v_n-v_{\mu }\right)dx-{\int }_{{\mathbb{R}}^3}|\nabla v_{\mu }{|}^{2}dx{\int }_{{\mathbb{R}}^3}\nabla v_{\mu }\nabla \left(v_n-v_{\mu }\right)dx\right]\\ -\mu {\int }_{{\mathbb{R}}^3}\left[\frac{h\left(G^{-1}\left(v_n\right)\right)}{g\left(G^{-1}\left(v_n\right)\right)}-\frac{h\left(G^{-1}\left(v_{\mu }\right)\right)}{g\left(G^{-1}\left(v_{\mu }\right)\right)}\right]\left(v_n-v_{\mu }\right)dx.\end{array}$(3.2)

We define $\phi :\mathbb{R}\to \mathbb{R}$ by φ(t) = G⁻¹(t)/g(G⁻¹(t)). Noting that l < g(t) ≤ 1 for $t\in \mathbb{R}$, jointly with [29], (2) of Lemma 2.1, we have

${\phi }^{\prime }\left(t\right)=\frac{1}{g^2\left(G^{-1}\left(t\right)\right)}\left[1-\frac{G^{-1}\left(t\right)g^{\prime }\left(G^{-1}\left(t\right)\right)}{g\left(G^{-1}\left(t\right)\right)}\right]\ge \frac{1}{g^2\left(G^{-1}\left(t\right)\right)}\ge 1.$

According to the mean value theorem, for any $x\in {\mathbb{R}}^3$, there exists a function ξ(x) between v_μ(x) and v_n(x) such that

$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^3}V\left(x\right)\left[\frac{G^{-1}\left(v_n\right)}{g\left(G^{-1}\left(v_n\right)\right)}-\frac{G^{-1}\left(v_{\mu }\right)}{g\left(G^{-1}\left(v_{\mu }\right)\right)}\right]\left(v_n-v_{\mu }\right)dx& ={\int }_{{\mathbb{R}}^3}V\left(x\right){\phi }^{\prime }\left(\xi \right)|v_n-v_{\mu }{|}^{2}dx\hfill \\ \hfill & \ge {\int }_{{\mathbb{R}}^3}V\left(x\right)|v_n-v_{\mu }{|}^{2}dx.\hfill \end{array}$(3.3)

It is easy to check that

$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^3}|\nabla v_n{|}^{2}dx{\int }_{{\mathbb{R}}^3}\nabla v_n\nabla \left(v_n-v_{\mu }\right)dx-\hfill \\ \hfill & {\int }_{{\mathbb{R}}^3}|\nabla v_{\mu }{|}^{2}dx{\int }_{{\mathbb{R}}^3}\nabla v_{\mu }\nabla \left(v_n-v_{\mu }\right)dx\hfill \\ \hfill & ={\int }_{{\mathbb{R}}^3}\left[|\nabla v_n{|}^2-|\nabla v_{\mu }{|}^2\right]dx{\int }_{{\mathbb{R}}^3}\nabla v_n\nabla \left(v_n-v_{\mu }\right)dx\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^3}|\nabla v_{\mu }{|}^{2}dx{\int }_{{\mathbb{R}}^3}|\nabla \left(v_n-v_{\mu }\right){|}^{2}dx\hfill \\ \hfill & \to 0,n\to \infty .\hfill \end{array}$(3.4)

Noting that [29], (3) of Lemma 2.1, we obtain

$\begin{array}{cc}\hfill & \left|{\int }_{{\mathbb{R}}^3}\left[\frac{h\left(G^{-1}\left(v_n\right)\right)}{g\left(G^{-1}\left(v_n\right)\right)}-\frac{h\left(G^{-1}\left(v_{\mu }\right)\right)}{g\left(G^{-1}\left(v_{\mu }\right)\right)}\right]\left(v_n-v_{\mu }\right)dx\right|\hfill \\ \hfill & \le C{\int }_{{\mathbb{R}}^3}\left(|v_n|+|v_n{|}^{q-1}+|v_{\mu }|+|v_{\mu }{|}^{q-1}\right)|v_n-v_{\mu }|dx.\hfill \end{array}$(3.5)

Therefore, v_n → v_μ in $\mathcal{H}$.

It is easy to check the following lemma.

Lemma 3.4: If $v\in \mathcal{H}$ is a critical point of J_b,μ(v), then v satisfies

$\begin{array}{cc}\hfill \frac{1}{2}{\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx& +\frac{3}{2}{\int }_{{\mathbb{R}}^3}V\left(x\right)|G^{-1}\left(v\right){|}^{2}dx+\frac{1}{2}{\int }_{{\mathbb{R}}^3}⟨\nabla V\left(x\right),x⟩|G^{-1}\left(v\right){|}^{2}dx\hfill \\ \hfill & +\frac{b}{2}{\left({\int }_{{\mathbb{R}}^3}|\nabla v{|}^{2}dx\right)}^2=3\mu {\int }_{{\mathbb{R}}^3}H\left(G^{-1}\left(v\right)\right)dx.\hfill \end{array}$

Up to this point, we can prove Theorem 1.3. In fact, it is deduced from Lemma 3.2 and 3.3 that there exists {μ_n} ⊂ [1, 2] such that $\underset{n\to \infty }{\lim }{\mu }_n=1,v_{{\mu }_n}\in \mathcal{H}$ satisfies $J_{b,{\mu }_n}(v_{{\mu }_n})=c_{{\mu }_n}>0,J_{b,{\mu }_n}^{\prime }(v_{{\mu }_n})=0$. Next, we prove $\{v_{{\mu }_n}\}$ is bounded in $\mathcal{H}$. Since the map μ → c_μ is non-increasing, combining with Lemma 3.4 and condition (V₂), we obtain

$\begin{array}{cc}\hfill M& \ge J_{b,{\mu }_n}\left(v_{{\mu }_n}\right)\hfill \\ \hfill & =\frac{1}{3}{\int }_{{\mathbb{R}}^3}|\nabla v_{{\mu }_n}{|}^{2}dx-\frac{1}{6}{\int }_{{\mathbb{R}}^3}⟨\nabla V\left(x\right),x⟩|G^{-1}\left(v_{{\mu }_n}\right){|}^{2}dx+\frac{b}{12}{\left({\int }_{{\mathbb{R}}^3}|\nabla v_{{\mu }_n}{|}^{2}dx\right)}^2\hfill \\ \hfill & \ge \frac{1}{3}{\int }_{{\mathbb{R}}^3}|\nabla v_{{\mu }_n}{|}^{2}dx.\hfill \end{array}$(3.6)

It is easy to check that

$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^3}|\nabla v_{{\mu }_n}{|}^{2}dx+{\int }_{{\mathbb{R}}^3}V\left(x\right)\frac{G^{-1}\left(v_{{\mu }_n}\right)}{g\left(G^{-1}\left(v_{{\mu }_n}\right)\right)}{v}_{{\mu }_n}dx+b{\left({\int }_{{\mathbb{R}}^3}|\nabla v_{{\mu }_n}{|}^{2}dx\right)}^2\hfill \\ \hfill & ={\mu }_n{\int }_{{\mathbb{R}}^3}\frac{h\left(G^{-1}\left(v_{{\mu }_n}\right)\right)}{g\left(G^{-1}\left(v_{{\mu }_n}\right)\right)}{v}_{{\mu }_n}dx\hfill \\ \hfill & \le \frac{\epsilon {\mu }_n}{l^2}{\int }_{{\mathbb{R}}^3}|v_{{\mu }_n}{|}^{2}dx+\frac{C_{\epsilon }{\mu }_n}{l^6}{\int }_{{\mathbb{R}}^3}|v_{{\mu }_n}{|}^{6}dx\hfill \\ \hfill & \le \frac{\epsilon {\mu }_n}{l^2}{\int }_{{\mathbb{R}}^3}|v_{{\mu }_n}{|}^{2}dx+\frac{C_{\epsilon }{\mu }_{n}S}{l^6}{\left({\int }_{{\mathbb{R}}^3}|\nabla v_{{\mu }_n}{|}^{2}dx\right)}^3\hfill \\ \hfill & \le \frac{\epsilon {\mu }_n}{l^2}{\int }_{{\mathbb{R}}^3}|v_{{\mu }_n}{|}^{2}dx+\frac{C_{\epsilon }{\mu }_{n}S}{l^6}{\left(3M\right)}^3.\hfill \end{array}$

Moreover, using (3) and (5) of Lemma 2.1 in [29], it is deduced from condition (V₁) that

$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^3}{V}_0|v_{{\mu }_n}{|}^{2}dx& \le {\int }_{{\mathbb{R}}^3}|\nabla v_{{\mu }_n}{|}^{2}dx+{\int }_{{\mathbb{R}}^3}V\left(x\right)\frac{G^{-1}\left(v_{{\mu }_n}\right)}{g\left(G^{-1}\left(v_{{\mu }_n}\right)\right)}{v}_{{\mu }_n}dx+b{\left({\int }_{{\mathbb{R}}^3}|\nabla v_{{\mu }_n}{|}^{2}dx\right)}^2\hfill \\ \hfill & \le \frac{\epsilon {\mu }_n}{l^2}{\int }_{{\mathbb{R}}^3}|v_{{\mu }_n}{|}^{2}dx+\frac{C_{\epsilon }{\mu }_{n}S}{l^6}{\left(3M\right)}^3.\hfill \end{array}$

Let $\epsilon =\frac{l^2{V}_0}{2{\mu }_n}$, then we obtain

${\int }_{{\mathbb{R}}^3}|v_{{\mu }_n}{|}^{2}dx\le \frac{2S{C}_{\epsilon }{\mu }_n}{l^6{V}_0}{\left(3M\right)}^3.$(3.7)

From Eqs 3.6, 3.7, we know that $\{v_{{\mu }_n}\}$ is bounded in $\mathcal{H}$.

A subsequence of $\{v_{{\mu }_n}\}$ is selected and also denoted by {v_n}, such that v_n ⇀ v in $\mathcal{H}$. Similar to the proof of Lemma 3.3, we obtain v_n → v in $\mathcal{H}$. It is well-known that μ↦c_μ is continuous from the left [ ([16], Theorem 4.1)]. So,

$\underset{n\to \infty }{\lim }{J}_b\left(v_{{\mu }_n}\right)=\underset{n\to \infty }{\lim }\left[J_{b,{\mu }_n}\left(v_{{\mu }_n}\right)+\left({\mu }_n-1\right){\int }_{{\mathbb{R}}^3}H\left(G^{-1}\left(v_{{\mu }_n}\right)\right)dx\right]=\underset{n\to \infty }{\lim }{c}_{{\mu }_n}=\tilde{c}.$

In addition,

$\underset{n\to \infty }{\lim }\langle J_b^{\prime }\left(v_{{\mu }_n}\right),\psi \rangle =\underset{n\to \infty }{\lim }\left[\langle J_{b,{\mu }_n}^{\prime }\left(v_{{\mu }_n}\right),\hspace{1em}\psi \rangle +\left({\mu }_n-1\right){\int }_{{\mathbb{R}}^3}\frac{h\left(G^{-1}\left(v_{{\mu }_n}\right)\right)}{g\left(G^{-1}\left(v_{{\mu }_n}\right)\right)}\psi dx\right]=0,$

for any $\psi \in C_0^{\infty }({\mathbb{R}}^3)$, which means that $J_b^{\prime }(v)=0$ satisfies $J_b(v)=\tilde{c}>0$. Let v⁻ = min{v, 0}. Using (3) and (5) of Lemma 2.1 in [29], we have

$\begin{array}{cc}\hfill 0=& ⟨J_b^{\prime }\left(v\right),v^{-}⟩\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^3}\left(|\nabla v^{-}{|}^2+V\left(x\right)\frac{G^{-1}\left(v^{-}\right)}{g\left(G^{-1}\left(v^{-}\right)\right)}{v}^{-}\right)dx\hfill \\ \hfill \ge & {\int }_{{\mathbb{R}}^3}\left(|\nabla v^{-}{|}^2+V\left(x\right)|v^{-}{|}^2\right)dx.\hfill \end{array}$(3.8)

It shows that v⁻≡ 0. Applying the strong maximum principle, we obtain v(x) > 0.

3.3 Proof of Theorem 1.3

This section studies the case $\frac{1}{4}h(t)t\ge H(t)$ for all t > 0 and without the Kirchhoff term ${\int }_{{\mathbb{R}}^3}{g}^2(u)|\nabla u{|}^{2}dx$. At first, let us check the mountain pass geometry of the functional J₀.

Lemma 3.5: Assume that (h₁)–(h₂) are satisfied, then

(i) there exists $v\in \mathcal{H}\backslash \{0\}$ such that J₀(v) < 0.

(ii) there exist ρ, α > 0 such that J₀(v) ≥ α, $\Vert v{\Vert }_{\mathcal{H}}=\rho$.

Proof (i) Motivated by Lemma 2.2 of [23], we need to study the following equation:

$-\mathrm{\Delta }v+V_{\infty }\frac{G^{-1}\left(v\right)}{g\left(G^{-1}\left(v\right)\right)}=\frac{h\left(G^{-1}\left(v\right)\right)}{g\left(G^{-1}\left(v\right)\right)},x\in {\mathbb{R}}^3.$(3.9)

The corresponding functional is J_0,∞(v). We also define the mountain pass min–max level

$c_{\infty }=\underset{\xi \in {\mathrm{\Gamma }}_{\infty }}{\inf }\underset{t\in \left[\mathrm{0,1}\right]}{\max }{J}_{0,\infty }\left(\xi \left(t\right)\right),$

where

${\mathrm{\Gamma }}_{\infty }=\left\{\xi \in C\left(\left[\mathrm{0,1}\right],\mathcal{H}:\xi \left(0\right)=0\ne \xi \left(1\right),J_{0,\infty }\left(\xi \left(1\right)\right)\right.<0\right\}.$

By the standard arguments, it shows that $w\in H^1({\mathbb{R}}^N)$ is a solution of Eq. 3.9, which satisfies J_0,∞(w) = c_∞. A continuous path $\alpha :[0,+\infty )\to \mathcal{H}$ is defined by α(t) (x) = w(x/t), if t > 0 and α(0) = 0. Taking the derivative, we know that

$\begin{array}{cc}\hfill \frac{d}{dt}{J}_{0,\infty }\left(\alpha \left(t\right)\right)=& \frac{1}{2}\underset{{\mathbb{R}}^3}{\int }|\nabla w{|}^{2}dx+\frac{3}{2}{t}^2{\int }_{{\mathbb{R}}^3}{V}_{\infty }|G^{-1}\left(w\right){|}^{2}dx\hfill \\ \hfill & -3t^2{\int }_{{\mathbb{R}}^3}H\left(G^{-1}\left(w\right)\right)dx.\hfill \end{array}$

Since w is a solution of Eq. 3.9, it satisfies the Pohožaev identity,

$\frac{1}{2}{\int }_{{\mathbb{R}}^3}|\nabla w{|}^{2}dx+\frac{3}{2}{\int }_{{\mathbb{R}}^3}{V}_{\infty }|G^{-1}\left(w\right){|}^{2}dx=3{\int }_{{\mathbb{R}}^3}H\left(G^{-1}\left(w\right)\right)dx.$

Therefore,

$\frac{d}{dt}{J}_{0,\infty }\left(\alpha \left(t\right)\right)=\frac{1}{2}\left(1-t^2\right){\int }_{{\mathbb{R}}^3}|\nabla w{|}^{2}dx.$

The map t ↦ J_0,∞(α(t)) achieves the maximum value at t = 1. By choosing L > 0 sufficiently large, we have J_0,∞(α(L)) < 0. Taking ζ(t) = α(tL), we have ζ ∈ Γ_∞. If ${\zeta }_y(t)≔w\left(\frac{\cdot -y}{tL}\right)$, noting that (V₁), we obtain

$J_0\left({\zeta }_y\left(1\right)\right)=J_{0,\infty }\left({\zeta }_y\left(1\right)\right)+\frac{1}{2}{\int }_{{\mathbb{R}}^3}\left(V\left(x+y\right)-V_{\infty }\right)|G^{-1}\left({\zeta }_y\left(1\right)\right){|}^{2}dx<0,\mathrm{f}\mathrm{o}\mathrm{r}|y|\text{is\hspace{0.17em}large}.$

Choosing e = ζ_y(1), we can obtain the result.

(ii) Similar to (ii) of Lemma 3.2, we obtain

$\begin{array}{cc}\hfill J_0\left(v\right)& \ge \frac{1}{2}{\int }_{{\mathbb{R}}^3}\left(|\nabla v{|}^2+V\left(x\right)|v{|}^2\right)dx-{\int }_{{\mathbb{R}}^3}\left(\frac{\epsilon }{2}|G^{-1}\left(v\right){|}^2+\frac{C_{\epsilon }}{q}|G^{-1}\left(v\right){|}^q\right)dx\hfill \\ \hfill & \ge \frac{C}{4}\Vert v{\Vert }_{\mathcal{H}}^2-\frac{C_1{C}_{\epsilon }}{q{l}^q}\Vert v{\Vert }_{\mathcal{H}}^q.\hfill \end{array}$

Hence, choosing $\Vert v{\Vert }_{\mathcal{H}}=\rho >0$ small enough, we can obtain the desired conclusion.

Therefore, there is a (PS) $c_0$ sequence $\{v_n\}\subset \mathcal{H}$ where c₀ is the mountain pass level of the J₀.

Lemma 3.6: {v_n} is bounded.

Proof: Since $G^{-1}(v_n)g(G^{-1}(v_n))\in \mathcal{H}$, jointly with (h₃) and [29], (2) of Lemma 2.1], we obtain

$\begin{array}{cc}\hfill c+o_n\left(1\right)=& J_0\left(v_n\right)-\frac{1}{4}⟨J_0^{\prime }\left(v_n\right),G^{-1}\left(v_n\right)g\left(G^{-1}\left(v_n\right)\right)⟩\hfill \\ \hfill \ge & \frac{1}{4}\Vert v_n{\Vert }_{\mathcal{H}}^2.\hfill \end{array}$

Hence, {v_n} is bounded in $\mathcal{H}$.

Similar to Lemma 3.3, we obtain the following result.

Lemma 3.7: Up to a subsequence, v_n → v in $\mathcal{H}$.

Proof of Theorem 1.3: It deduces from lemmas 3.5, 3.6, and 3.7 that Eq. 1.2 has a non-trivial solution v. Similar to Eq. 3.8, we know that $v(x)>0,x\in {\mathbb{R}}^3$.

4 Asymptotic properties of the positive radial solution

Proof of Theorem 1.4: If $v_{b_n}$ is a critical point of $J_{b_n}$, which is obtained in Theorem 1.2 for each $n\in \mathbb{N}$. Similar to the proof of Lemma 3.2, for b_n → 0, n → ∞, $\{v_{b_n}\}$ is a (PS)_c sequence, which is bounded in $\mathcal{H}$. There exists a subsequence of {b_n}, still denoted by {b_n}, such that $v_{b_n}\rightharpoonup v_0$ in $\mathcal{H}$. It is easy to obtain

$\begin{array}{cc}\hfill \Vert v_{b_n}-v_0{\Vert }_{\mathcal{H}}^2& \le ⟨J_{b_n}^{\prime }\left(v_{b_n}\right)-J_0^{\prime }\left(v_0\right),v_{b_n}-v_0⟩-b_n{\int }_{{\mathbb{R}}^3}|\nabla v_{b_n}{|}^{2}dx{\int }_{{\mathbb{R}}^3}\nabla v_{b_n}\nabla \left(v_{b_n}-v_0\right)dx+{\int }_{{\mathbb{R}}^3}\left[\frac{h\left(G^{-1}\left(v_{b_n}\right)\right)}{g\left(G^{-1}\left(v_{b_n}\right)\right)}-\frac{h\left(G^{-1}\left(v_0\right)\right)}{g\left(G^{-1}\left(v_0\right)\right)}\right]\left(v_{b_n}-v_0\right)dx\hfill \\ \hfill & =o_n\left(1\right).\hfill \end{array}$

On one hand, in view of (3) of Lemma 2.1 in [29], we can use the Lebesgue dominated convergence theorem to obtain

$\underset{n\to \infty }{\lim }{\int }_{{\mathbb{R}}^3}V\left(x\right)\frac{G^{-1}\left(v_{b_n}\right)\varphi }{g\left(G^{-1}\left(v_{b_n}\right)\right)}dx={\int }_{{\mathbb{R}}^3}V\left(x\right)\frac{G^{-1}\left(v_0\right)\varphi }{g\left(G^{-1}\left(v_0\right)\right)}dx,$

$\underset{n\to \infty }{\lim }{\int }_{{\mathbb{R}}^3}\frac{h\left(G^{-1}\left(v_{b_n}\right)\right)\varphi }{g\left(G^{-1}\left(v_{b_n}\right)\right)}dx={\int }_{{\mathbb{R}}^3}\frac{h\left(G^{-1}\left(v_0\right)\right)\varphi }{g\left(G^{-1}\left(v_0\right)\right)}dx.$

On the other hand, we have $⟨J_{b_n}^{\prime }(v_{b_n}),\varphi ⟩=o_n(1)$ and $⟨J_0^{\prime }(v_0),\varphi ⟩=o_n(1)$. Moreover,

$\underset{n\to \infty }{\lim }{\int }_{{\mathbb{R}}^3}\nabla v_{b_n}\nabla \varphi dx={\int }_{{\mathbb{R}}^3}\nabla v_0\nabla \varphi dx,$

$\underset{n\to \infty }{\lim }{b}_n{\int }_{{\mathbb{R}}^3}|\nabla v_{b_n}{|}^{2}dx{\int }_{{\mathbb{R}}^3}\nabla v_{b_n}\nabla \varphi dx=0.$

Thus,

${\int }_{{\mathbb{R}}^3}\nabla v_0\nabla \varphi dx+{\int }_{{\mathbb{R}}^3}V\left(x\right)\frac{G^{-1}\left(v_0\right)}{g\left(G^{-1}\left(v_0\right)\right)}\varphi dx={\int }_{{\mathbb{R}}^3}\frac{h\left(G^{-1}\left(v_0\right)\right)}{g\left(G^{-1}\left(v_0\right)\right)}\varphi dx.$

It shows that v₀ is a positive solution of Eq. 1.2.

Data availability statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author.

Author contributions

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

Funding

This work was supported by the National Natural Science Foundation of China (Grant Nos 12261075 and 12261076), Yunnan Local Colleges Applied Basic Research Projects (Grant Nos 202001BA070001-032 and 202101BA070001-280), the Technology Innovation Team of University in Yunnan Province (Grant No. 2020CXTD25), and Yunnan Fundamental Research Projects (Grant Nos 202201AT070018 and 202105AC160087). WW was supported in part by the Yunnan Province Basic Research Project for Youths (202301AU070001) and the Xingdian Talents Support Program of Yunnan Province.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

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References

1. Feit M, Fleck J, Steiger A. Solution of the Schrödinger equation by a spectral method. J Comput Phys (1982) 47:412–33. doi:10.1016/0021-9991(82)90091-2

CrossRef Full Text | Google Scholar

2. He J, He C, Saeed T. A fractal modification of chen-lee-liu equation and its fractal variational principle. Internat J Mod Phys. B (2021) 35:2150214. doi:10.1142/S0217979221502143

CrossRef Full Text | Google Scholar

3. Almutairi A. Stochastic solutions to the non-linear Schrödinger equation in optical fiber. Therm Sci (2022) 26:185–90. doi:10.2298/tsci22s1185a

CrossRef Full Text | Google Scholar

4. Deng Y, Peng S, Yan S. Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth. J Differential Equations (2015) 258:115–47. doi:10.1016/j.jde.2014.09.006

CrossRef Full Text | Google Scholar

5. Alves C, Wang Y, Shen Y. Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J Differential Equations (2015) 259:318–43. doi:10.1016/j.jde.2015.02.030

CrossRef Full Text | Google Scholar

6. Bouard A, Hayashi N, Saut J. Global existence of small solutions to a relativistic non-linear Schrödinger equation. Comm Math Phys (1997) 189:73–105. doi:10.1007/s002200050191

CrossRef Full Text | Google Scholar

7. Brizhik L, Eremko A, Piette B, Zakrzewski W. Static solutions of aD-dimensional modified non-linear Schr dinger equation. Non-linearity (2003) 16:1481–97. doi:10.1088/0951-7715/16/4/317

CrossRef Full Text | Google Scholar

8. Li G, Huang Y, Liu Z. Positive solutions for quasilinear Schrödinger equations with superlinear term. Complex Var. Elliptic Equ (2020) 65:936–55. doi:10.1080/17476933.2019.1636791

CrossRef Full Text | Google Scholar

9. Li Q, Wu X. Existence, multiplicity, and concentration of solutions for generalized quasilinear Schrödinger equations with critical growth. J Math Phys (2017) 58:041501. doi:10.1063/1.4982035

CrossRef Full Text | Google Scholar

10. Shen Y, Wang Y. Soliton solutions for generalized quasilinear Schrödinger equations. Nonlinear Anal (2013) 80:194–201. doi:10.1016/j.na.2012.10.005

CrossRef Full Text | Google Scholar

11. Shi H, Chen H. Positive solutions for generalized quasilinear Schrödinger equations with potential vanishing at infinity. Appl Math Lett (2016) 61:137–42. doi:10.1016/j.aml.2016.06.004

CrossRef Full Text | Google Scholar

12. Shen Y, Wang Y. Standing waves for a class of quasilinear Schrödinger equations. Complex Var. Elliptic Equ (2016) 61:817–42. doi:10.1080/17476933.2015.1119818

CrossRef Full Text | Google Scholar

13. Kirchhoff G. Mechanik (leipzig: Teubner) (1883).

Google Scholar

14. He J, El-dib Y. The enhanced homotopy perturbation method for axial vibration of strings. Facta Universitatis Ser Mech Eng (2021) 19:735–50. doi:10.22190/FUME210125033H

CrossRef Full Text | Google Scholar

15. Wang K. Construction of fractal soliton solutions for the fractional evolution equations with conformable derivative. Fractals (2023) 31:2350014. doi:10.1142/S0218348X23500147

CrossRef Full Text | Google Scholar

16. Chen J, Tang X, Cheng B. Existence and nonexistence of positive solutions for a class of generalized quasilinear Schrödinger equations involving a Kirchhoff-type perturbation with critical sobolev exponent. J Math Phys (2018) 59:021505. doi:10.1063/1.5024898

CrossRef Full Text | Google Scholar

17. Cheng B, Tang X. Ground state sign-changing solutions for asymptotically 3-linear Kirchhoff-type problems. Complex Var. Elliptic Equ (2017) 62:1093–116. doi:10.1080/17476933.2016.1270272

CrossRef Full Text | Google Scholar

18. Feng R, Tang C. Ground state sign-changing solutions for a Kirchhoff equation with asymptotically 3-linear nonlinearity. Qual Theor Dyn. Syst. (2021) 20:91–19. doi:10.1007/s12346-021-00529-y

CrossRef Full Text | Google Scholar

19. Li F, Zhu X, Liang Z. Multiple solutions to a class of generalized quasilinear Schrödinger equations with a Kirchhoff-type perturbation. J Math Anal Appl (2016) 443:11–38. doi:10.1016/j.jmaa.2016.05.005

CrossRef Full Text | Google Scholar

20. Shen L. Existence and nonexistence results for generalized quasilinear Schrödinger equations of Kirchhoff type in. Appl Anal (2020) 99:2465–88. doi:10.1080/00036811.2019.1569225$\mathbb{R}^{3}$

CrossRef Full Text | Google Scholar

21. Li G, Cheng B, Huang Y. Positive solutions for asymptotically 3-linear quasilinear Schrödinger equations. Electron J Differential Equations (2020) 2020:1–17. Available at: http://ejde.math.txstate.edu.

Google Scholar

22. Wang W, Yu Y, Li Y. On the asymptotically cubic fractional Schrödinger-Poisson system. Appl Anal (2021) 100:695–713. doi:10.1080/00036811.2019.1616695

CrossRef Full Text | Google Scholar

23. Lehrer R, Maia L, Squassina M. Asymptotically linear fractional Schrödinger equations. Complex Var. Elliptic Equ (2015) 60:529–58. doi:10.1080/17476933.2014.948434

CrossRef Full Text | Google Scholar

24. Chu C, Liu H. Existence of positive solutions for a quasilinear Schrödinger equation. Nonlinear Anal RWA (2018) 44:118–27. doi:10.1016/j.nonrwa.2018.04.007

CrossRef Full Text | Google Scholar

25. Colin M, Jeanjean L. Solutions for a quasilinear schrödinger equation: A dual approach. Nonlinear Anal (2004) 56:213–26. doi:10.1016/j.na.2003.09.008

CrossRef Full Text | Google Scholar

26. Liang Z, Gao J, Li A. Infinitely many solutions to a quasilinear Schrödinger equation with a local sublinear term. Appl Math Lett (2019) 89:22–7. doi:10.1016/j.aml.2018.09.015

CrossRef Full Text | Google Scholar

27. Shen Y, Wang Y. Standing waves for a relativistic quasilinear asymptotically Schrödinger equation. Appl Anal (2016) 95:2553–64. doi:10.1080/00036811.2015.1100296

CrossRef Full Text | Google Scholar

28. Severo U, Gloss E, Silva E. On a class of quasilinear Schrödinger equations with superlinear or asymptotically linear terms. J Differential Equations (2017) 263:3550–80. doi:10.1016/j.jde.2017.04.040

CrossRef Full Text | Google Scholar

29. Li G, Huang Y. Positive solutions for generalized quasilinear Schrödinger equations with asymptotically linear nonlinearities. Appl Anal (2021) 100:1051–66. doi:10.1080/00036811.2019.1634256

CrossRef Full Text | Google Scholar

30. Jeanjean L. On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝ^N. Proc Roy Soc Edin (1999) 129A:787–809. doi:10.1017/S0308210500013147$\mathbb{R}^{n}$

CrossRef Full Text | Google Scholar

31. Willem M. Minimax theorems, progress in nonlinear differential equations and their applications. Boston, MA: Birkhäuser Boston, Inc. (1996). p. 24.

Google Scholar