Infinitely many normalized solutions for a quasilinear Schrödinger equation

Assume $N\geq 2$ and consider the following quasilinear Schrödinger equation

which is related to the standing waves $\psi(x,t)=u(x)e^{-iEt}$ of the following Schrödinger equation

where $i$ denotes the imaginary unit, $\lambda=\mu-E$, $g(t)=h(|t|^{2})t$. The quasilinear Schrödinger equation (1.2) is derived as a model of several physical phenomena. For example, it was used for the superfluid film equation in plasma physics by Kurihara [20]. It also appears in the theory of Heisenberg ferromagnetism and magnons (see [19] and [36]), in dissipative quantum mechanics and the condensed matter theory [33]. In literature the study of standing waves of (1.1) has been pursed in two main directions, which opened two different challenging research fields.

The first topic regards the search for solutions of (1.1) with a prescribed frequency $\mu$ and free mass, which is called unconstrained problem. To this aim, we consider the associated energy functional to (1.1)

defined on the natural space

where $G(u)=\int_{0}^{u}g(s)ds.$ A function $u\in X$ is called a weak solution of (1.1) if

for all $\varphi\in C_{0}^{\infty}(\mathbb{R}^{N}).$ Recall that $22^{\ast}=\frac{4N}{N-2}$ corresponds to a critical exponent of (1.1) when $g(u)=|u|^{r-2}u$ and $N\geq 3$, see [26]. Different from the semilinear case that $\kappa=0$, there is a non-convex term $\Delta(u^{2})u$ appears in the quasilinear problem (1.1), and hence the search of solutions of such an equation must face a challenge: the functional

associated with the quasilinear term is non differentiable in the space $X$ when $N\geq 2$. Consequently, the standard critical point theory can not be applied directly. In order to overcome this difficulty, several approaches have been successfully developed in the last decade, such as the constraint minimization ([29]), the Nehari manifold method ([27]), the perturbation method ([32]) and the nonsmooth critical point theory ([6, 22]). It seems that the first contribution to the equation (1.1) via variational methods dues to the work [11]. After this work, a series of subsequent studies have been done concerning with the existence, multiplicity and qualitative properties of solutions to (1.1), see for example [1, 12, 18, 25, 30, 28, 31, 35].

The second topic of investigation of the problem (1.1) consists of prescribing the mass $m>0$, thus conserved by $\psi$ in time

and letting the frequency $\mu$ to be free. Such a problem is usually called constrained one. The constrained problem has a significant relevance in Physics, not only for the quantum probability normalization, but also because the mass may also have specific meanings, such as the power supply in nonlinear optics, or the total number of atoms in Bose-Einstein condensation. Moreover, the investigation of constrained problems can give a better insight of the dynamical properties, as the orbital stability of solutions of (1.2), see for example [9]. For the case $\kappa=0$, (1.1) is reduced to the semilinear Schrödinger equation whose normalized solutions are studied widely in recent years. We can refer the interested readers to [7, 9, 13, 15, 39] and the references therein.

In recent years, there has been an increasing interest in studying of normalized solutions to the quasilinear Schrödinger equation. To obtain normalized solutions, a natural method is to look for the minimizer of the following constrained problem:

where

and

It is proved in[10, Theorem 4.6] that each minimizer of $e(m)$, corresponds to a Lagrange multiplier $\lambda<0$ such that $(u,\lambda)$ solves weakly $\eqref{eqA1}$ when $g(u)=|u|^{r-2}u$. Moreover, it is easy to see that $e(m)>-\infty$ for $2<r<4+\frac{4}{N}$ and $e(m)=-\infty$ for $4+\frac{4}{N}<r<\frac{4N}{N-2}$. From this point of view, $r=4+\frac{4}{N}$ plays a critical exponent for the constrained problem. By this constrained argument, Colin, Jeanjean and Squassina [10] first prove the following result for $L_{2}$ -constrained problem (see also [16]).

Interestingly, by a combination of analytical and numerical arguments in dimension $N=3$, Caliari and Squassina [5] studied the explicit bounds on $m(r,N)$ and $m_{N}$ given in Theorem 1.1. In particular, for $r=\frac{13}{3}$, they find that there exists $\underline{m},~{}\overline{m}>0$ such that

Moreover, $\underline{m}\approx 19.73$ and $\overline{m}\approx 85.09$. However, the constrained minimization argument does not use much of smoothness of the variational functional, so it is not suitable for dealing with the existence of multiple solutions. Subsequently, Jeanjean, Luo and Wang [17] prove the existence of two normalized solutions for $L_{2}$ -subcritical case via the perturbation approach. More precisely, they proved the following result.

Very recently, Ye and Yu [47] study the $L_{2}$-critical problem to (1.1) if $g(u)=|u|^{r-2}r$ for $r=2+\frac{4}{N}$ and $r=4+\frac{4}{N}$. They prove the following result.

• $(2)$

  For $r=4+\frac{4}{N}$, there exists $m_{N}>0$ such that

  (i) $E(u)$ has no critical point on the constraint $S(m)$ for all $m\in\big{(}0,m_{N}\big{]}$;

  (ii) $E(u)$ has a critical point on the constraint $S(m)$ for all $m\in\big{(}m_{N},+\infty\big{)}$ when $N\leq 3$.

We also point out that some works focus on the existence and asymptotic behavior of normalized solutions related to some quasilinear elliptic equations with potential function, that is, $\mu$ is replaced by $V(x)$ satisfying different assumptions in (1.1), we can refer the interested readers to [2, 21, 50, 48, 49] and the references therein for this case.

However, to the best of our knowledge, the existing results in references of the constrained problem (1.1) mainly focus on the case where $g$ is a power function. It seems that there is no work considering the existence of normalized solutions for (1.1) when $g$ is a general nonlinearity. A natural question is whether the nonlinearity $g(u)=|u|^{r-2}u$ can be relaxed to a general nonlinearity $g$ or not? On the other hand, it seems that the multiplicity of normalized solutions for (1.1) only can be found in Jeanjean, Luo and Wang [17], where the authors partly obtained the existence of two normalized solutions for (1.1) when $g(u)=|u|^{r-2}u$. Since such a power nonlinearity is odd, another natural question is whether (1.1) has infinitely many normalized solutions or not? Moreover, we note from Theorem 1.2 that $r\in\big{(}2+\frac{4}{N},4+\frac{4}{N}\big{)}$ for $N\leq 4$ or $r\in\big{(}2+\frac{4}{N},2+\frac{4}{N-2})$ for $r\geq 5$ are required. However, it is well-known that the existence of solutions of (1.1) can be obtained for the unconstrained problem when $r\in\big{(}2,\frac{4N}{N-2}\big{)}$ and $N\geq 1$. So the last natural questions is whether the scope of $r$ and the dimension of the space can be relaxed to obtain multiple normalized solutions of (1.1)?

Motivated by the above results especially by [16] and [17], in the present paper, we are concerned with the existence and multiplicity of normalized solutions of the following quasilinear Schrödinger equation

where $g$ satisfies the well-known Berestycki–Lions condition, which is regarded as an almost optimal assumption of the existence of solutions for nonlinear Schrödinger equation (see [3, 4]). To state our main results, we need the following assumptions.

• $(g_{1})$

  $g\in C(\mathbb{R},\mathbb{R})$.

• $(g_{2})$

  $\lim\limits_{s\rightarrow 0}\frac{g(s)}{s}=0$.

• $(g_{3})$

  For $p=4+\frac{4}{N}$, $\lim\limits_{s\rightarrow\infty}\frac{|g(s)|}{|s|^{p-1}}=0$.

• $(g_{4})$

  There exists some $s_{0}>0$ such that $G(s_{0})>0$.

• $(g_{5})$

  $g(-s)=-g(s)$ for all $s\in\mathbb{R}$.

• $(g_{6})$

  For $q=2+\frac{4}{N}$, $\liminf\limits_{s\rightarrow 0}\frac{|g(s)|}{|s|^{q-2}s}=\infty$.

• $(g_{7})$

  $g(s)s\geq 0$ for all $s\in\mathbb{R}$.

Now, our main results of this paper can be stated as follows.

We immediately conclude the following result from Theorem 1.5:

To prove Theorems 1.4–1.5, we employ a new approach introduced by Hirata and Tanaka [14]. One advantage of this method is that it can be suitably adapted to derive multiplicity results of normalized solutions by using minimax and deformation arguments. Firstly, we consider a augmented functional $I^{m}(\lambda,u)$, which is not well defined in the normal Sobolev space $\mathbb{R}\times H^{1}(\mathbb{R}^{N})$. Consequently, one of the difficulties to deal with (1.1) stems from the fact that there is no suitable working space where the energy functional enjoys both the smoothness and compactness properties. In order to overcome this difficulty, inspired by [26], we make a change of variables and transform the quasilinear problem into a semilinear one.

Since $g$ only satisfies the Berestycki–Lions condition, it impossible for us to verify the boundedness of $(PS)$ or $(C_{c})$ sequence because of the absence of Ambrosetti-Rabinowize condition and monotonicity condition. Inspired by [14], we use a new deformation argument under a new version of the Palias–Samle condition related to a Pohozaev functional. Compared with the unconstrained case, we find that the corresponding functional seems to satisfies the compactness condition only at the negative energy level but not at positive energy level (see Lemma 2.4). Consequently, we have to construct a family of negative minimax values $b^{k}_{m}$ to prove Theorems 1.4–1.5. To this aim, a key problem is prove that none of $b^{k}_{m}$ is equal to $-\infty$ and hence $b^{k}_{m}$ is well defined. In [14], Hirata and Tanaka accomplished it by comparing $b^{k}_{m}$ and the least energy of the following classic problem

where $2<r<2^{\ast}.$ This argument heavily relies on scaling properties of functional. However, it seems to fail for the quasilinear case because of the nonhomogeneous properties of change of variables and nonlinearity. In this paper, we will borrow from a Pohozaev Mountain developed in [9] to overcome this difficulty. We point out that a new property of change of variables play a important role in proof, see Lemma 2.1–(14).

To look for a minimizer of $e(m)$, our work uses in particular some arguments of Steiner rearrangement (see [23, 24]) and a new scaling of [10]. We will see that a key point is show that $e(m)<0$. Indeed, for the homogeneous case that $g(u)=|u|^{r-2}u$, one can achieve this by using a scaling argument. However, for the nonhomogeneous case, it is not easy to see whether $e(m)<0$ holds or not. In this paper, by using minimax method, we first find a least energy normalized solution with negative energy to $(P_{\mu,m})$, which concludes that $e(m)<0$. From this perspective, we gives a strategy for finding a minimizer for constraint problems.

The paper is organized as follows. In Section 2, we will focus on the variational frame and deformation lemma. In Section 3, we will construct the negative minimax level and and analyze their behavior. In Section 4, we will devote to the proofs of main results.

Notation. For the sake of notational simplicity, we omit integral symbols $dx$ without causing confusion. Throughout this paper, $\rightarrow$ and $\rightharpoonup$ denote the strong convergence and the weak convergence, respectively. $\|\cdot\|$ denotes the norm in $H^{1}(\mathbb{R}^{N})$. $\|(\lambda,u)\|:=\sqrt{\lambda^{2}+\|u\|^{2}}$ denotes the norm of $(\lambda,u)$ in $\mathbb{R}\times H^{1}(\mathbb{R}^{N})$. For $A\subset\mathbb{R}\times H^{1}(\mathbb{R}^{N})$ and $\rho>0$, $N_{\rho}(A)=\{(\lambda,u):\hbox{dist}((\lambda,u),A)<\rho\}$. $\|\cdot\|_{p}$ denotes the norm in $L^{p}(\mathbb{R}^{N})$ for $1\leq p\leq\infty$. $\partial D_{k}=\{x\in\mathbb{R}^{k}:|x|=1\},~{}~{}D_{k}=\{x\in\mathbb{R}^{k}:|x|\leq 1\}.$ $\Sigma_{\rho}=\big{\{}v\in H^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\big{(}|\nabla v|^{2}+f^{2}(v)\big{)}\leq\rho\big{\}}$ and $\partial\Sigma_{\rho}=\big{\{}v\in H^{1}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}\big{(}|\nabla v|^{2}+f^{2}(v)\big{)}=\rho\big{\}}.$ $C_{0}$, $C$, $C_{i}$ denote various positive constants whose value may change from line to line but are not essential to the analysis of the proof.