Nodal Solution for a Kirchhoff-Type Problem in RN
Abstract: In this paper, we study the existence of nodal solutions of the following general Schödinger-Kirchhoff type problem: where a,b > 0, N ≥ 3, g : R → R+ is an even differential function and g''(s) ≥ 0 for all s ≥ 0, h : R → R is an odd differential function. These equations are related to the generalized quasilinear Schödinger equations: Because the general Schödinger-Kirchhoff type problem contains the nonlocal term, it implies that the equation (KP1) is no longer a pointwise identity and is very different from classical elliptic equations. By introducing a variable replacement, we first prove that (KP1) is equivalent to the following problem: whereand G-1 is the inverse of G. Next, we prove that (KP2) is equivalent to the following system with respect to : For every integer k > 0, radial solutions of (KP1) with exactly k nodes are obtained by dealing with the system (S) under some appropriate assumptions. Moreover, this paper established the nonexistence results if N ≥ 4 and b is sufficiently large.

$-\left(a+b{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x\right)\Delta v+\frac{{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}=\frac{h\left({G}^{-1}\left(v\right)\right)}{g\left({G}^{-1}\left(v\right)\right)},\text{in}{ℝ}^{N},$ (KP2)

$\left\{\begin{array}{l}-\Delta v+\frac{{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}=\frac{h\left({G}^{-1}\left(v\right)\right)}{g\left({G}^{-1}\left(v\right)\right)},\\ \mu -a-b{\mu }^{\frac{N-2}{2}}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x=0.\end{array}$ (S)

Keywords: 1. Introduction

In this paper, we consider the existence of nodal solutions for the following generalized quasilinear Schödinger-Kirchhoff type problem:

$\begin{array}{l}\left(a+b{\int }_{{ℝ}^{N}}\text{ }{g}^{2}\left(u\right){|\nabla u|}^{2}\text{d}x\right)\left[-div\left({g}^{2}\left(u\right)\nabla u\right)+{g}^{\prime }\left(u\right)g\left(u\right){|\nabla u|}^{2}\right]+V\left(x\right)u\\ =h\left(u\right),\text{in}{ℝ}^{N}\end{array}$ (1)

where $a,b>0$, $N\ge 3$. $g:ℝ\to {ℝ}^{+}$ is an even differential function and ${g}^{\prime }\left(s\right)\ge 0$ for all $s\ge 0$, $h:ℝ\to ℝ$ is an odd differential function, the potential $V\left(x\right):{ℝ}^{N}\to ℝ$ is positive in ${ℝ}^{N}$. It is necessary pointing out that we only consider the potential $V\left(x\right)=1$.

These equations are related to the quasilinear Schödinger (QS) equations

$i{\partial }_{t}z=-\Delta z+W\left(x\right)z-h\left(z\right)-\kappa \Delta l\left({|z|}^{2}\right){l}^{\prime }\left({|z|}^{2}\right)z,$ (QS)

where $z:ℝ×{ℝ}^{N}\to ℂ$, $W:{ℝ}^{N}\to ℝ$ is a given potential, $\kappa$ is a real constant and $h,l:ℝ\to ℝ$ are suitable functions. Set $z\left(t,x\right)=\mathrm{exp}\left(-iEt\right)u\left(x\right)$, where $E\in ℝ$ and u is a real function, (QS) can be reduce to the corresponding equation of elliptic type (see ):

$-\Delta u+V\left(x\right)u-\kappa \Delta l\left({|u|}^{2}\right){l}^{\prime }\left({|u|}^{2}\right)u=h\left(u\right),x\in {ℝ}^{N}.$

The form of the above equation has derived as models of several physical phenomena corresponding to various types of $l\left(s\right)$. For instance, the case $l\left(s\right)=s$ models the time evolution of the condensate wave function in super-fluid film and are called the superfluid film equation in the fluid mechanics by Kurihara

. In the case $l\left(s\right)={\left(1+s\right)}^{\frac{1}{2}}$ models the self-channeling of a high-power ultra

short laser in matter, the propagation of a high-irradiance laser in a plasma creates an optional index depending nonlinearity on the light intensity and leads to new interesting nonlinear wave equation (see   ). For more physical motivations and more references dealing with applications, we can refer to     and references therein.

If we take ${g}^{2}\left(u\right)=1+\frac{{\left({l}^{\prime }\left({u}^{2}\right)\right)}^{2}}{2}$, Equation (1) can derive the corresponding equations with nonlocal term. For instance, If we set ${g}^{2}\left(u\right)=1+2{u}^{2}$, i.e., $l\left(s\right)=s$, we get the superfluid film equation with nonlocal term in plasma physics:

$-\left(a+b{\int }_{{ℝ}^{N}}\left(1+2{u}^{2}\right){|\nabla u|}^{2}\text{d}x\right)\left[\Delta u+\Delta \left({u}^{2}\right)u\right]+V\left(x\right)u=h\left(u\right),x\in {ℝ}^{N}.$

If we set ${g}^{2}\left(u\right)=1+\frac{{u}^{2}}{2\left(1+{u}^{2}\right)}$, i.e., $l\left(s\right)={\left(1+s\right)}^{\frac{1}{2}}$, we get the equation:

$\begin{array}{l}-\left(a+b{\int }_{{ℝ}^{N}}\left(1+\frac{{u}^{2}}{2\left(1+{u}^{2}\right)}\right){|\nabla u|}^{2}\text{d}x\right)\left[\Delta u+\Delta {\left(1+{u}^{2}\right)}^{\frac{1}{2}}\frac{u}{2{\left(1+{u}^{2}\right)}^{\frac{1}{2}}}u\right]+V\left(x\right)u\\ =h\left(u\right),x\in {ℝ}^{N},\end{array}$

which is related to the model of the self-channeling of high-power ultrashort laser in matter.

Denote $H\left(u\right)={\int }_{0}^{u}\text{ }h\left(t\right)\text{d}t$. We observe that the natural variational functional corresponding to Equation (1)

$I\left(u\right)=\frac{a}{2}{\int }_{{ℝ}^{N}}{g}^{2}\left(u\right){|\nabla u|}^{2}\text{d}x+\frac{b}{4}{\left({\int }_{{ℝ}^{N}}{g}^{2}\left(u\right){|\nabla u|}^{2}\text{d}x\right)}^{2}+\frac{1}{2}{\int }_{{ℝ}^{N}}{u}^{2}\text{d}x-{\int }_{{ℝ}^{N}}H\left(u\right)\text{d}x$

may be not well defined in ${H}^{1}\left({ℝ}^{N}\right)$. Moreover, the set $\left\{u\in {H}^{1}\left({ℝ}^{N}\right):{\int }_{{ℝ}^{N}}{g}^{2}\left(u\right){|\nabla u|}^{2}\text{d}x<+\infty \right\}$ is not linear space. To overcome this difficulty, we make a change of variable constructed by Shen and Wang in , as $v=G\left(u\right):={\int }_{0}^{u}\text{ }g\left(t\right)\text{d}t$.

Then we get

$\begin{array}{l}I\left(u\right)=J\left(v\right)=\frac{a}{2}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x+\frac{b}{4}{\left({\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x\right)}^{2}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\frac{1}{2}{\int }_{{ℝ}^{N}}{|{G}^{-1}\left(v\right)|}^{2}\text{d}x-{\int }_{{ℝ}^{N}}H\left({G}^{-1}\left(v\right)\right)\text{d}x.\end{array}$

Under suitable assumptions on g and h, we conclude that J is well defined in ${H}^{1}\left({ℝ}^{N}\right)$ and $J\in {C}^{1}$.

If u is a nontrivial solution of (1), then it should satisfy

$\left(a+b{\int }_{{ℝ}^{N}}{|\nabla u|}^{2}\text{d}x\right){\int }_{{ℝ}^{N}}\left[{g}^{2}\left(u\right)\nabla u\nabla \varphi +{g}^{\prime }\left(u\right)g\left(u\right){|\nabla u|}^{2}\varphi +u\varphi -h\left(u\right)\varphi \right]\text{d}x=0,$ (2)

for all $\varphi \in {C}_{0}^{\infty }\left({ℝ}^{N}\right)$. Let $\varphi =\frac{1}{g\left(u\right)}\phi$, (2) is equivalent to

$\begin{array}{c}〈{J}^{\prime }\left(v\right),\phi 〉={\int }_{{ℝ}^{N}}\left(a\nabla v\nabla \phi +\frac{{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}\phi -\frac{h\left({G}^{-1}\left(v\right)\right)}{g\left({G}^{-1}\left(v\right)\right)}\phi \right)\text{d}x\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+b{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x{\int }_{{ℝ}^{N}}\nabla v\nabla \phi \text{d}x\\ =0,\end{array}$

for all $\phi \in {C}_{0}^{\infty }\left({ℝ}^{N}\right)$. Therefore, in order to find nodal solutions of (1), it suffices to study the following equation

$-\left(a+b{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x\right)\Delta v+\frac{{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}=\frac{h\left({G}^{-1}\left(v\right)\right)}{g\left({G}^{-1}\left(v\right)\right)},\text{in}{ℝ}^{N}.$ (3)

Now, we consider the existence of nodal solutions of (3). Nonlocal problems like (3) have drawn a great deal of attention in recent years (see     ). To begin with, Equation (3) can be derived as a nonlocal model for the vibrating string. It is related to the stationary analogue of equation

$\rho \frac{{\partial }^{2}u}{\partial {t}^{2}}-\left(\frac{{P}_{0}}{h}+\frac{E}{2L}{\int }_{0}^{L}|\frac{\partial u}{\partial x}|\text{d}x\right)\frac{{\partial }^{2}u}{\partial {x}^{2}}=f\left(x,u\right)$

proposed by Kirchhoff in  as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Here,

$\rho$ is the mass density, $\frac{{P}_{0}}{h}$ is the initial tension, E is related to the intrinsic

properties of the string, such as the Young’s modulus of material and L is the length of the string. In , it was pointed out that such problems as (3) may be applied to describe the growth and movement of a particular species.

Different from the above mentioned literatures, for example   , we provide a new viewpoint motivated by  for solving the generalized quasilinear Schödinger-Kirchhoff type problem. In this paper, some suitable algebraic techniques are used to find solutions. Precisely, we derive nodal solutions of (1) by transforming it into (3) and establish Equation (2) is equivalent to a system. Then, we occur the existence of nodal solutions of this system. We not only prove multiplicity result for $N=3$, but also give information about the case that $N\ge 4$. We achieve our purpose by solving the following system respect to $\left(v,\mu \right)\in {ℝ}^{N}×{ℝ}^{+}$ :

$\left\{\begin{array}{l}-\Delta v+\frac{{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}=\frac{h\left({G}^{-1}\left(v\right)\right)}{g\left({G}^{-1}\left(v\right)\right)},\\ \mu -a-b{\mu }^{\frac{N-2}{2}}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x=0.\end{array}$ (4)

Recall that a node of a radial solution of (1) is a radius $\rho >0$ such that $u\left(x\right)=0$ with $|x|=\rho$. The main purpose in this paper is to prove the equivalent of (3) and (4).

Proposition 1. Problem (3) has least one radial solution $v\in {H}^{1}\left({ℝ}^{N}\right)$ if and only if system (4) has at least one solution $\left(w,\mu \right)\in {H}^{1}\left({ℝ}^{N}\right)×{ℝ}^{+}$ such that w is radial. Moreover, w and v have the same number of nodes.

Proposition 2. Problem (1) has least one radial solution $u\in {H}^{1}\left({ℝ}^{N}\right)$ if and only if problem (3) has at least one solution $v\in {H}^{1}\left({ℝ}^{N}\right)$ such that v is radial. Moreover, v and u have the same number of nodes.

In order to state our main result, we need the following hypotheses:

(h1) $h\left(t\right)=o\left(t\right)$ as $t\to 0$.

(h2) There exists $C>0$ and $2 such that $|h\left(t\right)|\le C\left(1+g\left(t\right){|G\left(t\right)|}^{p-1}\right)$ for all $t\in ℝ$.

(h3) There exists $\delta >0$ such that for any $t>0$, there holds $\left(1+\delta \right)h\left(t\right)\le G\left(t\right){\left(\frac{h\left(t\right)}{g\left(t\right)}\right)}^{\prime }$, where $G\left(t\right)={\int }_{0}^{t}\text{ }g\left(s\right)\text{d}s$.

(h4) $h\in {C}^{1}\left(ℝ\right)$ is an odd function, $h\left(t\right)>0$ for $t>0$, $g\in {C}^{1}\left(ℝ\right)$ is an even positive function and ${g}^{\prime }\left(t\right)\ge 0$ for all $t\ge 0$, ${g}^{\prime }\left(t\right)t for all $t\in ℝ$.

Under the assumptions (h1)-(h2), g and h possess many important properties. Readers can find them in .

Applying Proposition 1 and 2, we can prove the following theorems.

Theorem 3. Suppose that (h1)-(h4) are satisfied. Then for any integer $k>0$ the following holds.

i) If $N=3$, (3) has a pair ${v}_{k}^{+}$ and ${v}_{k}^{-}$ of radial solutions for any $a,b>0$.

ii) If $N=4$, there exists ${b}_{0}\ge {b}_{k}>0$ with ${b}_{k\to {0}_{+}}$ such that (3) has a pair ${v}_{k}^{+}$ and ${v}_{k}^{-}$ of radial solutions for any $a>0$ and $0, has no nontrivial solution for any $a>0$ and $b\ge {b}_{0}$.

iii) If $N\ge 5$, there exists ${\alpha }_{0}\ge {\alpha }_{k}>0$, with ${\alpha }_{k}\to {0}^{+}$ such that (3) has two pair ${v}_{k,1}^{±}$ and ${v}_{k,2}^{±}$ of radial solutions if $a{b}^{\frac{2}{N-4}}<{\alpha }_{k}$, has a radial solution ${v}_{k}$ if $a{b}^{\frac{2}{N-4}}={\alpha }_{k}$, and has no nontrivial solution if $a{b}^{\frac{2}{N-4}}>{\alpha }_{0}$.

Moreover, each solution ${v}_{k}$ obtained in i)-iii) has exactly k nodes $0<{\rho }_{{v}_{k}}^{1}<{\rho }_{{v}_{k}}^{2}<\cdots <{\rho }_{{v}_{k}}^{k}<\infty$.

Theorem 4. Suppose that (h1)-(h4) are satisfied. Then for any integer $k>0$ the following holds.

i) If $N=3$, (1) has a pair ${u}_{k}^{+}$ and ${u}_{k}^{-}$ of radial solutions for any $a,b>0$.

ii) If $N=4$, there exists ${b}_{0}\ge {b}_{k}>0$ with ${b}_{k\to {0}_{+}}$ such that (1) has a pair ${u}_{k}^{+}$ and ${u}_{k}^{-}$ of radial solutions for any $a>0$ and $0, has no nontrivial solution for any $a>0$ and $b\ge {b}_{0}$.

iii) If $N\ge 5$, there exists ${\alpha }_{0}\ge {\alpha }_{k}>0$, with ${\alpha }_{k}\to {0}^{+}$ such that (1) has two pair ${u}_{k,1}^{±}$ and ${u}_{k,2}^{±}$ of radial solutions if $a{b}^{\frac{2}{N-4}}<{\alpha }_{k}$, has a radial solution ${u}_{k}$ if $a{b}^{\frac{2}{N-4}}={\alpha }_{k}$, and has no nontrivial solution if $a{b}^{\frac{2}{N-4}}>{\alpha }_{0}$.

Moreover, each solution ${u}_{k}$ obtained in i)-iii) has exactly k nodes $0<{\rho }_{{u}_{k}}^{1}<{\rho }_{{u}_{k}}^{2}<\cdots <{\rho }_{{u}_{k}}^{k}<\infty$.

2. Existence and Nonexistence of Solutions of (4)

In this section, we devote to solve system (4). Firstly, we will show an essential result which will be used to conclude Theorem 3.

Proposition 5. Suppose that (h1)-(h4) are satisfied. Then for any integer $k>0$ the following holds.

i) If $N=3$, (4) has a pair $\left({v}_{k}^{+},{\mu }_{k}^{+}\right)$ and $\left({v}_{k}^{-},{\mu }_{k}^{-}\right)$ of radial solutions for any $a,b>0$.

ii) If $N=4$, there exists ${b}_{0}\ge {b}_{k}>0$ with ${b}_{k\to {0}_{+}}$ such that (4) has a pair $\left({v}_{k}^{+},{\mu }_{k}^{+}\right)$ and $\left({v}_{k}^{-},{\mu }_{k}^{-}\right)$ of radial solutions for any $a>0$ and $0, has no nontrivial solution for any $a>0$ and $b\ge {b}_{0}$.

iii) If $N\ge 5$, there exists ${\alpha }_{0}\ge {\alpha }_{k}>0$, with ${\alpha }_{k}\to {0}^{+}$ such that (4) has two pair $\left({v}_{k}^{±},{\mu }_{k,1}^{±}\right)$ and $\left({v}_{k}^{±},{\mu }_{k,2}^{±}\right)$ of radial solutions if $a{b}^{\frac{2}{N-4}}<{\alpha }_{k}$, has a radial solution $\left({v}_{k},{\mu }_{k}\right)$ if $a{b}^{\frac{2}{N-4}}={\alpha }_{k}$, and has no nontrivial solution if $a{b}^{\frac{2}{N-4}}>{\alpha }_{0}$.

Moreover, the first component ${v}_{k}$ of every solution obtained in i)-iii) is radially symmetric and has exactly k nodes $0<{\rho }_{{v}_{k}}^{1}<{\rho }_{{v}_{k}}^{2}<\cdots <{\rho }_{{v}_{k}}^{k}<\infty$.

Clearly, $\left(0,a\right)$ is a trivial solution of (4). Here a solution $\left(v,\mu \right)\in {H}^{1}\left({ℝ}^{N}\right)×{ℝ}^{+}$ of (4) is called a nontrivial one if $\left(v,\mu \right)\ne \left(0,a\right)$.

In order to obtain conclusion of Proposition 5, we first establish the result about existence of nodal solutions for the following nonlinear elliptic equation:

$-\Delta v+\frac{{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}=\frac{h\left({G}^{-1}\left(v\right)\right)}{g\left({G}^{-1}\left(v\right)\right)},\text{in}{ℝ}^{N}.$ (5)

Lemma 1. Suppose that (h1)-(h4) are satisfied. Then for any integer $k>0$, there exists a pair ${v}_{k}^{+}$ and ${v}_{k}^{-}$ of radial solutions of (5) with ${v}_{k}^{-}\left(0\right)<0<{v}_{k}^{+}\left(0\right)$, having exactly k nodes $0<{r}_{1}^{±}<{r}_{2}^{±}<\cdots <{r}_{k}^{±}<\infty$.

Proof. This theorem was proved in , here we omit the detail.

Let $\mathcal{A}$ be the set of solutions of (5) and define

$\beta =\mathrm{inf}\left\{S\left(v\right):v\in \mathcal{A}\\left\{0\right\}\right\},$ (6)

where

$S\left(v\right)={\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x,\forall v\in {H}^{1}\left({ℝ}^{N}\right).$ (7)

According to Theorem 2.1, we obtain that $\mathcal{A}\\left\{0\right\}\ne \varnothing$ and $\beta$ is well defined.

Lemma 2. Suppose that (h1), (h2) and (h4) are satisfied. Then it holds $\beta >0$.

Proof. For any $v\in \mathcal{A}\\left\{0\right\}$, one has the Pohozaev identity

$\frac{N-2}{2N}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x+\frac{1}{2}{\int }_{{ℝ}^{N}}{|{G}^{-1}\left(v\right)|}^{2}\text{d}x={\int }_{{ℝ}^{N}}H\left({G}^{-1}\left(v\right)\right)\text{d}x.$

From assumptions (h1), (h2) and (h4), we obtain $|{G}^{-1}\left(s\right)g\left({G}^{-1}\left(s\right)\right)|\ge |s|$,

${\mathrm{lim}}_{|s|\to 0}\frac{h\left({G}^{-1}\left(s\right)\right)}{{G}^{-1}\left(s\right)}=0\text{and}{\mathrm{lim}}_{|s|\to \infty }\frac{h\left({G}^{-1}\left(s\right)\right)}{g\left({G}^{-1}\left(s\right)\right){|s|}^{{2}^{*}-1}}=0.$

For any $ϵ>0$, there exists ${C}_{ϵ}>0$ such that

$|\frac{h\left({G}^{-1}\left(s\right)\right)}{g\left({G}^{-1}\left(s\right)\right)}|\le ϵ|\frac{{G}^{-1}\left(s\right)}{g\left({G}^{-1}\left(s\right)\right)}|+{C}_{ϵ}{|s|}^{{2}^{*}-1},\forall s\in ℝ.$

Obviously, this implies that $S\left(v\right)\le C{\int }_{{ℝ}^{N}}{|v|}^{{2}^{*}}\text{d}x$ for some $C>0$. Applying Sobolev embedding theorem, we conclude $S\left(v\right)\le C{\left[S\left(v\right)\right]}^{\frac{N}{N-2}}$, which establish that there exists a constant $\gamma >0$ such that $S\left(v\right)\ge \gamma$. Thus $\beta \ge \gamma >0$.

Proof of Proposition 5. This result was proved in ( , Proposition 2.1). However, it plays a key role in this paper and for the sake of completeness and convenience ro reader, we here give the detail. For any $v\in {H}^{1}\left({ℝ}^{N}\right)\\left\{0\right\}$, define a function in ${ℝ}^{+}$ given by

${f}_{v}\left(\mu \right)=\mu -a-b{\mu }^{\frac{N-2}{2}}S\left(v\right),\forall \mu \in {ℝ}^{+}.$

Case 1: $N=3$. Theorem 5 implies that for any integer $k>0$, problem (5) has a pair ${v}_{k}^{+}$ and ${v}_{k}^{-}$ of radial solutions with exactly k nodes $0<{r}_{1}^{±}<{r}_{2}^{±}<\cdots <{r}_{k}^{±}<\infty$. Since $N=3$, we conclude

${f}_{{v}_{k}^{±}}\left(\mu \right)=\mu -a-b{\mu }^{\frac{1}{2}}S\left({v}_{k}^{±}\right),$

which implies that ${\mathrm{lim}}_{\mu \to +\infty }{f}_{{v}_{k}^{±}}\left(\mu \right)=+\infty$ for any $a,b>0$. Noting that ${f}_{{v}_{k}^{±}}\left(\mu \right)<0$ for all $\mu \in \left(0,a\right]$, there exists ${\mu }_{k}^{±}>a$ such that ${f}_{{v}_{k}^{±}}\left({\mu }_{k}^{±}\right)=0$. Thus $\left({v}_{k}^{±},{\mu }_{k}^{±}\right)$ is a pair of solutions of (3) and conclusion i) holds.

Case 2: $N=4$. In the case, we have

${f}_{v}\left(\mu \right)=\left(1-bS\left(v\right)\right)\mu -a,$

for all $\left(v,\mu \right)\in {H}^{1}\left({ℝ}^{4}\right)×{ℝ}^{+}$. For any integer $k>0$, let ${v}_{k}^{±}$ be a pair of radial solutions of (5) with exactly k nodes $0<{r}_{1}^{±}<{r}_{2}^{±}<\cdots <{r}_{k}^{±}<\infty$, ${b}_{k}=\mathrm{min}\left\{\frac{1}{S\left({v}_{k}^{+}\right)},\frac{1}{S\left({v}_{k}^{-}\right)}\right\}$, and ${b}_{0}=\frac{1}{\beta }$, where $\beta$ is given in Lemma 2. Then ${b}_{k}\le {b}_{0}$. Let ${B}_{i}=\left\{x\in {ℝ}^{4}:{r}_{i-1}<|x|<{r}_{i}\right\}$ and ${v}_{k,i}^{±}={v}_{k}^{±}$ in ${B}_{i}$, ${v}_{k,i}^{±}=0$ on ${ℝ}^{N}\{B}_{i},1=1,2,\cdots ,k$. Then ${v}_{k,i}^{±},i=1,2,\cdots ,k$, are nontrivial solutions of (5), which implies that $S\left({v}_{k}^{±}\right)={\sum }_{i=1}^{k}\text{ }S\left({v}_{k,i}^{±}\right)\ge k\beta$ and hence ${b}_{k}\to {0}^{+}$ as $k\to \infty$. For any $a>0$ and $0, taking ${\mu }_{k}^{±}=\frac{a}{1-bS\left({v}_{k}^{±}\right)}$, then one has ${\mu }_{k}^{±}>a$ and ${f}_{{v}_{k}^{±}}\left({\mu }_{k}^{±}\right)=0$. Thus $\left({v}_{k}^{±},{\mu }_{k}^{±}\right)$ solve (2). However, for any $a>0$ and $b\ge {b}_{0}$, we have

$\underset{v\in \mathcal{A}\\left\{0\right\}}{\mathrm{sup}}{f}_{v}\left(\mu \right)\le \left(1-{b}_{0}\beta \right)\mu -a=-a<0,$

for all $\mu \in {ℝ}^{+}$, where $\mathcal{A}$ is defined in (6). This establishes that (4) has no nontrivial solution if $a>0$ and $b\ge {b}_{0}$.

Case 3: $N=5$. For each $v\ne 0$, we see that $\frac{\text{d}}{\text{d}\mu }{f}_{v}\left(\mu \right)=1-\frac{N-2}{2}{\mu }^{\frac{N-4}{2}}S\left(v\right)$ and hence ${f}_{v}\left(\mu \right)$ has a unique maximum point

${\mu }_{v}={\left(\frac{2}{\left(N-2\right)bS\left(v\right)}\right)}^{\frac{2}{N-4}}>0.$

Let ${v}_{k}^{±}$ be a pair of radial solution of (5) with exactly k nodes. It is easy to check that ${f}_{{v}_{k}^{±}}\left(\mu \right)<0$ for $\mu \in \left(0,a\right]$, ${\mathrm{lim}}_{\mu \to +\infty }{f}_{{v}_{k}^{±}}\left(\mu \right)=-\infty$ and

$\underset{\mu \in {ℝ}^{+}}{\mathrm{max}}{f}_{{v}_{k}^{±}}\left(\mu \right)={f}_{{v}_{k}^{±}}\left({\mu }_{{v}_{k}^{±}}\right)=\left(\frac{N-4}{N-2}\right){\left(\frac{2}{\left(N-2\right)bS\left({v}_{k}^{±}\right)}\right)}^{\frac{2}{N-4}}-a.$

Define ${\alpha }_{k}=\mathrm{min}\left\{{\alpha }_{k}^{+},{\alpha }_{k}^{-}\right\}$, where

${\alpha }_{k}^{+}=\left(\frac{N-4}{N-2}\right){\left(\frac{2}{\left(N-2\right)S\left({v}_{k}^{±}\right)}\right)}^{\frac{2}{N-4}}.$

From Case 2, we have ${\alpha }_{k}\to {0}^{+}$ as $k\to \infty$. Clearly, it holds ${f}_{{v}_{k}^{±}}\left({\mu }_{{v}_{k}^{±}}\right)>0$ if $a{b}^{\frac{2}{N-4}}<{\alpha }_{k}$, which implies that there exists ${\mu }_{k,1}^{±}\in \left(a,{\mu }_{{v}_{k}^{±}}\right)$ and ${\mu }_{k,2}^{±}\in \left({\mu }_{{v}_{k}^{±}},+\infty \right)$ such that ${f}_{{v}_{k}^{±}}\left({\mu }_{k,1}^{±}\right)=0$ and ${f}_{{v}_{k}^{±}}\left({\mu }_{k,2}^{±}\right)=0$ if $a{b}^{\frac{2}{N-4}}<{\alpha }_{k}$.

Moreover, it is easy to verify that either ${f}_{{v}_{k}^{+}}\left({\mu }_{{v}_{k}^{+}}\right)=0$ or ${f}_{{v}_{k}^{-}}\left({\mu }_{{v}_{k}^{-}}\right)=0$ if $a{b}^{\frac{2}{N-4}}={\alpha }_{k}$.

Set

${\alpha }_{0}=\left(\frac{N-4}{N-2}\right){\left(\frac{2}{\left(N-2\right)\beta }\right)}^{\frac{2}{N-4}},$

where $\beta$ is given in (6). For any $a,b>0$ with $a{b}^{\frac{2}{N-4}}>{\alpha }_{0}$, we conclude that

$\underset{v\in \mathcal{A}\\left\{0\right\}}{\mathrm{sup}}{f}_{v}\left(\mu \right)\le \underset{v\in \mathcal{A}\\left\{0\right\}}{\mathrm{sup}}{f}_{v}\left({\mu }_{v}\right)\le \left(\frac{N-4}{N-2}\right){\left(\frac{2}{\left(N-2\right)b\beta }\right)}^{\frac{2}{N-4}}-a={\alpha }_{0}{b}^{\frac{-2}{N-4}}-a<0.$

for any $\mu \in {ℝ}^{+}$. This implies that (4) has no nontrivial solutions if $a{b}^{\frac{2}{N-4}}>{\alpha }_{0}$ and iii) holds.

3. Proofs of Proposition 1 and 2

Proof of Proposition 1. If (4) has a solution $\left(v,\mu \right)\in {H}^{1}\left({ℝ}^{N}\right)×{ℝ}^{+}$, then one has

$-\Delta v+\frac{{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}=\frac{h\left({G}^{-1}\left(v\right)\right)}{g\left({G}^{-1}\left(v\right)\right)},\text{in}{ℝ}^{N},$

at least in a weak sense, and

$\mu =a+b{\mu }^{\frac{N-2}{2}}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x.$

Make a change of variable, as $w\left(x\right)=v\left({\mu }^{-\frac{1}{2}}x\right)=v\left(y\right)$. Then, we conclude that

$\begin{array}{l}-\left(a+b{\int }_{{ℝ}^{N}}{|\nabla w|}^{2}\text{d}x\right)\Delta w\left(x\right)+\frac{{G}^{-1}\left(w\left(x\right)\right)}{g\left({G}^{-1}\left(w\left(x\right)\right)\right)}\\ =-{\mu }^{-1}\left(a+b{\mu }^{\frac{N-2}{2}}{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x\right)\Delta v\left(y\right)+\frac{{G}^{-1}\left(v\left(y\right)\right)}{g\left({G}^{-1}\left(v\left(y\right)\right)\right)}\\ =\frac{h\left({G}^{-1}\left(v\left(y\right)\right)\right)}{g\left({G}^{-1}\left(v\left(y\right)\right)\right)}=\frac{h\left({G}^{-1}\left(w\left(x\right)\right)\right)}{g\left({G}^{-1}\left(w\left(x\right)\right)\right)},\end{array}$

which implies that w is a solution of (3).

If (3) has a solution $w\in {H}^{1}\left({ℝ}^{N}\right)$, then one has

$-\left(a+b{\int }_{{ℝ}^{N}}{|\nabla w|}^{2}\text{d}x\right)\Delta w+\frac{{G}^{-1}\left(w\right)}{g\left({G}^{-1}\left(w\right)\right)}=\frac{h\left({G}^{-1}\left(w\right)\right)}{g\left({G}^{-1}\left(w\right)\right)},\text{in}{ℝ}^{N},$

at least in a weak sense.

Letting $\mu =a+b{\int }_{{ℝ}^{N}}{|\nabla w|}^{2}\text{d}x$ and $v\left(x\right)=w\left({\mu }^{\frac{1}{2}}x\right)=w\left(y\right)$, then we have

$\mu =a+b{\mu }^{\frac{N-2}{2}}{\int }_{{ℝ}^{N}}|\nabla v|\text{d}x$

and

$\begin{array}{l}-\Delta v\left(x\right)+\frac{{G}^{-1}\left(v\left(x\right)\right)}{g\left({G}^{-1}\left(v\left(x\right)\right)\right)}=-\mu \Delta w\left(y\right)+\frac{{G}^{-1}\left(w\left(y\right)\right)}{g\left({G}^{-1}\left(w\left(y\right)\right)\right)}\\ =-\left(a+b{\int }_{{ℝ}^{N}}{|\nabla w|}^{2}\text{d}x\right)\Delta w\left(y\right)+\frac{{G}^{-1}\left(w\left(y\right)\right)}{g\left({G}^{-1}\left(w\left(y\right)\right)\right)}\\ =\frac{h\left({G}^{-1}\left(w\left(y\right)\right)\right)}{g\left({G}^{-1}\left(w\left(y\right)\right)\right)}=\frac{h\left({G}^{-1}\left(v\left(x\right)\right)\right)}{g\left({G}^{-1}\left(v\left(x\right)\right)\right)},\end{array}$

which implies that $\left(v,\mu \right)\in {H}^{1}\left({ℝ}^{N}\right)×{ℝ}^{+}$ is a solution of (4). Moreover, it is evident that v and w have the same radial symmetry and sign.

Proof of Proposition 2. If (3) has a solution $v\in {H}^{1}\left({ℝ}^{N}\right)$, then one has

$-\left(a+b{\int }_{{ℝ}^{N}}{|\nabla v|}^{2}\text{d}x\right)\Delta v+\frac{{G}^{-1}\left(v\right)}{g\left({G}^{-1}\left(v\right)\right)}=\frac{h\left({G}^{-1}\left(v\right)\right)}{g\left({G}^{-1}\left(v\right)\right)},\text{in}{ℝ}^{N}.$

at least in a weak sense. Letting $u={G}^{-1}\left(v\right)$ and combining the assumptions on g and h, then we conclude that

$\left(a+b{\int }_{{ℝ}^{N}}{g}^{2}\left(u\right){|\nabla u|}^{2}\text{d}x\right)\left[-div\left({g}^{2}\left(u\right)\nabla u\right)+{g}^{\prime }\left(u\right)g\left(u\right){|\nabla u|}^{2}\right]+u=h\left(u\right),\text{in}{ℝ}^{N}.$

which implies that $u\in {H}^{1}\left({ℝ}^{N}\right)$ is a solution of (1). Moreover, by the definition of $G\left(t\right)$, it is obvious that u and v have the same radial symmetry and sign.

4. Conclusion

By the proof of Proposition 1 and 2, we establish problem (1) has least one radial solution $v\in {H}^{1}\left({ℝ}^{N}\right)$ if and only if system (4) has at least one solution $\left(w,\mu \right)\in {H}^{1}\left({ℝ}^{N}\right)×{ℝ}^{+}$ such that w is radial. And w and v have the same number of nodes. Proving the existence of nodal solutions of problem (1) is equivalent to solve problem (4). In Section 2, we show the existence and nonexistence of solutions of (4). Combining the above statements the conclusions of Theorem 3 follow directly from Proposition 1 and 5. Similarly, the conclusions of Theorem 4 follow directly from Proposition 1, 2 and 5.

Cite this paper: Sha, L. (2020) Nodal Solution for a Kirchhoff-Type Problem in RN. Applied Mathematics, 11, 42-52. doi: 10.4236/am.2020.111005.
References

   Cuccagna, S. (2009) On Instability of Excited States of Boblinear Schrödinger Equation. Physica D: Nonlinear Phenomena, 238, 38-54.
https://doi.org/10.1016/j.physd.2008.08.010

   Kurihara, S. (1981) Large-Ampliyude Quasi-Solitons in Superfluid Films. Journal of the Physical Society of Japan, 50, 3262-3267.
https://doi.org/10.1143/JPSJ.50.3262

   Brandi, H., Manus, C., Mainfray, G., Lehner, T. and Bannuad, G. (1993) Relativistic and Pondermotive Self-Focusing of a Laser Beam in a Radially Inhomogeneous Plasma. Physics of Fluids B: Plasma Physics, 5, 3539-3550.
https://doi.org/10.1063/1.860828

   De Bouard, A., Hayashi, N. and Saut, J. (1997) Global Existence of Small Solutions to a Relativistic Nonlinear Schrödinger Equation. Communications in Mathematical Physics, 189, 73-105.
https://doi.org/10.1007/s002200050191

   Ritchie, B. (1994) Relativistic Self-Focusing and Channel Formation in Laser-Plasma Interactions. Physical Review E, 50, R687(R).
https://doi.org/10.1103/PhysRevE.50.R687

   Bass, F.G. and Nasanov, N.N. (1990) Nonlinear Electromagnetic-Spin Waves. Physics Reports, 189, 165-223.
https://doi.org/10.1016/0370-1573(90)90093-H

   Brüll, L. and Lange, H. (1986) Solitary Waves for Quasilinear Schrödinger Equations. Expositiones Mathematicae, 4, 279-288.
https://doi.org/10.1080/00036818608839619

   Poppenberg, M., Schmitt, K. and Wang, Z. (2002) On the Existence of Soliton Solutions to Quasilinaer Schrödinger Equations. Calculus of Variations and Partial Differential Equations, 14, 329-344.
https://doi.org/10.1007/s005260100105

   Lange, H., Poppenberg, M. and Teismann, H. (1999) Nash-Moser Methods for the Solution of Quasilinaer Schrödinger Equations. Communications in Partial Differential Equations, 24, 1399-1418.
https://doi.org/10.1080/03605309908821469

   Shen, Y. and Wang, Y. (2013) Soliton Solutions for a Generalized Quasilinear Schrödinger Equations. Nonlinear Analysis, Theory, Methods and Applications, 80, 194-201.
https://doi.org/10.1016/j.na.2012.10.005

   Chen, C.Y., Kuo, Y.C. and Wu, T.F. (2011) The Nehari Manifold for a Kirchhoff Type Problem Involving Sign-Changing Weight Functions. Journal of Differential Equations, 250, 1876-1908.
https://doi.org/10.1016/j.jde.2010.11.017

   He, X. and Zou, W. (2016) Existence and Concentration Result for the Fractional Schrödinger Equations with Critical Nonlinearities. Calculus of Variations and Partial Differential Equations, 55, 1-39.
https://doi.org/10.1007/s00526-016-1045-0

   He, Y., Li, G. and Peng, S. (2014) Concentrating Bound States for Kirchhoff Type Problems in R3 Involving Critical Sobolev Exponents. Advanced Nonlinear Studies, 14, 483-510.
https://doi.org/10.1515/ans-2014-0214

   Li, G. and Ye, H. (2014) Existence of Positive Ground State Solutions for the Nonlinear Kirchhoff Type Equations in R3. Journal of Differential Equations, 257, 566-600.
https://doi.org/10.1016/j.jde.2014.04.011

   Tang, X.H. and Cheng, B. (2016) Ground State Sign-Changing Solutions for Kirchhoff Type Problems in Bounded Domains. Journal of Differential Equations, 261, 2384-2402.
https://doi.org/10.1016/j.jde.2016.04.032

   Kirchhoff, G. (1883) Vorlesungen Über Mechanik Birkhäuser Basel.

   Pucci, P., Xiang, M. and Zhang, B. (2015) Multiple Solutions for Nonhomogeneous Schrödinger-Kirchhoff Type Equations Involving the Fractional p-Laplacian in RN. Calculus of Variations and Partial Differential Equations, 54, 2785-2806.
https://doi.org/10.1007/s00526-015-0883-5

   Wu, K. and Zhou, F. (2019) Nodal Solutions for a Kirchhoff Type Problem in RN. Applied Mathematics Letters, 55, 58-63.
https://doi.org/10.1016/j.aml.2018.08.008

   Deng, Y., Peng, S. and Wang, J. (2014) Nodal Soliton Solutions for Generalized Quasilinear Schrödinger Equations. Journal of Mathematical Physics, 55, 349-381.
https://doi.org/10.1063/1.4874108

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