Existence of axially symmetric solutions for a kind of planar Schrödinger-Poisson system

1.   Introduction

We consider the following planar Schrödinger-Poisson system:

where $K$, $V$ and $f$ satisfy the following basic assumptions:

(V1) $V\in C({ \mathbb{R}}^{2}, (0, \infty))$, $V(x) = V(x_{1}, x_{2}) = V(|x_{1}|, |x_{2}|)$, $\forall\ x\in{ \mathbb{R}}^{2}$ and $\liminf_{|x|\rightarrow \infty}V(x) > 0$;

(K1) $K\in C({ \mathbb{R}}^{2}, (0, \infty))$, $K(x) = K(x_{1}, x_{2}) = K(|x_{1}|, |x_{2}|)$, $\forall\ x\in{ \mathbb{R}}^{2}$ and $\liminf_{|x|\rightarrow \infty}K(x) > 0$;

(F1) $f(u) = o(|u|)$ as $u\rightarrow 0$;

(F2) $f\in C({ \mathbb{R}}, { \mathbb{R}})$, there exists $c_{0} > 0$ and $p > 2$ such that $|f(u)|\leq c_{0}(1+|u|^{p-1}), \ \ \forall \ u\in { \mathbb{R}}$.

It is pointed out that (V1) and (K1) imply that $V(x)$ and $K(x)$ are both axially symmetric functions. As shown in ^[1], axially symmetric functions are widely existing in real world, but axially symmetric functions are less used in the existing works because of the lack of compact embedding from the subspace of $H^{1}({ \mathbb{R}}^{N})$ to $L^{s}({ \mathbb{R}}^{N})$ for $N\geq2$, where the elements of the subspace are axially symmetric functions. In recent years, the following nonlinear Schrödinger-Poisson equations have gained more attentions:

where $\mu\in{ \mathbb{R}}\backslash\{0\}$, $V\in C({ \mathbb{R}}^{N}, (0, \infty))$ and $f\in C({ \mathbb{R}}^{N}\times { \mathbb{R}}, { \mathbb{R}})$. It is easy to see that system (1.1) is a special form of system (1.2).

From ^[2], we know that system (1.2) comes from semiconductor theory and quantum mechanics theory. Recently, the $N = 2$ problem has attracted a lot of attention in relation to optical propagation in certain media. Paredes, Olivieri and Michinel ^[3] gave a detail review on nonlinear Schrödinger-Poisson systems. They mainly reviewed two kinds of Schrödinger-Poisson systems: nonlinear optics in thermo-optical media and the 1+2D Schrödinger-Poisson model; ultralight axion dark matter and 3D Schrödinger-Poisson systems. Specially, they presented families of stationary solutions, discussed the implications of the simulation of propagation dynamics and discussed some numerical methods to solve the system of time-dependent partial differential equations. In the physical aspects, the solution $\phi$ of $\Delta \phi = u^{2}$ in system (1.2) can be solved by $\phi = \Gamma_{N}\ast u^{2}$, where

is the fundamental solutions of the Laplacian, $\ast$ is the convolution in ${ \mathbb{R}}^{N}$ and $\omega_{N}$ is the volume of the unit $N$-ball. With this formal inversion, an integro-differential equation is obtained as follows

When $N = 2$ and $\mu\neq0$, there are only a few works dealing with system (1.2) or (1.3). Chen, Chen and Tang ^[4] investigated system (1.2) in the periodic and asymptotically periodic cases using the non-Nehari manifold method derived from ^[5]. Bernini and Mugnai ^[6] have rewritten a nonlinear planar Schrödinger-Poisson system as a nonlinear Hartree equation and obtained an existence result of radially symmetric solutions when $V(x)$ is a positive constant and $\mu = 1$. If $f(x, u) = f(u)$ and $\mu > 0$, the authors in ^[7,8] dealt with periodic case and constructed a variational setting for (1.3). Recently, the author in ^[9] improved and extended the main results obtained in ^[8] with $V(x) = 1$ and more general nonlinearity $f(u)$. Very recently, Chen and Tang ^[10] dealt with axially symmetric potential instead of the periodic case and developed a natural constraint function space for system (1.2). More recently, Chen and Tang ^[1] considered the case that the nonlinearity is sub-cubic growth at infinity. As pointed out in ^[1] that this case is more difficult and the methods used in ^[10] is no longer available since it is not sure whether $\{u_{n}\}$ are bounded in $H^{1}({ \mathbb{R}}^{N})$. Motivated by ^[1], Wen, Chen and Rădulescu ^[11] studied system (1.1) with $V(x) = 0$ and $K(x)$ is a axially symmetric function and obtained a main result.

When $\mu = 0$, system (1.2) reduces to Schrödinger equations. Many researchers investigated Schrödinger equations and obtained many existence results of nontrivial solutions, see ^{[12,13,14,15,16,17,18,19,20]} and references therein. It is pointed out that critical point theory is very important for studying Schrödinger equations and other kinds of elliptic systems ^{[21,22,23,24]}. However, most of the existing works of Schrödinger equations or Schrödinger-Poisson equations are dealt with one of the following two cases: i) $\inf_{x\in{ \mathbb{R}}^{N}} V(x) > 0$; ii) $V(x)$ and $K(x)$ vanish at infinity. There is a question: what will happen if $\inf_{x\in{ \mathbb{R}}^{2}} V(x) > 0$ and $\liminf_{|x|\rightarrow \infty}K(x) > 0$ in system (1.1). Moreover, the methods handling the case $N = 3$ are no longer available for $N = 2$ since the integral $\Gamma_{2} = \frac{\ln |x|}{2\pi}$ is sign-changing and unbounded, which causes the functional associated with system (1.1) is not well-defined on $H^{1}({ \mathbb{R}}^{2})$ even if $V\in L^{\infty}({ \mathbb{R}}^{2})$ and $\inf_{x\in{ \mathbb{R}}^{2}} V(x) > 0$. As far as we known, there seems no related works in the case of $\inf_{x\in{ \mathbb{R}}^{2}} V(x) > 0$ and $\liminf_{|x|\rightarrow \infty}K(x) > 0$. In this paper, motivated by the aforementioned works, we will give a positive answer and obtain an axially symmetric solution for system (1.1) by establishing a new variational setting and using some new analytic tricks.

To present our result, the following assumptions are needed.

(V2) $V\in C^{1}({ \mathbb{R}}^{2}, { \mathbb{R}})$, $t\mapsto t^{2}[2V(tx)-\nabla V(tx)(tx)]$ is nondecreasing on $(0, \infty)$ for all $x\in{ \mathbb{R}}^{2}$;

(K2) $K\in C^{1}({ \mathbb{R}}^{2}, { \mathbb{R}})$, $\nabla K(x)\cdot x\leq0$, $t\mapsto 4K(tx)-\nabla K(tx)(tx)$ is nonincreasing on $(0, \infty)$ for all $x\in{ \mathbb{R}}^{2}$;

(F3) the function $\frac{f(u)u-F(u)}{u^{3}}$ is nondecreasing on both $(-\infty, 0)$ and $(0, \infty)$, where and in the sequel, $F(u) = \int_{0}^{u}f(s)ds$.

The main result is as follows.

Theorem 1.1. Suppose that $V$, $K$ and $f$ satisfy (V1), (V2), (K1), (K2) and (F1)–(F3). Then (1.1) possesses an axially symmetric solution $\bar{u}$ satisfying

where $u_{t} = u_{t}(x) = u(tx)$, the definitions of $\Phi$, $E$ and $\mathcal {P}$ will be given in the next section.

In the next section, we will construct a variational setting and give some preliminaries. In Section 3, we give the proof of Theorem 1.1. Throughout this paper, $\|\cdot\|_{H^{1}}$ and $\|\cdot\|_{s}$ denote the norms of $H^{1}({ \mathbb{R}}^{2})$ and $L^{s}({ \mathbb{R}}^{2})$ for $1\leq s\leq\infty$, respectively. $C_{i}$ are different positive constants in different places.

2.   Variational setting and preliminaries

The following bilinear forms are given as

and

where $u, v:{ \mathbb{R}}^{2}\rightarrow { \mathbb{R}}$ are measurable functions. Since $u, v$ are measurable functions, $A_{1}(u, v)$, $A_{2}(u, v)$ and $A_{0}(u, v)$ are well defined in Lebesgue sense. From the Hardy-Littlewood-Sobolev inequality ^[25] and $0\leq\ln(1+t)\leq t$ for $t\geq0$, we have

where $C_{1}$ is a positive constant. In order to obtain the existence of ground state solution for system (1.1), we develop a new variational framework for system (1.1). The working function space is

where

and

Under (V1) and (K1), it is easy to see that the space $E$ is a suitable constraint to study system (1.1). The norm of $E$ is given by

where

The energy functional of system (1.1) on $E$ is given by

From (F1), (F2) and [1,(2.9)], we have $\Phi\in C^{1}(X, { \mathbb{R}})$ and the embedding $X\hookrightarrow L^{s}({ \mathbb{R}}^{2})$ is compact for $s\in[2, \infty)$, moreover,

Now, the Pohožaev functional associated to (1.1) is defined as follows:

Similar to ^[1], any solution $u$ of (1.1) satisfies $\mathcal {P}(u) = 0$. The following constraint is defined as:

where

Similar to ^[1,11], the following lemmas are obtained.

Lemma 2.1. Assume that (V1), (K1), (F1) and (F2) hold. If $u$ is a critical point of $\Phi$ restricted to $E$, then $u$ is a critical point of $\Phi$ on $X$.

Lemma 2.2. Assume that (V1) and (V2) hold. Then

and there exists a constant $\gamma > 0$ such that

Lemma 2.3. Assume that (V1), (V2), (K1), (K2), (F1)–(F3) hold. Then for all $t > 0$, $u\in{ \mathbb{R}}$ and $x\in{ \mathbb{R}}^{2}$,

Lemma 2.4. Assume that (V2) holds. Then

3.   Proof of Theorem 1.1

In this section, we first establish an energy estimate inequality related to $\Phi(u)$, $\Phi(t^{2}u_{t})$ and $I(u)$, where

Lemma 3.1. Assume that (V1), (V2), (K1), (K2) and (F1)–(F3) hold. Then

and

Proof. From (2.3), (2.5), (2.6) and (3.1), we have

Now, we prove that $K$ is bounded in ${ \mathbb{R}}^2$. From (K1), we know that $K\in C({ \mathbb{R}}^{2}, (0, \infty))$, then $K$ is bounded in bounded domain. Now we need to prove that $K$ is also bounded at infinity. From (K2), we have

For every fixed $x\in{ \mathbb{R}}^2$, if letting $t\rightarrow0$, then $|t^{-1}x|\rightarrow \infty$ and if letting $t\rightarrow \infty$, then $|t^{-1}x|\rightarrow0$. Hence, from (K1) and (3.5), we have $K(\infty) = \lim_{t\rightarrow0}K(t^{-1}x)\leq\lim_{t\rightarrow \infty}K(t^{-1}x) = K(0) < \infty$, which implies that $K$ is also bounded at infinity. Hence, $K$ is bounded in ${ \mathbb{R}}^2$, that is

From (V2) and (2.6), we have

Let $t\rightarrow0$ in (3.7), we obtain

From (K1), (K2), (2.3), (2.4), (3.8) and (2.5) when $t\rightarrow0$ with the boundedness of $K(x)$, we get

It follows from (3.4) and (3.9) that Lemma 3.1 holds.

From (3.2) and the fact that $1+4t^{4}\ln t-t^{4}\geq0$ for $t > 0$, the following corollary is obtained.

Corollary 3.2. Assume that (V1), (V2), (K1), (K2) and (F1)–(F3) hold. Then $\Phi(u) = \max_{t > 0}\Phi(t^{2}u_{t})$ for all $u\in\mathscr{M}$.

Lemma 3.3. Assume that (V1), (V2), (K1), (K2) and (F1)–(F3) hold. Then for any $u\in E\backslash\{0\}$, there exists a constant $t_{u} > 0$ such that $t_{u}^{2}u_{t_{u}}\in\mathscr{M}$.

Proof. Fix $u\in E\backslash\{0\}$ and define $\xi(t): = \Phi(t^{2}u_{t})$ on $(0, \infty)$. From (2.4) and (3.1), one has

From (F1)–(F3), we have

By (3.10), we obtain

From (3.11), we get

From (3.12), we have

and

From (F3), we have

and

By (V2), we have

and

By (K2), we obtain

It follows from (K1), (K2), (3.5), (3.6), (3.13), (3.15), (3.17), (3.18) and (3.20) that

From (K1), (K2), (3.5), (3.6), (3.11), (3.14), (3.16), (3.17), (3.19) and (3.20), we have

Then, from (3.21) and (3.22), for $t\in(0, 1)$ small enough, one has $\xi'(t) > 0$ and for $t > 1$ large enough, $\xi'(t) < 0$. Hence, there exists $t_{u} > 0$ such that $\xi'(t_{u}) = 0$ and $t_{u}^{2}u_{t_{u}}\in\mathscr{M}$.

From Corollary 3.2 and Lemma 3.3, we get the following lemma.

Lemma 3.4. Assume that (V1), (V2), (K1), (K2) and (F1)–(F3) hold. Then

Lemma 3.5. Assume that (V1), (V2), (K1), (K2) and (F1)–(F3) hold. Then $c = \inf_{u\in\mathscr{M}}\Phi(u) > 0$.

Proof. By a standard argument, by (F1), (F2) and $I(u) = 0$ for $u\in \mathscr{M}$, one can easily show that there exists $\sigma > 0$ such that $\|u\|_{H^{1}}\geq\sigma$, $\forall \ u\in \mathscr{M}$. Let $\{u_{n}\}\subset\mathscr{M}$ be such that $\Phi(u_{n})\rightarrow c$. We consider two cases:

Case 1. $\inf_{n\in{ \mathbb{N}}}\|u_{n}\|_{2}: = \sigma_{1} > 0$. From (3.3), we get

Case 2. $\inf_{n\in{ \mathbb{N}}}\|u_{n}\|_{2}: = 0$. Since $\|u\|_{H^{1}}\geq\sigma$ for all $u\in \mathscr{M}$, passing to a subsequence, one obtains

From (2.1) and the Gagliardo-Nirenberg inequality, we get

From (3.24), we have

From (F1) and (F2), there exists a positive constant $C_{5}$ such that

Let $t_{n} = \|\nabla u_{n}\|_{2}^{-1/2}$. Since $I(u_{n}) = 0$ and the boundedness of $K$, from (3.1), (3.24), (3.25), (3.26), (3.27) and Corollary 3.2, we have

It follows from the above two cases that $c = \inf_{u\in\mathscr{M}}\Phi(u) > 0$.

Lemma 3.6. Assume that (V1), (V2), (K1), (K2) and (F1)–(F3) hold. Then $c$ is achieved. Moreover, if $\bar{u}\in \mathscr{M}$ and $\Phi(\bar{u}) = c$, then $\bar{u}$ is a critical point of $\Phi$ in $E$.

Proof. Let $\{u_{n}\}\subset\mathscr{M}$ be such that $\Phi(u_{n})\rightarrow c$. From (3.9) and $I(u_{n}) = 0$, it yields

From (3.28), we know that $\{\|u_{n}\|_{2}\}$ is bounded. It is needed to prove that $\{\|\nabla u_{n}\|_{2}\}$ is bounded too. Arguing by indirectly, assume that $\|\nabla u_{n}\|_{2}\rightarrow \infty$ as $n\rightarrow \infty$. Let $t_{n} = (2\sqrt{c}/\|\nabla u_{n}\|_{2})^{1/2}$, then $t_{n}\rightarrow0$ as $n\rightarrow \infty$. Hence, $t_{n}^{4}\ln t_{n}\rightarrow0$ as $n\rightarrow \infty$. From (K1), (F1), (F2), (2.1), (3.1), (3.25), (3.27), the boundedness of $K$ and Corollary 3.2, we have

a contradiction, hence, we have that $\{\|\nabla u_{n}\|_{2}\}$ is bounded too, so $\{u_{n}\}$ is bounded in $H^{1}({ \mathbb{R}}^{2})$. Thanks to (K1), (F1), (F2), (2.1) and (2.2), we know that $\{\|u_{n}\|\}$ and $A_{1}(u_{n}^{2}, u_{n}^{2})$ are both bounded. From [1,Lemma 3.5], one has

which together with Lemma 2.2 shows that $\{\|u_{n}\|_{*}\}$ is bounded. Then $\{u_{n}\}$ is bounded in $E$. Passing to a subsequence, one may assume that $u_{n}\rightharpoonup \bar{u}$ in $E$, $u_{n}\rightarrow \bar{u}$ in $L^{s}({ \mathbb{R}}^{2})$ for $s\in[2, \infty)$, $u_{n}\rightarrow \bar{u}$ a.e. on ${ \mathbb{R}}^{2}$. By a standard argument, we have

It follows from (3.30) and (3.31) that $\bar{u}\neq0$. From Lemma 3.3, there exists $\bar{t} > 0$ such that $\bar{t}^{2}\bar{u}_{\bar{t}}\in \mathscr{M}$ and $\Phi(\bar{t}^{2}\bar{u}_{\bar{t}})\geq c$. Hence, by (2.2), (2.4), (3.30), Fatou's Lemma, Lebesgue's dominated convergence theorem and the fact $1+4t^{4}\ln t-t^{4}\geq0$ for $t > 0$, we obtain

From (3.32), we have $I(\bar{u}) = 0$ and $\Phi(\bar{u}) = c$. Similar to [1,Lemma 4.11], we can obtain that $\bar{u}$ is a critical point of $\Phi$ in $E$.

From Lemmas 2.1, 3.4 and 3.6, it is easy to get Theorem 1.1. The proof is complete.

4.   Conclusions

This paper studies a kind of Schrödinger-Poisson system in ${ \mathbb{R}}^{2}$. By constructing a new variational framework and using some new analytic techniques, we obtain an axially symmetric solution. Specially speaking, we first establish an energy estimate inequality related to $\Phi(u)$, $\Phi(t^{2}u_{t})$ and $I(u)$, which is useful in the proof of the main result. Secondly, we prove that for any $u\in E\backslash\{0\}$, there exists a constant $t_{u} > 0$ such that $t_{u}^{2}u_{t_{u}}\in\mathscr{M}$, and then prove $\inf_{u\in\mathscr{M}}\Phi(u): = c = \inf_{u\in E\backslash\{0\}}\max_{t > 0}\Phi(t^{2}u_{t})$ and $c = \inf_{u\in\mathscr{M}}\Phi(u) > 0$. Finally, we prove that $c$ is achieved. Moreover, if $\bar{u}\in \mathscr{M}$ and $\Phi(\bar{u}) = c$, then $\bar{u}$ is a critical point of $\Phi$ in $E$. Our result improves and extends some existing works.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 11961014, No. 61563013) and Guangxi Natural Science Foundation (2018GXNSFAA281021, 2018GXNSFBA281019, 2019GXNSFAA245031). The authors would like to thank the editors and referees for their useful suggestions and comments.

Conflict of interest

All authors declare no conflict of interest in this paper.