The aim of this paper is to establish the existence of ground state solutions to the following generalized quasilinear Schrödinger–Poisson system:
$$ \textstyle\begin{cases} -\operatorname{div}(g^{2}(u)\nabla u)+g(u)g^{\prime}(u)|\nabla u|^{2}+a(x)u+\phi G(u)g(u)=k(x,u), & x\in \mathbb {R}^{3}, \\ -\Delta\phi=G^{2}(u), & x\in \mathbb {R}^{3}, \end{cases} $$
(1.1)
where \(g:\mathbb {R}\to \mathbb {R}^{+}=[0,\infty)\) is an even differential function, \(g^{\prime}(s)\geq0\) for all \(s\geq0\), and \(G(t)=\int_{0}^{t}g(s)\,ds\), \(k:\mathbb {R}^{3}\times \mathbb {R}\to \mathbb {R}\) is continuous, \(a(x):\mathbb {R}^{3}\to \mathbb {R}^{+}\) is continuous.
If \(\phi=0\) in (1.1), solutions of this type are related to the existence of solitary wave solutions for quasilinear Schrödinger equations of the form
$$ i\partial_{t}z=-\Delta z+W(x)z-k(x,z)-\omega\Delta l \bigl(|z|^{2}\bigr)l^{\prime }\bigl(|z|^{2}\bigr)z,\quad x\in \mathbb {R}^{N}, $$
(1.2)
where ω is a real constant, \(N\geq3\), \(z:\mathbb {R}\times \mathbb {R}^{N}\to \mathbb{C}\), \(W:\mathbb {R}^{N}\to \mathbb {R}\) is a given potential, \(l:\mathbb {R}\to \mathbb {R}\) and \(k:\mathbb {R}^{N}\times \mathbb {R}\to \mathbb {R}\) are suitable functions.
The semilinear case corresponding to \(\omega=0\) has been studied extensively by many scholars in recent years (see [1–5]). Quasilinear equations of the form (1.2) have been derived as models of several physical phenomena corresponding to various types of \(l(s)\). For instance, the case \(l(s)=s\) models the time evolution of the condensate wave function in a superfluid film [6, 7], and it is called the superfluid film equation in fluid mechanics by Kurihara [6]. In the case \(l(s)=(1+s)^{1/2}\), problem (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 optical index depending nonlinearly on the light intensity and this leads to an interesting new nonlinear wave equation (see [8–11]). Equation (1.2) also appears in plasma physics and fluid mechanics [12–15], in dissipative quantum mechanics [16] and in condensed matter theory [17].
Recently, Deng–Peng–Yan [18] introduced a class of generalized quasilinear critical Schrödinger equations,
$$ -\operatorname{div}\bigl(g^{2}(u)\nabla u \bigr)+g(u)g^{\prime}(u)|\nabla u|^{2}+a(x)u =k(x,u),\quad x\in \mathbb {R}^{N} , $$
(1.3)
to study the existence of positive soliton solutions. The reason we call Eq. (1.3) a generalized quasilinear Schrödinger equation is that if we take
$$g^{2}(u)=1+\frac{ [l^{\prime}(u^{2}) ]^{2}}{2}, $$
then the following quasilinear equation:
$$-\Delta u+a(x)u-\Delta l\bigl(u^{2}\bigr)l^{\prime} \bigl(u^{2}\bigr)u=k(x,u),\quad x\in \mathbb {R}^{N}, $$
turns into it (see [18, 19]). Equation (1.3) also arises in biological models and propagation of laser beams when \(g(u)\) is a positive constant. If we set \(g^{2}(u) =1 +2u^{2}\), i.e. \(l(s) =s\), we get the superfluid film equation in plasma physics:
$$-\Delta u+V(x)u-\Delta\bigl(u^{2}\bigr)u=k(x,u),\quad x\in \mathbb {R}^{N}. $$
If we set \(g^{2}(u) =1 + \frac{u^{2}}{2(1+u^{2})}\), i.e. \(l(s)=(1+s)^{1/2}\), we get the equation
$$-\Delta u+V(x)u- \bigl[\Delta\bigl(1+u^{2}\bigr)^{\frac{1}{2}} \bigr] \frac {1}{2(1+u^{2})^{\frac{1}{2}}}u=k(x,u), \quad x\in \mathbb {R}^{N}, $$
which models the self-channeling of a high-power ultrashort laser in matter. For the related and important results on quasilinear Schrödinger equations, we refer the reader to [19–26] and the references therein.
We call problem (1.1) the generalized quasilinear Schrödinger–Poisson system because of the coupling of the Poisson equation with (1.3). Indeed, if we choose \(g(t)=1\) for all \(t\in \mathbb {R}\), then (1.1) transforms to the following classical Schrödinger–Poisson system:
$$\textstyle\begin{cases} -\Delta u+a(x)u+\phi u=k(x,u), & x\in \mathbb {R}^{3}, \\ -\Delta\phi=u^{2}, & x\in \mathbb {R}^{3}, \end{cases} $$
proposed by Benci–Fortunato [27, 28] to represent solitary waves for nonlinear Schrödinger type equations and look for the existence of standing waves interacting with an unknown electrostatic field. We refer the reader to [29–34] for some related and important results. In view of this, it is also reasonable to consider the generalized quasilinear Schrödinger–Poisson system.
According to Ruiz [35], for any \(u\in H^{1}(\mathbb {R}^{3})\) we can define
$$\phi_{u}(x)=\frac{1}{4\pi} \int_{\mathbb {R}^{3}}\frac{u^{2}(y)}{|x-y|}\, dy, $$
which is a weak solution to \(-\Delta\phi=u^{2}\) in \(\mathbb {R}^{3}\). Therefore the weak solution of \(-\Delta\phi=G^{2}(u)\) can be represented as
$$\phi_{G(u)}(x)=\frac{1}{4\pi} \int_{\mathbb {R}^{3}}\frac{G^{2}(u(y))}{|x-y|}\,dy $$
and then (1.1) can be reduced to a single equation:
$$ -\operatorname{div}\bigl(g^{2}(u)\nabla u \bigr)+g(u)g^{\prime}(u)|\nabla u|^{2}+a(x)u+\phi _{G(u)} G(u)g(u)=k(x,u),\quad x\in \mathbb {R}^{3}. $$
(1.4)
In this paper, we establish the existence of ground state solutions for problem (1.1). To this end, we assume \(k(x,t)=b(x)|G(u)|^{p-2}G(u)g(u)-c(x)|G(u)|^{q-2}G(u)g(u)\). Hence the problem (1.4) can be rewritten in the following form:
$$\begin{aligned}& -\operatorname{div}\bigl(g^{2}(u)\nabla u\bigr)+g(u)g^{\prime}(u) \vert \nabla u \vert ^{2}+a(x)u+\phi _{G(u)} G(u)g(u) \\& \quad =b(x) \bigl\vert G(u) \bigr\vert ^{p-2}G(u)g(u)-c(x) \bigl\vert G(u) \bigr\vert ^{q-2}G(u)g(u), \end{aligned}$$
(1.5)
whose corresponding variational functional is given by
$$\begin{aligned} I(u) =&\frac{1}{2} \int_{\mathbb {R}^{3}}g^{2}(u) \vert \nabla u \vert ^{2}\,dx+\frac{1}{2} \int _{\mathbb {R}^{3}}a(x)u^{2}\,dx +\frac{1}{4} \int_{\mathbb {R}^{3}}\phi_{G(u)}G^{2}(u)\,dx \\ &{}-\frac{1}{p} \int_{\mathbb {R}^{3}}b(x) \bigl\vert G(u) \bigr\vert ^{p} \,dx+\frac{1}{q} \int_{\mathbb {R}^{3}}c(x) \bigl\vert G(u) \bigr\vert ^{q} \,dx. \end{aligned}$$
Unfortunately, the above functional I may be not well defined in \(H^{1}(\mathbb {R}^{3})\). To overcome this difficulty, we make a change of variable constructed by Shen–Wang [19],
$$v=G(u)= \int^{u}_{0}g(\tau)\,d\tau. $$
Then we get
$$\begin{aligned} J(v) =&\frac{1}{2} \int_{\mathbb {R}^{3}} \vert \nabla v \vert ^{2}\,dx+ \frac {1}{2} \int_{\mathbb {R}^{3}}a(x) \bigl\vert G^{-1}(v) \bigr\vert ^{2}\,dx + \frac{1}{4} \int_{\mathbb {R}^{3}}\phi_{v}v^{2}\,dx \\ &{}-\frac{1}{p} \int_{\mathbb {R}^{3}}b(x) \vert v \vert ^{p}\,dx+ \frac{1}{q} \int_{\mathbb {R}^{3}}c(x) \vert v \vert ^{q} \,dx. \end{aligned}$$
(1.6)
Since g is a nondecreasing positive function, we get \(|G^{-1}(v)|\leq |v|/g(0)\). It is clear that J is well defined in \(H^{1}(\mathbb {R}^{3})\) and \(J\in C^{1}\) if assumption (H1) holds.
If u is a nontrivial solution of (1.5), then it should satisfy
$$\begin{aligned}& \int_{\mathbb {R}^{3}} \bigl[g^{2}(u)\nabla u\nabla \varphi+g(u)g^{\prime }(u) \vert \nabla u \vert ^{2} \varphi+a(x)u\varphi +\phi_{G(u)}G(u)g(u)\varphi \\& \quad {}-b(x) \bigl\vert G(u) \bigr\vert ^{p-2}G(u)g(u)\varphi+c(x) \bigl\vert G(u) \bigr\vert ^{q-2}G(u)g(u)\varphi \bigr]\,dx=0, \end{aligned}$$
for any \(\varphi\in C^{\infty}_{0}(\mathbb {R}^{3})\). Let \(\varphi=\psi/g(u)\), we know that the above formula is equivalent to
$$\begin{aligned} \bigl\langle J^{\prime}(v),\psi\bigr\rangle =& \int_{\mathbb {R}^{3}} \biggl[ \nabla v \nabla\psi+a(x)\frac{G^{-1}(v)}{g(G^{-1}(v))} \psi+\phi_{v} v\psi -b(x)|v|^{p-2}v\varphi+c(x)|v|^{q-2}v \varphi \,dx \biggr] \\ =&0,\quad \forall\psi\in C^{\infty}_{0}\bigl(\mathbb {R}^{3} \bigr). \end{aligned}$$
Therefore, in order to find the nontrivial solutions of (1.5), it suffices to study the existence of the nontrivial solutions of the following equations:
$$ -\Delta v+a(x)\frac{G^{-1}(v)}{g(G^{-1}(v))}+\phi_{v} v=b(x)|v|^{p-2}v-c(x)|v|^{q-2}v. $$
(1.7)
It is easy to verify that the problem (1.5) is equivalent to problem (1.7) and the nontrivial critical points of \(J(v)\) are the nontrivial solutions of problem (1.7). Inspired by all the work described above, particularly, by the results in [18, 25], we intend to show the existence of ground state solutions of problem (1.7). To this end, we first give some assumptions on g, a, b and c.
- (g):
-
\(g\in C^{1}(\mathbb {R})\)
is an even positive function and
\(g^{\prime}(t)\geq0\)
for all
\(t\geq0\)
and
\(g(0) =1\);
- (H1):
-
\(a(x)\),
\(b(x)\)
and
\(c(x)\)
are continuous and nonnegative and bounded;
- (H2):
-
\(a(x)\leq\lim_{|x|\to\infty}a(x)\triangleq a_{\infty}\),
\(b(x)\geq\lim_{|x|\to\infty}b(x)\triangleq b_{\infty}\)
and
\(c(x)\leq\lim_{|x|\to\infty}c(x)\triangleq c_{\infty}\)
and one of these inequalities is strict on a set of positive measure.
Our main result is as follows.
Theorem 1.1
Suppose (g) and (H1)–(H2) hold. Problem (1.1) admits at least a ground state solution if
\(2< q<4<p<6\).
To prove our main theorem, we need to introduce the limiting equation at infinity related to problem (1.7)
$$ -\Delta v+a_{\infty}\frac{G^{-1}(v)}{g(G^{-1}(v))}+\phi_{v} v=b_{\infty}|v|^{p-2}v-c_{\infty}|v|^{q-2}v, $$
(1.8)
which plays a vital role.
The outline of this paper is as follows. In Sect. 2, we introduce and provide several lemmas. In Sect. 3, we prove the limiting equation (1.8) has a ground state solution. The proof of Theorem 1.1 is completed in Sect. 4.
Notations
Throughout this paper we shall denote by C and \(C_{i}\) (\(i=1, 2,\ldots \)) various positive constants whose exact value may change from line to line but are not essential to the analysis of the problem. \(L^{p}(\mathbb {R}^{3})\) (\(1\leq p\leq+\infty\)) is the usual Lebesgue space with the standard norm \(|u|_{p}\). We use “→” and “⇀” to denote the strong and weak convergence in the related function space, respectively. For any \(\rho>0\) and any \(x\in \mathbb {R}^{3}\), \(B_{\rho}(x)\) denotes the ball of radius ρ centered at x, that is, \(B_{\rho}(x):=\{y\in \mathbb {R}^{3}:|y-x|<\rho \}\).