Abstract
In this paper, we consider the existence of multiple positive solutions for Kirchhoff-Schrödinger-Poisson system with the nonlinear term containing both general singularity and quasicritical nonlinearity. By combining the variational method with the perturbation method, we obtain the existence of two positive solutions with the parameter λ small enough. One of the solutions is the local minimum of the corresponding functional, and the other is the limit of the mountain pass type solution to the perturbation problem.
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1 Introduction and main results
In this paper, we are interested in discussing the existence and multiple positive solutions to the following general singular Kirchhoff-Schrödinger-Poisson system:
where \(\Omega \subset \mathbb{R}^{3}\) is a smooth bounded domain with boundary ∂Ω, \(a>0\), \(b\geqslant 0\), \(\lambda >0\) is a parameter, f, g, h satisfy the following assumptions:
- (f):
-
\(f\in C((0,\infty ),\mathbb{R}_{+})\) is nonincreasing and \(\int_{0}^{1}f(s)\,ds<\infty \). Moreover, there exists \(\gamma \in (0,1)\) such that
$$ \lim_{s\to 0^{+}}f(s)s^{\gamma }=\infty ; $$ - (g1):
-
\(g\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})\), \(g(s)=o(s)\) as \(s\to 0\) and g has a ‘quasicritical growth’, namely
$$ \lim_{s\to +\infty }\frac{g(s)}{s^{5}}=0; $$ - (g2):
-
\(\lim_{s\to +\infty }\frac{g(s)}{s^{3}}=\infty \);
- (g3):
-
\(g(s)s\geqslant 4G(s)\), where \(G(s)=\int_{0}^{s} g(t)\,dt\), \(s\in {\mathbb{R}}_{+}\);
- (h):
-
\(h\in L^{2}(\Omega )\) with \(h(x)>0\) a.e. \(x\in \Omega \).
Recently, the following singular Kirchhoff type problem has been studied extensively in [1–5]
where \(\Omega \subset {\mathbb{R}}^{n}\), \(n\geqslant 3\) is a smooth bounded domain with boundary \(\partial \Omega , \gamma \in (0,1), \lambda , \mu \geqslant 0\) are parameters. In [5], by using the method of Nehari manifold, Liu and Sun discussed the existence of two positive solutions to (1.2) with \(g(x,s)=g(x)\frac{s^{p}}{ \vert x \vert ^{t}},t \in [0,1), p\in [3,5-2t)\) for the parameter \(\lambda >0\) small enough. Lei, Liao and Tang in [1] combined the variational method with the perturbation argument to discuss (1.2) with \(n=3\) and g being critical term: \(g(s)=s^{5}\) and obtained two positive solutions to this problem. In [3, 4], the authors discussed the existence and multiple positive solutions to (1.2) with \(n=4\) and g being critical term: \(g(s)=s^{3}\). By using the Nehari manifold method and analyzing the relations between the parameters λ, μ and the first eigenvalue to the Kirchhoff type problem, the authors in [3] obtained multiple positive solutions to this problem. In [4], the authors obtained the existence of two positive solutions to (1.2) with \(h(x)=\frac{1}{ \vert x \vert ^{\beta }}, \beta \in (0,3)\) by using the variational method and the perturbation method. The existence of unique positive solution to (1.2) with \(g(x,s)=-s^{p}, p\in (0,2^{*}-1)\) was obtained in [2] by using the variational method. Meanwhile, the singular \((p,q)\) Kirchhoff type system was also considered in [6].
In [7], the author of the present paper considered the following singular Schrödinger-Poisson system:
where \(\Omega \subset {\mathbb{R}}^{3}\) is a smooth bounded domain with boundary \(\partial \Omega , \eta =\pm 1, r\in (0,1)\) is a constant, \(\mu >0\) is a parameter. The existence of unique positive solution was obtained for \(\eta =1\) and any \(\mu >0\) by using the variational method. The multiple positive solutions were also obtained for \(\eta =-1\) and \(\mu >0\) small enough by combining the variational method with the Nehari manifold method. Recently, the Schrödinger-Poisson system with singular potential was also considered in [8]. The existence of system (1.1) with \(h=0, g(s)=-s^{p}\) was considered in [9].
The Kirchhoff-Schrödinger-Poisson system with general singularity f and −g (\(g\geqslant 0\)) in (1.1) was firstly considered in our recent paper [10], where f, g, h satisfy the more weaker assumptions:
(f0) \(f\in C((0,\infty ),{\mathbb{R}}_{+})\) satisfies that there exists \(\delta >0\) such that f is nonincreasing on \((0,\delta ]\), \(\int_{0}^{\delta } f(s)\,ds<\infty \), and there exist \(\alpha , \gamma \in (0,1)\) such that
(g0) \(g\in C({\mathbb{R}}_{+},{\mathbb{R}}_{+})\) and there exists \(c_{1}>0\) such that
(h0) \(h\in L^{6/(5-\gamma )}(\Omega )\) with \(h(x)>0\) a.e. \(x\in \Omega \).
Under the weaker assumptions (f0), (g0) and (h0), the corresponding functional is well defined and is coercive. By using the variational method, the negative global minimum is obtained and is the unique positive solution to this problem. Based on our work [7, 10], recently, Mu and Lu in [11] considered the existence and multiplicity of positive solutions for system (1.1) with \(f(s)=s^{-\gamma }\), \(\gamma \in (0,1)\). A natural question is whether there exist multiple positive solutions to system (1.1) with the nonlinear term containing both general singular nonlinearity and the quasicritical nonlinearity.
Motivated by the above reference, especially by [1, 4, 5], and based on our work [10], in this paper, we would like to continue to study the existence of multiple solutions to the general singular Kirchhoff-Schrödinger-Poisson system (1.1).
Throughout this paper, let \(H_{0}^{1}(\Omega )\) be the usual Sobolev space with the inner product and the norm
We denote the norm of \(L^{p}(\Omega )\) by \(\vert u \vert _{p}=(\int_{\Omega } \vert u \vert ^{p})^{1/p}\). By the Sobolev embedding theorem, \(H_{0}^{1}(\Omega )\) can be compactly embedded into \(L^{p}(\Omega )\) for all \(p\in [1,6)\) and the embedding \(H_{0}^{1}( \Omega )\hookrightarrow L^{6}(\Omega )\) is continuous.
For any given \(u\in H^{1}_{0}(\Omega )\), by using the Lax-Milgram theorem, the Dirichlet boundary problem \(-\Delta \phi =u^{2}\) in Ω has a unique solution \(\phi_{u}\in H^{1}_{0}(\Omega )\). Substitute \(\phi_{u}\) to the first equation of system (1.1), then system (1.1) can be transformed into the following variable equation:
Some necessary properties of \(\phi_{u}\) are given in Lemma 2.1.
Since we only consider the positive solution to system (1.1), we can assume that \(f(s)=0\) and \(g(s)=0\) for all \(s\in (-\infty ,0)\). By (f), for \(s\geqslant 1/2\),
Since \(F(s)\leqslant F(1/2), s\in [0,1/2]\), then there exist \(c_{1}, c_{2}>0\) such that
It is obvious that F is continuous on \(\mathbb{R}\). By (g1), we easily obtain that for any \(\varepsilon >0\), there exist \(C_{\varepsilon }>0, p\in (3,5)\) such that
and
Thus, by (1.4), (1.6) and (h), the energy functional corresponding to (1.3)
is well defined and continuous on \(H_{0}^{1}(\Omega )\).
As we know, under the general singular assumption (f) or (f0), the functional J fails to be Fréchet differentiable because of the singular term. We then cannot apply the critical point theory to obtain the existence of solution directly. In general, a function \(u\in H _{0}^{1}(\Omega )\) is called a solution of (1.3), that is, \((u,\phi_{u})\) is a solution of (1.1) and \(u(x)>0\) a.e. in Ω satisfying
In fact, under the weaker singular assumption (f0), from (1.4), (1.6), (1.7), we easily deduce that the functional J has a negative local minimum around the neighborhood of origin with the parameter \(\lambda >0\) small enough. With the two skilled lemmas (Lemmas 2.3, 2.4 in [10]) on the properties of the singular term f, we can show that the negative local minimum point is a solution of problem (1.3). In order to obtain the second solution of system (1.1), here we assume that (f) holds, that is, f is singular at 0 and nonincreasing on \((0,\infty )\). It is obvious that assumption (f) implies that (f0) holds. Assumption (f) was first introduced in [12] to consider the singular semilinear elliptic equation. To obtain the second solution of problem (1.3), motivated by [1, 4], we also consider the perturbation problem
where \(\alpha >0\). The functional corresponding to problem (1.9) is as follows:
Under the assumptions of (f), (g1)-(g3) and (h), we can show \(J_{\alpha }\in C^{1}(H_{0}^{1}(\Omega ),{\mathbb{R}})\) and problem (1.9) has a mountain pass type solution \(u_{\alpha }\). Finally, we can prove that the limit \(v_{0}\) of a family of solutions \(\{u_{\alpha }\}\) of problem (1.9) is the second solution of problem (1.3). In the proof, the monotonic property of f and a result from [13] are crucial to showing the uniform boundedness of \(\{u_{\alpha }\}\) and the convergence of \(u_{\alpha } \to v_{0}\) as \(\alpha \to 0\).
Our main result can be described as follows.
Theorem 1.1
If \(a>0\), \(b\geqslant 0\), and assumptions (f), (g1)-(g3) and (h) hold, then there exists \(\lambda^{*}>0\) such that system (1.1) has at least two solutions for each \(\lambda \in (0,\lambda^{*})\).
Remark 1.2
There are a number of functions which satisfy (f), (g1)-(g3), (h) respectively. For example,
-
(i)
\(f_{1}(s)= [ s^{\alpha }\arctan (1+s) ] ^{-1}\) for all \(s\in (0,\infty )\), where \(0<\gamma <\alpha <1\);
-
(ii)
\(f_{2}(s)=\sqrt{1+s^{2\beta }}/s^{\alpha }\) for all \(s\in (0, \infty )\), where \(0<\max \{\gamma ,\beta /2\}<\alpha <1\).
It is easy to verify that the functions \(f_{1}\), \(f_{2}\) satisfy condition (f).
Let \(g_{1}(s)=s^{2+\beta }\ln (1+s), s>0\) with \(\beta \in (0,1)\) and \(g_{2}(s)=s^{p}, s>0\) with \(p\in (3,5)\). Let \(\rho \in C^{1}({\mathbb{R} ^{+}},[0,1])\) be a cut-off function verifying \(s\rho '(s)\leqslant 0\), \(\vert \rho '(s) \vert \leqslant 2\),
Set \(G(s)=\rho (s)G_{1}(s)+(1-\rho (s))G_{2}(s), g(s)=G'(s)\), where \(G_{i}(s)=\int_{0}^{s}g_{i}(t)\,dt\). Then it is easy to verify that g satisfies conditions (g1)-(g3).
Take some \(x_{0}\in \Omega \) and let \(h(x)= \vert x-x_{0} \vert ^{-\beta }\) for all \(x\in \Omega \setminus \{x_{0}\}\), where \(\beta \in [0,3/2)\). It is obvious that h satisfies condition (h).
This paper is organized as follows. In Section 2, we give the existence of a negative local minimum of the functional J for \(\lambda >0\) small enough and show that it is a solution of problem (1.3). In Section 3, we firstly discuss the existence of the mountain pass type solution to the perturbation problem (1.9). Furthermore, by approximation, the second solution of problem (1.3) is obtained.
In this paper, c, \(c_{i}\), \(C_{i}\) denote various positive constants, which may vary from line to line.
2 Existence of the first solution to system (1.1)
Let us first collect some properties of \(\phi_{u}\). We refer the readers to [7, 14–16], etc.
Lemma 2.1
For each \(u\in H^{1}_{0}(\Omega )\), there exists a unique solution \(\phi_{u}\in H^{1}_{0}(\Omega )\) of
The following properties hold for the solution \(\phi_{u}\):
-
(i)
\(\Vert \phi_{u} \Vert ^{2}=\int_{\Omega }\phi_{u}u^{2}\);
-
(ii)
\(\phi_{u}\geqslant 0\). Moreover, \(\phi_{u}>0\) in Ω when \(u\neq 0\);
-
(iii)
for each \(t\ne 0\), it holds that \(\phi_{tu}=t^{2}\phi _{u}\);
-
(iv)
if \(u_{n}\rightharpoonup u\) in \(H^{1}_{0}(\Omega )\), then we have
$$\begin{aligned} &\phi_{u_{n}}\to \phi_{u} \quad \textit{in } H^{1}_{0}( \Omega ), \\ & \int_{\Omega }\phi_{u_{n}}u_{n}\phi \to \int_{\Omega }\phi_{u}u \phi ,\quad \phi \in H_{0}^{1}(\Omega ), \\ & \int_{\Omega }\phi_{u_{n}}u_{n}(u_{n}-u) \to 0; \end{aligned}$$ -
(v)
\(\phi_{u}\in W_{\mathrm{loc}}^{2,3}(\Omega )\cap C^{0}(\bar{\Omega })\);
-
(vi)
\(\phi_{u}=\phi_{u^{+}}+\phi_{u^{-}}\), where \(u^{\pm }= \pm \max \{\pm u,0\}\).
Under the assumptions of (f), (g1) and (h), we can show that the functional J defined in (1.7) has a negative local minimum for small \(\lambda >0\). In fact, we have the following lemma.
Lemma 2.2
Under the assumptions of (f), (g1) and (h), there exist \(\lambda^{*}>0\) and \(r, \rho >0\) such that for any \(\lambda \in (0, \lambda^{*})\), we have
where \(B_{r}=\{u\in H_{0}^{1}(\Omega ): \Vert u \Vert < r\}, S_{r}=\partial B _{r}\).
Proof
For any \(u\in H^{1}_{0}(\Omega )\), by (1.4), (1.6) with \(0<\varepsilon <\frac{a}{4}\mu_{1}\), and (1.7), where \(\mu_{1}>0\) is the first eigenvalue of the operator −Δ in \(H_{0}^{1}(\Omega )\), we have
Let \(m(t)=\frac{a}{4}t-c_{3}t^{5}-c_{4}t^{p}\), since \(p>3\), there exists \(r>0\) such that \(m(r)=\max_{t\geqslant 0}m(t)\). We choose \(\lambda _{1}, \lambda_{2}>0\) respectively such that \(\lambda_{1}c_{0} \vert h \vert _{2}= \frac{1}{2}m(r), \lambda_{2}c_{2} \vert h \vert _{1}=\frac{1}{4}rm(r)\). Thus, when \(0<\lambda <\lambda^{*}=\min \{\lambda_{1},\lambda_{2}\}\), for any \(u\in S_{r}\), we have
Hence, for any \(\lambda \in (0,\lambda^{*})\), there exist \(r,\rho >0\) such that \(J|_{S_{r}}\geqslant \rho \).
On the other hand, by assumption (f), there exists \(\delta >0\) such that
Choose a nonnegative function \(\varphi \in C^{\infty }_{0}(\Omega ) \backslash \{0\}\) with \(\max_{\Omega }\varphi \leqslant \delta \). Then, for any \(t\in (0,1]\), by Lemma 2.1(iii), (2.2), we have
Since \(1-\gamma \in (0,1)\) and \(h(x)>0\), a.e. \(x\in \Omega \), we get that \(J(t\varphi )<0\) for \(t>0\) small enough. Hence, it follows from (2.1) that \(m=\inf_{\bar{B}_{r}}J<0\). □
In order to prove that the local minimum m can be obtained by some \(u_{0}\in H_{0}^{1}(\Omega )\) and to prove that \(u_{0}\) is a solution of problem (1.3), we need the following two skilled lemmas which can be found in [10].
Lemma 2.3
Assume that (f0) holds, for \(a_{0}\), \(b_{0}\geqslant 0\), one has that \(\lim_{t\to 0^{+}}\frac{1}{t}[F(a_{0}+tb_{0})-F(a_{0})]=f(a_{0})b_{0}\), which equals ∞ if \(a_{0}=0\) and \(b_{0}>0\).
Lemma 2.4
Assume that (f0) holds. Then, for any \(u\in H^{1}_{0}(\Omega )\) with \(u(x)>0\), a.e. \(x\in \Omega \), we have
Theorem 2.5
Assume that (f), (g1) and (h) hold. Then, for \(\lambda \in (0, \lambda^{*})\), problem (1.3) possesses a solution \(u_{0}\) with \(J(u_{0})=m\).
Proof
According to the definition of m, there exists a sequence \(\{u_{n}\}\subset \bar{B}_{r}\) such that \(\lim_{n\to \infty }J(u_{n})=m\). Then \(\{u_{n}\}\) is bounded in \(H^{1}_{0}(\Omega )\). Going if necessary to a subsequence, still denoted by \(\{u_{n}\}\), there exists \(u_{0}\in H^{1}_{0}(\Omega )\) such that
as \(n\to \infty \). By (1.4) and the Sobolev embedding theorem, we see that \(\{F(u_{n})\}\) is bounded in \(L^{2}(\Omega )\). Moreover, it follows from the continuity of F that \(F(u_{n}(x))\to F(u_{0}(x))\), a.e. \(x\in \Omega \). Thus, we obtain that \(F(u_{n})\rightharpoonup F(u _{0})\) in \(L^{2}(\Omega )\). By \(h\in L^{2}(\Omega )\), it follows that
By (1.6), we easily deduce \(\int_{\Omega }G(u_{n}) \to \int_{\Omega }G(u_{0})\). Then, by the weak lower semi-continuity of the norm and Lemma 2.1(iv), (2.3), we have
On the other hand, \(u_{n}\rightharpoonup u_{0}\) implies that \(\Vert u_{0} \Vert \leqslant \liminf_{n\to \infty } \Vert u_{n} \Vert \leqslant r\), then \(J(u_{0})\geqslant m\). Hence \(J(u_{0})=m\).
To show that \(u_{0}\) is a solution of problem (1.3), we need to show that \(u_{0}(x)>0\), a.e. \(x\in \Omega \) and \(u_{0}\) satisfies (1.8). The proof is similar to the proof of Theorem 1.1 in [10], for completeness, here we give the details.
By Lemma 2.1(vi), we obtain that \(m\leqslant J(u_{0}^{+}) \leqslant J(u_{0})=m\), then \(J(u_{0}^{+})=J(u_{0})=m\). Thus we may assume \(u_{0}\geqslant 0\). While \(m<0\), then \(u_{0}\neq 0\). Now we divide the proof into two steps. For convenience, we denote \(l_{0}=a+b \Vert u_{0} \Vert ^{2}\).
Firstly, we prove that \(u_{0}(x)>0\), a.e. \(x\in \Omega \). In fact, for each \(v\in H^{1}_{0}(\Omega )\) with \(v\geqslant 0\) and \(t>0\) small enough, we have that
This implies that
Thus, by Fatou’s lemma and Lemma 2.3, we have
Now let \(e_{1}\in H^{1}_{0}(\Omega )\) be the first eigenfunction of the operator −Δ in \(H_{0}^{1}(\Omega )\) and \(e_{1}(x)>0\) for all \(x\in \Omega \). Taking \(v=e_{1}\) in (2.4), one gets that
which implies that \(u_{0}(x)>0\), a.e. \(x\in \Omega \) by assumption (h). If not, there exists \(E\subset \Omega \) such that \(m(E)>0\) and \(u_{0}(x)=0\) for all \(x\in E\). Then, by Lemma 2.3,
it is a contradiction.
Secondly, we shall prove that \(u_{0}\) is a solution of problem (1.3), namely, \(u_{0}\) satisfies the following:
For this purpose, we define a function \(\Phi :\mathbb{R}\to \mathbb{R}\) by \(\Phi (t)=J(u_{0}+tu_{0})\), that is,
Then Φ attains its local minimum at \(t=0\). It follows from Lemma 2.4 that Φ is differentiable at \(t=0\) and \(\Phi '(0)=0\), that is,
For each \(v\in H^{1}_{0}(\Omega )\) and \(\varepsilon >0\), let us define \(v_{\varepsilon }=u_{0}+\varepsilon v\) and
Then \(v_{\varepsilon }^{-} |_{\Omega_{+}}=0\) and \(v_{\varepsilon } ^{-} |_{\Omega_{-}}=u_{0}+\varepsilon v\). Inserting \(v_{\varepsilon }^{+}\) into (2.4) and using (2.6), we obtain that
which implies that
Now let \(E_{n}=\{x\in \Omega :u_{0}(x)>0,v(x)>-\infty ,u_{0}(x)+v(x)/n<0 \}\) for all n. Then \(\{E_{n}\}\) is a nonincreasing sequence of measurable sets and
Thus we have
Select \(\varepsilon =1/n\). Then \(\Omega_{-}\subset \{x\in \Omega :u _{0}(x)\leqslant 0\}\cup \{x\in \Omega :v(x)=-\infty \}\cup E_{n}\) and \(m(\Omega_{-})=m(E_{n})\to 0\) as \(n\to \infty \). Letting \(\varepsilon =1/n\to 0\) in (2.7), we have
According to the arbitrariness of \(v\in H_{0}^{1}(\Omega )\), this inequality also holds for −v. Thus, (2.5) holds. Therefore, \(u_{0}\) is a solution of system (1.3) with \(J(u_{0})=m\). □
3 Proof of Theorem 1.1
In order to overcome the difficulty caused by the singular term and to obtain the second solution of problem (1.3) for \(\lambda >0\) small enough, in this section, we firstly consider the following perturbation problem:
where \(\alpha >0\). We define the functional corresponding to problem (3.1)
It is obvious that \(J_{\alpha }\) is a \(C^{1}\) functional defined on \(H_{0}^{1}(\Omega )\). The solution of problem (3.1) corresponds to the critical point of the functional \(J_{\alpha }\). That is, if \(u\in H_{0}^{1}(\Omega )\) is a solution of problem (3.1), it satisfies
For any \(s>0\), since f is nonincreasing, we have
by \(F(s)=0\) if \(s\leqslant 0\), (3.3) holds for all \(s\in {\mathbb{R}}\). Then, for any \(u\in H_{0}^{1}(\Omega )\), we have
where \(I(u)=\frac{a}{2} \Vert u \Vert ^{2}+\frac{b}{4} \Vert u \Vert ^{4}+\frac{1}{4} \int_{\Omega }\phi_{u}u^{2}-\int_{\Omega }G(u)\), \(u\in H_{0}^{1}(\Omega)\).
In order to show that \(J_{\alpha }\) satisfies the mountain pass geometry and to estimate the mountain pass critical level, we firstly consider the functional I. In fact, under the assumptions of (g1) and (g2), we have the following lemma.
Lemma 3.1
Under the assumptions of (g1) and (g2), there exist \(r_{0},\rho_{0}>0\) such that the functional I satisfies
-
(i)
\(I|_{S_{r_{0}}}\geqslant \rho_{0}\);
-
(ii)
there exists \(u_{1}\in H_{0}^{1}(\Omega )\) with \(\Vert u_{1} \Vert >r_{0}\) such that \(I(u_{1})<0\).
Proof
(i) By (1.6) with \(\varepsilon >0\) small enough, for any \(u\in H_{0}^{1}(\Omega )\), we have
it is obvious that the conclusion (i) holds.
(ii) It follows from (g1) and (g2), for any given \(M>0\), there exists \(R>0\) such that \(g(t)\geqslant Mt^{3}\), \(t>R\) and
Then there exists \(C>0\) such that \(g(t)-Mt^{3}\geqslant -Ct\), \(t\in [0,R]\) and \(g(t)\geqslant Mt^{3}-Ct\), \(t\geqslant 0\). For G, we also have
Thus, for any \(u\in H_{0}^{1}(\Omega )\setminus \{0\}\), \(\int_{\Omega }G(tu) \geqslant \frac{M}{4}t^{4} \vert u \vert _{4}^{4}-\frac{C}{2}t^{2} \vert u \vert _{2}^{2}\). It follows that
Thus
it follows from (3.5) that \(\lim_{t\to \infty }I(tu)=-\infty \), hence, there exists \(t>0\) large enough such that \(\Vert u_{1} \Vert = \Vert tu \Vert >r _{0}\) and \(I(u_{1})<0\). □
Now, we define
It follows from Lemma 3.1 that \(c\geqslant \rho_{0}>0\). For any given \(\alpha >0\), \(J_{\alpha }\) also has the mountain pass geometry. In fact, we have the following lemma.
Lemma 3.2
Assume \(\lambda \in (0,\lambda^{*})\), under the assumptions of (f), (g1), (g2) and (h), for \(r,\rho >0\) (where \(\lambda^{*}\), r, ρ are given in Lemma 2.2), the functional \(J_{\alpha }\) satisfies the following:
-
(i)
\(J_{\alpha }|_{S_{r}}\geqslant \rho \);
-
(ii)
there exists \(v\in H_{0}^{1}(\Omega )\) such that \(J_{\alpha }(v)<0\).
Proof
(i) By (3.4) and Lemma 2.2, the conclusion holds.
(ii) From (3.4) and (ii) of Lemma 3.1, we choose \(v=u_{1}\) in Lemma 3.1 and the conclusion holds. □
We also can define the mountain pass critical level
From (3.4) and (i) of Lemma 3.2, for \(\lambda \in (0, \lambda^{*})\),
In the following, we give the existence of the mountain pass type solution to system (3.1).
Lemma 3.3
Suppose that (f), (g1)-(g3) and (h) hold, \(\lambda \in (0, \lambda^{*})\). Then there exists \(u_{\alpha }\in H_{0}^{1}(\Omega )\) such that
Proof
By Lemma 3.2 and the mountain pass lemma, there exists a sequence \(\{u_{n}\}\subset H_{0}^{1}(\Omega )\) such that \(J_{\alpha }(u _{n})\to c_{\alpha }\), \(J'_{\alpha }(u_{n})\to 0\). By (3.3), (1.4) and (g3), for n large enough, we have
Then \(\{u_{n}\}\) is bounded. Up to a subsequence, there exists \(u_{\alpha }\in H_{0}^{1}(\Omega )\) such that \(u_{n}\rightharpoonup u _{\alpha }\) in \(H_{0}^{1}(\Omega )\) and
It follows from \(J'_{\alpha }(u_{n})\to 0\) that
Since \(hf(u_{n}^{+}+\alpha )(u_{n}-u_{\alpha })\to 0\) a.e. in Ω and
by the dominated convergence theorem, we have
By (1.5), we can deduce that
and
From (3.7), using (3.8), (3.9), Lemma 2.1(iv) and the boundedness of \(\{u_{n}\}\), we get \(\Vert u_{n} \Vert \to \Vert u_{\alpha } \Vert \). This combined with \(u_{n}\rightharpoonup u_{\alpha }\) implies that \(u_{n}\to u_{\alpha }\) in \(H_{0}^{1}(\Omega )\). Consequently, we have \(J_{\alpha }(u_{\alpha })=c_{\alpha }> \rho , J'_{\alpha }(u_{\alpha })=0\), that is, \(u_{\alpha }\) is a nontrivial solution to problem (3.1). Then \(u_{\alpha }\) satisfies (3.2), taking the test function \(\phi =u_{\alpha }^{-}\) in (3.2), it follows that \(\Vert u_{\alpha }^{-} \Vert =0\). Thus, we have \(u_{\alpha }\geqslant 0, u_{\alpha }\neq 0\) and \(J_{\alpha }(u_{ \alpha })=c_{\alpha }\geqslant \rho_{1}\). Hence, by the strong maximum principle, \(u_{\alpha }\) is a positive solution of the perturbation problem (3.1). □
In order to consider the convergence of \(\{u_{\alpha }\}\) as \(\alpha \to 0\) and to obtain the second solution of problem (1.3), we need the following result, which can be found in [13].
Lemma 3.4
Brezis and Nirenberg [13]
Let Ω be a bounded domain in \({\mathbb{R}}^{n}\) with smooth boundary ∂Ω. Let \(u\in L_{\mathrm{loc}}^{1}(\Omega )\) and assume that, for some \(k\geqslant 0\), u satisfies, in the sense of distributions,
Then either \(u\equiv 0\), or there exists \(C>0\) such that
Remark 3.5
By Lemma 3.3, (3.2) and Lemma 2.1(v), we have
where \(K>0\). Then, by Lemma 3.4, there exists \(C>0\) such that \(u_{\alpha }(x)\geqslant C\operatorname{dist}(x,\partial \Omega ), x\in \Omega \).
Finally, let \(\alpha \to 0\), we shall prove that the limit of a family of solutions \(\{u_{\alpha }\}\) of the perturbation problem (3.1) is the second solution of problem (1.3) with \(\lambda \in (0,\lambda^{*})\), where \(\lambda^{*}\) is defined in Lemma 2.2.
Theorem 3.6
Suppose that (f), (g1)-(g3) and (h) hold, \(\lambda \in (0, \lambda^{*})\). Then problem (1.3) has a solution \(v_{0}\) satisfying \(J(v_{0})>0\).
Proof
Let \(\alpha \to 0\) and \(u_{\alpha }\geqslant 0\) is the solution of problem (3.1), that is, \(J_{\alpha }(u_{\alpha })=c_{\alpha }\), \(J'_{ \alpha }(u_{\alpha })=0\). Then, by (g3), (3.3), (3.6) and (1.4), we have
then \(\{u_{\alpha }\}\) is bounded in \(H_{0}^{1}(\Omega )\). Up to a subsequence, there exists \(v_{0}\in H_{0}^{1}(\Omega )\) such that \(u_{\alpha }\rightharpoonup v_{0}\) in \(H_{0}^{1}(\Omega )\) and
Firstly, we show that \(v_{0}(x)>0\) a.e. in Ω. For that purpose, we denote \(w_{\alpha }=u_{\alpha }-v_{0}\) and \(l=\lim_{\alpha \to 0} \Vert w_{\alpha } \Vert \). Taking \(\phi \in H_{0}^{1}(\Omega )\) with \(\phi \geqslant 0\) in (3.2), we have
By using Fatou’s lemma, Lemma 2.1(iv) and (3.10), we have
Similar to the proof of \(u_{0}(x)>0\) in \(x\in \Omega \) in Theorem 2.5, we can show that \(v_{0}(x)>0\) a.e. in Ω.
Next, we show that \(u_{\alpha }\to v_{0}\) in \(H_{0}^{1}(\Omega )\) and \(v_{0}\) is the solution of problem (1.3), that is, we need to show that \(l=0\) and \(v_{0}\) satisfies (1.8).
We take \(\phi \in C_{0}^{\infty }(\Omega )\) with \(\operatorname{supp}\phi =\Omega _{1}\Subset \Omega \) in (3.2). By Remark 3.5, for \(x\in \Omega_{1}\), we have
where \(k_{0}=\min_{x\in \Omega_{1}}\operatorname{dist}(x,\partial \Omega )>0\). Since \(hf(u_{\alpha }+\alpha )\phi \to hf(v_{0})\phi \) a.e. in Ω, then by the dominant convergence theorem, we have
By \((J'_{\alpha }(u_{\alpha }),\phi )=0\), using \(\int_{\Omega } \phi_{u_{\alpha }}u_{\alpha }\phi \to \int_{\Omega }\phi_{v_{0}}v_{0} \phi \) and (3.10), we get that
Since \(C_{0}^{\infty }(\Omega )\) is dense in \(H_{0}^{1}(\Omega )\), then for \(\phi \in H_{0}^{1}(\Omega )\), there exists a sequence \(\{\phi _{n}\}\subset C_{0}^{\infty }(\Omega )\) such that \(\phi_{n}\to \phi \) as \(n\to \infty \). For \(n,m\in {\mathbb{N}}\) large enough, replacing ϕ with \(\phi_{n}-\phi_{m}\) in (3.11), we obtain that
Since \(\phi_{n}\to \phi \), from (3.12), we can deduce that \(\{hf(v_{0})\phi_{n}\}\) is a Cauchy sequence in \(L^{1}(\Omega )\), hence there exists \(v\in L^{1}(\Omega )\) satisfying \(hf(v_{0})\phi_{n} \to v\) in \(L^{1}(\Omega )\), which means that \(hf(v_{0})\phi_{n}\to v\) in measure. By Riesz’s theorem, \(\{hf(v_{0})\phi_{n}\}\) has a subsequence, still denoted by \(\{hf(v_{0})\phi_{n}\}\), such that \(hf(v_{0})\phi_{n}\to v\) a.e. in Ω. On the other hand, \(hf(v_{0})\phi_{n}\to hf(v_{0})\phi \) a.e. in Ω. So \(v=hf(v_{0})\phi \), that is, \(\int_{\Omega }hf(v_{0})\phi_{n}\to \int_{\Omega }hf(v_{0})\phi \) as \(n\to \infty \). Then, taking the test function \(\phi_{n}\) in (3.11) and passing to the limit as \(n\to \infty \), we obtain that (3.11) holds for any \(\phi \in H_{0}^{1}(\Omega )\). We take \(\phi =v_{0}\) in (3.11) and obtain that
On the other hand, by \((J'_{\alpha }(u_{\alpha }),u_{\alpha })=0\), we have
Since f is nonincreasing on \((0,\infty )\), we have
Then, by (1.4), we have
Then, combining with \(hf(u_{\alpha }+\alpha )\to hf(v_{0})v_{0}\) a.e. in Ω and using the dominant convergence theorem, we have
Thus, from (3.14), by (3.10) and Lemma 2.1(iv), we obtain that
Combining with (3.13) and (3.15), we get
from \(a>0, b\geqslant 0\), it implies that \(l=0\), that is, \(u_{\alpha }\to v_{0}\). Hence, from (3.11) with \(l=0\) and \(\phi \in H_{0} ^{1}(\Omega )\), we get that \(v_{0}\) is the solution of problem (1.3) for \(\lambda \in (0,\lambda^{*})\) and \(J(v_{0})= \lim_{\alpha \to 0}J_{\alpha }(u_{\alpha })\geqslant \rho >0\). □
Proof of Theorem 1.1
By Theorems 2.5 and 3.6, for \(\lambda \in (0,\lambda^{*})\), there exist two solutions \(u_{0},v_{0}\in H_{0}^{1}(\Omega )\) to problem (1.3) with \(J(v_{0})>0>J(u_{0})\), that is, system (1.1) possesses at least two solutions for each \(\lambda \in (0,\lambda^{*})\). □
4 Conclusion
In this paper, by using the variational method and the perturbation method, we consider the existence and multiple solutions to the singular Kirchhoff-Schrödinger-Poisson system (1.1). The nonlinear terms contain the quasicritical nonlinearity g, which satisfies assumptions (g1)-(g3), and the general singularity f, which satisfies (f). The general singular assumption derives from our previous work [10], in which we consider the uniqueness of solution to Kirchhoff-Schrödinger-Poisson system. Therefore, the results in this paper are the continuation of our research in [10]. Our results also improve the results in [11], in which the authors considered the existence of Kirchhoff-Schrödinger-Poisson system with the singular term \(f(s)=s^{-r}\), \(r\in (0,1)\).
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The author thanks the anonymous referee for the careful reading and some helpful comments.
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The paper is supported by the National Natural Science Foundation of China (Grant No. 11571209, 11671239), Science Council of Shanxi Province (2015021007), Scientific and Technological Higher Education Institutions in Shanxi (2016106).
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Zhang, Q. Multiple positive solutions for Kirchhoff-Schrödinger-Poisson system with general singularity. Bound Value Probl 2017, 127 (2017). https://doi.org/10.1186/s13661-017-0858-x
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DOI: https://doi.org/10.1186/s13661-017-0858-x