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Ground state solutions of Nehari–Pohožaev type for a kind of nonlinear problem with general nonlinearity and nonlocal convolution term
Boundary Value Problems volume 2019, Article number: 150 (2019)
Abstract
In this paper, we consider the following nonlinear problem with general nonlinearity and nonlocal convolution term:
where \(a\in (0,3)\), \(q\in [1+\frac{\alpha }{3},3+\alpha )\), \(I_{\alpha }:\mathbb{R}^{3}\rightarrow \mathbb{R}\) is the Riesz potential, \(V\in \mathcal{C}(\mathbb{R}^{3},[0,\infty ))\), \(f\in \mathcal{C}(\mathbb{R},\mathbb{R})\) and \(F(t)=\int _{0}^{t}f(s)\,ds\) satisfies \(\lim_{|t|\to \infty }F(t)/|t|^{\sigma }=\infty \) with \(\sigma =\min \{2,\frac{2\beta +2}{\beta }\}\) where \(\beta =\frac{ \alpha +2}{2(q-1)}\). By using new analytic techniques and new inequalities, we prove the above system admits a ground state solution under mild assumptions on V and f.
1 Introduction
In this paper we are concerned with the existence of ground state solutions for the nonlinear problem:
where \(\alpha \in (0,3)\), \(q\in [1+\frac{\alpha }{3},3+\alpha )\), \(V\in \mathcal{C}(\mathbb{R}^{3},[0,\infty ))\), \(f\in \mathcal{C}( \mathbb{R},\mathbb{R})\) and \(I_{\alpha }:\mathbb{R}^{3}\rightarrow \mathbb{R}\) is the Riesz potential of order \(\alpha \in (0,3) \), given by
When \(q=2\) and \(\alpha =2\), the convolution term of (1.1) can be considered as analogous to the internal potential ϕ in Schrödinger–Poisson system like the following equation:
System (1.2) was first introduced in [1] as a model describing solitary waves for the nonlinear stationary Schrödinger equation interacting with the electrostatic field. We note that system (1.2) is also called a Schrödinger–Maxwell equation system, for more details on the physical aspects of this problem, we refer to [1] and the references therein.
In recent years, the existence, multiplicity and concentration of nontrivial solutions of (1.2) have been the subject of extensive mathematical studies, for example, [2,3,4,5,6,7,8]. Problem (1.2) with \(V(x)\equiv 1\) or being radially symmetric, has been widely studied under various conditions on f; see for example [9,10,11,12,13,14,15,16,17]. When \(V=1\) and \(f(u)=|u|^{p-2}u\), by introducing a new manifold that is defined by a condition which is a combination of the Nehari equation and the Pohožaev equality, Ruiz [16] showed that (1.2) admits a positive radial solution if \(3< p<6\), but does not have a nontrivial solution for \(2< p\leq 3\). Under the same assumptions, based on Ruiz’s approach in [16], Azzollini and Pomponio [3] obtained the existence of ground state solutions for (1.2) by using a concentration-compactness argument. When \(f(u)=|u|^{p-2}u\) and V satisfies the following assumptions:
- (V1):
-
\(V\in \mathcal{C}(\mathbb{R}^{3},[0,\infty ))\) and \(V_{\infty }:=\lim_{|y|\rightarrow \infty }V(y)\geq V(x)\) for \(x\in \mathbb{R}^{3}\);
- (V2′):
-
\(V(x)\) is weakly differentiable, and satisfies \(\nabla V(x) \cdot x \in L^{\infty }(\mathbb{R}^{3})\cup L^{3/2}( \mathbb{R}^{3})\), and
$$ 2V(x)+\nabla V(x) \cdot x \geq 0 \quad \mbox{a.e. } x\in \mathbb{R}^{3}, $$
Zhao and Zhao [8] established the existence of ground state solutions for (1.2) by using the Jeanjean’s monotonicity trick [18]. In a recent paper [19], Tang and Chen introduced some new tricks to generalize and improve the results in [3, 8, 16] to the more general case where V satisfies (V1) and (V2′) and f satisfies the following assumptions:
-
(F1)
\(f\in \mathcal{C}(\mathbb{R},\mathbb{R})\), and there exist constants \(C>0\) and \(q\in (2,6)\) such that
$$ \bigl\vert f(t) \bigr\vert \leq C\bigl(1+ \vert t \vert ^{q-1} \bigr), \quad \forall t\in \mathbb{R}; $$ -
(F2)
\(f(t)=o(t)\) as \(t\rightarrow 0\);
-
(F3)
\(\lim_{|t|\rightarrow \infty }\frac{F(t)}{|t|^{3}}=\infty \), where \(F(t)=\int _{0}^{t}f(s)\,ds\);
-
(F4)
\([2f(t)t-3F(t)]/t^{3}\) is nondecreasing on \((-\infty ,0) \cup (0,+\infty )\).
There is also other work about ground state solutions for (1.2); we refer to [20, 21]. Motivated by the above work and [22,23,24,25,26,27,28,29,30], in the present paper, we shall extend the results concerning the existence of ground state solutions for (1.2) in [23] to (1.1). Compared with (1.2), it is more difficult to deal with (1.1) for the reason that \(q \in [1+\frac{ \alpha }{3},3+\alpha )\). Because of the changing of q, the competing effect of \((I_{\alpha }\ast |u|^{q})|u|^{q-2}u\) and \(f(u)\) is also changing.
For any \(\varepsilon >0\), it follows from (F1) and (F2) that there exists \(C_{\varepsilon } >0\) such that
Under assumptions (V1), (F1), (F2) and (1.3), the functional
is well defined in \(H^{1}(\mathbb{R}^{3})\) and \(\varPhi \in \mathcal{C} ^{1}(H^{1}(\mathbb{R}^{3}),\mathbb{R})\), where \(F(t)=\int _{0}^{t}f(s) \,ds\). Moreover, for any \(u,\upsilon \in H^{1}(\mathbb{R}^{3})\),
Hence, the solution of (1.1) are critical points of \(\varPhi (u)\). A solution is called a ground state solution if its energy is minimal among all nontrivial solutions.
In this paper, let \(\beta =\frac{\alpha +2}{2(q-1)}\), and in addition to (V1), (F1) and (F2), we also need to introduce the following assumptions:
- (V2):
-
\(V\in \mathcal{C}^{1}(\mathbb{R}^{3})\), the set \(\{x\in \mathbb{R}^{3}:|\nabla V(x)\cdot x|\geq \epsilon \}\) has finite Lebesgue measure for every \(\epsilon >0\), and the function \(t\mapsto t^{2}[(2\beta -3)V(tx)-\nabla V(tx)\cdot (tx)]\) is increasing on \((0,+\infty )\) for every \(x\in \mathbb{R}^{3}\);
- (V3):
-
\(V\in \mathcal{C}^{1}(\mathbb{R}^{3})\), \(\nabla V(x) \cdot x \in L^{\infty }(\mathbb{R}^{3})\) and there exists \(\varrho >0\) such that
$$ 2V(x)+\nabla V(x)\cdot x\geq \varrho , \quad \forall x\in \mathbb{R}^{3}; $$ - (V3′):
-
\(V\in \mathcal{C}^{1}(\mathbb{R}^{3})\), \(\nabla V(x) \cdot x \in L^{\infty }(\mathbb{R}^{3})\), \(2V(x)+\nabla V(x)\cdot x \geq \varrho \), \(\forall \ x\in \mathbb{R}^{3} \) and there exists \(\mu > \frac{2\beta +2}{\beta }\) such that
$$ f(t)t-\mu F(t)\geq 0,\quad \forall t\in \mathbb{R} ; $$ - (F5):
-
\(\lim_{|t|\rightarrow \infty } \frac{F(t)}{|t|^{\frac{2 \beta +2}{\beta }}}=\infty \);
- (F6):
-
\([\beta f(t)t-3F(t)]/t|t|^{\frac{\beta +2}{\beta }}\) is nondecreasing on both \((-\infty ,0)\) and \((0,+\infty )\).
To state our results, we define the Nehari–Pohožaev manifold as follows:
where
which is associated with the Pohožaev identity \(\mathcal{P}(u)=0\) of (1.1) that can be obtained by the same argument as in [13, 31]. And
Throughout the paper we use the following notations:
-
\(H^{1}(\mathbb{R}^{3})\) denotes the usual Sobolev space equipped with the inner product and norm
$$ (u,v)= \int _{\mathbb{R}^{3}}(\nabla u\cdot \nabla v+uv)\,dx, \quad \Vert u \Vert =(u,u)^{\frac{1}{2}},\quad \forall u,v\in H^{1}\bigl( \mathbb{R}^{3}\bigr). $$ -
\(L^{s}(\mathbb{R}^{3})(1\leq s <\infty )\) denotes the Lebesgue space with the norm \(\|u\|_{s}=(\int _{\mathbb{R}^{3}}|u|^{s}\,dx)^{1/s}\);
-
for any \(u\in H^{1}(\mathbb{R}^{3}) \setminus \{0\}\), \(u_{t}(x):=u(tx)\) for \(t > 0\);
-
for any \(x\in \mathbb{R}^{3}\) and \(r > 0\), \(B_{r}(x):=\{y\in \mathbb{R}^{3}:|y-x| < r\}\);
-
\(C_{1},C_{2},\ldots\) denotes positive constants possibly different in different places.
Next,we state the main results of this paper.
Theorem 1.1
Assume that V and f satisfy (V1), (V2), (F1), (F2), (F5) and (F6) hold. Then problem (1.1) has a ground state solution \(u_{0}\in H^{1}(\mathbb{R}^{3})\) such that \(\varPhi (u_{0})= \inf_{\mathcal{M}}\varPhi =\inf_{u\in H^{1}(\mathbb{R}^{3})\setminus \{0\}}\max_{t > 0} \varPhi (t^{\beta }u_{t}) > 0 \).
Theorem 1.2
Assume that V and f satisfy (V1), (V3), (F1), (F2), (F5) and (F6) hold. Then problem (1.1) has a positive ground state solution.
Theorem 1.3
Assume that V and f satisfy (V1), (V3′), (F1), (F2), (F5) and (F6) hold. Then problem (1.1) has a positive ground state solution.
Inspired by [19, 32], we shall prove Theorem 1.1 following this scheme:
-
Step (i)
we verify \(\mathcal{M}\neq \emptyset \) and establish the minimax characterization of \(m=\inf_{\mathcal{M}} \varPhi >0\);
-
Step (ii)
we prove that m can be obtained;
-
Step (iii)
we show that the minimizer of Φ on \(\mathcal{M}\) is a critical point.
Although we mainly follow the procedure of [19, 32], we have to face many new difficulties due to the mutual competing effect between \(f(u)\) and \((I_{\alpha }\ast |u|^{q})|u|^{q-2}u\). More precisely, in Step (i), we first establish a key inequality in Lemma 2.2 by using some properties about nonlinearity term, and also we use Nehari–Pohožaev manifold to remove the influence of the term \((I_{\alpha }\ast |u|^{q})|u|^{q-2}u\) which needs a kind of computational technique; it is worth mentioning that [23, Proposition 2.7] gives us an excellent numerical property as regards \(V(x)\) which is convenient for us to eliminate the potential term \(V(x)\) in some inequalities. Then we construct a saddle point structure with respect to the fibre \(\{t^{\beta }u_{t}:t>0\} \subset H^{1}(\mathbb{R}^{3})\) for \(u\in H^{1}(\mathbb{R}^{3}) \setminus \{0\}\); see Lemma 2.6, finally based on these constructions we obtain the minimax characterization of m; see Lemma 2.8. In Step (ii), we first choose a minimizing sequence \(\{u_{n}\}\) of Φ on \(\mathcal{M}\), and show that \(\{u_{n}\}\) is bounded in \(H^{1}(\mathbb{R}^{3})\), then with the help of the key inequality established in Lemma 2.2 and a concentration-compactness argument, we prove that there exist \(\hat{u} \in H^{1}(\mathbb{R}^{3})\) and \(\hat{t} >0\) such that \(u_{n}\rightharpoonup \hat{u} \) in \(H^{1}(\mathbb{R}^{3})\) up to translations and extraction of a subsequence, and \(\hat{t}^{\beta }\hat{u}_{\hat{t}} \in \mathcal{M}\) is a minimizer of \(\inf_{\mathcal{M}} \varPhi \); see Lemma 2.13 and Lemma 2.14. Step (iii) is similar to [19, Lemma 2.10].
Motivated by [8, 19], we use the Jeanjean’s monotonicity trick [18] to prove Theorems 1.2 and 1.3, which can helps us to construct a bounded (PS) sequence. The difficulty in the proof is to overcome the lack of compactness, and a more careful analysis is needed to consider the relationship between the mountain pass level for Φ and the least energy of the functional associated “limit problem” of (1.1) which is used to recover the compactness; see Lemma 3.4. By using Theorem 1.1 and applying the global compactness lemma and (V1) and (V3) (or (V3′)), we can prove Theorems 1.2 and 1.3; see Lemma 3.5 and Lemma 3.6.
2 Proof of Theorem 1.1
First, by a simple calculation, we establish some key inequalities.
Lemma 2.1
Assume that (F1) and (F6) hold, Then
Proof
It is evident that (2.1) holds for \(\tau =0\). For \(\tau \neq 0\), let
Then from (F4), one has
It follows that \(g(t)\geq g(1)=0\) for \(t \geq 0\). This, together with (2.2) implies (2.1) holds. □
Define
It is easy to check that (V2) implies
Lemma 2.2
Assume that (V1), (V4), (F1), (F2) and (F6) hold, Then
Proof
Note that
Thus, by (1.4), (1.8), (2.1), (2.5) and (2.7), one has
This shows that (2.6) holds. □
Remark that (2.6) with \(t\rightarrow 0\) implies
To overcome the lack of compactness of Sobolev space embedding in \(\mathbb{R}^{3}\), we define the following energy functional:
Corresponding to (1.6) and (1.8), we define
and
From Lemma 2.2, we have the following two corollaries.
Corollary 2.3
Assume that (F1), (F2) and (F6) hold. Then
Corollary 2.4
Assume that (V1), (V2), (F1) and (F6) hold. Then for \(u\in \mathcal{M}\)
From [23, Proposition 2.7], we can obtain the following lemma.
Lemma 2.5
Assume that (V1) and (V2) hold. Then there exist two constants \(\rho _{1}\), \(\rho _{2} > 0\) such that
and
Lemma 2.6
Assume that (V1), (V2), (F1), (F2) and (F6) hold. Then, for any \(u\in H^{1}(\mathbb{R}^{3})\setminus \{0\}\), there exists a unique \(t_{u}>0\) such that \(t_{u}^{\beta }u_{t_{u}}\in \mathcal{M}\).
Proof
Let \(u\in H^{1}(\mathbb{R}^{3})\setminus \{0\}\) be fixed and define a function \(\zeta (t):=\varPhi (t^{\beta }u_{t})\) on \((0,\infty )\). Clearly, by (1.8) and (2.5), we have
By (V1), (F1) and (F5), we have \(\lim_{t\rightarrow 0^{+}}\zeta '(t)=0\), \(\zeta '(t)>0\) for \(t>0\) small and \(\zeta '(t)<0 \) for t large. Therefore \(\max_{t\in [0,+\infty )}\zeta (t)\) is obtained at \(t_{u}>0\) so that \(\zeta '(t_{u})=0\) and \(t_{u}^{\beta }u_{t_{u}}\in \mathcal{M}\).
Next we claim that \(t_{u}\) is unique for any \(u\in H^{1}(\mathbb{R} ^{3})\setminus \{0\}\). In fact, for any given \(u\in H^{1}(\mathbb{R} ^{3})\setminus \{0\}\), let \(t_{1},t_{2}>0 \) such that \(t_{1}^{\beta }u _{t_{1}},t_{2}^{\beta }u_{t_{2}}\in \mathcal{M}\). Then \(J(t_{1}^{ \beta }u_{t_{1}})=J(t_{2}^{\beta }u_{t_{2}})=0\). Jointly with (2.9), we have
Combining (2.18) and (2.19), we have \(t_{1}=t_{2}\). Therefore, \(t_{u}>0\) is unique for any \(u\in H^{1}(\mathbb{R}^{3})\setminus \{0 \}\). □
Corollary 2.7
Assume that (F1), (F5) and (F6) hold. Then, for any \(u\in H^{1}(\mathbb{R}^{3})\setminus \{0\}\), there exists a unique \(t_{u}>0\) such that \(t_{u}^{\beta }u_{t_{u}}\in \mathcal{M}^{\infty }\).
Combining Corollary 2.4 with Lemma 2.6, we have the following lemma.
Lemma 2.8
Assume that (V1), (V2), (F1), (F2), (F5) and (F6) hold. Then
Lemma 2.9
Assume that (F1) and (F2) hold. If \(u_{n}\rightharpoonup \bar{u} \) in \(H^{1}(\mathbb{R}^{3})\), then along a subsequence of \(\{u_{n}\}\)
From [33, Proposition 4.3], we can obtain the following Lemma.
Lemma 2.10
Let \(\alpha \in (0,3)\) and \(q \in [1,3+\alpha )\) hold. If \(u_{n} \rightharpoonup \bar{u} \) in \(H^{1}(\mathbb{R}^{3})\), then along a subsequence of \(\{u_{n}\}\)
Lemma 2.11
Assume that (V1), (V2), (F1), (F5) and (F6) hold. Then
-
(i)
there exists \(\rho > 0\) such that \(\|u\| \geq \rho \), \(\forall u\in \mathcal{M}\);
-
(ii)
\(m=\inf_{\mathcal{M}} \varPhi >0\).
Proof
(i) Since \(J(u)=0\), \(\forall u\in \mathcal{M}\), by (F1), (F2), (1.8), (2.16) and due to the Sobolev embedding theorem, one has
which implies
(ii) Let \(\{u_{n}\} \subset \mathcal{M}\) be such that \(\varPhi (u_{n}) \rightarrow m\). There are two possible cases:
-
(1)
\(\inf_{n\in \mathbb{N}}\|u_{n}\|_{2}>0\) and
-
(2)
\(\inf_{n\in \mathbb{N}}\|u_{n}\| _{2}=0\).
Case (1) \(\inf_{n\in \mathbb{N}}\|u_{n}\|_{2}:=\varrho _{1}>0\). In this case, by (2.9) and (2.15), one has
Case (2) \(\inf_{n\in \mathbb{N}}\|u_{n}\|_{2}:=0\), by (2.23), passing to a subsequence, we have
Note that (F1) implies that, for any \(\varepsilon >0\), there exists \(C_{\varepsilon }>0\) such that
By (2.26) and the Sobolev embedding inequality, we have
Let \(t_{n}=\|\nabla u_{n}\|_{2}^{-\frac{2}{2\beta -1}}\), then (2.25) implies that \(\{t_{n}\}\) is bounded. Since \(J(u_{n})=0\), it follows from (2.6), (2.7), (2.25) and (2.27) that
Case (1) and Case (2) show that \(m=\inf_{\mathcal{M}}\varPhi >0\). □
Lemma 2.12
Assume that (V1), (F1) and (F2) hold. If \(u_{n} \rightharpoonup \bar{u}\) in \(H^{1}(\mathbb{R}^{3})\), then along a subsequence
Lemma 2.13
Assume that (V1), (V2), (F1), (F2), (F5) and (F6) hold. Then \(m^{\infty }:=\inf_{\mathcal{M}^{\infty }}\varPhi ^{\infty } \geq m\).
Proof
In view of Lemma 2.2 and Corollary 2.4, we have \(\mathcal{M}^{\infty }\neq \emptyset \). Arguing indirectly, we assume that \(m>m^{\infty }\). Let \(\varepsilon :=m-m^{\infty }\), then there exists \(u_{\varepsilon }^{\infty }\) such that
In view of Corollary 2.7, there exists \(t_{\varepsilon }>0\) such that \(t_{\varepsilon }^{\beta }(u_{\varepsilon }^{\infty })_{t_{\varepsilon }}\in \mathcal{M}\). Thus, it follows from (V1), (1.4), (2.10), (2.13) and (2.30) that
This contradiction shows that \(m^{\infty }>m\). □
Lemma 2.14
Assume that (V1), (V2), (F1), (F2), (F5) and (F6) hold. Then m is obtained.
Proof
In view of Lemma 2.6 and Lemma 2.11, we have \(\mathcal{M}\neq \emptyset \) and \(m>0\). Let \(\{u_{n}\}\subset \mathcal{M}\) be such that \(\varPhi (u_{n})\rightarrow m\). Since \(J(u_{n})=0\), if follows from (2.9) and (2.15) that
This shows that \({\|u_{n}\|_{2}}\) is bounded. Next, we prove that \({\|\nabla u_{n}\|_{2}}\) is also bounded. Arguing by contradiction, suppose that \(\|\nabla u_{n}\|_{2}\rightarrow \infty \). By (1.3) and the Sobolev embedding inequality, for \(u\in H^{1}(\mathbb{R}^{3})\) one has
Let \(t_{n}=(8m/\|\nabla u_{n}\|_{2}^{2})^{\frac{1}{2\beta -1}}\). Since \(J(u_{n})=0\), it follows from (1.4), (2.6) and (2.32) that
This contradiction shows that \(\{\|\nabla u_{n}\|_{2}^{2}\}\) is also bounded, and so \(\{u_{n}\}\) is bounded in \(H^{1}(\mathbb{R}^{3})\). Passing to a subsequence, we have \(u_{n}\rightharpoonup \bar{u}\) in \(H^{1}(\mathbb{R}^{3})\). Then \(u_{n}\rightarrow \bar{u}\) in \(L_{\mathrm{loc}}^{s}(\mathbb{R}^{3})\) for \(2 \leq s <6\) and \(u_{n}\rightarrow \bar{u}\) a.e. in \(\mathbb{R}^{3}\). There are two possible cases: (i) \(\bar{u}=0\) and (ii) \(\bar{u}\neq 0\).
Case (i) \(\bar{u}=0\). i.e. \(u_{n}\rightharpoonup 0\) in \(H^{1}( \mathbb{R}^{3})\). Then \(u_{n}\rightarrow 0\) in \(L_{\mathrm{loc}}^{s}( \mathbb{R}^{3})\) for \(2 \leq s <2^{*}\) and \(u_{n}\rightarrow 0\) a.e. in \(\mathbb{R}^{3}\). Using (V1) and (V2), it is easy to show that
From (1.4), (1.8), (2.10), (2.12) and (2.35), one can get
Note that (F1) and (F2) imply that, for any \(\varepsilon >0\), there exists \(C_{\varepsilon }>0\) such that
By (1.3), (1.8), (2.10), (2.37) and Lemma 2.11(i), one has
Using (2.38) and Lion’s concentration compactness principle [34, Lemma 1.21], we can prove that there exists \(\delta >0\) and \(y_{n}\in \mathbb{R}^{3}\), such that \(\int _{B_{1}(y_{n})}|u_{n}|^{2}\,dx> \delta \). Let \(\hat{u}_{n}(x)=u_{n}(x+y_{n})\). Then we have \(\|\hat{u}_{n}\|=\|u_{n}\|\) and
Therefore, there exists \(\hat{u} \in H^{1}(\mathbb{R}^{3})\setminus \{0\}\) such that, passing to a subsequence,
Let \(w_{n}=\hat{u}_{n}-\hat{u}\). then (2.40) and Lemma 2.12 yield
We define the functional \(\varPsi _{\infty }:H^{1}(\mathbb{R}^{3})\rightarrow \mathbb{R}\) for \(u\in H^{1}(\mathbb{R}^{3})\) by
From (2.5), (2.7), (2.39), (2.41) and (2.42), one has
If there exists a subsequence \(\{w_{n_{i}}\}\) of \(w_{n}\) such that \(w_{n_{i}}=0\), then we have
Next, we assume that \(w_{n}\neq 0 \), we claim that \(J^{\infty }(w_{n}) \leq 0\). Otherwise, if \(J^{\infty }(\hat{u})>0\), then (2.43) implies \(J^{\infty }(w_{n})< 0\) for large n. In view of Corollary 2.7, there exists \(t_{n}>0\) such that \(t_{n}^{\beta }(w_{n})_{t_{n}}\in \mathcal{M}^{\infty }\) for large n. From (2.5), (2.12), (2.13), (2.43) and Lemma 2.13, we obtain
which is a contradiction due to \(\varPsi ^{\infty }(\hat{u})> 0\). This shows that \(J^{\infty }(\hat{u})\leq 0\). In view of Lemma 2.6, there exists \(t_{\infty }>0\) such that \(t_{\infty }^{\beta }\hat{u}_{t_{\infty }} \in \mathcal{M}^{\infty }\). By (2.5), (2.7), (2.36), (2.39), (2.42), Fatou’s lemma and Lemma 2.13, we have
which implies (2.44) holds also. In view of Lemma 2.6, there exists \(\hat{t}>0\) such that \(\hat{t}^{\beta }\hat{u}_{\hat{t}} \in \mathcal{M}\), moreover, it follows from (V1), (1.3), (2.7), (2.44) and Corollary 2.4 that
This shows that m is obtained at \(\hat{t}^{\beta }\hat{u}_{\hat{t}} \in \mathcal{M}\).
Case (ii) \(\bar{u}\neq 0\). In this case, analogous to the proof of (2.44), by using Φ and J instead of \(\varPhi ^{\infty }\) and \(J^{\infty }\), we can deduce that \(\varPhi (\bar{u})=m\) and \(J(\bar{u})=0\). □
In the same way as [19] or [35], we can obtain the following lemma.
Lemma 2.15
Assume that (V1), (V2), (F1), (F2), (F5) and (F6) hold. If \(\bar{u}\in \mathcal{M}\) and \(\varPhi (\bar{u})=m\), then ū is a critical point of Φ.
Proof of Theorem 1.1
In view of Lemma 2.14 and 2.15, there exists \(\bar{u}\in \mathcal{M}\) such that
This shows that ū is a ground state solution of (1.1) such that \(\varPhi (\bar{u})=m=\inf_{\mathcal{M}}\varPhi \). □
3 Proofs of Theorem 1.2 and 1.3
Since we are looking for positive solutions to (1.1), without loss of generality, we suppose that \(f(t)=0\) for \(t<0\) in this section.
To use the Jeanjean’s monotonicity trick [18, Theorem 1.1], for \(\lambda \in [1/2,1]\) we introduce two families of \(\mathcal{C}^{1}\)-functions on \(H^{1}(\mathbb{R}^{3})\) defined by
and
In the same way as [13, 31], we can obtain the following lemma.
Lemma 3.1
Assume that (V1), (V3) (or(V3′)), (F1), (F2), (F5) and (F6) hold. Let u be a critical point of \(\varPhi _{\lambda }\) in \(H^{1}(\mathbb{R}^{3})\), then we have the following Pohožaev type identity:
We set \(J_{\lambda }(u):= \beta \langle \varPhi _{\lambda }^{\prime }(u),u \rangle -P_{\lambda }(u)\), then, for \(\lambda \in [1/2,1]\),
Correspondingly, for \(\lambda \in [1/2,1]\) we also let
Set
By Corollary 2.3, we have the following lemma.
Lemma 3.2
Assume that (F1), (F5) and (F6) hold. Then
Since \(f(t)=0\) for \(t<0\), from Theorem 1.1, the strong maximum principle and standard arguments, we can deduce that \(\varPhi _{1}^{\infty }\) has a minimizer \(u_{1}^{\infty }>0\) on \(\mathcal{M}_{1}^{\infty }\), i.e.
Lemma 3.3
Under the assumptions of Theorem 1.2 or Theorem 1.3, we have
-
(i)
There exists \(T>0\) independent of λ such that \(\varPhi _{\lambda }(T^{\beta }(u_{1}^{\infty })_{T})<0\) for all \(\lambda \in [1/2,1]\);
-
(ii)
there exists a positive constant \(\kappa _{0}\) independent of λ such that, for all \(\lambda \in [1/2,1]\),
$$ c_{\lambda }=\inf_{\gamma \in \tau }\max _{t\in [0,1]} \varPhi _{\lambda }\bigl( \gamma (t)\bigr) \geq \kappa _{0} > \max \bigl\{ \varPhi _{\lambda }(0),\varPhi _{\lambda }\bigl(T^{\beta }\bigl(u_{1}^{\infty } \bigr)_{T}\bigr)\bigr\} , $$(3.9)where \(\tau =\{\gamma \in \mathcal{C}([0,1],H^{1}(\mathbb{R}^{3})): \gamma (0)=0,\gamma (1)=T^{\beta }(u_{1}^{\infty })_{T}\}\);
-
(iii)
\(c_{\lambda }\) and \(m_{\lambda }^{\infty }\) are non-increasing on \(\lambda \in [1/2,1]\).
The proof of Lemma 3.3 is standard, so we omit it.
Lemma 3.4
Under the assumptions of Theorem 1.2 or Theorem 1.3, there exists \(\bar{\lambda }\in [1/2,1)\) such that \(c_{\lambda }< m_{\lambda }^{ \infty }\) for \(\lambda \in (\bar{\lambda },1]\).
Proof
It is easy to see that \(\varPhi _{\lambda }(t^{\beta }(u_{1}^{\infty })_{t})\) is continuous on \(t\in (0,\infty )\). Hence for any \(\lambda \in [1/2,1)\), we can choose \(t_{\lambda }\in (0,T)\) such that \(\varPhi _{\lambda }(t_{\lambda }^{ \beta }((u_{1}^{\infty })_{t_{\lambda }}))\). Let \(\beta _{0}= \inf_{\lambda \in [1/2,1]}t_{\lambda }\). If \(\beta _{0}=0\), then there exists a sequence \(\{\lambda _{n}\}\subset [1/2,1]\) such that \(\lambda _{n}\rightarrow \lambda _{0} \in [1/2,1]\) and \(t_{\lambda _{n}} \rightarrow 0\), and so by (3.1) and Lemma 3.3(iii), one has
This contradiction shows \(\beta _{0}>0\). Thus \(0<\beta _{0}\leq t_{ \lambda }<T\) for all \(\lambda \in [1/2,1]\). Let
Then \(1/2 \leq \bar{\lambda } <1\). From (3.1), (3.2), (3.7), (3.11) and Lemma 3.3(iii), we derive
□
Lemma 3.5
Under the assumptions of Theorem 1.2 or Theorem 1.3, for almost every \(\lambda \in (\bar{\lambda },1]\), there exists \(u_{\lambda }\in H^{1}( \mathbb{R}^{3})\setminus \{0\}\) such that.
Proof
In view of the Jeanjean’s monotonicity trick [36, Theorem 1.1] and Lemma 3.3, for almost every \(\lambda \in [1/2,1]\), there exists a bounded sequence \(\{u_{n}(\lambda )\}\subset H^{1}(\mathbb{R}^{3})\), for simplicity, we denote it by \(\{u_{n}\}\) instead of \(\{u_{n}(\lambda ) \}\) such that
Using Lemma 2.12, we can deduce that there exists \(u_{\lambda }\in H ^{1}(\mathbb{R}^{3})\), an integer \(l\in \mathbb{N} \cup \{0\}\), a sequence \(\{y_{n}^{k}\} \subset \mathbb{R}^{3}\) and \(w^{k}\in H^{1}( \mathbb{R}^{3})\) for \(1\leq k \leq l\) such that \(u_{n}\rightharpoonup u_{\lambda }\) in \(H^{1}(\mathbb{R}^{3})\), \(\varPhi _{\lambda }^{\prime }(u_{ \lambda })=0\), \((\varPhi _{\lambda }^{\infty })^{\prime }(w^{k})=0\) and \(\varPhi _{\lambda }^{\infty }(w^{k})\geq m_{\lambda }^{\infty }\) for \(1 \leq k \leq l\),
Since \(\varPhi _{\lambda }^{\prime }(u_{\lambda })=0\), \(J_{\lambda }(u_{\lambda })=0\). It follows from (3.1) and (3.4) that
If \(l\neq 0\), then
which is a contradiction by Lemma 3.4. Thus \(l=0\), and (3.15) implies that \(u_{n}\rightarrow u_{\lambda }\) in \(H^{1}(\mathbb{R}^{3})\) and \(\varPhi _{\lambda }(u_{\lambda })=c_{\lambda }\) for almost every \(\lambda \in (\bar{\lambda },1]\). □
Lemma 3.6
Under the assumptions of Theorem 1.2 or Theorem 1.3, there exists \(\bar{u}>0\) such that
Proof
Under the assumptions of Theorem 1.2 or 1.3, in view of Lemma 3.5, there exists two sequences of \(\{\lambda _{n}\} \subset (\bar{\lambda },1]\) and \(\{u_{\lambda _{n}}\} \subset H^{1}(\mathbb{R}^{3})\), denoted \(\{u_{n}\}\), such that
By (3.1), (3.4), (3.19) and Lemma 3.3(iii), one has
To prove the boundedness of \(\{\|u_{n}\|\}\), we distinguish two cases: (1) (V3) holds; (2) (V3′) holds.
Case (1) (V3) holds. In this case, by (F6) and (3.20), one has
which implies that \(\{\|u_{n}\|_{2}\}\) is bounded. Next, we prove that \(\{\|\nabla u_{n}\|_{2}\}\) is also bounded. Arguing by contradiction, suppose that \(\|\nabla u_{n}\|_{2} \rightarrow \infty \). By (V1), (V3), (3.21) and Lemma 3.3(iii), one has
for some constant \(M_{0}\). Let \(t_{n}=\min \{1,2(M_{0}/\|\nabla u_{n} \|_{2}^{2})^{1/{2\beta -1}}\}\), then \(t_{n}\rightarrow 0\). Thus, it follows from (3.1), (3.2), (3.4), (3.5) and (3.22) that
Analogous to the proof of (2.34), we can deduce a contradiction by using (3.23). Hence, \(\{u_{n}\}\) is bounded in \(H^{1}(\mathbb{R} ^{3})\) under the assumptions of Theorem 1.2.
Case (2) (V3′) holds. In this case, (V3′) and (3.20) imply
Then it follows from (V1), (3.1), and (3.24) that
where \(\gamma _{0}\) is a positive constant. Hence, \(\{u_{n}\}\) is bounded in \(H^{1}(\mathbb{R}^{3})\) under the assumptions of Theorem 1.3. Similar to the proof of Lemma 3.5, there exists \(\bar{u}\in H^{1}(\mathbb{R} ^{3})\setminus \{0\}\) such that (3.18) holds. Moreover, by the strong maximum principle and a standard argument, we can conclude that \(\bar{u}>0\). □
Proofs of Theorem 1.2 and Theorem 1.3 . Let
Then Lemma 3.6 shows that \(K \neq \emptyset \) and \(\hat{m} \leq c_{1}\). For any \(u\in K\), (1.8), (3.4) and Lemma 3.1 imply \(J(u)=J_{1}(u)=\beta \langle \varPhi '(u),u\rangle -\mathcal{P}(u)=0\). As in (3.16), we have \(\varPhi (u)=\varPhi _{1}(u)\geq 0 \) for any \(u\in K\), and so \(\hat{m}\geq 0\). Let \(\{u_{n}\} \subset K\) such that \(\varPhi '(u_{n})=0\) and \(\varPhi (u_{n})\rightarrow \hat{m}\). In view of Lemma 3.4, \(\hat{m}\leq c_{1}\leq m_{1}^{\infty }\). Similar to the proof of Lemma 3.6, we can deduce that there exists \(\hat{u}>0\) such that \(\varPhi '(\hat{u})=0\) and \(\varPhi (\hat{u})=\hat{m}\). This shows that \(\hat{u}\in H^{1}(\mathbb{R}^{3})\) is a positive ground state solution of (1.1).
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This work was supported by the NNSF (11701487, 11626202), Hunan Provincial Natural Science Foundation of China (2016JJ6137), Scientific Research Fund of Hunan Provincial Education Department (15B223).
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Yuan, S., Liao, F. Ground state solutions of Nehari–Pohožaev type for a kind of nonlinear problem with general nonlinearity and nonlocal convolution term. Bound Value Probl 2019, 150 (2019). https://doi.org/10.1186/s13661-019-1264-3
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DOI: https://doi.org/10.1186/s13661-019-1264-3