Ground state solutions of Nehari-Pohozaev type for Schrödinger-Poisson problems with general potentials

This paper is dedicated to studying the following Schrodinger-Poisson problem \begin{document} $\left\{ \begin{array}{ll} -\triangle u+V(x)u+φ u=f(u), \ \ \ \ x∈ \mathbb{R}^{3},\\ -\triangle φ=u^2, \ \ \ \ x∈ \mathbb{R}^{3}, \end{array} \right. $ \end{document} where \begin{document}$V(x)$\end{document} is weakly differentiable and \begin{document}$f∈ \mathcal{C}(\mathbb{R}, \mathbb{R})$\end{document} . By introducing some new tricks, we prove the above problem admits a ground state solution of Nehari-Pohozaev type and a least energy solution under mild assumptions on \begin{document}$V$\end{document} and \begin{document}$f$\end{document} . Our results generalize and improve the ones in [D. Ruiz, J. Funct. Anal. 237 (2006) 655-674], [J.J. Sun, S.W. Ma, J. Differential Equations 260 (2016) 2119-2149] and some related literature.

Problem (1.1), also known as the nonlinear Schrödinger-Maxwell problem, was first introduced in [4] as a model describing solitary waves for the nonlinear stationary Schrödinger equations interacting with the electrostatic field. It has a strong physical meaning because it appears in quantum mechanics models (see e.g. [6,7,22]) (1.2) Moreover, under our assumptions it is standard to see that (1.2) is variational and its solutions are the critical points of the functional defined in H 1 (R 3 ) by Hence if u ∈ H 1 (R 3 ) is a critical point of I, then the pair (u, φ u ) is a solution of (1.1) (see [12] for more details). For the sake of simplicity, in many cases we just say u ∈ H 1 (R 3 ), instead of (u, φ u ) ∈ H 1 (R 3 ) × D 1,2 (R 3 ), is a weak solution of (1.1).
In recent years, there has been increasing attention to problems like (1.1) on the existence of positive solutions, ground state solutions, multiple solutions and semiclassical states, see e.g. [2, 3, 8-10, 15-19, 27, 32, 34, 35]. The greatest part of the literature focuses on the study of Problem (1.1) with V (x) ≡ 1 or V (x) =V (|x|), and f satisfying the following assumptions of Ambrosetti-Rabinowitz type and 4superlinear In fact, under (AR) and (SF), it is easy to verify the Mountain Pass geometry and the boundedness of (PS) or (C) c sequences for I.
For the following special form of (1.1) its functional defined in H 1 (R 3 ) by (1.5) where F (u) = |u| p /p associated with (1.4), there are more results on the existence of solutions. For example, in [11,12] a radial positive solution of (1.4) is found for 4 < p < 6. To do that they use the mountain pass theorem. It is easy to show that Φ attains a local minimum at zero. Moreover, in [12] it is pointed out that Φ is unbounded below even for p > 3. Furthermore, in [13] a related Pohozaev equality is found. With this equality in hand, the authors can prove that there does not exist nontrivial solutions of (1.4) for p ≤ 2 or p ≥ 6.
In the case 2 < p ≤ 4, it is very difficult to verify the Mountain Pass geometry and the boundedness of (PS) or (C) c sequences for Φ. By introducing Nehari-Pohozaev manifold and establishing a key inequality, Ruiz [27] proved that Problem (1.4) admits a positive radial solution if 3 < p ≤ 4, but does not have a nontrivial solution for 2 < p ≤ 3. Obviously, this result fills the gap p ∈ (2,4] which is left in the previous results. Ruiz ' approach in [27] consists of minimizing Φ on the Nehari-Pohozaev manifold M, which is defined by a condition which is a combination of the Nehari equation and the Pohozaev equality, since the usual method of Nehari manifold becomes invalid in this case. To the best of our knowledge, this approach is entirely new in the literature. In fact, the positive radial solution obtained in [27] is not a minimizer of Φ on M, but on M ⊂ M which consists of radially symmetric functions in M. Based on Ruiz' approach in [27] and a concentration-compactness argument on suitable measures, Azzollini and Pomponio [3] obtained a minimizer of Φ on M for (1.4) (it will be called a ground solution of Nehari-Pohozaev type) under the same assumption 3 < p ≤ 4 as in [27].
The approaches used in [3,27] are successful for (1.4), however, they are no longer applicable for the following problem with more general nonlinearity (1.6) Even for the case when f (u) = |u| p1−2 u + |u| p2−2 u with 3 < p 1 < p 2 ≤ 4, it is difficult to obtain a ground solution of (1.6) by using the approaches in [3,27].
Applying an alternative method, Sun and Ma [31] proved that (1.6) admits a least energy solution if f satisfies (F1), (F2) and the following assumption of Ambrosetti-Rabinowitz type To do that, Sun and Ma [31] used Jeanjeans monotonicity trick [20] to construct a special (PS) sequence, and by using a Pohozaev identity and a global compactness lemma, they proved the boundedness of the special (PS) sequence and hence, got a nontrivial critical point.
Since the radial solutions are critical points of Φ in H 1 r (R 3 ) which is the set of radially symmetric functions in H 1 (R 3 ), and the embedding H 1 r (R 3 ) → L s (R 3 ) is compact for 2 < q < 6. In this case, it is easy to verify the following condition . Hence, the global compactness lemma used in [31] is not necessary in [29].
We point out that the global compactness lemma is very crucial to prove the above result in [31]. The first global compactness lemma was proved by Struwe [30] in 1984. In the proof, it is essential the following equality reasoning from condition u n u 0 in H 1 (R N ). In order to show (1.7), it is necessary to show It is standard to show (1.8) under (F2) and the following assumption (F1 ) f ∈ C 1 (R, R) and there exist constants C 0 > 0 and p ∈ (2, 2 * ) such that which is assumed in [30]. However, it is difficult to show (1.8) if one uses (F1) with p ∈ (2, 2 * ) instead of (F1 ). It is worth pointing out that the crucial equality (1.7) is not presented in the proof of the global compactness lemma of [31], see [31,Lemma 5.6].
In [34], the authors proved that the following nonlinear Schrödinger-Poission problem admits a least energy solution if p ∈ (3, 6) and V satisfies (V1) and the following assumptions To prove the above result, the authors in [34] first applied Ruiz' result to the "limit problem" associated with (1.9), and got a minimizer u ∞ λ on the Nehari-Pohozaev manifold M ∞ λ for λ ∈ [1/2, 1], see Section 4. Then they used Jeanjeans monotonicity trick [20] to show that there exists a bounded (PS) sequence {u n (λ)} at the level c λ for almost every λ ∈ [1/2, 1). Then by comparing c λ with the energy m ∞ λ of the minimizer u ∞ λ and using a global compactness lemma, they can get a nontrivial critical point u λ which possesses energy c λ . Finally, with a Pohozaev identity, they proved that (1.9) admits a least energy solution by a standard argument. It is worth pointing out that it seems to be not sufficient the proof of the inequality c λ < m ∞ λ for λ ∈ [1/2, 1) in [34], see Remark 4.6 in present paper. Now, a natural question is whether Ruiz' result (or the one in [3]) on the existence of ground solution for (1.4) with 3 < p ≤ 4 can be generalized to (1.1)?
To answer the above question, we must overcome three main difficulties: 1) verifying the boundedness of the minimizing sequence of I on the Nehari-Pohozaev manifold M; 2) showing inf M I can be achieved (due to the lack of compactness of the Sobolev spaces embeddings in the unbounded domain R 3 ); 3) proving the minimizer of I on M is a critical point (because it is not assumed that f is differentiable, M may not be a C 1 -manifold in H 1 (R 3 )).
Motivated by the above works, in the present paper, we will introduce some new tricks to obtain a ground solution of Nehari-Pohozaev type for (1.1) (i.e. a minimizer of I on the Nehari-Pohozaev manifold M). This answers the above question affirmatively. In addition, we also prove the existence of the least energy solutions of (1.1) and (1.6), which make a substantial improvement to the main results in [31,34]. In particular, we give a proof of (1.8) under (F1) and (F2), see Lemma 2.7.
To state our results, we make the following assumptions on the potential V and the nonlinearity f . (V5) V (x) is weakly differentiable, and satisfies (∇V (x), x) ∈ L ∞ (R 3 ), and for some 0 > 0 and there exist κ > 3/2 and C 1 > 0 such that where γ 0 is Sobolev imbedding constant such that γ 0 u 2 2 ≤ u 2 for u ∈ H 1 (R 3 ).
then it follows from (V2) and (V3) that V (x) ≥ 0 for all x ∈ R 3 , and is a norm which is equivalent to the usual norm of H 1 (R 3 ). Hence, the second part of (V1) holds. To state our results, we define two functionals on H 1 (R 3 ) as follows: Now, we state our results of this paper. Remark 1.9. Theorems 1.4 and 1.5 answer the question mentioned above. Theorems 1.7 and 1.8 make a substantial improvement to the main results in [31,34].
The paper is organized as follows. In Section 2, we give some notation and preliminaries, and give the proof of Theorem 1.4. In Section 3, we complete the proof of Theorems 1.5. Section 4 is devoted to finding a least energy solution for (1.1) and (1.6). Theorems 1.6-1.8 will be proved in this section.
Throughout this paper, we let u t (x) := u(tx) for t > 0, and denote the norm of : |y − x| < r}, and positive constants possibly different in different places, by C 1 , C 2 , · · · .
2. Ground state solutions for (1.6). Hereafter, H 1 (R 3 ) is the usual Sobolev space with the standard scalar product and norm equipped with the norm defined by It is easy to show that (1.1) can be reduced to a single equation (1.2) with a non-local term. Namely, for any u 2 ∈ L 1 loc (R 3 ) such that belongs to D 1,2 (R 3 ) and is the unique weak solution in D 1,2 (R 3 ) (see e.g. [28] for more details), and Moreover, φ u (x) > 0 when u = 0. By Hardy-Littlewood-Sobolev inequality (see [23] or [24, page 98]), we have the following inequality: Formally, the solutions of (1.1) are then the critical points of the reduced functional (1.5). Indeed, (V1), (F1), (F2) and (2.4) imply that I and Φ are two well-defined classes of C 1 functional, and that Proof. It is evident that (2.7) holds for τ = 0. For τ = 0, let Then from (F4), one has It follows that g(t) ≥ g(1) = 0 for t > 0, which, together with (2.8), implies (2.7) holds.
From Lemma 2.2, we have the following corollary immediately.
Lemma 2.7. Assume that (F1) and (F2) hold. If u n ū in H 1 (R 3 ), then along a subsequence of {u n }, Proof. This lemma has been proved in [1,Appendix] and [14,. Here, we give a simpler and more direct proof.
To this end, we define two new functionals as follows:
From Lemma 3.2, we have the following two corollaries.
Analogous to the proof of Lemma 2.6, we can prove the following lemma by using Lemmas 3.2 and 3.8 instead of Lemma 2.2.  This shows that { u n 2 } is bounded. Next, we prove that { ∇u n 2 } is also bounded. Arguing by contradiction, suppose that ∇u n 2 → ∞. Let t n = ( √ 6m / ∇u n 2 ) 2/3 . Hence, using (V2), (F1), (F2), (3.4), (3.6), (3.14) and J(u n ) = 0, one has This contradiction shows that { ∇u n 2 } is also bounded, and so {u n } is bounded in H 1 (R 3 ). Passing to a subsequence, we have u n ū in H 1 (R 3 ). Then u n →ū in L s loc (R 3 ) for 2 ≤ s < 6 and u n →ū a.e. in R 3 . Next, we proveū = 0. Arguing by contradiction, suppose thatū = 0, i.e. u n 0 in H 1 (R 3 ). Then u n → 0 in L s loc (R 3 ) for 2 ≤ s < 6 and u n → 0 a.e. in   Let w n =û n −v. Then (3.19) and Lemma 2.8 yield and The rest of the proof is the same as the one of Lemma 2.9, so we omit it.
In the same way as the proof of Lemma 2.10, we can prove the following lemma by using Lemmas 3.2 and 3.5 instead of Lemmas 2.2 and 2.4, respectively.  [20] Let X be a Banach space and let J ⊂ R + be an interval. We consider a family {Φ λ } λ∈J of C 1 −functional on X of the form where B(u) ≥ 0, ∀u ∈ X, and such that either A(u) → +∞ or B(u) → +∞, as u → ∞. We assume that there are two points v 1 , v 2 in X such that  Then, for almost every λ ∈ J , there is a bounded (PS) c λ sequence for Φ λ , that is, there exists a sequence such that (i). {u n (λ)} is bounded in X; (ii). Φ λ (u n (λ)) → c λ ; (iii). Φ λ (u n (λ)) → 0 in X * , where X * is the dual of X.