Berestycki-Lions conditions on ground state solutions for a Nonlinear Schr\"odinger equation with variable potentials

This paper is dedicated to studying the nonlinear Schr\"odinger equations of the form \begin{equation*}\label{KE} \left\{ \begin{array}{ll} -\triangle u+V(x)u=f(u),&x\in \R^N; u\in H^1(\R^N), \end{array} \right. \end{equation*} where $V\in \mathcal{C}^1(\R^N, [0, \infty))$ satisfies some weak assumptions, and $f\in \mathcal{C}(\R, \R)$ satisfies the general Berestycki-Lions assumptions. By introducing some new tricks, we prove that the above problem admits a ground state solution of Poho\u{z}aev type and a least energy solution. These results generalize and improve some ones in [L. Jeanjean, K. Tanka,Indiana Univ. Math. J. 54 (2005), 443-464], [L. Jeanjean, K. Tanka, Proc. Amer. Math. Soc. 131 (2003) 2399-2408], [H. Berestycki, P.L. Lions, Arch. Rational Mech. Anal. 82 (1983) 313-345] and some other related literature. In particular, our assumptions are"almost"necessary when $V(x)\equiv V_{\infty}>0$, moreover, our approach could be useful for the study of other problems where radial symmetry of bounded sequence either fails or is not readily available, or where the ground state solutions of the problem at infinity are not sign definite.


Introduction
In this paper, we consider the nonlinear Schrödinger equations of the form: (1.1) where N ≥ , V : R N → R and f : R → R satisfy the following basic assumptions: f ∈ C(R, R) and there exists a constant C > such that (F2) f (t) = o(t) as t → and |f (t)| = o |t| (N+ )/(N− ) as |t| → +∞.
If the potential V(x) ≡ V∞, then (1.1) reduces to the following autonomous form: its energy functional is as follows: It is well known that every solution u(x) of (1.3) satis es the following Pohozaev type identity [9]: (1.6) Berestycki-Lions [1] proved that (1.3) has a radially symmetric positive solution provided f satis es (F1), (F2) and the following two assumptions: (F0) f is odd; (F3) there exists s > such that F(s ) > V∞s .
To prove the above result, Berestycki-Lions [1] considered the following constrained minimization problem min ∇u : u ∈ S , (1 (1.8) they rst showed that by the Pólya-Szegö inequality for the Schwarz symmetrization, the minimum can be taken on radial and radially nonincreasing functions. Then they showed the existence of a minimizer w ∈ H (R N ) by the direct method of the calculus of variations. With the Lagrange multiplier Theorem, they concluded thatū(x) :=ŵ(x/tŵ) with tŵ = N− N ∇ŵ is a least energy solution of (1.3). By noting the oneto-one correspondence between S and M ∞ , Jeanjean-Tanaka [6] proved thatū is also a ground state solution of Pohozaev type for (1.3), i.e.ū ∈ M ∞ and satis es (1.9) By using a di erent way, Shatah [12] showed that there existsũ ∈ M ∞ r such that Obviously, (F1)-(F3) are satis ed by a very wide class of nonlinearities. In particular only conditions on f (t) near , ∞ and the point s are required. Moreover, in view of [1, 2.2], (F1) is "almost" necessary, and (F2) and (F3) are necessary for the existence of a nontrivial solution of problem (1.3). For more related results under Berestycki-Lions conditions, we refer to [3,4,11,18,19].
When V(x) ̸ ≡ V∞, the approach used in [1] does not work any more for nonautonomous equation (1.1), since the Schwarz symmetrization can only be applied to autonomous problems. In a di erent way, Rabinowitz [10] proved that (1.1) has a nontrivial solution if V satis es (V1) and (V2) and f does (F1), (F2), the Nehari monotonic condition: (Ne) f (t)/|t| is strictly increasing on (−∞, ) ∪ ( , ∞); and the global growth Ambrosetti-Rabinowitz condition: (Ne) and (AR) are used to recover the compactness and to get the boundedness of Palais-Smale sequences, respectively, see [17] for more details. By means of Jeanjean's monotonicity trick, developed in [5], which is a generalization of the Struwe's one (see [13]), consisting in a suitable approximating method, Jeanjean and Tanaka [7] derived an existence result using two weaker conditions instead of (Ne) and (AR). More precisely, Jeanjean and Tanaka proved that (1.1) has a least energy solution if f satis es (F1), (F2) and the following nonnegativity condition: and the superlinear growth condition: and V does (V1), (V2) and the decay condition: Clearly, (NG) and (SL) are stronger than (F3), moreover, (Vd) puts relatively strict constrains on the decay of |∇V(x)|. For example, V(x) = a − b +|x| α does not satisfy (Vd) for a, b > and < α ≤ N. Motivated by the work of [1,2,7,12,16], we shall develop a more direct approach (the least energy squeeze approach) to show that (1.1) has a solutionū ∈ M such that I(ū) = inf M I under (F1)-(F3), (V1), (V2) and an additional decay condition on V: and is the Pohozaev functional associated with (1.1) (see [7]). To prove the above conclusion, we shall divide our arguments into three steps: i Step ii) is the most di cult due to lack of global compactness and adequate information on I (un). Since (1.1) is nonautonomous, the radial compactness does not work for M. To overcome this di culty, we establish a crucial inequality related to I(u), I(u t ) and P(u) (the IIP inequality in short, see Lemma 2.2), where u t (x) = u(x/t), it plays an important role in many places of this paper. With the help of the IIP inequality, we then can complete Step ii) by using Lions' concentration compactness, the least energy squeeze approach and some subtle analysis. In particular, we only use Lions' concentration compactness in our arguments, the radial and other compactness are not required, see the proofs of Lemmas 2.12 and 2.13. Moreover, such an approach could be useful for the study of other problems where radial symmetry of bounded sequence either fails or is not readily available. In Step iii), usually, one uses the Lagrange multipliers Theorem to show that the minimizerū is a critical point of I, but it is impossible to verify P (u) ≠ for all u ∈ M under (V1)-(V3) and (F1)-(F3). To overcome this di culty, we employ the combination of the IIP inequality and a deformation lemma, see Lemma 2.14. We believe that our approach could be applied to deal with many similar problems, such as quasilinear Schrödinger equations, Choquard equations and fractional Schrödinger equations.
As a consequence of Theorem 1.2, we can prove the following theorem.
Remark 1.4. We point out that, as a consequence of Theorem 1.2, the least energy value m := inf M I has a minimax characterization m = inf u∈Λ max t> I(u t ) which is much simpler than the usual characterizations related to the Mountain Pass level.
In the second part of the paper, we are interested in the existence of the least energy solutions for (1.1) under (F1)-(F3). In this case, we can replace (V3) by the following weaker decay assumption on ∇V: As in Jeanjean and Tanaka [6], for λ ∈ [ / , ] we consider a family of functionals I λ : H (R N ) → R de ned by (1.14) These functionals have a Mountain Pass geometry, and denoting the corresponding Mountain Pass levels by c λ . Corresponding to (1.14), we also let and is not sign de nite, it prevents us from employing Jeanjean's monotonicity trick [5]. Thanks to the work of Jeanjean-Toland [8], Di erent from the arguments in the existing literature, by means of u ∞ and the IIP inequality, we can ndλ ∈ [ / , ) and then directly prove the following crucial inequality Remark 1.6. Relative to (Vd), there seem to be more functions satisfying (V4). For example, it is easy to verify that However, it does not satisfy (Vd) when ≤ α ≤ N.
Applying Theorem 1.5 to the following perturbed problem: where V∞ is a positive constant and the function h ∈ C (R N , R) veri es: Then we have the following corollary.

Corollary 1.7. Assume that h and f satisfy (H1), (H2) and (F1)-(F3). Then there exists a constant ε > such that problem (1.19) has a least energy solutionūε
Classically, in order to show the existence of solutions for (1.1), one compares the critical level of I with the one of I ∞ (i.e. the energy functional corresponds to the problem at in nity). To this end, it is necessary to establish a strict inequality similar to ). Clearly, γ (t) > is a natural requirement under (V1) and (V2). But we only need γ (t) ≠ in our arguments. Therefore, our approach could be useful for the study of other problems where paths or the ground state solutions of the problem at in nity are not sign de nite.
Throughout the paper we make use of the following notations: ♠ H (R N ) denotes the usual Sobolev space equipped with the inner product and norm ♠ C , C , · · · denote positive constants possibly di erent in di erent places.
The rest of the paper is organized as follows. In Section 2, we give some preliminaries, and give the proof of Theorem 1.2. Section 3 is devoted to nding a least energy solution for (1.1) and Theorem 1.5 will be proved in this section.

Ground state solutions for (1.1)
In this section, we give the proofs of Theorem 1.2. To this end, we give some useful lemmas. Since V(x) ≡ V∞ satis es (V1)-(V3), thus all conclusions on I are also true for I ∞ . For (1.3), we always assume that V∞ > . By a simple calculation, we can verify Lemma 2.1.
Lemma 2.1. The following inequality holds: Moreover (V3) implies the following inequality holds: Proof. According to Hardy inequality, we have Note that This shows that (2.3) holds.
From Lemma 2.2, we have the following two corollaries.
Next we claim that tu is unique for any u ∈ Λ. In fact, for any given u ∈ Λ, let t , t > such that u t , u t ∈ M. Then P (ut ) = P (ut ) = . Jointly with (2. and  The following lemma is a known result which can be proved by a standard argument (see [14,15]).
which implies ii). For u ∈ H (R N ), by the Sobolev inequality, one has S u * ≤ ∇u . By (V2), there exists R > such that V(x) ≥ V∞ for |x| ≥ R. It follows from (F1) and (F2) that there exists C > such that This shows that m = inf u∈M I(u) > .
This contradiction shows the conclusion of Lemma 2.12 is true.
which implies (2.40) also holds. Arguing by contradiction, suppose that there exist ε > and a sequence {tn} such that From (2.43) and (2.44), one has and In view of Lemma 2.2, one has By Corollary 2.4, I (ūt) ≤ I(ū) = m for t > , then it follows from (2.48) and ii) that On the other hand, by iii) and (2.49), one has This shows thatū is a ground state solution of Pohozaev type for (1.1).

The least energy solutions for (1.1)
In this section, we give the proof of Theorem 1.5. Proposition 3.1. [8] Let X be a Banach space and let J ⊂ R + be an interval, and be a family of C -functional on X such that B maps every bounded set of X into a set of R bounded below; iii) there are two points v , v in X such that Then, for almost every λ ∈ J, there exists a sequence {un(λ)} such that lim sup λ→λ c λ ≤ c λ for λ ∈ ( . , ]. Since Proof. It is easy to see that I λ (u ∞ ) t is continuous on t ∈ ( , ∞). Hence for any λ ∈ [ / , ], we can choose Then γ ∈ Γ de ned by Lemma 3.4 (ii). Moreover I λ γ (t) ≥ c λ . Since P ∞ (u ∞ ) = , then R N F(u ∞ )dx > . Let Then it follows from (2.1), (3.5) and (3.8) that / ≤λ < . We have two cases to distinguish: Case ii). t λ ∈ ( , − ζ ) ∪ ( + ζ , T). From (V2), (1.14), (1.15), (2.1), (3.3), (3.4), (3.6), (3.9) and Lemma 3.4 (iv), we have In both cases, we obtain that c λ < m ∞ λ for λ ∈ (λ, ]. where we agree that in the case l = the above holds without w k . Since I λ (u λ ) = , then it follows from Lemma 3.2 that Since un , we deduce from (3.13) and (3.14) that if u λ = then l ≥ and which contradicts with Lemma 3.5. Thus u λ ≠ . It follows from (1.14), (2.4), (3.15) and (V4) that This shows that { ∇un } is bounded. Next, we demonstrate that {un} is bounded in H (R N ). According to (V1) and (V2), it is easy to show that there exists a constant γ > such that This shows thatū is a nontrivial least energy solution of (1.1).