Ground state solutions to a class of critical Schrödinger problem

and study the existence of semiclassical ground state solutions of Nehari-Pohoz̆aev type to (SKε), where f (u) may behave like |u|q−2u for q ∈ (2, 4] which is seldom studied. With some decay assumption on V, we establish an existence result which improves some exiting works which only handle q ∈ (4, 6). With some monotonicity condition on V, we also get a ground state solution v̄ε and analysis its concentrating behaviour around global minimum xε of V as ε → 0. Our results extend some related works.

Problem (SKε) is related to the stationary analogue of the following equation which is proposed by Kirchho [20] as an extension of classical D'Alembert's wave equation. It has been applied widely to model various physics problems and appears in some biological systems. For more details and backgrounds, we refer the reader to [2][3][4] and references therein. Owing to the presence of the term ( R |∇u| dx)∆u, problem (SKε) is no longer a pointwise identity, which makes the study of this question very complicated. However, it is worth mentioning that the pioneering work of Lions [24] introduced an abstract framework, and since then Kirchho type problem has received more and more attention from the mathematical community by using variational methods.
Making the scaling u(x) = v(εx), (SKε) is transformed to u resolves (PKε) i v resolves (SKε). It follows from (V1) and (F1) that (PKε) has a variational structure, which means the weak solutions of (PKε) are the critical points of the C functional Iε : H R → R de ned by For simplicity, inspired by [6,39], we set Most recently, many authors are concerned with semiclassical problems like (SKε), i.e. the parameter ε goes to zero. For ε > small, the solutions are called semiclassical states, which possess an important physical interest in describing the translation from quantum to classical mechanics. There are some valuable results on semiclassical solutions for Kirchho -type problems like (SKε), we refer to [13,[15][16][17]37]. Set ε = , V(x) ≡ , and replace R with a bounded domain Ω, then problem (SKε) is related to the following problem By minimax methods and invariant sets of descent ow, Mao and Zhang [28], Perera and Zhang [30] proved the existence of sign-changing solutions. At the same time, [1] obtained the existence of positive solution using variational methods when f is critical growth. When ε > , the existence and qualitative properties of solutions to have been extensively studied; see for example [6,9,10,18,26,34,37]. If f (t) is super-linear at t = and super-cubic at t = ∞, He and Zou [17] rstly studied ( SKε) via the Mountain Pass Theorem and the Nehari manifold approach, under the condition that f ∈ C (R + , R + ) satis es the Ambrosetti-Rabinowitz condition ((AR) in short) ∃ µ > , < µF(t) ≤ f (t)t, t ≠ and the monotonicity condition((MN) in short) f (t)/t is strictly increasing for t ∈ (−∞, ) ∪ ( , +∞).
If f (t) is not super-cubic at t = ∞, following the idea of Ruiz [31], Li and Ye [22] proved the special case that V = and f (u) = |u| s− u for < s < has a positive solution by using a new manifold related to Nehari equation and Pohozaev equality. Then, Guo [14] and Tang and Chen [36] improve the above results with more general V and f , which handles the case where f (u) behaves like |u| s− u for < s ≤ .
For the critical case, Wang, Tian, Xu and Zhang [37] considered the problem under the assumption that g ∈ C (R + , R + ) is subcritical growth, g(t) = o(t ) as t → and g(t)/t is strictly increasing on ( , ∞). Inspired by [17], they obtained the existence, multiplicity and concentration of solutions when ε > small enough and λ > is large enough, in addition, they extended the results of [17] to the critical case. In [18], He and Zou also obtained the similar results relaying on (AR) and (MN). Based on the work of [11,17], Liu and Guo [22] obtained the existence and concentration of positive ground state solution for (SKε), We would like to emphasize that the previous work depends heavily on the condition (MN) or (AR) and can be applied to the case where f (u) ∼ |u| q− u for ≤ q < . Obviously, the approaches adopted in them do not work when f satis es neither (MN) nor (AR). Therefore, there are very few results concerning semiclassical ground state solutions for (SKε) where f (u) behaves like |u| q− u for q ∈ ( , ]. The rst purpose of this paper is to consider this case and improve some previous results. To state our results, we need introduce the following decay assumption on V (V2) V ∈ C R , R , and ∇V(x) · x (p − )V(x) for all x ∈ R , where p is given by (F3).
Next we consider the concentration of ground solution of (SKε) as ε → . Furthermore, we establish the exponential decay property of the solution obtained in the following theorem. For this reason, inspired by [36], we introduce the following monotonicity condition on V is nonincreasing on ( , +∞) for any x ∈ R \{ } which is di erent from the following monotonicity condition used in [6]: For ε ≥ , we de ne the Pohozaev type functional Pε as follows Based on the fact that any solution u of (PKε) satis es Pε(u) = and motivated by [22], de ne the following Nehari-Pohozaev functional on H R and set i.e., the Nehari-Pohozaev manifold of Iε. So every non-trivial solution of (PKε) is contained in Mε. A nontrivial solutionū of (PKε) is called a ground state solution of Nehari-Pohozaev type ifū satis es Iε(ū) = inf u∈Mε Iε(u).
Our second result is given as follows.
Moreover, the following statements hold (i) for ε ∈ ( , ε ], there is a maximum point xε of |vε| which satis es that (ii) there exist Π , κ independent of ε ∈ ( , ε ] such that the maximum point xε of |vε| satis es that (iii) for any sequence εn → ,vε n (εn x + xε n ) converges in H R to a ground state solution of the following problem It's worthy noting that (F3) (V3) are di erent from that in [6], and unlike [36], we just assume in (V1) that Owing to the critical term, we have to face the lack of compactness. To resolve the obstacle caused by the lack of compactness, we compare the mountain pass level with the minimax level of the associated limiting problem. For this purpose, we study the existence of ground solutions to the following equation where V is a parameter with < V ≤ Vmax := sup x∈R V(x). We have the following statement.
where (u) t (x) := u(x/t) for all x ∈ R and t > , and The main di culties lies in two aspects: (i) The fact that f (u) does not satisfy (AR) condition nor (MN) condition prevents us from obtaining a bounded (PS) sequence and from using the Nehari manifold. (ii) The unboundedness of R and the presence of critical term u result in the lack of compactness. Motivated by [6], we rstly consider (K V ) and prove Theorem 1.3 to nd semiclassical solutions of (SKε). Based on the general minimax principle [21, Proposition 2.8], we construct a Cerami sequence {un} with I V (un) → c V and with the extra property that J V (un) → which is crucial to deduce the boundedness of {un}, even the (AR) condition is not satis ed. By using some new estimates and subtle analysis introduced in [6], we show that c V < abS + b S + (b S + aS) / (see Lemma 3.3). More speci cally, we determine the lower bound µ in Theorem 1.3 (see Lemma 3.3) unlike [15].
To prove the existence of the semiclassical solutions, following the idea of Jeanjean [19](the so-called monotonicity trick) and using a new global compactness lemma of critical type developed by [22,36], we construct a sequence {un} of the exact critical points of nearby functionals I ε,λn which satis es λn → and I ′ ε,λn (un) = , where and show that I ε,λ satis es the Palais-Smale condition because of c ε,λ < m ∞ λ , as proved in Lemma 4.4. The fact that un is the exact critical point provides additional information related to Pohozaev identity, which is important to ensure the boundedness of {un}. Remark 1.4 As mentioned above, unlike [36], we just assume inf x∈R V(x) < V∞ instead of V(x) ≤ V∞ for all x ∈ R , which makes it di cult to show cε < m ∞ ε . To overcome this obstacle, we use some new energy inequalities and some subtle analysis and nd two constants ε * > andλ ∈ [ / , ) determined by V and f (see Lemma 4.4) such that c ε,λ < m ∞ λ , ∀λ ∈ (λ, ], ε ∈ [ , ε * ].

Remark 1.5
To obtain the concentration phenomenon for ground state solution to (SKε) as ε → , we introduce some new proof techniques due to [6] to overcome the obstacles caused by the lack of (AR) and (MN), which is di erent from the previous work. Our work extended the results of [23] to critical case. In Sect. 2, we give some preliminaries and necessary lemmas. Sect. 3 is devoted to show the existence of the ground state solution for the limited problem (K V ), and give the proof of Theorem 1.3. The proof of theorem 1.1 is given in Sect. 4. We investigate the existence and concentration of the ground state solution of Nehari-Pohozaev type and complete the Proof of Theorem 1.2 in Sect. 5 and Sect. 6.

Preliminaries
In this section, we give some preliminaries. We will make use of the following notations.
• H (R ) denotes the usual Sobolev space equipped with the inner product and norm (u, v) = R (∇u · ∇v)dx, u = (u, u) / for all u, v ∈ H (R ).
• C, C i denote (possibly di erent) various positive constant.
Next, we claim that tu is unique for any u ∈ H (R )\{ }. If not, we can assume that there exist positive This contradiction shows that tu > is unique for any u ∈ H (R )\{ }. Combining Corollary 2.1 with Lemma 2.3, we have the following lemma.
As the proof of [23, Lemma 2.9], we have the following statement.

Ground state solution for the limited problem
In this section, by using the following general minimax principle, we show the existence of the ground state solution for (K V ), and give the proof of Theorem then, for every σ ∈ ( , (c − b)/ ), δ > and γ ∈Γ satisfying In the following, we apply Lemma 3.1 to obtain a Cerami sequence for the functional I V with J V (un) → . where Proof. By (F1), one has I V (tu) → −∞, as t → ∞ for every u ∈ H (R )\{ }. A standard argument shows that Γ ≠ ∅ and c V < ∞. Moreover, it is easy to see that there exist constants ρ > and σ > such that Clearly γ( ) = and I V (γ( )) < for every γ ∈ Γ. Hence (3.2) implies that γ( ) > ρ . There exists tγ ∈ ( , ) such that γ(tγ ) = ρ . Thus, we have Let us de ne the continuous map h : ( / ) . We consider the following auxiliary functional It is easy to see thatĨ V ∈ C (R × H (R ), R), and Then, we apply Lemma 3.
From J V ( ) = , (F1) and J V (γ( )) < , there exist ρ ∈ ( , γ( ) ) and σ ≥ such that J V (u) ≥ σ for all u = ρ , which implies that there exist t V ∈ ( , ) such that J V (γ(t V )) ≥ σ . Thus, for every γ ∈ Γ V has to crossM V , and c V ≥m V . This shows that c V =m V . Proof of Theorem 1.3. In view of Lemma 3.2, there exist a sequence {un} ⊂ H (R ) satisfying (3.1). By (3.1), (3.22) and (3.23), we have If not, then Lions' concentration compactness principle implies that un → in L s (R ) for all s ∈ ( , ). By (F1), we have for every ε > , there exist constant Cε such that Thus, as n → ∞ and Since {un} is bounded in H (R ) and c V > , up to a subsequence, we may assume there exists constants l , l > such that a ∇un + V un → l , b ∇un , n → ∞.
Together with (3.27), (3.28) and Sobolev inequality, we have Letting n → ∞ in the above two inequalities, we achieve that l ≥ (l + l ) / .
Moreover,û is nontrivial, andû satis es where A := lim n→∞ ∇ûn and ∇û ≤ A . Hence, we have the following equalities ( . ) Next, we show that J V (û) = . From (1.8) and (3.34), we have If J V (û) < , it follows from Lemma 2.3, which is also true for I V and J V , that there exist a unique t ∈ ( , ) such that J V (t / (û t )) = . Combining with ( which is a contradiction. Hence we have J V (û) = and I V (û) =m V which, together with Lemma 2.4 and Lemma 2.9, implies This completes the proof.

Existence of the ground state solutions
In this section, by using the Jeanjean's monotonicity trick [ and In the same way as in [14], we can obtain the following lemma.

Lemma 4.2. Assume (F1), (F3) hold. Then
For λ ∈ [ / , ], let It is easy to check that there exists T > such that By (4.8) and simple calculation, we can derive the following lemma. (ii) there exist a constant κ independent of λ and ε such that for all λ ∈ [ / , ] and ε ≥ ,

From (F2) and (3.22), we can deduce that the function F(t)/t|t| p− is nondecreasing on
Then / ≤λ < . We have two cases to distinguish.

Ground state solutions of Nehari-pohožaev type
In this section, we consider the existence of ground state solution of (PKε). By and In view of Lemma 2.3, there exists t > such that Let Using (F1), (F2) and (4.8), it is easy to check that there exists T > such that In view of (5.6) and Lemma 2.3, for any ε > , there exists tε ∈ ( , T ) such that is independent of ε > .  Proof. In view of Lemma 2.3 and Lemma 2.6, we have that Mε ≠ ∅ and mε > for ε ∈ ( , ε ]. For any xed ε ∈ ( , ε ], let {un} ⊂ Mε be such that Iε(un) → mε. Since Jε(un) = , then it follows from (2.9) that

Proof. By (5.3) and (5.4),
which implies {un} is bounded in H (R ) together with (2.11). Passing to a subsequence, we can assume that un ū in H (R ), un →ū in L s loc (R ) for all s, with ≤ s < and un →ū a.e. on R .
Next, we claim thatū ≠ . Arguing by contradiction, suppose thatū = , then un → in L s loc (R ) for ≤ s < and un → a.e. on R . In view of Lemma 2.3, there exists tn > such that t / n (un) tn ∈MV for every n ∈ N. We claim that there exist two positive constant T < T such that T ≤ tn ≤ T , ∀n ∈ R.

Concentration of ground state solutions of (SK ε )
In this section, we consider the concentration of ground state solutions of (SKε) and give the proof of Theorem Lemma 6.2. There exists a constant K > independent of ε such that u ≤ K for all u ∈ Λ.
In the following we prove that (i) or (ii) hold.
( . ) In view of Theorem 1.3, there existsû β ∈M β such that Note that mε n = Iε n (uε n ) = a ∇ûε n + It is easy to check that for any bounded set Ω ⊂ R , Proof. Suppose to the contrary that δ = . Then there is a sequence {uε n } ⊂ Λ such that lim n→∞ u ∞ = .
By the maximum principle, we can conclude that w(x) ≤ for |x − yu| ≥ R i.e., Therefore, the claim (6.32) holds.