Existence , multiplicity and nonexistence results for Kirchho type equations

We consider rst the case thatΩ ⊂ R3 is a bounded domain. Existence of at least one or two positive solutions for above equation is obtained by using the monotonicity trick. Nonexistence criterion is also established by virtue of the corresponding Pohožaev identity. In particular, we show nonexistence properties for the 3sublinear case as well as the critical case. Under general assumption on the nonlinearity, existence result is also established for the whole space case that Ω = R3 by using property of the Pohožaev identity and some delicate analysis.


Introduction and main results
This paper is concerned with following Kirchho type equation: where Ω ⊂ R is a bounded domain or Ω = R , ≤ h ∈ L (Ω) and f ∈ C(R, R). Problem like (1.1) is associated with the stationary analogue of the wave equation arising in the study of string or membrane vibrations: Such type equation proposed rst by Kirchho [1] is used to describe the transversal oscillations of a stretched string. The nonlocal term is also used to model suspension bridges [2] and to describe the growth and movement of a particular species in biological systems [3]. More mathematical and physical background and applications of such problems can be found in [1,[4][5][6] and the references therein.
When f is critical growth at in nity, particularly, behaving like following form: f (x, t) = t + λg(x, t), and < ϱ t g(x, s)ds ≤ g(x, t)t, (1.4) mountain pass solution for (1.3) was established by Alves et al. in [23] when < ϱ < and the parameter λ > is su ciently large. This result was improved by Figueiredo [9] by extending the range of ϱ to ( , ), moreover, existence of positive solution was also obtained by them using truncation technique. In [22], two distinct solutions of (1.3) were obtained by Xie et al. for case that < ϱ < and λ = in (1.4). For the case that ϱ ∈ ( , ), two positive solutions of (1.3) were constructed by Lei et al. [11] via the concentration compactness argument provided that g(x, t) ≡ t q and the parameters b, λ are su ciently small. See also [10] where multiple results were obtained via the perturbation method. In recent paper [13], the case f (x, t) = Q(x)|t| t + λ|t| q− t ( < q < ) was considered by Qin et al, and existence of positive ground state solutions was established there based on energy estimates and the Nehari manifold method, moreover, by minimizing the corresponding energy functional on some proper subsets of the Nehari manifold associated with the barycenter map, the relation between the number of positive solutions for (1.3) and the number of maxima of Q was also studied there [13].
When f has subcritical growth at in nity, i.e., lim sup t→+∞ f (x, t) t = uniformly in x, existence and uniqueness properties for parabolic problem (1.2) and elliptic equation (1.3) were studied by Chipot et al. [5]. Corrêa [6] found positive solutions to (1.3) for all N ≥ with the aid of xed point theorems.
Assuming the well known Ambrosetti-Rabinowitz type condition or following monotonic condition, (AR) there exist η > and r > such that |t| is nondecreasing on (−∞, ) ∪ ( , ∞), He and Zou [20] obtained in nitely many large energy solutions for (1.1) via the fountain theorems. See [21] for multiplicity results concerning an oscillating behavior of f . In [16], following eigenvalue problem was studied by Perera and Zhang, and the corresponding eigenvalues are < µ ≤ µ ≤ . . ., Particularly, µ is simple and isolated, and it can be expressed by (see [16,Section 3] and [18]) They [16] mainly concerned with the asymptotically 3-linear case, precisely, f satis es Existence of nontrivial solution to (1.3) was established by them via critical group and Yang index when λ ∈ (λ l , λ l+ ) and µ ∈ (µm , µ m+ ) with l ≠ m, where Taking advantage of the invariant sets of descent ow, the 3-sublinear and 3-superlinear cases((AR) was assumed there) were also studied by Zhang and Perera [18] (see also [15]), moreover, existence of sign-changing solutions was constructed by them.
By developing the Nehari manifold method, the asymptotically 3-linear case was also considered by Qin et al. [12] where existence of ground state solution to (1.3) was established under following weak version of the monotonic condition (F1) (see [12,Lemma 3.9]).
(F2) for all θ ≥ and t ∈ R, In [8], Cheng and Wu studied (1.3) by assuming (F1) and following condition on f , where ≤ p, q ∈ L ∞ (Ω) and p ∞ < λ . Positive solutions were obtained for the 3-linear case and 3superlinear case, respectively. In particular, nonexistence result was showed there by virtue of the monotonic condition (F1). See also [19]. In recent paper [3], Chen et al. considered (1.3) with concave-convex nonlinearity having the form: , and the continuous functions h, g satisfy h + := max{h, } ≠ and g + := max{g, } ≠ . Taking advantage of the Nehari manifold approach, they got some existence results for the case p > , p = and p < , respectively. Recently, Shuai [17] studied (1.3) under the monotonic condition (F1), moreover, they succeeded in nding a least energy sign-changing solution via the degree theory and quantitative deformation lemma. Such result was improved later by Tang et al. [14] based on the Non-Nehari manifold method.
Inspired by above works [8,12,19,28], we consider rst the bounded domain case and focus our attention on the subcritical case, precisely, the 3-sublinear, 3-linear growth and 3-superlinear are considered separately in this paper. Moreover, existence of at least one or two positive solutions for the equation (1.1) are obtained correspondingly. Based on the Pohožaev identity, we also establish a nonexistence criterion, in particular, we show a nonexistence property for the 3-sublinear case. In order to nd one solution at the mountain pass level, we have to overcome the di culty caused by the lack of a priori bounds on the Palais-Smale sequences. By combining the monotonicity trick used in [29] and some techniques, as well as the strategy used in [12], we manage to obtain the boundedness of the sequence. By the compactness of the Sobolev embedding, then we show the convergence of the Palais-Smale sequence. The hypotheses imposed on f and h are generic and natural.
For the case that Ω ⊂ R is a bounded domain, recall that the working space to (1.1) is H (Ω), which is a Hilbert space equipped with following inner product (1.10) and the corresponding norm is u = (u, u) / . As well known, the embedding from H (Ω) to L s (Ω) is continuous for s ∈ [ , * ], and is compact for s ∈ [ , * ) (see [30,Theorem 1.9]), here * = since N = . Then we nd the existence of γs > satisfying where · s stands for the usual L s (Ω) norm.
Let u ∈ H (Ω) be a solution of (1.1), then we have following useful Pohožaev identity(see [30]): where ν is the unit outward normal to ∂Ω. Multiplying (1.1) by u and calculating each side of the equation, one has By above two equalities, we obtain following non-existence criterion.
is a solution of (1.1), then following identities hold: In particular, this solution is trivial, i.e., u ≡ if Consider rst the 3-sublinear case. Assume that f (t) = νg(t) with ν > and that g : R → R is a continuous function satisfying following assumptions: Since we focus our attention only on non-negative solutions of (1.1), we assume, without loss of generality, that g(t) = , ∀ t ≤ . Our results for the 3-sublinear case read as follows.

1) has no nontrivial solution if < ν < ν and (g3) is satis ed in addition.
Consider now the 3-linear and 3-superlinear cases. For the rst eigenvalues λ , µ > given by (1.6) and (1.8), respectively, we make use of following assumptions on the continuous function f : [ , ∞) → R: Now we are ready to state our results for these two cases. Next, we give a result for the critical growth case, precisely, the nonlinearity has the form f (t) = |t| t + ν|t| q− t.

In particular, when Ω is a smooth starshaped bounded domain, problem (1.1) has no nontrivial solution if h ≡ and no positive solution if h
Finally, we consider the whole space case, i.e., following problem: where V ∈ C (R , ( , +∞)) and f ∈ C([ , ∞), R) satisfying following assumptions: Problem (1.13) has been widely investigated in the literature, see, for example, [31] where positive ground state solution was obtained via the Nehari-Pohožaev manifold when f (t) = |t| p− t, < p < , and [32] where the monotonicity trick [29] is well applied in order to nd the ground states of (1.13) and the rang of p is extended to ( , ) under a stronger assumption of (V1) and some smooth restrictions on f . Later, these results were partially improved by Tang and Chen in recent papers [33,34]. For other results and related Kirchho problem, we refer the readers to [28,[35][36][37][38][39][40][41] and the references therein.
For convenience of the veri cation of compactness, as in [39], we consider following working space for problem (1.13), 14) and assume that (V2) E ⊂ H (R ) such that the following embedding is compact, De ne the inner product by we also get the compact embedding E → L s (R )( ≤ s < ) (see [42,Lemma 3.4]).
By using property of the Pohožaev identity associated to problem (1.13) and some delicate analysis, we establish following existence result.
Now, we give some nonlinear examples to illustrate our assumptions. Assume that α > , l > bµ and ≤ β < , following functions g, f satisfy all assumptions of Theorems 1.2 and 1.3.
Remaining of the paper is organized as follows. In Section 2, some preliminary lemmas are presented. We consider the bounded domain case in Section 3 and study the 3-sublinear case and the rest cases respectively, then we show Theorems 1.2 and 1.3 by using some energy estimates and the monotonicity trick. In the last section, we consider the whole space case and give the proof of Theorem 1.5. In this paper, we use C, C , C , C · · · to denote di erent positive constants in di erent places. L s (Ω)( ≤ s < ∞) denotes the Lebesgue space with the norm u s = Ω |u| s dx /s .

Preliminaries
In this section, we give some preliminaries. De ne rst u+ = max{u, } and u− = max{−u, }. Let us consider following auxiliary problem: The corresponding functional is de ned as follows where F(t) := t f (s)ds. Under (g1) or (f2), the functional Φ is of class C and From the fact that Ω f (u+)u−dx = and Ω hu−dx ≥ , we deduce that which implies that u ≥ , hence it is a solution of (1.1). When h ≢ , we get that u ≥ is nontrivial, thus by the strong maximum principle, u > for x ∈ Ω.
Proof. Since the embedding H (Ω) → L s (Ω) is compact for s ∈ [ , * ), by virtue of the weakly sequentially lower semicontinuous property of the norm, we can certify the rst part (i).
In view of (g1), we have Then by (1.11) and (4.2), we have which shows that Φ is coercive and bounded from below.
In order to nd a Palais-Smale sequence at the mountain pass level, we introduce following Proposition established by Jeanjean [29, Theorem 1.1], which helps us to nd a second solution.

Proposition 2.3. Assume that X is a Banach space with the norm · and J ⊂ R+ is an interval. Let
be a family of C -functions on X such that B(u) ≥ , ∀ u ∈ X, and either A(u) → +∞ or B(u) → +∞ as u → ∞, moreover, there are two points (v , v ) in X satisfying

Proofs of main results for the bounded domain case
In this section, we consider the 3-sublinear, 3-linear and 3-superlinear cases, respectively, and show Theorems 1.2 and 1.3 by using the monotonicity trick and some analytical techniques.

Proof. For any xed compact set M ⊂ Ω, using the Tietze's extension theorem, we nd the existence of map
which implies that Ω G(v)dx > when |M| approaches |Ω| close enough. Thus for ν su ciently large, The proof is completed.  Proof. By (f1) and (f2), there exists C > such that F(u+) ≤ aλ u + + Cu p+ + .
To apply Proposition 2.3, de ne rst the continuous functions f i : [ , ∞) → R, i = , as follows, Clearly, f (t), f (t) ≥ for all t ≥ , and (f3) implies that f (t) > for t large, this yields f (t) = for t large. Set Then F (t) ≥ , and ≤ F (t) ≤ C for some C > .

If we let
then functional Φρ(u) can be written as Note that B(u) ≥ for all u ∈ H (Ω), and then we see that A(u) → ∞ as u → ∞. Since Lemmas 3.3 and 3.5 hold also for Φρ with ρ ∈ J = ( − α, + α) for some α > su ciently small. Thus all the assumptions of Proposition 2.3 are satis ed, and following result holds.
Lemma 3.6. There exists an increasing sequence {ρn} such that ρn → and for any n ∈ N, Φρ n has a bounded Palais-Smale sequence at the mountain pass level cρ n .

Lemma 3.7. Every bounded Palais-Smale sequence {un} ⊂ H (Ω) for Φρ with ρ ∈ J has a convergent subsequence.
Proof. We assume, passing to a subsequence, that un u in H (Ω) and un → u in L q (Ω) for all q ∈ [ , ). By (f2) and Lebesgue dominated convergence theorem, we have Then by Φ ρ (un), un − u → and Φ ρ (u), un − u → , we get Thus un → u in H (Ω), we complete the proof.
By the proof of [29, Lemma 2.3], we see that the map ρ → cρ is continuous from the left. Thus cρ n → c as n → ∞. Proof. By (ii) above, for n large we have c + ≥ cρ n = Φρ n (un) and un is solution of the problem Arguing by contradiction, suppose that un → +∞ as n → ∞. Set vn = un un , then vn = ∇vn = . Up to a subsequence, we may assume that vn v in H (Ω), vn → v in L q (Ω) for all q ∈ [ , ), and vn → v ≥ a.e. on Ω.
If (f5) is satis ed, then there exists t > , C > such that Note that for n large enough, modifying t if necessary, there holds By Choosing t = t (M) > large enough, we have F (un) = on Ω n := {x ∈ Ω : |un| ≥ t }, thus by (3.4), there holds If v = , then vn → in L q (Ω), ∀ q ∈ [ , ). From above equality, we deduce that b = l limn→∞ vn = , which is a contradiction. Thus v ≠ . It follows from (3.5) that This shows that ∇v = = ∇vn , which together with vn v implies that vn → v in H (Ω). Then by which, together with the fact that Φ ρn (un) = , implies that For r given in Lemma 3.3, we set tn = r un . Then for n large, tn < and by (3.4) and Lemma 3.3, c + ≥ Φρ n (un) This contradiction shows that {un} is bounded in H (Ω).

Proof of main result for the whole space R
In this section, we consider problem (1.13) and show Theorem 1.5 by virtue of the Pohožaev identity and some delicate analysis.
As in section 2, we consider following auxiliary problem: (4.1) Note that any solution u ∈ E of problem (4.1) is non-negative, and it is a solution to (1.13) The corresponding functional of problem (4.1) is By (f ) and (f ), there exists t > large such that f (t ) = t and f (t) > t for t > t . Then similar to section 3, we can de ne continuous functions f i : [ , ∞) → R, i = , as follows, Note that, f (t), f (t) ≥ for all t ≥ , and f (t) = for t ≥ t . Set Then Similarly, we consider the family of C -functionals: where ρ ∈ J := ( − α, + α) and α > is small. (ii) if (f ) is satis ed, then Φρ(tu) < for all t ≥ t .
Moreover, boundedness of {un} can be derived from following lemma. Then we get a contradiction from above inequality and (4.8). Thus { ∇un } is bounded, then by (4.3), (f ) and (f2), we show that { un } is bounded, so we get the boundedness of {un}.
By the same proof of Lemma 3.9, we show that {un} is a bounded Palais-Smale sequence for Φ at the level c , then by Lemma 3.7 we get a convergence subsequence of {un}, i.e., un k → u in E as k → ∞, thus Φ (u) = and Φ(u) = c ≥ κ > by Lemma 3.3. This implies that u is a positive solution of (1.13), and the proof of Theorem 1.5 is completed.