Multiplicity results for some quasilinear elliptic problems with concave nonlinearities

The purpose of this paper is to investigate the existence of ﬁve nontrivial solutions and the existence of inﬁnitely many nontrivial solutions for a class of ( p, 2)-Laplacian equations with subcritical polynomial growth and subcritical (critical) exponential growth. Some existence results for nontrivial solutions are established by using the variational methods


Introduction
In this paper we are interested in investigating the existence of multiple nontrivial solutions for the following quasilinear elliptic boundary problem where Ω is a bounded domain in R N (N ≥ 1) with smooth boundary ∂Ω, 2 < p < ∞, △ p denotes the p-Laplacian operator defined by ∆ p u = div(|∇u| p−2 ∇u), µ is a real parameter, 1 < s < 2, λ ≥ 0 is a parameter, h ∈ L ∞ (Ω), h(x) ≥ 0, h(x) ≡ 0, and f ∈ C( Ω × R, R).One knows that when the nonlinearity f (x, u) satisfies some suitable conditions the nontrivial solutions of problem (1.1) correspond to the nonzero critical points of the C 1 -energy functional for all u ∈ W 1,p 0 (Ω), where F (x, t) = t 0 f (x, s)ds.In the case of p > 2, λ = 0 and µ > 0, there have been many works on finding the existence of nontrivial solutions of (1.1).For example, under the following conditions where m ≥ 1 and C is a constant, by the three critical point theorems the authors in [3,7] proved that (1.1) has at least two nontrivial solutions.Here 0 < µ 1 < µ 2 < • • •, µ i (i = 1, 2 • ••) denotes the eigenvalues of −∆ in H 1 0 (Ω), and λ 1 is the first eigenvalue of −∆ p in W 1,p 0 (Ω) (see [15] ).For Eq. (1.1) with right-hand side having p-linear growth at infinity, i.e., lim |t|→∞ f (x,t) |t| p−2 t = λ ∈ σ(−∆ p ), the spectrum of −∆ p in W 1,p 0 (Ω), the papers [4,5] studied the existence of nontrivial solution.In [30], the author generalized the results in [3,7] under the general asymptotically linear condition.The authors in [16] obtained a nontrivial solution under a new decomposition about the Banach space W 1,p 0 (Ω) by Morse theory.In [21], the authors studied the existence of a greatest negative, a smallest positive, and a nodal weak solution for problem (1.1) by using variational methods and truncation techniques.The authors in [11] obtained two positive solutions for problem (1.1) by some linking theorems and critical groups estimates.In [29], the authors obtained two solutions for problem (1.1) by using variational methods and critical theorems and also studied the positivity, which was shown by applying a generalized version of the strong maximum principle.The authors in [24] considered Ambrosetti-Prodi problems to equation (1.1) and proved a multiplicity theorem generating five nontrivial solutions.
In the case of 1 < p < 2, λ > 0 and µ > 0, the authors in [13] gave some qualitative results of critical groups of an isolated critical point for problem (1.1).
In the case of p > 2, λ > 0 and µ > 0, in [23] the authors considered problem (1.1) with nonlinearity f (x, u) being resonant with respect to the principle eigenvalue of the Dirichlet p-Laplacian and obtained six nontrivial smooth solutions under λ is small enough by using variational methods together with truncation and comparison techniques and Morse theory (critical groups).The authors in [12,17] did some similar work.
In this paper, we study problem (1.1) from two aspects.One is that we will prove two multiplicity results under the nonlinearity f being p-superlinear at infinity and having the standard subcritical polynomial growth but without satisfying the Ambrosetti-Rabinowitz condition, the other is we can obtain some existence results of multiple solutions when the nonlinearity f has the exponential growth but still not satisfy the Ambrosetti-Rabinowitz condition.In the first case, although some old methods for the verification of the compactness condition will fail, we will overcome it by using the functional analysis methods, i.e., Hahn − Banach Theorem combined the Resonance Theorem.In the last case, although the original mountain pass theorem of Ambrosetti-Rabinowitz [1] is not directly used for our purpose, we will adopt a suitable version of mountain pass theorem and some new techniques to achieve our goal.
When p < N , there have been essential many works (such as [4,5,11,12,13,17,21,23,24,29,30]) to deal with the existence of nontrivial solutions or the existence of nodal solutions for problem (1.1).Moreover, almost all of the works involve the nonlinearity f (x, u) with standard subcritical polynomial growth, say, (SCP) : There exist positive constants c 0 and q where p * = N p/(N − p) denotes the critical Sobolev exponent.In this case, people can study problem (1.1) variationally in the Sobolev space W 1,p 0 (Ω) owing to the some critical point theory, such as, Morse theory, minimax theorem and so on.One of the crucial condition used in many works is the so-called Ambrosetti-Rabinowitz condition: (AR) There exist θ > p and R > 0 such that 0 < θF (x, t) ≤ f (x, t)t, for x ∈ Ω and |t| ≥ R.
It is easy to see that there exist Thus problem (1.1) is called superlinear if the nonlinearity f is p-superlinear at infinity.Note that (AR) condition act as an important role in verifying the boundedness of Palais-Smale sequences.However, there are generous nonlinearities which are superlinear but do not satisfy above (AR) condition such as f (x, t) = |t| p−2 t ln(1 + |t| p ).In the past twenty years many authors made the effort to study problem (1.1) (λ = 0, µ = 0) where (AR) does not hold.Instead, they consider the condition (WSQC) the following limit holds with extra assumptions (see [6,14,19,20] and the references therein).In the most of them, there are some kind of monotonicity restrictions on the functions F (x, t) or f (x,t) |t| p−2 t , or some convex property for the function tf (x, t) − pF (x, t).
In the case of p = N , inspired by the Trudinger-Moser inequality, there are a few works devoted to study the existence of nontrivial solutions for problem (1.1) with the nonlinearity f involving the exponential growth, besides [25,27] and the references therein.
Let λ 1 be the first eigenvalue of (−∆ p ) and ϕ 1 (x) > 0 for every x ∈ Ω be the λ 1 eigenfunction.Throughout this paper, we denote by | • | p the L p (Ω) norm, the norm of u in W 1,p 0 (Ω) will be defined by , and always assume that µ = 1 in (1.1).
Let us now state our main results: Suppose f (x, t) satisfies: In the case of p < N , our results are stated as follows: Theorem 1.1 Suppose that conditions (SCP) and (H 1 ) -(H 3 ) hold.If f 0 < λ 1 and h(x) ≥ h 0 (h 0 is a positive constant ), then there exists Λ * > 0 such that for λ ∈ (0, Λ * ), problem (1.1) has five nontrivial solutions.Theorem 1.2 Suppose that conditions (SCP), (H 2 ) and Remark.In fact, when µ = 0 and p = 2, our this result still holds.Furthermore, our nonlinearity f (x, u) satisfies more general condition (H 3 ) compared with the classical condition (AR).Thus, our Theorem 1.2 completely extends Theorem 3.20 in [33].
In case of p = N , we have p * = +∞.In this case, every polynomial growth is admitted, but one knows easy examples that W 1,N 0 (Ω) L ∞ (Ω).Hence, one is led to look for a function g(s) : R → R + with maximal growth such that sup It was shown by Trudinger [31] and Moser [22] that the maximal growth is of exponential type.We must redefine the subcritical exponential growth and the critical exponential growth as follows: (SCE): f has subcritical exponential growth on Ω, i.e., lim When p = N and f has the subcritical exponential growth (SCE), our work is still to study problem (1.1) where the nonlinearity f does not satisfy the (AR)-condition at infinity.To our knowledge, this problem is rarely studied by other people for (p, 2)-Laplacian equations with concave nonlinearity.Hence, our results are new and our methods are technique since we successfully verified the compactness condition by using the Resonance Theorem combined Trudinger-Moser inequality and the truncated technique.In fact, the new idea comes from our work [26].Our results are as follows: Theorem 1.3 Suppose that conditions (SCE) and (H 1 )-(H 3 ) hold.If f 0 < λ 1 and h(x) ≥ h 0 (h 0 is a positive constant ), then there exists Λ * > 0 such that for λ ∈ (0, Λ * ), problem (1.1) has five nontrivial solutions.
Remark.Let F (x, t) = t p ln(1+|t|), ∀(x, t) ∈ Ω×R.Then it satisfies that our conditions (H 1 )-(H 3 ) but not satisfy the condition (AR).We also give a function satisfying conditions (H 1 )-(H 3 ) and condition (AR), such as 2λ1p .It's worth noting that we do not impose any monotonicity condition on f (x,t) |t| p−2 t or some convex property on tf (x, t) − pF (x, t).Hence, our Theorem 1.3 completely extends some results contained in [18,27] When p = N and f has the critical exponential growth (CG), the study of problem (1.1) becomes more difficult than in the case of subcritical exponential growth.Similar to the case of the critical polynomial growth in R N (N ≥ 3) for the standard Laplacian studied by Brezis and Nirenberg in their pioneering work [2], our Euler-Lagrange functional does not satisfy the Palais-Smale condition at all level anymore.For the class standard Laplacian problem, the authors [10] used the extremal function sequences related to Trudinger-Moser inequality to overcome the verification of compactness of Euler-Lagrange functional at some suitable level.The idea of choosing the testing functions firstly appeared in [2].Here, we still adopt it to study problem (1.1) without (AR) condition.Our result is as follows: Theorem 1.5 Suppose that conditions (CG) and (H 1 )-(H 3 ) hold.Furthermore, assume that -dimensional surface of the unit sphere) and If f 0 < λ 1 , then there exists Λ * > 0 such that for λ ∈ (0, Λ * ), problem (1.1) has at least four nontrivial solutions.
Remark.For N -Laplacian problem, Lam and Lu [18] have recently obtained the existence of nontrivial nonnegative solutions when the nonlinearity f has the critical exponential growth of order exp(α|u| N N −1 ) but without satisfying the Ambrosetti-Rabinowitz condition.However, for mixed operator type problem (1.1) involving critical exponential growth and the concave term, there are few works to study it.Hence our result is new and interesting.
The paper is organized as follows.In Section 2, we present some necessary preliminary knowledge and some important lemmas.In Section 3, we give the proofs for our main results.
2 Preliminaries and some lemmas Definition 2.1.Let ( E, • E ) be a real Banach space with its dual space (E * , • E * ) and there is a subsequence {u n k } such that {u n k } converges strongly in E.
Then, there exists a sequence {u n } ⊂ E such that The inequality is optimal: for any growth exp(α|u| Proof.We only prove the case of I + λ .The arguments for the case of I − λ are similar.Let {u n } ⊂ W 1,p 0 (Ω) be a (PS) c * sequence, i.e. (2.4) Step 1.We prove that {u n } is bounded in W 1,p 0 (Ω).In fact, suppose that Then, v n = 1, ∀n ∈ N and then, it is possible to select a subsequence (denoted also by {v n }), which converges weakly to v in W 1,p 0 (Ω), converges strongly in L p1 (Ω)(1 ≤ p 1 < p * ) and converges v a.e.x ∈ Ω. Letting v − = max{−v, 0} and v − n = max{−v n , 0}, from condition (H 1 ), it is easy to see that v − = 0 by dividing (2.4) with u n p−1 and choosing Dividing both sides of (2.2) by u n p , we obtain Set Using (H 3 ) , we deduce that If |Ω + | is positive, from Fatou's lemma, we imply which contradicts with (2.5).Dividing both sides of (2.4) by u n p−1 , for any ϕ ∈ W 1,p 0 (Ω), then there exists a positive constant C(ϕ) such that Then from (SCP), we conclude that {f n } is a family bounded linear functionals defined on W 1,p 0 (Ω).Using (2.7) and the famous Resonance Theorem, we know that {|f n |} is bounded, where |f n | represents the norm of f n .This reveals that (2.8) Since E ⊂ L p * p * −q (Ω), by the Hahn -Banach Theorem, there exists a continuous functional fn defined on L p * p * −q (Ω) such that fn is an extension of f n , and ) where fn p * q is the norm of fn (ϕ) in L p * q (Ω) which is defined on L p * p * −q (Ω).
On the other hand, by the definition of the linear functional on L p * p * −q (Ω), we get that there exists a function (2.11) Thus, from (2.9) and (2.11), we get which means that From the basic lemma of variational, we can conclude that So, from (2.8) and (2.10), we know Now, by choosing ϕ = v n − v in (2.4), we have where A : W 1,p 0 (Ω) → W 1,p 0 (Ω) * defined by From the Hölder inequality and (2.12), we get So, from (2.13), we can deduce that This leads to a contradiction since v n = 1 and v = 0. Thus, {u n } is bounded in W 1,p 0 (Ω).
Step 2. We prove that {u n } has a convergence subsequence.In fact, we can assume that Now, since f satisfies the condition (SCP) and (H 2 ) then there exist two positive constants c 3 , c 4 > 0 such that Similarly, since u n ⇀ u in W 1,p 0 (Ω), Ω |u n − u| p dx → 0 and Ω |u n − u| p * p * −q dx → 0. Thus, from (2.4) and the formula above, we obtain So, we get u n → u .Thus we have u n → u in W 1,p 0 (Ω) which implies that I + λ satisfies (PS) c * .
We can show that there exists t * such that In fact, solving h ′ (t) = 0, we have From the knowledge of mathematical analysis, h(t) has a minimum at t = t * .Denote On the other hand, from (2.15), we get Similarly, we get I − λ (t(−ϕ 1 )) → −∞, as t → +∞.Thus part (ii) holds.
On the other hand, from (2.17), we get Similarly, we obtain I − λ (t(−ϕ 1 )) → −∞, as t → +∞.Thus part (ii) holds.Proof.We only deal with the case of I + λ .The arguments for the case of I − λ are similar.Let {u n } ⊂ W 1,N 0 (Ω) be a (PS) c * sequence satisfying the formulas (2.2)-(2.4) in Lemma 2.3.Now, in the light of the previous section of Step 1 of the proof of Lemma 2.3, we also get that the formula (2.7) holds.Let Then from for any u ∈ W 1,N 0 (Ω), e α1|u| N N −1 ∈ L 1 (Ω) for all α 1 > 0, we can deduce a conclusion that {f n } is a family bounded linear functionals defined on W 1,N 0 (Ω).By (2.7) and the famous Resonance Theorem, we obtain that {|f n |} is bounded, where |f n | expresses the norm of f n .It indicates that the formula (2.8) (see the proof of Lemma 2.4) holds.
Because of W 1,N 0 (Ω) ⊂ L q2 (Ω) for some q 2 > 1, by the Hahn-Banach Theorem, there exists a continuous functional fn defined on L q2 (Ω) such that fn is an extension of f n , and (2.18) where fn q * 2 denotes the norm of fn (ϕ) in L q * 2 (Ω) which is defined on L q2 (Ω) and q * 2 is the conjugate number of q 2 .
From the definition of the linear functional on L q2 (Ω), we get that there exists a function Since the condition (SCE), there is a constant So, by the Trudinger-Moser inequality (see Lemma 2.2), where k > 1 and k ′ is the conjugate number of k.Similar to the last proof of Lemma 2.4, we have Lemma 2.7 .Assume conditions (SCP) and (H 3 ) hold.Then I λ satisfies (PS) c * . ) Step 1.To show that {u n } has a convergence subsequence, we first need to claim that it is a bounded sequence.To do this, argue by contradiction that Without loss of generality, suppose u n ≥ 1 for all n ∈ N and let Clearly, v n = 1, ∀n ∈ N and then, it is possible to choose a subsequence (expressed also by {v n }) converging weakly to v in W 1,p 0 (Ω), converging strongly in L p2 (Ω)(1 ≤ p 2 < p * ) and convergeing v a.e.x ∈ Ω.
Dividing both sides of (2.21) by u n p , we get Let Using (H 3 ) , we conclude that If |Ω 0 | is positive, by Fatou's lemma, we get which contradicts with (2.23).Dividing both sides of (2.22) by u n p−1 , for any ϕ ∈ W 1,p 0 (Ω), then there exists a positive constant C(ϕ) such that Write Thus, from (SCP), we reach that {f n } is a family bounded linear functionals defined on W 1,p 0 (Ω).Sine (2.25) and the famous Resonance Theorem, we obtain that {|f n |} is bounded, where , by the Hahn -Banach Theorem, there exists a continuous functional fn defined on L p * p * −q (Ω) such that fn is an extension of f n , and where fn p * q represents the norm of fn (ϕ) in L p * q (Ω) which is defined on L p * p * −q (Ω).Remained proof is essentially identical to the last proof of Lemma 2.3, we omit it here.where ν N denotes the volume of the unit ball.Since (2.30), the inequality above holds if, and only if (2.33) (2.34) The last integral in (2.34), write it ξ n , is evaluated as follows: Hence, from (2.34) we have α N α 0 This leads to a contradiction with (H 4 ).

Proofs of the main results
Proof of Theorem 1.1.For I ± λ , we first claim that the existence of local minimum v ± with I ± λ (v ± ) < 0. We only prove the case of I + λ .The arguments for the case of I − λ are similar.For ρ determined in Lemma 2.4, we denote Since Lemma 2.4, we get for 0 < λ < Λ * , In addition, we know that I + λ ∈ C 1 ( B(ρ), R).Hence, I + λ is lower semi-continuous and bounded from below on B(ρ).Write with φ > 0, and for t > 0, we obtain for all t > 0 small enough.Therefore c * 1 < 0. By Ekeland's variational principle and Lemma 2.4, for any n > 1, there exists u n with u n < ρ such that Hence, there exists a subsequence still denoted by {u n } such that u n → v + , I + ′ λ (v + ) = 0. Thus v + is a weak solution of problem (1.1) and I + λ (v + ) < 0. Besides, from the general maximum principle, we know v + > 0. Similarly, we get a negative solution v − with I − λ (v − ) < 0. On the other hand, by Lemmas 2.3-2.4,the functional I + λ has a mountain pass-type critical point u + with I + λ (u + ) > 0. Again using the general maximum principle, we have u + > 0. Hence, u + is a positive weak solution of problem (1.1).Similarly, we also get a negative mountain passtype critical point u − for the functional I − λ .Thus, we have proved that problem (1.1) has four different nontrivial solutions.Next, we adapt to the idea developed in [32] to obtain the fifth solution for problem (1.1).We can suppose that v + and v − are isolated local minima of I λ .Let us define by b λ the mountain pass critical level of I λ with base points v + , v − : We will demonstrate that b λ < 0 if λ is small enough.To this end, we consider We declare that there exists δ > 0 such that I λ (tv ± ) < 0, ∀t ∈ (0, 1), ∀λ ∈ (0, δ). (3.1) If not, then there exists t 0 ∈ (0, 1) such that I λ (t 0 v ± ) ≥ 0 for λ small enough.Similarly, we get I λ (tv ± ) < 0 for t > 0 small enough.Let ρ 0 = t 0 v ± and č± * = inf{I ± λ (u), u ∈ B(ρ 0 )}.By previous arguments, we obtain a solution v * ± such that I λ (v * ± ) < 0, a contradiction.Therefore, (3.1) holds.Now, let us consider the 2-dimensional plane Π 2 containing the straightlines tv − and tv + , and take v ∈ Π 2 with v = ǫ.Note that for such v one has v s = c s ǫ.By Sobolev embedding inequality, we have where S * > 0 is a constant.Hence, for small ǫ, Proof of Theorem 1.2.For the reflexive and separable Banach space W 1,p 0 (Ω), define Y k , Z k as in (2.35).By Lemma 2.7, we know that I λ satisfies the (PS) c condition.Moreover, from our assumption, we also easily see that I λ (−u) = I λ (u).It remains to verify the conditions (i) and (ii) of Lemma 2.10.The verification of (i) is quite standard, therefore it is omitted here.Now we claim that for every k ∈ N, By (H 3 ), there exists large enough M * such that So, for any u ∈ Y k , we have Hence, our claim holds, i.e. (ii) of Lemma 2.10 is verified.
Next we use the dual fountain theorem (Lemma 2.11) to prove the case of b).Since Lemma 2.7, we know that the functional I λ satisfies (PS) * c condition.Next, we just need to prove the conditions (i)-(iii) of Lemma 2.11.
Proof of Theorem 1.3.As our assumptions, similar to previous section of the proof of Theorem 1.1, we get that the existence of local minimum v ± with I ± λ (v ± ) < 0. Furthermore, by Lemmas 2.5,2.6, for I ± λ , we also can find two mountain pass type critical points u + and u − with positive energy.Similar to the last section of the proof of Theorem 1.1, we can also get another solution u 3 , which is different from v ± and u ± .Thus, our theorem is proved.
Proof of Theorem 1.4.We first use the symmetric mountain pass theorem to prove the case of a).Since our assumptions, we know that the functional I λ is even.By the condition (SCE), we easily get that (I ′ 1 ) of Theorem 9.12 in [28] holds.Actually, similar to the proof of (i) of Lemma 2.5, we can verify it.In addition, from condition (H 3 ), we easily show that (I ′ 2 ) of Theorem 9.12 also holds.Thus, by Lemma 2.8, our result holds.
Next we use the dual fountain theorem (Lemma 2.11) to prove the case of b).By Lemma 2.8, we know that the functional I λ satisfies (PS) * c condition.So, we just need to verify the conditions (i)-(iii) of Lemma 2.11.
First, we prove (i) of Lemma 2.11.Set By the conditions (SCE), (H 2 ) and Lemma 2.2, we obtain, for u ∈ Z k , u ≤ R, Here, R is a positive constant small enough and ǫ > 0 small enough.We choose ρ k = (2pλ There exists k 0 such that ρ k ≤ R when k ≥ k 0 .Hence, for k ≥ k 0 , u ∈ Z k and u = ρ k , we conclude I λ (u) ≥ 0 and (i) holds.The proof of (ii) and (iii) is standard, we omit it here.
Proof of Theorem 1.5.By our assumptions, similar to previous section of the proof of Theorem 1.1, we get that the existence of local minimum v ± with I ± λ (v ± ) < 0. Now, we demonstrate that I + λ has a positive mountain pass type critical point.By Lemma 2.5 and Lemma 2.9, then there exists a (C) cM sequence {u n } at the level 0 < c M ≤ α N α0 N −1 1 N .By same arguments as previous section of the proof of Lemma 2.6, we can show that (C) cM sequence {u n } is bounded in W 1,N 0 (Ω).Now, we can assume that u n ⇀ u + in W 1,N 0 (Ω).
.20)Similarly to the last section of the Step 1 of the proof of Lemma 2.3, we can reach that (C) c * sequence {u n } is bounded in W 1,N 0 (Ω).Next, we show that {u n } has a convergence subsequence.Without loss of generality, provide that and break the integral in (2.31) into a sum of integrals over A * n and B n .By simple computation, we get α