Abstract
In this article, we deal with existence and multiplicity of solutions to the pLaplacian system of the type
where Ω ⊂ ℝ^{N }is a bounded domain with smooth boundary ∂Ω, Δ_{p}u = div(∇u^{p2}∇u) is the pLaplacian operator, denotes the Sobolev critical exponent, is a homogeneous function of degree p*. By using the variational method and LjusternikSchnirelmann theory, we prove that the system has at least cat_{Ω}(Ω) distinct nonnegative solutions.
AMS 2010 Mathematics Subject Classifications: 35J50; 35B33.
Keywords:
pLaplacian system; LjusternikSchnirelmann theory; critical exponent; multiple solutions1 Introduction and main results
In this article, we consider the existence and multiplicity of solutions for the following critical pLaplacian system:
where Ω ⊂ ℝ^{N }is a bounded domain with smooth boundary ∂Ω, Δ_{p}u = div(∇u^{p2}∇u) is the pLaplacian operator, denotes the Sobolev critical exponent, is a homogeneous function of degree and λ, δ are positive parameters.
The starting point on the study of the system (1.1) is its scalar version:
with 2 ≤ p ≤ q < p*. In a pioneer work Brezis and Nirenberg [1] showed that, if p = q = 2, the equation (1.2) has at least one positive solution provided N ≥ 4 and 0 < λ < λ_{1}, where λ_{1 }is the first eigenvalue of the operator . In particular, the first multiplicity result for (1.2) has been achieved by Rey [2] in the semilinear case. Precisely Rey proved that if N ≥ 5, p = q = 2, for λ small enough equation (1.2) has at least cat_{Ω}(Ω) solutions, where cat_{Ω}(Ω) denotes the LjusternikSchnirelmann category of Ω in itself. Furthermore, Alves and Ding [3] obtained the existence of cat_{Ω}(Ω) positive solutions to equation (1.2) with p ≥ 2, p ≤ q < p*.
In recent years, more and more attention have been paid to the elliptic systems. In particular, Ding and Xiao [4] concerned the case F(x, u, v) = 2u^{α}v^{β},α > 1, β >1 satisfying α + β = p*, i.e., the following elliptic system
Using standard tools of the variational theory and the LjusternikSchnirelmann category theory, Ding and Xiao [4] have proved that system (1.3) has at least cat_{Ω}(Ω) positive solutions if λ, δ satisfied a certain condition. Hsu [5] obtained the existence of two positive solutions of system (1.3) with the sublinear perturbation of 1 < q < p < N. Recently, Shen and Zhang [6] extended the results in [5] to the case (1.1) with 1 < q < p < N and obtained similar results. In this article, we study (1.1) and complement the results of [5,6] to the case 2 ≤ p ≤ q < p*, also extend the results of [4,7]. To the best of our knowledge, problem (1.1) has not been considered before. Thus it is necessary for us to investigate the critical pLaplacian systems (1.1) deeply. For more similar problems, we refer to [817], and references therein.
Before stating our results, we need the following assumptions:
(F_{2}) are strictly increasing functions about u and v for all u, v > 0.
The main results we get are the following:
Theorem 1.1. Suppose N ≥ p^{2 }and F satisfies (F_{0})(F_{2}), then the problem (1.1) has at least one nonnegative solution for 2 ≤ p < q < p* and λ, δ > 0, or q = p and λ, δ ∈ (0, Λ_{1}), where Λ_{1 }is the first eigenvalue of .
Theorem 1.2. Suppose N ≥ p^{2}, 2 ≤ p ≤ q < p* and F satisfies (F_{0})(F_{2}), then there exists Λ > 0 such that the problem (1.1) has at least cat_{Ω}(Ω) distinct nonnegative solutions for λ, δ ∈ (0,Λ).
Remark 1.1. Theorem 1 in [4]is the special case of our Theorem 1.2 corresponding to F(x,u,v) = 2u^{α}v^{β},α > 1,β > 1,α + β = p*. There are functions F(x,u,v) satisfying the conditions of our Theorems 1.1 and 1.2. Some typical examples are:
where . Obviously, F(x, u, v) satisfies (F_{0})(F_{2}).
This article is organized as follows. In Section 2, some notations and the MountainPass levels are established and the Theorem 1.1 is proved. We present some technical lemmas which are crucial in the proof of the Theorem 1.2 in Section 3. Theorem 1.2 is proved in Section 4.
2 Notations and proof of Theorem 1.1
Throughout this article, C, C_{i }will denote various positive constants whose exact values are not important, → (respectively ⇀) denotes strong (respectively weak) convergence. O(ε^{t}) denotes O(ε^{t})/ε^{t }≤ C, o_{m}(1) denotes o_{m}(1) → 0 as m → ∞. L^{s}(Ω)(1 ≤ s < +∞) denotes Lebesgue spaces, the norm L^{s }is denoted by  · _{s }for 1 ≤ s < + ∞. Let B_{r}(x) denotes a ball centered at x with radius r, the dual space of a Banach space E will be denoted by E^{1}. We define the product space endowed with the norm , and the norm .
Using assumption of (F_{1}), we have the socalled Euler identity
In addition, we can extend the function F(x,u,v) to the whole by considering , where u^{+ }= max{u,0}. It is easy to check that is of class C^{1 }and its restriction to coincides with F(x,u,v). In order to simplify the notation we shall write, from now on, only F(x,u,v) to denote the above extension.
A pair of functions (u, v) ∈ E is said to be a weak solution of problem (1.1) if
Thus, by (2.1) the corresponding energy functional of problem (1.1) is defined on E by
Using (F_{0})(F_{2}), we can verify I_{λ, δ}(u, v) ∈ C^{1}(E, ℝ) (see [6]). It is well known that the weak solutions of problem (1.1) are the critical points of the energy functional I_{λ, δ}(u, v).
The functional I ∈ C^{1}(E, ℝ) is said to satisfy the (PS)_{c }condition if any sequence {u_{m}} ⊂ E such that as m → ∞, I(u_{m}) → c, I'(u_{m}) → 0 strongly in E^{1 }contains a subsequence converging in E to a critical point of I. In this article, we will take I = I_{λ, δ}(u, v) and .
As the energy functional I_{λ,δ }is not bounded below on E, we need to study I_{λ,δ }on the Nehari manifold
Note that contains every nonzero solution of problem (1.1), and define the minimax c_{λ,δ }as
Next, we present some properties of c_{λ,δ }and . Its proofs can be done as [18, Theorem 4.2]. First of all, we note that there exists ρ > 0, such that
It is standard to check that I_{λ,δ }satisfies MountainPass geometry, so we can use the homogeneity of F to prove that c_{λ,δ }can be alternatively characterized by
where Γ = {γ ∈ C([0, 1],E) : γ(0) = 0,I_{λ,δ}(γ(1)) < 0}. Moreover, for each (u, v) ∈ E\{(0,0)}, there exists a unique t* > 0 such that . The maximum of the function t ↦ I_{λ,δ}(t(u, v)), for t ≥ 0, is achieved at t = t*.
In this section, we will find the range of c where the (PS)_{c }condition holds for the functional I_{λ,δ}. First let us define
Lemma 2.1. If N ≥ p^{2 }and F satisfies (F_{0})(F_{2}), then the functional I_{λ,δ }satisfies the (PS)_{c }condition for all , provide one of the following conditions holds
(i) 2 ≤ p < q < p* and λ, δ > 0;
(ii) q = p, and λ, δ ∈ (0, Λ_{1}), where Λ_{1 }> 0 denotes the first eigenvalue of .
Proof. Let {(u_{m}, v_{m})} ⊂ E such that and . Now, we first prove that {(u_{m}, v_{m})} is bounded in E. If the above item (i) is true it suffices to use the definition of I_{λ,δ }to obtain C_{1 }> 0 such that
The above expression implies that {(u_{m}, v_{m})} ⊂ E is bounded. When (ii) occurs, in this case, it follows that
and therefore we get
Since λ, δ ∈ (0,Λ_{1}) the boundedness of {(u_{m}, v_{m})} follows as the first case.
So, {(u_{m}, v_{m})} is bounded in E. Going if necessary to a subsequence, we can assume that
as m → ∞. Clearly, we have
Moreover, a standard argument shows that . Thus we get
Let , then by BrezisLieb Lemma in [19] implies
By the same method of [8, Lemma 5] (or [6, Lemma 3.4]), we obtain
By (2.4)(2.7) and the weak convergence of (u_{m}, v_{m}), we have
By using and (2.4), (2.6), and (2.7), we get
Recalling that , we can use the above equality and (2.8) to obtain
where k is a nonnegative number.
In view of the definition of S_{F}, we have that
Taking the limit we get . So, if k > 0, we conclude that and therefore
which is a contradiction. Hence k = 0 and therefore (u_{m}, v_{m}) → (u, v) strongly in E.
Before presenting our next result we recall that, for each ε > 0, the function
satisfies
where S is the best constant of the Sobolev embedding . Thus, using [8, Lemma 3] and the homogeneity of F, we obtain A, B > 0 such that
from which and (2.10) it follows that
We define a cutoff function such that ϕ(x) = 1 if x ≤ R; ϕ(x) = 0 if x ≥ 2R and 0 ≤ ϕ(x) ≤ 1, where B_{2R}(0) ⊂ Ω, set , where U_{ε }was defined in (2.9). So that . Then, we can get the following results from [[20], Lemma 11.1]:
where A ≈ B means C_{1}B ≤ A ≤ C_{2}B.
Lemma 2.2. Suppose that F satisfies (F_{0})(F_{2}), 2 ≤ p < q < p* and λ > 0, δ > 0, then . The same result holds if q = p and λ, δ ∈ (0,Λ_{1}), where Λ_{1 }> 0 denotes the first eigenvalue of .
Proof. We can use the homogeneity of F to get, for any t ≥ 0,
We shall denote by h(t) the righthand side of the above equality and consider two distinct cases.
Case 1. 2 ≤ p < q < p*.
From the fact that and h(t) > 0 when t is close to 0, there exists t_{ε }> 0 such that
Let
and notice that the maximum value of g(t) occurs at the point
So, for each t ≥ 0,
and therefore
We claim that, for some C_{2 }> 0, there holds
Indeed, if this is not the case, we have that for some sequence ε_{m }→ 0^{+}, then,
which is a contradiction. So, the claim holds and we infer from (2.15) and (2.11)(2.13) that
where . We know if N ≥ p^{2}. By N ≥ p^{2 }and 2 ≤ p < q < p* we obtain . Thus from the above inequality we conclude that, for each ε > 0 small, there holds
Case 2. q = p.
In this case, we have that h'(t) = 0 if and only if,
Since we suppose λ, δ ∈ (0,Λ_{1}), we can use Poincaré's inequality to obtain
Thus, there exists t_{ε }> 0 satisfying (2.14).
Arguing as in the first case we conclude that, from (2.16) for ε > 0 small, there holds
Because if N > p^{2 }and if N = p^{2}, then ε^{p1 }= o(ε^{p1} ln ε). If N > p^{2}, then , so . Choosing ε > 0 small enough, we have
This concludes the proof.
By Lemmas 2.1 and 2.2 we can prove our first result.
Proof of Theorem 1.1.
Since I_{λ,δ }satisfies the geometric conditions of the MountainPass theorem, there exists {(u_{m}, v_{m})} ⊂ E such that . It follows from Lemmas 2.1 and 2.2 that {(u_{m}, v_{m})} converges, along a subsequence, to a nonzero critical point (u,v) ∈ E of I_{λ,δ}. Then, if we denote by u^{ }= max{u,0} and v^{ }= max{v,0} the negative part of u and v, respectively, we get
it follows that (u^{},v^{}) = (0,0). Hence, u,v ≥ 0 in Ω. The Theorem 1.1 is proved.
We finalize this section with the study of the asymptotic behavior of the minimax level c_{λ,δ }as both the parameters λ, δ approach zero.
Proof. We first prove the second equality. It follows from λ = δ = 0 that λu^{q }+ δv^{q }≡ 0. If A, B, u_{ε}, g_{ε}, and t_{ε }are the same as those in the proof of Lemma 2.2, we have that . Thus
Taking the limit as ε →0^{+ }and using (2.11), we conclude that .
In order to obtain the reverse inequality we consider {(u_{m}, v_{m})} ⊂ E such that I_{0,0 }(u_{m}, v_{m}) → c_{0,0 }and . It is easy to show that the sequence {(u_{m}, v_{m})} is bounded in E and therefore . It follows that
Taking the limit in the inequality we conclude, as in the proof of Lemma 2.1, that . Hence,
We proceed now with the calculation of . Let {λ_{m}},{δ_{m}} ⊂ ℝ^{+ }such that λ_{m}, δ_{m }→ 0^{+}. Since λ_{m}, δ_{m }are positive, we have that whenever (u, v) is nonnegative. Thus, for this kind of function, we have that .
It follows that
in the last equality, we have used the infimum c_{0,0 }which can be attained at a nonnegative solution. The above inequality implies that
On the other hand, it follows from Theorem 1.1 that there exists {(u_{m}, v_{m})} ⊂ E such that
Since is bounded, the same argument performed in the proof of Lemma 2.1 implies that {(u_{m}, v_{m})} is bounded in E. Since
Let t_{m }> 0 be such that . Since , we have that
If {t_{m}} is bounded, we can use the above estimate and (2.18) to get
This and (2.17) we get
It remains to check that {t_{m}} is bounded. A straightforward calculation shows that
Hence , and therefore from the above expression it follows that ∫_{Ω }F(x, u_{m}, v_{m})dx ≥ C_{5 }> 0. Thus, the boundedness of {(u_{m}, v_{m})} and (2.19) imply that {t_{m}} is bounded. This completes the proof.
3 Some technical lemmas
In this section, we denote by the Banach space of finite Radon measures over Ω equipped with the norm . A sequence is said to converge weakly to provided σ_{m}(φ) → σ(φ) for all φ ∈ C_{0}(Ω). By [18, Theorem 1.39], every bounded sequence contains a weakly convergent subsequence.
The next lemma is a version of the second concentrationcompactness lemma of Lions [21]. It is also inspired by [18, Lemma 1.40] and [[22], Lemma 2.4].
Lemma 3.1. Suppose that the sequence {(u_{m},v_{m})} ⊂ D^{1,p}(ℝ^{N}) × D^{1,p}(ℝ^{N}) satisfies
and define
then it follows that
Moreover, if (u,v) = (0,0) and , then the measures μ,ν, and σ are concentrated at a single point, respectively.
Proof. We first recall that, in view of the definition of S_{F}, for each nonnegative function we have
Moreover, arguing as [8, Lemma 5], we have that
Since F is p*homogeneous, we can use the two above expressions and argue along the same line of the proof of Lemma 1.40 in [18] to conclude that (3.2)(3.5) hold. If (u, v) = (0,0) and , the same argument of step 3 of the proof of Lemma 1.40 in [18] implies that the measures μ, ν and σ are concentrated at a single point, respectively.
Remark 3.1. We notice that the last conclusion of the above result holds even if (u, v) ≢ (0,0). Indeed, in this case we can define and notice that
Since and therefore , and , where μ,σ, and ν are the same as those in Lemma 3.1. Thus, if we also have that and the result follows from the last part of Lemma 3.1.
Now, we introduce the following Lemma.
Lemma 3.2. Suppose {(u_{m}, v_{m})} ⊂ E such that ∫_{Ω }F(x, u_{m}, v_{m})dx = 1 and . Then there exist {r_{m}} ⊂ (0, +∞) and {y_{m}} ⊂ ℝ^{N }such that
contains a convergent subsequence denoted again by such that in D^{1,p}(ℝ^{N}) × D^{1,p}(ℝ^{N}). Moreover, as m → ∞, we have r_{m }→ 0 and .
Proof. For each r > 0, we consider the Lévy concentration functions
Since for every m,
there exist r_{m }> 0 and a sequence satisfying
Recalling that , we conclude that is bounded. Hence, up to a subsequence, and we obtain
We shall prove that the above sequences {r_{m}} and {y_{m}} satisfy the statements of the lemma. First notice that
By (3.6), a straightforward calculation provides
Hence, we can apply Lemma 3.1 to obtain (ω_{1},ω_{2}) ∈ D^{1,p}(ℝ^{N}) × D^{1,p }(ℝ^{N}) satisfying
The second equality in (3.8) implies that . If one of these values belongs to the open interval (0,1), we can use (3.8), and (3.9) to get
which is a contradiction. Thus . Actually, it follows from (3.7) that for any R > 1. Thus, we conclude that ν_{∞ }= 0.
Let us prove that ν = 0. Arguing by contradiction, then ν = 1. It follows from the first equality in (3.8) that S_{F }≥ μ + σ. On the other hand, the first inequality in (3.9) provides μ + σ ≥ S_{F}. Hence, we conclude that μ + σ = S_{F}. Since we suppose that ν = 1 we obtain . It follows from Remark 3.1 that for some x_{0 }∈ ℝ^{N}. Thus, from (3.7), we get
This contradiction proves that ∥ν∥ = 0.
Since ∥ν∥ = ν_{∞ }= 0, we have that . This and (3.8) provide
So, and therefore strongly in D^{1,p}(ℝ^{N}) × D^{1,p}(ℝ^{N}) and for a.e. x ∈ ℝ^{N}. In order to conclude the proof we notice that
Since {(u_{m}, v_{m})} is bounded and (ω_{1}, ω_{2}) ≢ (0,0), we infer from the above equality that, up to a subsequence, r_{m }→ r_{0 }≥ 0. If y_{m} → ∞, for each fixed x ∈ ℝ^{N}, we have that there exists m_{x }∈ N such that r_{m}x + y_{m }∉ Ω for m ≥ m_{x}. For such values of m we have that . Taking the limit and recalling that x ∈ ℝ is arbitrary, we conclude that (ω_{1}, ω_{2}) = (0,0), which is a contradiction. So, along a subsequence, y_{m }→y ∈ ℝ^{N}.
We claim that r_{0 }= 0. Indeed, suppose by contradiction that r_{0 }> 0. Then, as m becomes large, the set Ω_{m }= (Ωy_{m})/r_{m }approaches Ω_{0 }= (Ω y)/r_{0 }≠ ℝ^{N}. This implies that ω_{1},ω_{2 }has compact support in ℝ^{N}. On the other hand, since (ω_{1},ω_{2}) achieves the infimum in (2.3) and F is homogeneous, we can use the Lagrange Multiplier Theorem to conclude that ( ω_{1}, ω_{2}) satisfies
for . It follows from (F_{2}) and the maximum principle that at least one of the functions ω_{1},ω_{2 }is positive in ℝ^{N}. But this contradicts supp (ω_{1},ω_{2}) ⊂ Ω_{0}. Hence, we conclude that r_{0 }= 0. Finally, if we obtain r_{m}x + y_{m }∉ Ω for large values of m, and therefore we should have (ω_{1}, ω_{2}) ≡ (0, 0) again. Thus, and the proof is completed.
Up to translations, we may assume that 0 ∈ Ω, since Ω is a smooth bounded domain of ℝ^{N}, we can choose r > 0 small enough such that B_{r }= B_{r}(0) = {x ∈ ℝ^{N }: d(x, 0) < r} ⊂ Ω and the sets
are homotopically equivalent to Ω. Let
and
We define the functional
and set
where
Clearly, m_{λ},_{δ }is nonincreasing in λ, δ. Note that m_{λ},_{δ }> 0 for all λ, δ > 0.
Arguing as in the proof of Lemma 2.3 and Theorem 1.1, we obtain the following result.
Lemma 3.3. Suppose F satisfies (F_{0})(F_{2}), then the infimum m_{λ,δ }is attained by a nonnegative radial function (u_{λ,δ}, v_{λ,δ}) ∈ E_{rad }whenever 2 ≤ p < q < p* and λ,δ > 0, or q = p and λ,δ ∈ (0,Λ_{1,rad}), where Λ_{1,rad }> 0 is the first eigenvalue of the operator . Moreover,
We introduce the barycenter map as follows
This map has the following property.
Lemma 3.4. If N ≥ p^{2},2 ≤ p ≤ q < p* and F satisfies (F_{0})(F_{2}), then there exists λ* > 0 such that whenever and I_{λ,δ}(u, v) ≤ m_{λ,δ}.
Proof. By way of contradiction, we suppose that there exist {λ_{m}}, {δ_{m}} ⊂ ℝ^{+ }and such that λ_{m}, δ_{m }→ 0^{+ }as but .
From and we have that {(u_{m}, v_{m})} is bounded in E. Moreover,
Since λ_{m}, δ_{m }→ 0^{+}, we can use the boundedness of {(u_{m}, v_{m})} to get
from which it follows that
Notice that
Recalling that and both converge to , we can use the above expression and ∫_{Ω}(λ_{m}u_{m}^{q }+ δ_{m}v_{m}^{q})dx → 0 again to conclude that , that is,
Let and notice that t_{m}(u_{m}, v_{m}) satisfies the hypotheses of Lemma 3.2. Using Lemma 3.2, there exist sequences {r_{m}} ⊂ (0,+∞) and {y_{m}} ⊂ ℝ^{N }satisfying we have that in D^{1,p }(ℝ^{N}) × D^{1,p }(ℝ^{N}).
The definition of β(u, v), (3.10), the strong convergence of and Lebesgue's theorem provide
Since and ∫_{Ω }F(x,ω_{1},ω_{2})dx = 1, the above expression implies that
According to Lemma 3.3, for each λ, δ > 0 small the infimum m_{λ,δ }is attained by a nonnegative radial function . We consider
and define the function by setting, for each ,
A change of variables and straightforward calculations show that the map γ is well defined. Since σ_{λ,δ }is radial, we have that . Hence, for each , we obtain
Along the way of proving Lemma 3.4 we can check easily the following
Lemma 3.5. If λ,δ → 0^{+}, α_{λ,δ }→ 1.
Proof. By Lemma 3.3, we have
As before . Thus, , the above expression and the same arguments used in the proof of Lemma 3.3 imply that
The above equality and the definition of α_{λ,δ }imply that α_{λ,δ }→ 1. The lemma is proved.
We have the following
Lemma 3.6. if F satisfies (F_{0})(F_{2}), then there exists λ** > 0 such that
for all λ, δ ∈ (0, λ**).
Proof. Arguing by contradiction, we suppose that there exist sequences {λ_{m}},{δ_{m}} ⊂ ℝ^{+ }and such that λ_{m}, δ_{m }→ 0^{+}, as m → ∞, and for all m. Up to a subsequence t_{m }→ t_{0 }∈ [0, 1]. Moreover, the compactness of and Lemma 3.4 imply that, up to a subsequence, . From Lemma 3.5 . So, we can use the definition of H_{λ,δ }to conclude that , which is a contradiction. The lemma is proved.
4 Proof of Theorem 1.2
We begin with the following lemma.
Lemma 4.1. If (u, v) is a critical point of I_{λ,δ }on , then it is a critical point of I_{λ,δ }in E.
Proof. The proof is almost the same as that [4, Lemma 4.1] and is omitted here.
Lemma 4.2. Suppose F satisfies (F_{0})(F_{2}), then any sequence such that and contains a convergent subsequence for λ,δ > 0 if q > p and λ,δ ∈ (0, λ*) if q = p for some small λ* > 0.
Proof. By hypothesis there exists a sequence θ_{m }∈ ℝ such that as m → ∞, where . Thus
Recall that
Consequently ∥(u_{m},v_{m})∥_{E }→ 0.
On the other hand, if it follows that
for some C > 0. Hence we arrive at a contradiction if λ, δ > 0 and q > p or λ, δ ∈ (0, λ*) for small λ* > 0 when q = p. Thus we may assume that . Since , we conclude that θ_{m }= 0, consequently, . Using this information we have
so by Lemma 2.1 the proof is completed.
Below we denote by the restriction of I_{λ,δ }on .
Lemma 4.3. Suppose N ≥ p^{2},2 ≤ p ≤ q < p* and F satisfies (F_{0})(F_{2}), let Λ = min{λ*,λ**} > 0, λ, δ ∈ (0,Λ), then , where λ*, λ** given by Lemmas 34 and 3.6, respectively.
Proof. Assume that , where A_{j},j = 1,2,...,m, are closed and contractible sets in , i.e., there exists such that
where ϑ ∈ A_{j }is fixed. Consider B_{j }= γ1(A_{j}), 1 ≤ j ≤ m. The sets B_{j }are closed and
We define the deformation g_{j }: [0, 1] × B_{j }by setting
for λ,δ ∈ (0,Λ). Note that
implies
and g_{j}(1,y) = H_{λ,δ}(1, h_{j}(1,γ (y))) = β(h_{j}(1,γ(y))) implies
Thus the sets B_{j }are contractible in . It follows that .
Proof of Theorem 1.2.
Using Lemmas 2.1, 2.2, and 3.3 we know that for λ,δ ∈ (0,Λ). Moreover, by Lemma 4.2, satisfies the (PS)_{c }condition for all . Therefore, by Lemma 4.3, a standard deformation argument implies that, for contains at least cat_{Ω}(Ω) critical points of the restriction of I_{λ,δ }on . Now Lemma 4.1 implies that I_{λ,δ }has at least cat_{Ω}(Ω) critical points, and therefore at least cat_{Ω}(Ω) nontrivial solutions of (1.1). As Theorem 1.1, the obtained solutions are nonnegative in Ω. The proof is completed.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author would like to thank the referees for carefully reading this article and making valuable comments and suggestions. This study was supported by the Youth Foundation of Hubei Engineering University (No. Z2012003).
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