In this article, we study the nonlinear elliptic problem of 2-q- Laplacian type −∆u−µ∆qu=−λ|u|r−2u+au+b(u+)θ−1 in Ω, u= 0 on∂Ω, where Ω⊂RN is a bounded domain

Electronic Journal of Differential Equations, Vol. 2014 (2014), No. 170, pp. 1–13.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu

NONLINEAR ELLIPTIC PROBLEM OF 2-q-LAPLACIAN TYPE WITH ASYMMETRIC NONLINEARITIES

DANDAN YANG, CHUANZHI BAI

Abstract. In this article, we study the nonlinear elliptic problem of 2-q- Laplacian type

−∆u−µ∆qu=−λ|u|^r−2u+au+b(u⁺)^θ−1 in Ω, u= 0 on∂Ω,

where Ω⊂R^N is a bounded domain. Forais between two eigenvalues, we show the existence of three nontrivial solutions.

1. Introduction

In this article, we are interested in finding the multiple nontrivial weak solutions to the nonlinear elliptic problem of 2-q-Laplacian type,

−∆u−µ∆qu=−λ|u|^r−2u+au+b(u⁺)^θ−1 in Ω,

u= 0 on∂Ω, (1.1)

where Ω⊂R^N is a bounded domain with samooth boundary∂Ω,λ, µ >0 are two parameters,N >2, 1<min{q, r} ≤max{q, r}<2< θ≤2^∗=_N^2N₋₂,a∈R,b >0, andu⁺= max{u,0}. ∆_qu= div(|∇u|^q−2∇u) is theq-Laplacian ofu.

Paiva and Presoto [12] studied the semilinear elliptic problem with asymmetric nonlinearities,

−∆u=−λ|u|^q−2u+au+b(u⁺)^p−1 in Ω,

u= 0 on∂Ω. (1.2)

WhereN ≥3, 1< q <2< p≤2^∗,a∈R,b >0 andλis a positive parameter.

Problem (1.2) is also closely related to the class of superlinear Ambrosetti-Prodi problems [6],

−∆u=au+ (u⁺)^p+f(x) in Ω. (1.3) Further results for problem (1.3) can be found in [4, 5, 11, 14] and references cited therein.

2000Mathematics Subject Classification. 35J60, 35B38.

Key words and phrases. Quasilinear elliptic equations withq-Laplacian; critical exponent;

asymmetric nonlinearity; weak solution.

c

2014 Texas State University - San Marcos.

Submitted July 9, 2014. Published August 11, 2014.

1

Marano and Papageorgiou [10] obtained the existence of three solutions of the (p, q)-Laplacian problem

−∆pu−µ∆_qu=f(x, u) in Ω,

u= 0 on∂Ω, (1.4)

by using variational methods and truncation arguments. Nonlinear elliptic problems involving the p-q-Laplacian operator is an active are of research; see [8, 9, 13, 15, 17, 18] and the references therein.

Motivated by the above works, we shall extend the results of problem (1.2) to problem (1.1). By using variational methods, we obtain three solutions to (1.1).

We say thatg is asymmetric whengsatisfies the Ambrosetti-Prodi type condition g₋:= lim

t→−∞

g(t)

t < λ_k< g₊:= lim

t→+∞

g(t) t .

Since problem (1.1) involves−∆ and−∆q, the arguments will be more compli- cated, and more analysis and estimates are needed.

The eigenvalue problem of the Laplacian, in Ω⊂R^N, has the form

−∆u=λu in H₀¹(Ω). (1.5)

By the Ljusternik-Schnirelman principle it is well known that there exists a non- decreasing sequence of nonnegative eigenvalues 0< λ1 < λ2≤ · · · ≤λj ≤. . . and a correspondent eigenfunctionsϕj. Also, the first eigenvalueλ1 is simple and the eigenfunctions associated withλ1do not change sign.

Now we are ready to state our main results.

Theorem 1.1. LetN ≥3,1<min{q, r} ≤max{q, r}<2< θ <2^∗ andλ_k < a <

λ_k+1. Then, for λ > 0 and µ > 0 small enough, problem (1.1) has at least three nontrivial solutions.

Theorem 1.2. LetN ≥4,1<min{q, r} ≤max{q, r}<2< θ= 2^∗ andλk < a <

λk+1. Then, for λ > 0 and µ > 0 small enough, problem (1.1) has at least three nontrivial solutions.

This article is organized as follows. In Section 2, we show some geometric con- ditions to establish the Mountain-Pass levels and give a technical lemma which is crucial in the proof of our main results. In Section 3, we establish the existence of three nontrivial solutions for the nonlinear elliptic problem (1.1).

2. Preliminaries

In this article,k·kpand|·|pdenote the norms onW₀^1,p(Ω) andL^p(Ω), respectively;

kukp=Z

Ω

|∇u|^pdx^1/p

, |u|p=Z

Ω

|u|^pdx^1/p .

For convenience, we substitutek · kfork · k₂. The best Sobolev constant S of the embeddingH₀¹(Ω),→L^2∗(Ω) is denoted by

S= inf

u∈H₀¹(Ω)\{0}

kuk²

|u|²₂∗

.

It is known thatS is independent of Ω and is never achieved except when Ω =R^N (see [16]). Consider the energy functionalIλ,µ defined onH₀¹(Ω) given by

Iλ,µ(u) =1

2kuk²+µ

qkuk^q_q+λ r Z

Ω

|u|^rdx−a 2

Z

Ω

|u|²dx−b θ

Z

Ω

(u⁺)^θdx. (2.1) It is easy to know thatIλ,µis of classC²and there exists a one to one correspon- dence between the weak solutions of (1.1) and the critical points ofIλ,µ onH₀¹(Ω).

By a weak solution of (1.1) we mean thatu∈H₀¹(Ω) satisfying hI_λ,µ⁰ (u), vi=

Z

Ω

[∇u∇v+µ|∇u|^q−2∇u∇v]dx+λ Z

Ω

|u|^r−2uvdx

−a Z

Ω

uvdx−b Z

Ω

(u⁺)^θ−1vdx= 0 for allv∈H₀¹(Ω).

Denote by ϕi a normalized eigenvector relative to eigenvalue λi of (1.5). Let Vk =hϕ1, . . . , ϕkiand Wk = V_k^⊥. Without loss of generality, we suppose 0∈ Ω, andm∈Nlarge enough so thatB2/m⊂Ω, where B2/mdenotes the ball of radius 2/mwith center in 0. Consider the functions introduced in [7],

ζ_m(x) =







0 ifx∈B_1/m,

m|x| −1 ifx∈Am=B_2/m\B_1/m, 1 ifx∈Ω\B_2/m.

Setϕ^m_i =ζmϕi,

V_k^m=hϕ^m₁, ϕ^m₂, . . . , ϕ^m_ki

and W_k^m = (V_k^m)^⊥. For each m ∈ N, define a positive cut-off function η ∈ C_c^∞(B_1/m) such that η ≡ 1 in B_1/2m, η ≤ 1 in B_1/m and k∇ηk_∞ ≤ 4m; take ϕ^m_k+1=ηϕk+1. Then

suppu∩suppϕ^m_k+1=∅ (2.2)

wheneveru∈V_k^m. By [7], it is easy to check the following Lemma.

Lemma 2.1. As m→ ∞we have

ϕ^m_i →ϕi in H₀¹(Ω) and max

u∈V_k^m:R

Ω|u|²=1

kuk²≤λk+ckm^2−N. Corollary 2.2. Formlarge enough

V_k^m⊕W_k =H₀¹. (2.3)

As an easy consequence of Lemma 2.1 we have the following decomposition of H₀¹.

Lemma 2.3. Assume λ1 < a, 1 < min{q, r} ≤ max{q, r} < 2 < θ ≤ 2^∗ and λ, µ >0. Then every(PS)sequence ofIλ,µ is bounded.

Proof. Suppose{un} ⊂H₀¹(Ω) is a (PS) sequence ofI_λ,µ; i.e., it satisfies

1

2kunk²+µ

qkunk^q_q+λ r Z

Ω

|un|^rdx−a 2 Z

Ω

|un|²dx−b θ

Z

Ω

(u⁺_n)^θdx

≤C, (2.4)

Z

Ω

[∇un∇v+µ|∇un|^q−2∇un∇v]dx+λ Z

Ω

|un|^r−2unvdx

−a Z

Ω

unvdx−b Z

Ω

(u⁺_n)^θ−1vdx

≤nkvk, ∀v∈H₀¹(Ω),

(2.5)

wheren →0 asn→ ∞. By (2.4) and (2.5), we obtain C+_nkunk

≥

I_λ(u_n)−1

2hI_λ⁰(u_n), u_ni

=

µ q −µ

2

kunk^q_q+ λ r −λ

2

Z

Ω

|un|^rdx+ b 2 −b

θ

Z

Ω

(u⁺_n)^θdx

≥ b 2 −b

θ

Z

Ω

(u⁺_n)^θdx.

(2.6)

Thus, we have

Z

Ω

(u⁺_n)^θdx≤C+nkunk. (2.7) Moreover, by H¨older inequality, we have

Z

Ω

(u⁺_n)²dx≤ |Ω|^θ−2θ Z

Ω

(u⁺_n)^θdx^2/θ

. (2.8)

On the other hand, by (2.5) we have

|hI_λ,µ⁰ (un), u⁻_ni|=

ku⁻_nk²+µku⁻_nk^q_q+λ|u⁻_n|^r_r−a|u⁻_n|²₂

≤nku⁻_nk, (2.9) withu⁻= max{−u,0}. It follows from (2.4), (2.7), (2.8) and (2.9) that

1

2ku⁺_nk²≤ µ 2 −µ

q

ku⁻_nk^q_q+ λ 2 −λ

r

Z

Ω

|u_n|^r

+a 2

Z

Ω

(u⁺_n)²dx+b θ

Z

Ω

(u⁺_n)^θdx+1

2|hI_λ,µ⁰ (u_n), u⁻_ni|+C

≤a 2 Z

Ω

(u⁺_n)²dx+ b θ

Z

Ω

(u⁺_n)^θdx+_nku⁻_nk+C

≤nkunk+nku⁻_nk+C.

(2.10)

Firstly, we show that (u⁺_n) is bounded inH₀¹(Ω). Suppose by contradiction that ku⁺_nk → ∞, by (2.10), we know that (u⁻_n) is also unbounded. Letwn =un/kunk.

Since{wn} is bounded inH₀¹(Ω), there existsw∈H₀¹(Ω) such that w_n * w inH₀¹(Ω),

w_n→w in L^s, ∀1≤s <2^∗, wn→w a.e. in Ω.

From (2.10), there existsσ >0 satisfying

ku⁻_nk ≥σku⁺_nk² (2.11)

whenevernis large. Notice that w_n⁺= u⁺_n

kunk = u⁺_n

(ku⁺nk²+ku⁻nk²)^1/2 ≤ u⁺_n

(ku⁺nk²+σ²ku⁺nk⁴)^1/2, which implies thatw≤0. Furthermore, by

w⁻_n = u⁻_n

kunk = u⁻_n

(ku⁺nk²+ku⁻nk²)^1/2 = u⁻_n

ku⁻nk · ku⁻_nk (ku⁺nk²+ku⁻nk²)^1/2

and (2.11), we obtainkw⁻_nk →1. Hence, by (2.9),

−λ|u⁻_n|^r_r

ku⁻nk² +µku⁻_nk^q_q

ku⁻nk² +a|u⁻_n|²

ku⁻nk² →1. (2.12) Recalling thatq, r <2, we obtain

ku⁻_nk^q_q

ku⁻nk² ≤ |Ω|^2−q2 ku⁻_nk^q−2→0, (2.13)

|u⁻_n|^r_r

ku⁻nk² ≤ |Ω|^{2∗ −r2∗} S^−22r∗ku⁻_nk^2r∗−2→0. (2.14) Moreover, by (2.11) andkw_n⁻k →1, we have

u⁻_n

ku⁻nk− u⁻_n

ku_nk = u⁻_n ku_nk

kunk ku⁻nk −1

→0 inH₀¹(Ω).

Thus we may exchangeku⁻_nkforkunkin (2.12), and substituting (2.13) and (2.14) into it, we obtain|w⁻_n| →1/√

a, thenw6= 0. Takingv=ϕ₁in (2.5), one has Z

Ω

[∇wn∇ϕ1dx+µkunk^q−1_q kunk

Z

Ω

|∇wn|^q−2∇wn∇ϕ1dx

+ λ

ku_nk Z

Ω

|un|^r−2unϕ1dx−a Z

Ω

wnϕ1dx− b ku_nk

Z

Ω

(u⁺_n)^θ−1ϕ1dx→0;

that is,

(λ1−a) Z

Ω

wnϕ1dx+µku_nk^q−1_q kunk

Z

Ω

|∇wn|^q−2∇wn∇ϕ1dx

+ λ

kunk Z

Ω

|u_n|^r−2u_nϕ₁dx− b kunk

Z

Ω

(u⁺_n)^θ−1ϕ₁dx→0.

(2.15)

Since the second, the third and the fourth term above approach zero, it follows that (λ1−a)

Z

Ω

wϕ1dx= 0,

which is a contradiction, as w ≤0, w 6= 0 andλ1 < a, so that (u⁺_n) is bounded.

Finally, assume that kunk → ∞ and ku⁺_nk ≤C for all n∈N. Taking v =wn in (2.5), by (2.13) and

1 kunk

Z

Ω

(u⁺_n)^θdx→0,

forθ≤2^∗, we obtain a|wn|²₂→1, so thatwn →win L²(Ω) withw6= 0. Then by (2.5) we obtain

Z

Ω

∇w∇vdx−a Z

Ω

wvdx= 0 for allv∈H₀¹(Ω),

with w 6= 0 andw ≤0, which is a contradiction, as a is not the first eigenvalue.

Hence, we conclude that{un}must be bounded inH₀¹(Ω).

In the subcritical case, 1≤θ <2^∗, we can easily know according to the lemma above,I_λ,µ satisfies the (PS) condition at every level.

Lemma 2.4. Let λ₁ < a and θ = 2^∗. For each λ, µ >0, I_λ,µ satisfies the (PS) condition at levelc withc < _N¹b^2−N2 S^N/2.

Proof. Let{un} ⊂H₀¹(Ω) be a sequence satisfying

I_λ,µ(u_n)→c and |hI_λ,µ⁰ (u_n), vi| ≤_nkvk_p, ∀v∈H₀¹(Ω), (2.16) withn→0 asn→ ∞. By Lemma 2.3 we obtain that {un}is bounded. Thus, by passing to a subsequence, we have

un* u inH₀¹(Ω), un→u inL^s, ∀1≤s <2^∗,

un →u a.e. in Ω.

(2.17)

Since{u⁺_n}is bounded inH₀¹(Ω), from the Gagliardo-Nirenberg inequality it follows that {u⁺_n} is also bounded in L^2∗. By passing to a subsequence again, we have u⁺_n * u⁺ in L^2∗. Hence, we obtain by [11, Lemma 2.3] that

−∆u−µ∆_qu=−λ|u|^r−2u+au+b(u⁺)^2∗−1, in Ω

u= 0 on∂Ω, (2.18)

Thus, by (2.18) we have Iλ,µ(u) = µ

q −µ 2

kuk^q_q+ λ r −λ

2

Z

Ω

|u|^rdx+ b 2− b

2^∗

Z

Ω

(u⁺)^2∗dx≥0. (2.19) Setwn =un−u. It is easy to check that

|u⁺_n −u⁺|^s_s≤ |(u_n−u)⁺|^s_s=|w⁺_n|^s_s, 1≤s≤2^∗. (2.20) By (2.16) and the Brezis-Lieb Lemma, we have

kw_nk²+µkw_nk^q_q+λ|w_n|^r_r−a|w_n|²₂−b|u⁺_n −u⁺|²₂^∗∗

=ku_nk²− kuk²+µ(ku_nk^q_q− kuk^q_q) +λ(|u_n|^r_r− |u|^r_r)

−a(|un|²₂− |u|²₂)−b |u⁺_n|²₂^∗∗− |u⁺|²₂^∗∗

+on(1)

=hI_λ,µ⁰ (un), uni − hI_λ,µ⁰ (u), ui+o(1), which implies that

n→∞lim

kwnk²+µkwnk^q_q+λ|wn|^r_r−a|wn|²₂−b|u⁺_n −u⁺|²₂^∗∗

= 0. (2.21) Moreover, by (2.17) we havew_n→0 inL^randL². Thus, we have from (2.20) and (2.21) that

kwnk²+µkwnk^q_q =b|u⁺_n −u⁺|²₂^∗∗+o(1)≤b|w_n⁺|²₂^∗∗+o(1). (2.22) Without loss of generality, we assume that

kwnk²=d+o(1), kwnk^q_q =h+o(1). (2.23) By (2.22), (2.23) and Sobolev inequality, we obtain

d≥Sd+µh b

^2/2∗

≥Sb^−2/2∗d^2/2∗. (2.24) Ifd= 0, then we complete the proof. Otherwise, (2.24) implies that

d≥S^N/2b^2−N2 . (2.25)

Then by (2.16), (2.19) and the Brezis-Lieb Lemma, we conclude c≥c−I_λ,µ(u) =I_λ,µ(u_n)−I_λ,µ(u) +o(1)

= 1

2 kunk²− kuk² +µ

q kunk^q_q− kuk^q_q +λ

r(|un|^r_r− |u|^r_r)−a

2 |un|²₂− |u|²₂

− b

2^∗ |u⁺_n|²₂^∗∗− |u⁺|²₂^∗∗

+o(1)

= 1

2kwnk²+µ

qkwnk^q_q+λ

r|wn|^r_r−a

2|wn|²₂− b

2^∗|u⁺_n −u⁺|²₂^∗∗+o(1).

(2.26)

Letn→ ∞in (2.26), we obtain by (2.22), (2.23), (2.25) andw_n →0 inL^randL² that

c≥ d 2 +µh

q −d+µh 2^∗

= 1 2− 1

2^∗

d+ µ q − µ

2^∗ h

≥ 1 2− 1

2^∗ d

≥ 1

NS^N/2b^2−N2 ,

which is a contradiction.

3. Main result

Firstly, we consider the existence of the nonnegative solution of (1.1) . Define the functionalI_λ,µ⁺ :H₀¹(Ω)→Ras follows

I_λ,µ⁺ (u) =1

2kuk²+µ

qkuk^q_q+λ r Z

Ω

(u⁺)^rdx−a 2

Z

Ω

(u⁺)²dx−b θ

Z

Ω

(u⁺)^θdx. (3.1) It follows that I_λ,µ⁺ ∈ C¹ and the critical points u+ of I_λ,µ⁺ satisfy u+ ≥ 0 and so are critical points of Iλ,µ as well, actually, (I_λ,µ⁺ )⁰(u+)[(u+)⁻] = −k(u+)⁻k²− µk(u+)⁻k^q_q = 0.

Similar to the proofs of Lemma 2.3 and Lemma 2.4, we can show that I_λ,µ⁺ satisfies the (PS) condition.

Lemma 3.1. Let 2< θ≤2^∗. If λ, µ >0, thenI_λ,µ⁺ satisfies the(PS)condition at level cwith c < _N¹S^N/2b^2−N2 .

Lemma 3.2. The trivial solution u ≡ 0 is a local minimizer for I_λ,µ⁺ , for all λ, µ >0.

Proof. It suffices to show that 0 is a local minimizer of I_λ,µ⁺ in the topology (see [3]). Foru∈C₀¹(Ω), we have

I_λ,µ⁺ (u) = 1

2kuk²+µ

qkuk^q_q+λ r Z

Ω

(u⁺)^rdx−a 2 Z

Ω

(u⁺)²dx−b θ

Z

Ω

(u⁺)^θdx

≥ λ r Z

Ω

(u⁺)^rdx−a 2

Z

Ω

(u⁺)²dx−b θ

Z

Ω

(u⁺)^θdx

≥λ r −a

2|u|^2−r_C0 −b θ|u|^θ−r_C0

Z

Ω

(u⁺)^rdx≥0

whenever

a

2|u|^2−r_C0 +b

θ|u|^θ−r_C0 ≤ λ r.

Lemma 3.3. There existst0>0such thatI_λ,µ⁺ (t0ϕ1)≤0, for allλ, µin a bounded set.

Proof. Letϕ₁ be the positive eigenfunction associated toλ₁, fort >0, we have I_λ,µ⁺ (tϕ₁) =t²

2kϕ₁k²+t^qµ

q kϕ₁k^q_q+t^rλ r

Z

Ω

ϕ^r₁dx−t²a 2

Z

Ω

ϕ²₁dx−t^θb θ

Z

Ω

ϕ^θ₁dx

=t²

2(λ1−a) Z

Ω

ϕ²₁dx+t^qµ

q kϕ1k^q_q+t^rλ r

Z

Ω

ϕ^r₁dx−t^θb θ

Z

Ω

ϕ^θ₁dx

Sinceλ₁< aandq, r <2< θ, there exists a choice oft₀>0 such thatI_λ,µ⁺ (t₀ϕ₁)≤

0 forλ, µ in a bounded set.

Define

c⁺_λ,µ= inf

γ∈Γ⁺ sup

t∈[0,1]

I_λ,µ⁺ (γ(t)), where

Γ⁺={γ∈ C([0,1], γ(0) = 0, γ(1) =t0ϕ1}.

On the other hand, by the proof of Lemma 3.3, we obtain I_λ,µ⁺ (tϕ1)≤ t^qµ

q kϕ1k^q_q+t^rλ r

Z

Ω

ϕ^r₁dx.

Then, if λand µ are small enough, c⁺_λ,µ < _N¹S^N/2b^2−N2 , consequently, by means of the Mountain Pass Theorem,c⁺_λ,µ is a critical value ofI_λ,µ⁺ . Thus, we have the following result.

Lemma 3.4. Let N >2,1<min{q, r} ≤max{q, r}<2< θ≤2^∗ andλ1< a. If λ, µ are small enough, then (1.1)has at least a nontrivial positive solution.

To obtain the negative solution, consider the functionalI_λ,µ⁻ :H₀¹(Ω)→Rgiven by

I_λ,µ⁻ (u) = 1

2kuk²+µ

qkuk^q_q+λ r Z

Ω

(u⁻)^rdx−a 2 Z

Ω

(u⁻)²dx. (3.2) Again, I_λ,µ⁻ ∈ C¹ and the critical points u₋ of I_λ,µ⁻ satisfy u₋ ≤ 0 and so are critical points ofI_λ,µas well. We will apply once again the mountain pass theorem to obtain a critical point ofI_λ,µ⁻ .

Lemma 3.5. The trivial solution u ≡ 0 is a local minimizer for I_λ,µ⁻ , for all λ, µ >0.

Proof. It suffices to show that 0 is a local minimizer ofI_λ,µ⁻ in the topology. For u∈C₀¹(Ω), we have

I_λ,µ⁻ (u) = 1

2kuk²+µ

qkuk^q_q+λ r Z

Ω

(u⁻)^rdx−a 2 Z

Ω

(u⁻)²dx

≥ λ r Z

Ω

(u⁻)^rdx−a 2 Z

Ω

(u⁻)²dx

≥ λ r −a

2|u|^2−r_C0

Z

Ω

(u⁻)^rdx≥0

whenever ^a₂|u|^2−r_C0 ≤λ/r.

Lemma 3.6. There exists t0 > 0 such that I_λ,µ⁻ (−t0ϕ1) ≤ 0, for all λ, µ in a bounded set.

Proof. Fort >0, we have I_λ,µ⁻ (−tϕ1) = t²

2kϕ1k²+t^qµ

q kϕ1k^q_q+t^rλ r

Z

Ω

ϕ^r₁dx−t²a 2

Z

Ω

ϕ²₁dx

= t²

2(λ1−a) Z

Ω

ϕ²₁dx+t^qµ

q kϕ1k^q_q+t^rλ r

Z

Ω

ϕ^r₁dx.

Sinceλ₁< aand r, q <2, there exists a choice oft₀ >0 which proves the lemma.

As in the nonnegative solution case, we obtain a critical value

c⁻_λ,µ= inf

γ∈Γ⁻ sup

t∈[0,1]

I_λ,µ⁻ (γ(t)), where

Γ⁻={γ∈ C([0,1] :γ(0) = 0, γ(1) =−t0ϕ₁}.

Similar to the proof of Lemma 3.5, we obtain the estimate c⁻_λ,µ≤ max

s∈[0,1]I_λ,µ⁻ (−st₀ϕ₁)≤ t^q₀µ

q kϕ₁k^q_q+t^r₀λ r

Z

Ω

ϕ^r₁dx,

which implies that if λ, µ are small enough, then we obtain the estimate c⁻_λ,µ <

1

NS^N/2b^2−N2 , consequently, by the Mountain Pass Theorem, c⁻_λ,µ is a critical value ofI_λ,µ⁻ . Hence, we obtain another important result.

Lemma 3.7. Let N >2,1<min{q, r} ≤max{q, r}<2< θ≤2^∗ andλ1< a. If λ, µ small enough, then (1.1)has at least a nontrivial negative solution.

ForWk andV_k^m are as in Section 2, we now consider the existence of the third solution.

Lemma 3.8. There exist α >0 andρ >0 such that Iλ,µ(u)≥α wheneveru∈Wk andkuk=ρ.

Proof. Ifu∈Wk, then Iλ,µ(u) = 1

2kuk²+µ

qkuk^q_q+λ r Z

Ω

|u|^rdx−a 2 Z

Ω

|u|²dx− b θ

Z

Ω

(u⁺)^θdx

≥ 1

2kuk²−a 2 Z

Ω

|u|²dx−b θ

Z

Ω

(u⁺)^θdx

≥ 1 2 − a

2λk+1

kuk²−b θ|u|^θ_θ

≥ kuk² A−Bkuk^θ−2 ,

withA, B >0. Then it suffices to takeρ <(A/B)^θ−21 .

Lemma 3.9. Given λ0>0 andµ0>0, there existm0∈NandR > ρ such that Iλ,µ(u)≤ µ

qkuk^q_q+λ r Z

Ω

|u|^rdx,

whenever u∈∂Qm, where Qm = (BR∩V_k^m)⊕[0, Rϕ^m_k+1], m≥m0, λ≤λ0 and µ≤µ0. Henceforth∂ means the boundary relative to subspaceV_k^m.

Proof. Letmbe large enough anda_k < asuch that

λ_k+c_km^2−N ≤a_k < a. (3.3) Foru∈V_k^m, by Lemma 2.1 and (3.3) one can obtain

I_λ,µ(u) =1

2kuk²+µ

qkuk^q_q+λ r Z

Ω

|u|^rdx−a 2 Z

Ω

|u|²dx− b θ

Z

Ω

(u⁺)^θdx

≤ 1 2 − a

2ak

kuk²+µ

qkuk^q_q+λ r Z

Ω

|u|^rdx− b θ

Z

Ω

(u⁺)^θdx

≤µ

qkuk^q_q+λ r Z

Ω

|u|^rdx,

(3.4)

and

Iλ,µ(ξϕ^m_k+1)

=ξ²

2 kϕ^m_k+1k²+µξ^q

q kϕ^m_k+1k^q_q+λξ^r r

Z

Ω

|ϕ^m_k+1|^rdx

−aξ² 2

Z

Ω

|ϕ^m_k+1|²dx−bξ^θ θ

Z

Ω

((ϕ^m_k+1)⁺)^θdx

≤ξ²

2 kϕ^m_k+1k²+µ0ξ^q

q kϕ^m_k+1k^q_q+λ0ξ^r r

Z

Ω

|ϕ^m_k+1|^rdx−bξ^θ θ

Z

Ω

((ϕ^m_k+1)⁺)^θdx.

(3.5)

Sinceϕ^m_k+1 →ϕk+1 in W₀^1,2(Ω) asm→ ∞, ϕk+1 changes of sign, andθ >2, q, r, there existm0∈Nand R >0 such that

Iλ,µ(Rϕ^m_k+1)≤0 ∀m≥m0. (3.6) Then combining (2.2), (3.4) and (3.6) leads to

Iλ,µ(u)≤ µ

qkuk^q_q+λ r Z

Ω

|u|^rdx, (3.7)

wheneveru∈V_k^m∪(V_k^m⊕Rϕ^m_k+1). By (3.5), there existsβ >0 satisfying

I_λ,µ(ξϕ^m_k+1)≤β, (3.8)

for allξ≥0 andm≥m0. Sincea > λk, we may takeR >0 such that Iλ,µ(u)≤ 1

2 − a 2λ_k

kuk²+µ

qkuk^q_q+λ r Z

Ω

|u|^rdx

≤ −β+µ

qkuk^q_q+λ r Z

Ω

|u|^rdx.

(3.9)

Hence, by (2.2), (3.8) and (3.9) we obtain

I_λ,µ(u+ξϕ^m_k+1) =I_λ,µ(u) +I_λ,µ(ξϕ^m_k+1)≤µ

qkuk^q_q+λ r Z

Ω

|u|^rdx (3.10) for allm≥m0 andu∈∂(BR∩V_k^m). Thus, by (3.7) and (3.10), we complete the

proof.

Proof of Theorem 1.1. For the subcritical case, ifθ <2^∗,αis given by Lemma 3.8.

Takeλandµsmall enough in order that µ

qkuk^q_q+λ r Z

Ω

|u|^rdx < α for allu∈∂Qm. Then by Lemma 3.9 we have

Iλ,µ(u)< α

wheneveru∈∂Qm andm≥m0. Applying the Linking Theorem,Iλ,µpossesses a critical pointuat level cλ,µ, where

c_λ,µ= inf

Γ max

u∈Qm

I_λ,µ(η(u)),

Γ ={η∈ C(Qm, W₀^1,p(Ω));η=Id on∂Qm},

Finally, sincecλ,µ≥α,Iλ,µ(u)≥α >0 andc^±_λ,µ→0 asλ, µ→0. Hence, ifλ, µare small enough c^±_λ,µ < α≤cλ,µ, and we know that umay be neither of the critical points found above forI_λ,µ⁺ andI_λ,µ⁻ ; that is,uis the third solution of (1.1). Thus, combining Lemmas 3.4 and 3.7, we conclude that (1.1) has at least three nontrivial

solutions.

Proof of Theorem 1.2. For the critical case,θ = 2^∗. Consider the family of func- tions taken from [1]:

u= CN^(N−2)/2

(²+|x|²)^(N−2)/2, >0, where

CN = (N(N−2))^(N−2)/4.

Let u^m = ηu, where η is given as section 2, and Q_m = (B_R∩V_k^m)⊕[0, Ru^m ].

Replacingu^m byϕ^m_k+1 in Lemma 3.7, we obtain I_λ,µ(u)≤µ

qkuk^q_q+λ r Z

Ω

|u|^rdx, ∀u∈∂Q_m

wheneverm is large. Hence, to conclude the proof of Theorem 1.2, it remains to show that

sup

u∈Q_m

Iλ,µ(u)< 1

NS^N/2b^2−N2 (3.11)

for all,λandµsmall enough. Let J(u) =1

2kuk²−a 2 Z

Ω

|u|²dx− b 2^∗

Z

Ω

(u⁺)^2∗dx.

Then, we have

Iλ,µ(u) =J(u) +λ r Z

Ω

|u|^rdx+µ qkuk^q_q.

It is sufficient to prove that there existm0>0 and0>0 such that sup

u∈Q_m

J(u)< 1

NS^N/2b^2−N2

for allm≥m0and < 0. It is not difficult to obtain the following expressions [2]:

Z

Ω

|∇u^m |²dx=S^N/2+O(^N−2), (3.12)

Z

Ω

|u^m |^2∗dx=S^N/2+O(^N). (3.13) Moreover, we obtain

Z

Ω

|u^m |²dx= Z

B(0,1/m)

|u|²dx+O(^N−2)

≥ Z

B(0,)

C_N^2N−2 [2²]^N−2 +

Z

<|x|<1/m

C_N^2N−2

[2|x|²]^N−2 +O(^N−2)

=

(d²|ln|+O(²), ifN= 4, d²+O(^N−2), ifN≥5,

(3.14)

where dis a positive constant. If N = 4, according (3.12), (3.13) and (3.14), one has

ku^m k²−a|u^m |²

|u^m |²₂∗

≤ S²−ad²|ln|+O(²) (S²+O(⁴))^1/2

=S−ad²|ln|S⁻¹+O(²)< S, for >0 sufficiently small. And similarly, ifN≥5, we obtain

ku^m k²−a|u^m |²

|u^m |²₂∗

≤ S^N/2−ad²+O(^N−2) (S^N/2+O(^N))^2/2∗

=S−ad²S^(2−N)/2+O(^N−2)< S,

for >0 sufficiently small. Let u=v+tu^m ∈ Q_m. By simple computation, we obtain

maxt≥0 J(tu^m ) = b^2−N2 N

ku^m k²−a|u^m |²

|u^m |²₂∗

N/2

< 1

NS^N/2b^2−N2 . (3.15) Fixm0>0 such thatλk+ckm^2−N₀ ≤σ < a. Then, form≥m0, we obtain

J(v) =1

2kvk²−a 2 Z

Ω

|v|²dx− b 2^∗

Z

Ω

(v⁺)^2∗dx

≤1

2kvk²−a

2|v|²≤σ

2|v|²−a

2|v|²≤0.

(3.16)

From (3.15) and (3.16), we obtain

J(u) =J(v+tu^m ) =J(v) +J(tu^m )≤J(tu^m )< 1

NS^N/2b^2−N2 .

So, (3.11) holds.

Lettingµ→0 in Theorem 1.1 and Theorem 1.2, we easily show that Theorems 1.1 and 1.2 extend the main results in Paiva and Presoto [12].

Acknowledgments. The authors want to thank the anonymous referees for their valuable comments and suggestions. This work is supported by Natural Science Foundation of Jiangsu Province (BK2011407) and Natural Science Foundation of China (11271364 and 10771212).

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Dandan Yang

School of Mathematical Science, Huaiyin Normal University, Huaian, Jiangsu 223300, China

E-mail address:[email protected]

Chuanzhi Bai

School of Mathematical Science, Huaiyin Normal University, Huaian, Jiangsu 223300, China

E-mail address:[email protected]