p, andR >0, we have 0&lt

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

MULTIPLE SOLUTIONS FOR p-LAPLACIAN PROBLEMS INVOLVING GENERAL SUBCRITICAL GROWTH IN BOUNDED

DOMAINS

NGUYEN THANH CHUNG, PHAM HONG MINH, TRAN HONG NGA

Abstract. Using variational methods, we study the existence of multiple so- lutions for a class ofp-Laplacian problems with concave-convex nonlinearities in bounded domains. Our result improves those in [8, 9] stated only for sub- critical growth.

1. Introduction

In this article, we are interested in the existence of solutions for p-Laplacian problems of the form

−∆pu=g(x, u), x∈Ω,

u= 0, x∈∂Ω, (1.1)

where Ω⊂R^N (N ≥2) is a smooth bounded domain,g: Ω×R→Ris a continuous function satisfying subcritical growth condition.

Problem (1.1) has been studied extensively for many years. Since Ambrosetti and Rabinowitz proposed the mountain pass theorem in 1973 (see [1]), critical point theory has become one of the main tools for finding solutions to elliptic equations and systems of variational type. To apply this theorem, the authors introduced one of very important conditions (Ambrosetti and Rabinowitz type condition) on the nonlinear termg as follows:

(AR) For someθ > p, andR >0, we have

0< θG(x, t)≤g(x, t)t, ∀|t| ≥R, a.e. x∈Ω, where G(x, t) = Rt

0g(x, s)ds. This condition ensures that the energy functional associated to the problem satisfies the Palais-Smale condition ((PS) condition for short). Clearly, if the condition (AR) is satisfied then there exist two positive constantsd₁, d₂ such that

G(x, t)≥d1|t|^µ−d2, ∀(x, t)∈Ω×R.

2010Mathematics Subject Classification. 35D05, 35J60.

Key words and phrases. p-Laplacian problems; general subcritical growth;

concave-convex nonlinearities; variational method.

c

2016 Texas State University.

Submitted February 26, 2016. Published March 18, 2016.

1

This means thatg isp-superlinear at infinity in the sense that lim

|t|→+∞

G(x, t)

|t|^p = +∞.

In recent years, there have been many authors considering problem (1.1) without the (AR) type condition, we refer to some interesting papers on this topic [3, 6, 7, 10, 11, 12, 14, 15] and the references cited there. Miyagaki et al [12], studied problem (1.1) in the semilinear case p= 2 by proposing the following non-global condition on the superlinear termg(x, t): There existst0>0 such that

g(x, t)

t is increasing fort≥t0 and decreasing fort≤ −t0, ∀x∈Ω.

Using the mountain pass theorem with the (PS) condition in [1], the authors ob- tained the existence of a non-trivial weak solution. This result was extended to the p-Laplace operator−∆puby Li et al [10]. It should be noticed that in [10, 12], the authors need the following subcritical growth condition

(A0’) |g(x, t)| ≤C(1+|t|^r−1) for allt∈R, a.e. x∈Ω,r∈[1, p^∗), wherep^∗=_N^{N p}_−p if 1< p < N andp^∗= +∞ifp≥N.

Recently Lan et al [8, 9] studied problem (1.1) by introducing a general type of subcritical growth condition, wherer =p^∗. Using mountain pass theorem [1], they obtained the existence of at least one nontrivial weak solution of (1.1) without (AR) condition. In this article, we consider (1.1) wheng(x, u) =λ|u|^q−2u+f(x, u), i.e.

−∆pu=λ|u|^q−2u+f(x, u), x∈Ω,

u= 0, x∈∂Ω, (1.2)

where Ω⊂R^N (N ≥2) is a smooth bounded domain, 1< q < p, λ is a positive parameter,f : Ω×R→Ris a continuous function satisfying the following general subcritical growth condition

(A0) lim_|t|→+∞f(x, t)/|t|^p∗−1= 0 uniformly a.e. x∈Ω.

In particular, as in [8, 9], we do not use the (AR) condition for the nonlinear term f, see condition (A4) as well as some examples and comments in the papers [8, 9].

Using the mountain pass theorem [1] combined with Ekeland variational principle [5], we will obtain the existence of at least two nontrivial weak solutions for problem (1.1). Our result introduced here is a natural extension from the previous ones for elliptic problems with concave-convex nonlinearities [2, 13]. Regarding this interesting topic, we refer the readers to the paper [4], in which the authors studied elliptic problems with local superlinearity and sublinearity.

To state the main result of this paper, let us introduce the following conditions on the functionf:

(A1) There exists a positive constantt >0 such thatF(x, t)≥0 a.e. x∈Ω and allt∈[0, t], whereF(x, t) :=Rt

0f(x, s)ds.

(A2) lim sup_|t|→0^F(x,t)_|t|p < λ1 uniformly a.e. x∈Ω, where λ1 is the first eigen- value of−∆p.

(A3) lim_|t|→+∞^F_|t|^(x,t)_p = +∞uniformly a.e. x∈Ω.

(A4) There exist constantsθ≥1,C_∗>0 such that θH(x, t) +C_∗≥H(x, st)

for allt∈R,x∈Ω,s∈[0,1], whereH(x, t) =f(x, t)t−pF(x, t).

It should be noticed that the functionf(x, t) =|t|^p−2tlog(1 +|t|) satisfies (A1)–

(A4). We refer the readers to [8, 9] for more details. In this article, we look for weak solutions to (1.2) in the usual Sobolev spaceW₀^1,p(Ω) which is equipped with the normkuk= R

Ω|∇u|^pdx^1/p .

Definition 1.1. We say thatu∈W₀^1,p(Ω) is a weak solution of (1.2) if Z

Ω

|∇u|^p−2∇u∇v dx−λ Z

Ω

|u|^q−2uv dx− Z

Ω

f(x, u)v dx= 0 for allv∈W₀^1,p(Ω).

Our main result is given by the following theorem.

Theorem 1.2. Suppose that (A0)–(A4) are satisfied. Then, there exists λ^∗ > 0 such that for anyλ∈(0, λ^∗), problem (1.2)has two nontrivial weak solutions.

2. Multiple solutions

In this section, we prove our main result. Let us denote byc_i general positive constants. As we will see, in order to obtain the existence of at least two weak solutions for problem (1.2) we use variational methods (mountain pass theorem and Ekeland variational principle).

We look for the weak solutions of (1.2) which are the same as the critical points of the functionalJ :W₀^1,p(Ω)→Rdefined by

J(u) = 1 p Z

Ω

|∇u|^pdx−λ q Z

Ω

|u|^qdx− Z

Ω

F(x, u)dx.

We can see thatJ ∈C¹(W₀^1,p(Ω),R) and J⁰(u)(v) =

Z

Ω

|∇u|^p−2∇u∇v dx−λ Z

Ω

|u|^q−2uv dx− Z

Ω

f(x, u)v dx for allu, v∈W₀^1,p(Ω).

Lemma 2.1. There exists λ^∗ > 0 such that for any λ ∈ (0, λ^∗), we can choose α, ρ >0 so thatJ(u)≥αfor allu∈W₀^1,p(Ω) withkuk=ρ.

Proof. From (A0) and (A2), for any > 0, there exists c()>0 depending on , such that

F(x, t)≤ 1

p(λ1−)|t|^p+c()|t|^p∗ (2.1) for allt∈Rand a.e. x∈Ω. Hence, using Sobolev’s embedding, we have

J(u) = 1 p

Z

Ω

|∇u|^pdx−λ q Z

Ω

|u|^qdx− Z

Ω

F(x, u)dx

≥ 1

pkuk^p−λ

qc₁kuk^q−1

p(λ₁−) Z

Ω

|u|^pdx−c() Z

Ω

|u|^p∗dx

≥ 1 p

1−λ₁− λ1

kuk^p−λ

qc1kuk^q−c()kuk^p∗

=

pλ₁ −λ

qc₁kuk^q−p−c()kuk^p∗−p kuk^p,

(2.2)

wherec() andc₁ are positive constants.

For eachλ >0, we consider the functionγλ: (0,+∞)→Rdefined by γ_λ(t) =λ

qc₁t^q−p−c()t^p∗−p. (2.3) It is clear thatγλ(t) is a continuous function on (0,+∞). Sincep^∗> p > q >1, it follows that

lim

t→0⁺γλ(t) = lim

t→+∞γλ(t) = +∞. (2.4)

Hence, we can findt_∗>0 such that 0< γλ(t_∗) = min_t∈(0,+∞)γλ(t), in which t_∗ is defined by the equation

0 =γ⁰_λ(t_∗) = λc1

q (q−p)t^q−p−1_∗ +c()(p^∗−p)t^p_∗^∗−p−1 or

t_∗= λc₁(p−q) qc()(p^∗−p)

p∗ −q¹

. Some simple computations show that

γλ(t∗) =c2.λ^{p∗ −pp∗ −q} →0 as λ→0⁺. (2.5) From relations (2.3), (2.4) and (2.5), there exists λ^∗ > 0 such that for any λ ∈ (0, λ^∗), we can chooseα >0 andρ >0 so that J(u)≥α >0 for allu∈W₀^1,p(Ω)

withkuk=ρ.

Lemma 2.2. There existsφ∈W₀^1,p(Ω),φ >0such thatJ(tφ)→ −∞ast→+∞.

Proof. (ii) From (A3), it follows that for anyM >0 there exists a constantcM = c(M)>0 depending onM, such that

F(x, t)≥M|t|^p+−cM, for a.e. x∈Ω, ∀t∈R. (2.6) Takeφ∈C₀^∞(Ω) withφ >0, from (2.6) and the definition ofJ, we obtain

J(tφ) =1 p

Z

Ω

|∇tφ|^pdx−λ Z

Ω

1

q|tφ|^qdx− Z

Ω

F(x, tφ)dx

≤1

pktφk^p−M Z

Ω

|tφ|^pdx−λ q Z

Ω

|tφ|^qdx+c_M|Ω|

≤t^p1

pkφk^p−M Z

Ω

|φ|^pdx

−λt^q q

Z

Ω

|φ|^qdx+c_M|Ω|,

(2.7)

wheret >0 and|Ω|denotes the Lebesgue measure of Ω.

From (2.7) and the fact that 1< q < p, ifM is large enough such that 1

pkφk^p−M Z

Ω

|φ|^pdx <0,

then we have lim_t→+∞J(tφ) =−∞.

Lemma 2.3. There exists ψ∈W₀^1,p(Ω),ψ >0 such thatJ(tψ)<0 for all t >0 small enough.

Proof. Takeψ∈C₀^∞(Ω) withψ >0, from the definition ofJ and condition (A1), for allt∈

0,_kψk^t

L∞(Ω)

small enough, we obtain J(tψ) =1

p Z

Ω

|∇tψ|^pdx−λ q Z

Ω

|tψ|^qdx− Z

Ω

F(x, tψ)dx

≤t^p

pkψk^p−λt^q q

Z

Ω

|ψ|^qdx.

(2.8)

From this inequality, taking

0< δ <λpR

Ω|ψ|^qdx qkψk^p

we conclude that J(tψ) < 0 for all 0 < t < min{δ^p−q1 ,_kψk^t

L∞(Ω)}. The proof of

Lemma 2.3 is complete.

Lemma 2.4. The functional J satisfies the (Ce) condition.

Proof. Let{um} ⊂W₀^1,p(Ω) be a (Cc) sequence of the functionalJ, that is, J(um)→c, kJ⁰(um)k_∗(1 +kumk)→0 asm→ ∞, which shows that

c=J(um) +o(1), J⁰(um)(um) =o(1), (2.9) whereo(1)→0 asm→ ∞.

We prove that{um}is bounded inW₀^1,p(Ω). Indeed, by contradiction, we assume that kumk → +∞ as m → ∞. Let wm = _ku^um

mk we obtain wm ∈ W₀^1,p(Ω) with kwmk= 1. Then there existsw∈W₀^1,p(Ω) such that{wm}converges weakly tow inW₀^1,p(Ω) and

w_m(x)→w(x), a.e. in Ω, m→ ∞, (2.10) wm→w strongly in L^r(Ω), m→ ∞, 1≤r < p^∗, (2.11)

|wm|^p_p^∗∗≤c3. (2.12)

Let Ω₆₌ := {x ∈ Ω : w(x) 6= 0}. If x ∈ Ω₆₌ then it follows from (2.10) that lim_m→∞wm(x) = lim_m→∞^u_ku^m(x)

mk =w(x) and thus|um(x)|=|wm(x)|kumk →+∞

asm→ ∞for a.e. x∈Ω₆₌. Using (A3) we have

m→∞lim

F(x, um(x))

|um(x)|^p = +∞, a.e. x∈Ω6=. (2.13) This implies

m→∞lim

F(x, um(x))

|u_m(x)|^p |wm(x)|^p= +∞, a.e. x∈Ω₆₌. (2.14) Using condition (A3) again, there existst0>0 such that

F(x, t)

|t|^p >1 (2.15)

for all x∈Ω and |t| > t₀ >0. SinceF(x, t) is continuous on Ω×[−t₀, t₀], there exists a positive constantc4such that

|F(x, t)| ≤c₄ (2.16)

for all (x, t)∈Ω×[−t0, t0]. From (2.15) and (2.16) there existsc5∈Rsuch that

F(x, t)≥c5 (2.17)

for all (x, t)∈Ω×R. From (2.17), for allx∈Ω andm, we have F(x, um(x))−c5

ku_mk^p ≥0 or

F(x, um(x))

|um(x)|^p |wm(x)|^p− c5

kumk^p ≥0, ∀x∈Ω, ∀m. (2.18) Using (2.9) and the Sobolev embedding, there existsc₆>0 such that

c=J(um) +o(1)

= 1 p Z

Ω

|∇um|^pdx−λ q Z

Ω

|um|^qdx− Z

Ω

F(x, u_m)dx+o(1)

≥ 1

pkumk^p−λc₆

q kumk^q− Z

Ω

F(x, um)dx+o(1);

since 1< q < p, this implies Z

Ω

F(x, um)dx≥ 1

pkumk^p−λc6

q kumk^q−c+o(1)→+∞ asm→ ∞. (2.19) Also we have

kumk^p=p Z

Ω

F(x, um)dx+λp q

Z

Ω

|um|^qdx+pc−o(1)

≥p Z

Ω

F(x, um)dx+pc−o(1)>0 formlarge enough.

(2.20)

Next, we claim that|Ω₆₌|= 0. In fact, if|Ω₆₌| 6= 0, then by relations (2.18), (2.19), (2.20) and the Fatou lemma, we have

+∞= (+∞)|Ω6=|

= Z

Ω6=

lim inf

m→∞

F(x, um(x))

|um(x)|^p |wm(x)|^pdx− Z

Ω6=

lim sup

m→∞

c5

kumk^pdx

= Z

Ω6=

lim inf

m→∞

F(x, um(x))

|um(x)|^p |wm(x)|^p− c5

kumk^p

dx

≤lim inf

m→∞

Z

Ω6=

F(x, um(x))

|u_m(x)|^p |wm(x)|^p− c5

ku_mk^p

dx

≤lim inf

m→∞

Z

Ω

F(x, um(x))

|u_m(x)|^p |wm(x)|^p− c5

ku_mk^p

dx

= lim inf

m→∞

Z

Ω

F(x, u_m(x))

kumk^p dx−lim sup

m→∞

Z

Ω

c₅ kumk^p dx

= lim inf

m→∞

Z

Ω

F(x, u_m(x)) kumk^p dx

≤lim inf

m→∞

R

ΩF(x, um(x))dx pR

ΩF(x, um)dx+pc−o(1).

(2.21)

From (2.19) and (2.21), we obtain

+∞ ≤1 p,

which is a contradiction. This shows that|Ω₆₌|= 0 and thusw(x) = 0 a.e. in Ω.

Since the functiont7→J(tu_m) is continuous int∈[0,1], for eachmthere exists t_m∈[0,1] such that

J(tmum) := max

t∈[0,1]J(tum), m= 1,2, . . . . (2.22) It is clear that tm > 0 and J(tmum) ≥ c > 0 = J(0) = J(0.um). If tm <

1 then _dt^dJ(tu_m)|t=tm = 0 which gives J⁰(t_mu_m)(t_mu_m) = 0. If t_m = 1, then J⁰(u_m)(u_m) =o(1). So we always have

J⁰(tmum)(tmum) =o(1). (2.23) Now, we fix a big integerk≥1 and define the sequence {vm}by

vm= (2pkukk^p)^1/pwm, m= 1,2, . . . . (2.24) From the dominated convergence theorem and sincew= 0 we have

m→∞lim Z

Ω

|vm|^qdx= 0. (2.25)

Furthermore, by (A0), for every >0, there existsc()>0 such that

|F(x, t)| ≤ 1 c3

|t|^p∗+c(), ∀t∈R, a.e. x∈Ω.

Letδ= _2c() >0,E⊆Ω,|E|< δ we have

Z

E

F(x, v_m)dx ≤

Z

E

|F(x, v_m)|dx

≤ Z

E

c()dx+ 1 2c3

Z

E

|v_m|^p∗dx

≤ 2 +

2, hence {R

ΩF(x, vm)dx : m ∈ N} is equi-absolutely-continuous. It follows easily from Vitali convergence theorem that

Z

Ω

F(x, vm)dx→ Z

Ω

F(x,0)dx= 0 asm→ ∞.

Sincekumk →+∞asm→ ∞, we can find m_k≥ksuch that 0< (2pku_kk^p)^1/p

kumk <1, ∀m > mk. (2.26)

Hence, using relations (2.22), (2.24)-(2.26), it follows that J(t_mu_m)

≥J(2pkukk^p)^1/p kumk um

=J(vm)

=1 p

Z

Ω

|∇vm|^pdx−λ q Z

Ω

|vm|^qdx− Z

Ω

F(x, vm)dx

≥1 p

Z

Ω

kukk^p.(2p)^pp.|∇wm|^p

dx−λ q Z

Ω

|vm|^qdx− Z

Ω

F(x, vm)dx

≥2kukk^p−λ q Z

Ω

|vm|^qdx− Z

Ω

F(x, vm)dx

≥ kukk^p

(2.27)

for anym > mk ≥klarge enough.

On the other hand, using condition (A4) and relation (2.23), for allm > mk > k large enough, we have

J(t_mu_m)

=J(t_mu_m)−1

pJ⁰(t_mu_m)(t_mu_m) +o(1)

= 1 p Z

Ω

|∇tmum|^pdx−λ q Z

Ω

|tmum|^qdx− Z

Ω

F(x, tmum)dx

−1 p

Z

Ω

|∇tmum|^pdx+λ p Z

Ω

|tmum|^qdx +1

p Z

Ω

f(x, tmum)tmumdx+o(1)

=λ 1 p−1

q

Z

Ω

|tmum|^qdx+1 p Z

Ω

H(x, tmum)dx

≤ 1 p Z

Ω

θH(x, um) +C∗

dx+o(1)

=θ1 p

Z

Ω

|∇um|^pdx−λ q Z

Ω

|um|^qdx− Z

Ω

F(x, um)dx

−θ p

Z

Ω

|∇um|^pdx−λ Z

Ω

|um|^qdx− Z

Ω

f(x, um)umdx +λθ 1

q−1 p

Z

Ω

|um|^qdx+θC∗|Ω|

p +o(1)

=θJ(um)−θ

pJ⁰(um)(um) +λθ 1 q−1

p

Z

Ω

|um|^qdx+θC∗|Ω|

p +o(1)

≤θJ(um)−θ

pJ⁰(um)(um) +λθc7

1 q −1

p

kumk^q+θC_∗|Ω|

p +o(1).

(2.28)

From (2.27) and (2.28), we deduce that for allm > mk> klarge enough, ku_kk^p≤θJ(u_m)−θ

pJ⁰(u_m)(u_m) +λθc₇ 1 q−1

p

ku_mk^q+θC_∗|Ω|

p +o(1)

or

ku_kk^p−λθc₇ 1 q−1

p

ku_mk^q ≤θJ(u_m)−θ

pJ⁰(u_m)(u_m) +θC_∗|Ω|

p +o(1) (2.29) Recall that k ≥ 1 is an arbitrarily big integer and m > mk > k. In (2.29), let k→ ∞we have m→ ∞and the left hand side of (2.29) tends to +∞sinceq < p.

In the right hand side of (2.29), J(u_m) → c and ^θ_pJ⁰(u_m)(u_m) → 0 as m → ∞.

Thus, we have a contradiction. This proves that the sequence{um}is bounded in W₀^1,p(Ω).

Now, since the Banach spaceW₀^1,p(Ω) is reflexive, there existsu∈W₀^1,p(Ω) such that passing to a subsequence, still denoted by {um}, it converges weakly to u in W₀^1,p(Ω) and converges strongly to u in L^r(Ω), 1 ≤ r < p^∗. Moreover, {um} converges weakly to uin L^p∗(Ω) and we have |um|^p_p^∗∗ ≤c₈. From (A0), for every >0, there existsc()>0 such that

|f(x, t)t| ≤ 1 2c8

|t|^p∗+c(), ∀t∈R, a.e. x∈Ω.

Letδ= _2c() >0,E⊆Ω,|E|< δ we have

Z

E

f(x, um)umdx ≤

Z

E

|f(x, um)um|dx

≤ Z

E

c()dx+ 1 2c₈

Z

E

|um|^p∗dx

≤ 2+

2, hence{R

Ωf(x, um)umdx :m∈ N} is equi-absolutely-continuous. It follows easily from Vitali convergence theorem that

Z

Ω

f(x, u_m)u_mdx→ Z

Ω

f(x, u)u dx asm→ ∞. (2.30) Using (A0) again, for any >0 there existsc()>0 such that

|f(x, t)| ≤ 1

2c₉c₁₀|t|^p∗−1+c(), ∀t∈R, a.e. x∈Ω, where

c9≥Z

Ω

|um|^p∗dx^{p∗ −1}_p∗

, ∀m; c10:=Z

Ω

|u|^p∗dx1/p^∗

. From the H¨older inequality, for everyE⊆Ω, we have

Z

E

c()|u|dx≤c()|E|^{p∗ −1p∗} Z

E

|u|^p∗dx1/p^∗

=c()|E|^{p∗ −1p∗} c10, Z

E

|u_m|^p∗−1|u|dx≤Z

E

|u_m|^p∗dx^{p∗ −1}_p∗ Z

E

|u|^p∗dx^1/p∗

≤c₉c₁₀. Letδ= (_2c

10c()) ^p

∗

p∗ −1 >0,E⊆Ω,|E|< δ we have

Z

E

f(x, um)u dx ≤

Z

E

|f(x, um)u|dx

≤ Z

E

c()|u|dx+ 1 2c9c10

Z

E

|um|^p∗−1|u|dx

≤ 2+

2, hence{R

Ωf(x, u_m)u_mdx :m∈ N} is equi-absolutely-continuous. It follows easily from Vitali convergence theorem that

Z

Ω

f(x, u_m)u dx→ Z

Ω

f(x, u)u dx asm→ ∞. (2.31) From (2.30) and (2.31) we have

Z

Ω

f(x, u_m)(u_m−u)dx→0 asm→ ∞. (2.32) We also have

Z

Ω

|u_m|^q−2u_m(u_m−u)dx≤ Z

Ω

|u_m|^q−1|u_m−u|dx

≤Z

Ω

|u_m|^qdx^q−1_q Z

Ω

|u_m−u|^qdx1/q

→0

(2.33)

as m → ∞. Since J⁰(um)(um−u) → 0 as m → ∞, we deduce from (2.32) and (2.33) that

Z

Ω

|∇u_m|^p−2∇u_m(∇u_m− ∇u)dx→0 asm→ ∞,

which gives us that{u_m}converges strongly touin W₀^1,p(Ω) and the functionalJ

satisfies the (Ce) condition.

Proof Theorem 1.2. By Lemmas 2.1, 2.2 and 2.4, there existsλ^∗>0 such that for anyλ∈(0, λ^∗), the functionalJ satisfies all the assumptions of the mountain pass theorem. Then we deduceu1as a non-trivial critical point of the functionalJ with J(u1) =c >0 and thus a non-trivial weak solution of problem (1.2).

We now prove that there exists a second weak solutionu2∈W₀^1,p(Ω) such that u2 6= u1. Indeed, by (2.2), the functional J is bounded from below on the ball B_ρ(0).

Applying the Ekeland variational principle in [5] to the functionalJ :B_ρ(0)→R, it follows that there existsu∈B_ρ(0) such that

J(u)< inf

u∈Bρ(0)

J(u) +, J(u)< J(u) +ku−uk, u6=u. By Lemmas 2.1 and 2.2, we have

u∈∂Binf_ρ(0)J(u)≥R >0 and inf

u∈Bρ(0)

J(u)<0.

Let us choose >0 such that 0< < inf

u∈∂Bρ(0)

J(u)− inf

u∈Bρ(0)

J(u).

Then,J(u)<inf_{u∈∂Bρ(0)}J(u) and thus,u∈Bρ(0).

Now, we define the functionalI:B_ρ(0)→RbyI(u) =J(u) +ku−uk. It is clear thatuis a minimum point of Iand thus

I(u+tv)−I(u)

t ≥0

for allt >0 small enough and allv∈Bρ(0). The above information shows that J(u+tv)−J(u)

t +kvk ≥0.

Lettingt→0⁺, we deduce thathJ⁰(u), vi ≥ −kvk. It should be noticed that−v also belongs toB_ρ(0), so replacingvby−v, we obtain

hJ⁰(u),−vi ≥ −k −vk,hJ⁰(u), vi ≤kvk, which helps us to deduce thatkJ⁰(u)k_∗≤.

Then, there exists a sequence{um} ⊂Bρ(0) such that J(um)→c= inf

u∈Bρ(0)

J(u)<0, J⁰(um)→0 in W^−1,p(Ω) asm→ ∞. (2.34) From Lemma 2.4, the sequence{um}converges strongly to someu₂∈W₀^1,p(Ω) as m→ ∞. Moreover, since J ∈C¹(W₀^1,p(Ω),R), by (2.9) it follows that J(u2) =c andJ⁰(u2) = 0. Thus,u2is a non-trivial weak solution of (1.2).

Finally, we point out thatu16=u2 sinceJ(u1) =c >0> c=J(u2). The proof

of Theorem 1.2 is complete.

Acknowledgments. The authors would like to thank the referees for their sugges- tions and helpful comments which improved the presentation of the original manu- script. This work is supported by Quang Binh University (Grant N. CS.05.2016).

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Nguyen Thanh Chung

Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam

E-mail address:[email protected]

Pham Hong Minh

Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam

E-mail address:[email protected]

Tran Hong Nga

Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Viet Nam

E-mail address:[email protected]