Electronic Journal of Differential Equations, Vol. 2006(2006), No. ??, pp. 1–15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)
A SEMILINEAR ELLIPTIC PROBLEM INVOLVING NONLINEAR BOUNDARY CONDITION AND SIGN-CHANGING POTENTIAL
TSUNG-FANG WU
Abstract. In this paper, we study the multiplicity of nontrivial nonnegative solutions for a semilinear elliptic equation involving nonlinear boundary con-dition and sign-changing potential. With the help of the Nehari manifold, we prove that the semilinear elliptic equation:
−∆u + u = λf (x)|u|q−2u in Ω, ∂u
∂ν = g(x)|u|
p−2u on ∂Ω,
has at least two nontrivial nonnegative solutions for λ is sufficiently small.
1. Introduction
In this paper, we consider the multiplicity of nontrivial nonnegative solutions for the following semilinear elliptic equation
−∆u + u = λf (x)|u|q−2u in Ω,
∂u
∂ν = g(x)|u|
p−2u on ∂Ω, (1.1)
where 1 < q < 2 < p < 2(N −1)N −2 , λ > 0, Ω is a bounded domain in RN with smooth boundary, ∂
∂ν is the outer normal derivative and f, g : Ω → R are continuous functions which change sign in Ω. Associated with (1.1), we consider the energy functional Jλ in H1(Ω), Jλ(u) =1 2kuk 2 H1− λ q Z Ω f |u|qdx −1 p Z ∂Ω g|u|pds.
where ds is the measure on the boundary and kuk2 H1 =
R
Ω|∇u|2+ u2dx. It is well known that Jλis of C1 in H1(Ω) and the solutions of equation (1.1) are the critical points of the energy functional Jλ.
The fact that the number of solutions of equation (1.1) is affected by the non-linear boundary conditions has been the focus of a great deal of research in recent years. Garcia-Azorero, Peral and Rossi [10] have investigated (1.1) when f ≡ g ≡ 1.
2000 Mathematics Subject Classification. 35J65, 35J50, 35J55. Key words and phrases. Semilinear elliptic equations; Nehari manifold; Nonlinear boundary condition.
c
°2006 Texas State University - San Marcos. Submitted July 6, 2006. Published ??.
Partially supported by the National Science Council of Taiwan(R.O.C.).
They found that there exist positive numbers Λ1, Λ2with Λ1≤ Λ2such that equa-tion (1.1) admits at least two positive soluequa-tions for λ ∈ (0, Λ1) and no positive solu-tion exists for λ > Λ2. Also see Chipot-Chlebik-Fila-Shafrir [4], Chipot-Shafrir-Fila [5], Flores-del Pino [8], Hu [11], Pierrotti-Terracini [14] and Terraccini [16] where problems similar to equation (1.1) have been studied.
The purpose of this paper is to consider the multiplicity of nontrivial nonnegative solutions of equation (1.1) with sign-changing potential. We prove that equation (1.1) has at least two nontrivial nonnegative solutions for λ is sufficiently small. Theorem 1.1. There exists λ0> 0 such that for λ ∈ (0, λ0), equation (1.1) has at
least two nontrivial nonnegative solutions.
Among the other interesting problems which are similar of equation (1.1), Ambro-setti-Brezis-Cerami [3] have investigated the equation
−∆u = λ|u|q−2u + |u|p−2u in Ω,
u = 0 on ∂Ω, (1.2)
where 1 < q < 2 < p ≤ 2N
N −2. They proved that there exists λ0> 0 such that (1.2) admits at least two positive solutions for λ ∈ (0, λ0), has a positive solution for
λ = λ0, and no positive solution for λ > λ0. Actually, Adimurthi-Pacella-Yadava [1], Damascelli-Grossi-Pacella [6], Ouyang-Shi [13] and Tang [17] proved that there exists λ0 > 0 such that equation (1.2) in the unit ball BN(0; 1) has exactly two positive solutions for λ ∈ (0, λ0), has exactly one positive solution for λ = λ0 and no positive solution exists for λ > λ0. Generalizations of the result of equation (1.2) were done by Ambrosetti-Azorero-Peral [2], de Figueiredo-Gossez-Ubilla [9] and Wu [18].
This paper is organized as follows. In section 2, we give some notation and pre-liminaries. In section 3, we prove that (1.1) has at least two nontrivial nonnegative solutions for λ is sufficiently small.
2. Notation and Preliminaries
Throughout this section, we denote by Sp, Cp the best Sobolev embedding and trace constant for the operators H1(Ω) ,→ Lp(Ω), H1(Ω) ,→ Lp(∂Ω), respectively. Now, we consider the Nehari minimization problem: For λ > 0,
αλ= inf{Jλ(u) : u ∈ Mλ}, where Mλ= {u ∈ H1(Ω)\{0} : hJλ0(u), ui = 0}. Define
ψλ(u) = hJλ0(u), ui = kuk2H1− λ
Z Ω f |u|qdx − Z ∂Ω g|u|pds. Then for u ∈ Mλ, hψ0 λ(u), ui = 2kuk2H1− λq Z Ω f |u|qdx − p Z ∂Ω g|u|pds.
Similarly to the method used in Tarantello [15], we split Mλ into three parts: M+λ = {u ∈ Mλ: hψλ0(u), ui > 0},
M0λ= {u ∈ Mλ: hψλ0(u), ui = 0}, M−λ = {u ∈ Mλ: hψλ0(u), ui < 0}. Then, we have the following results.
Lemma 2.1. There exists λ1> 0 such that for each λ ∈ (0, λ1) we have M0λ= φ.
Proof. We consider the following two cases.
Case (I): u ∈ Mλ and R ∂Ωg|u|pds ≤ 0. We have λ Z Ω f |u|qdx = kuk2 H1− Z ∂Ω g|u|pds. Thus, hψ0λ(u), ui = 2kuk2H1− λq Z Ω f |u|qdx − p Z ∂Ω g|u|pds = (2 − q)kuk2H1+ (q − p) Z ∂Ω g|u|pds > 0 and so u ∈ M+ λ. Case (II): u ∈ Mλ and
R
∂Ωg|u|pds > 0. Suppose that M0λ 6= φ for all λ > 0. If
u ∈ M0 λ, then we have 0 = hψ0 λ(u), ui = 2kuk2H1− λq Z Ω f |u|qdx − p Z ∂Ω g|u|pds = (2 − q)kuk2 H1− (p − q) Z ∂Ω g|u|pds. Thus, kuk2H1= p − q 2 − q Z ∂Ω g|u|pds (2.1) and λ Z Ω f |u|qdx = kuk2 H1− Z ∂Ω g|u|pds =p − 2 2 − q Z ∂Ω g|u|pds. (2.2) Moreover, (p − 2 p − q)kuk 2 H1 = kuk2H1− Z ∂Ω g|u|pds = λ Z Ω f |u|qdx ≤ λkf kLp∗kukqLp ≤ λkf kLp∗SqpkukqH1, where p∗= p p−q. This implies kukH1 ≤ h λ(p − q p − 2)kf kLp∗S q p i1/(2−q) . (2.3) Let Iλ: Mλ→ R be given by Iλ(u) = K(p, q) ³ kuk2(p−1) H1 R ∂Ωg|u|pds ´1/(p−2) − λ Z Ω f |u|qdx,
where K(p, q) = (2−qp−q)(p−1)/(p−2)(p−2
2−q). Then Iλ(u) = 0 for all u ∈ M0λ. Indeed, from (2.1) and (2.2) it follows that for u ∈ M0
λ we have Iλ(u) = K(p, q) ³ kuk2(p−1) H1 R ∂Ωg|u|pds ´1/(p−1) − λ Z Ω f |u|qdx =¡ 2 − q p − q ¢ p p−1(p − 2 2 − q) ³ (p−q 2−q)p−1 ¡ R ∂Ωg|u|pds ¢p−1 R ∂Ωg|u|pds ´ 1 p−2 −p − 2 2 − q Z ∂Ω g|u|pds = 0. (2.4)
However, by (2.3), the H¨older and Sobolev trace inequality, for u ∈ M0 λ Iλ(u) ≥ K(p, q) ³ kuk2(p−1) H1 R ∂Ωg|u|pds ´1/(p−2) − λSqpkf kLp∗kukqH1 ≥ kukqH1 ³ K(p, q) ³ kuk2(p−1) H1 Cppkgk∞kukp+q(p−2)H1 ´1/(p−2) − λSq pkf kLp∗ ´ ≥ kukqH1 © K(p, q)C p 2−p p λ 1−q 2−qh¡p − q p − 2 ¢ kf kLp∗Spq i1−q 2−q − λSq pkf kLp∗ ª .
This implies that for λ sufficiently small we have Iλ(u) > 0 for all u ∈ M0λ, this contradicts (2.4). Thus, we can conclude that there exists λ1 > 0 such that for
λ ∈ (0, λ1), we have M0λ= φ. ¤
By Lemma 2.1, for λ ∈ (0, λ1) we write Mλ= M+λ ∪ M−λ and define
α+ λ = inf u∈M+ λ Jλ(u); α−λ(Ω) = inf u∈M− λ Jλ(u).
The following lemma shows that the minimizers on Mλare “usually” critical points for Jλ.
Lemma 2.2. For λ ∈ (0, λ1). If u0 is a local minimizer for Jλ on Mλ, then
J0
λ(u0) = 0 in H∗(Ω).
Proof. If u0 is a local minimizer for Jλ on Mλ, then u0 is a solution of the opti-mization problem
minimize Jλ(u) subject to ψλ(u) = 0.
Hence, by the theory of Lagrange multipliers, there exists θ ∈ R such that
J0
λ(u0) = θψλ0(u0) in H∗(Ω). Thus,
hJ0
λ(u0), u0iH1 = θhψλ0(u0), u0iH1. (2.5)
By Lemma 2.1, u0∈ M+λ∪ M−λ, we have hψλ0(u0), u0iH1 6= 0 and so by (2.5) θ = 0.
This completes the proof. ¤
Lemma 2.3. (i) If u ∈ M+ λ, then
R
Ωf |u|qdx > 0; (ii) If u ∈ M−λ, thenR∂Ωg|u|pds > 0.
Proof. (i) Case (I):R∂Ωg|u|pds ≤ 0. We have λ Z Ω f |u|qdx = kuk2H1− Z ∂Ω g|u|pds > 0.
Case (II):R∂Ωg|u|pds > 0. We have
kuk2H1− λ Z Ω f |u|qdx − Z ∂Ω g|u|pds = 0 and kuk2 H1> p − q 2 − q Z ∂Ω g|u|pds. Thus, λ Z Ω f |u|qdx = kuk2H1− Z ∂Ω g|u|pds > p − 2 2 − q Z ∂Ω g|u|pds > 0. (ii) Since (2 − q)kuk2H1− (p − q) Z ∂Ω g|u|pds = hψλ0(u), ui < 0.
It follows thatR∂Ωg|u|pds > 0. This completes the proof. ¤ For each u ∈ M−λ, we write
tmax= ³ (2 − q)kuk2 H1 (p − q)R∂Ωg|u|pds ´1/(p−2) < 1.
Then we have the following lemma. Lemma 2.4. Let p∗ = p p−q and λ2= ( p−2 p−q)( 2−q p−q) 2−q p−2C p(2−q) 2−p p Sp−qkf k−1Lp∗. Then for each u ∈ M− λ and λ ∈ (0, λ2), we have (i) ifRΩf |u|qdx ≤ 0, then J
λ(u) = supt≥0Jλ(tu) > 0;
(ii) ifRΩf |u|qdx > 0, then there is a unique 0 < t+= t+(u) < t
max such that
t+u ∈ M+ λ and
Jλ(t+u) = inf 0≤t≤tmax
Jλ(tu), Jλ(u) = sup t≥tmax
Jλ(tu).
Proof. Fix u ∈ M−λ. Let
h(t) = t2−qkuk2
H1− tp−q
Z ∂Ω
g|u|pds for t ≥ 0.
We have h(0) = 0, h(t) → −∞ as t → ∞, h(t) achieves its maximum at tmax, increasing for t ∈ [0, tmax) and decreasing for t ∈ (tmax, ∞). Moreover,
h(tmax) =³ (2 − q)kuk 2 H1 (p − q)R∂Ωg|u|pds ´2−q p−2 kuk2 H1− ³ (2 − q)kuk2 H1 (p − q)R∂Ωg|u|pds ´p−q p−2 Z ∂Ω g|u|pds = kukqH1 h (2 − q p − q) 2−q p−2− (2 − q p − q) p−q p−2 i³ kukp H1 R ∂Ωg|u|pds ´2−q p−2 ≥ kukqH1( p − 2 p − q)( 2 − q p − q) 2−q p−2C p(2−q) 2−p p or h(tmax) ≥ kukqH1( p − 2 p − q)( 2 − q p − q) 2−q p−2C p(2−q) 2−p p . (2.6)
(i): RΩf |u|qdx ≤ 0. There is a unique t− > t
max such that h(t−) = λ R Ωf |u|qdx and h0(t−) < 0. Now, (2 − q)kt−uk2 H1− (p − q) Z ∂Ω |t−u|pds = (t−)1+qh(2 − q)(t−)1−qkuk2 H1− (p − q)(t−)p−q−1 Z ∂Ω g|u|pdsi = (t−)1+qh0(t−) < 0, and hJ0 λ(t−u), t−ui = (t−)2kuk2H1− (t−)qλ Z Ω f |u|qdx − (t−)p Z ∂Ω g|u|pds = (t−)q h h(t−) − λ Z Ω f |u|qdx i = 0. Thus, t−u ∈ M−
λ or t− = 1. Since for t > tmax, we have (2 − q)ktuk2 H1− (p − q) Z ∂Ω g|tu|pds < 0, d2 dt2Jλ(tu) < 0, d dtJλ(tu) = tkuk 2 H1− λtq−1 Z Ω f |u|qdx − tp−1 Z ∂Ω g|u|pds = 0 for t = t−. Thus, Jλ(u) = supt≥0Jλ(tu). Moreover,
Jλ(u) ≥ Jλ(tu) ≥ t 2 2kuk 2 H1− tp p Z ∂Ω
g|u|pds for all t ≥ 0. By routine computations, g(t) = t2
2kuk2H1−t p
p R
∂Ωg|u|pds achieves its maximum at
t0= (kuk2H1/ R ∂Ωg|u|pds)1/(p−2). Thus, Jλ(u) ≥p − 2 2p ³ kukp H1 R ∂Ωg|u|pds ´ 2 p−2 > 0.
(ii): RΩf |u|qdx > 0. By (2.6) and
h(0) = 0 < λ Z Ω f |u|qdx ≤ λkf k Lp∗SpqkukqH1 < kukqH1( p − 2 p − q)( 2 − q p − q) 2−q p−2C p(2−q) 2−p p ≤ h(tmax) for λ ∈ (0, λ2), there are unique t+ and t− such that 0 < t+ < t
max< t−, h(t+) = λ Z Ω f |u|qdx = h(t−), h0(t+) > 0 > h0(t−).
We have t+u ∈ M+
λ, t−u ∈ M−λ, and Jλ(t−u) ≥ Jλ(tu) ≥ Jλ(t+u) for each
t ∈ [t+, t−] and J
λ(t+u) ≤ Jλ(tu) for each t ∈ [0, t+]. Thus, t−= 1 and
Jλ(u) = sup
t≥0Jλ(tu), Jλ(t
+u) = inf 0≤t≤tmax
Jλ(tu).
This completes the proof. ¤
Next, we establish the existence of nontrivial nonnegative solutions for the equa-tion
−∆u + u = λf (x)|u|q−2u in Ω,
u = 0 on ∂Ω, (2.7)
Associated with equation (2.7), we consider the energy functional
Kλ(u) = 1 2kuk 2 H1− λ q Z Ω f |u|qdx and the minimization problem
βλ= inf{Kλ(u) : u ∈ Nλ},
where Nλ= {u ∈ H01(Ω)\{0} : hKλ0(u), ui = 0}. Then we have the following result. Theorem 2.5. Suppose that λ > 0. Then equation (2.7) has a nontrivial
nonneg-ative solution vλ with Kλ(vλ) = βλ< 0.
Proof. First, we need to show that Kλis bounded below on Nλ and βλ< 0. Then for u ∈ Nλ, kuk2 H1 = λ Z Ω f |u|qdx ≤ λkf k Lq∗S −q 2 p kukqH1. where p∗= p p−q. This implies kukH1 ≤ (λkf kLp∗S− q 2 p ) 1 2−q. (2.8) Hence, Kλ(u) =1 2kukH1− λ q Z Ω f |u|qdx =¡ 1 2 − 1 q ¢ kuk2H1 ≤¡ 1 2 − 1 q ¢¡ λkf kLp∗S− q 2 p ¢ 1 2−q
for all u ∈ Nλand βλ< 0. Let {vn} be a minimizing sequence for Kλon Nλ. Then by (2.8) and the compact imbedding theorem, there exist a subsequence {vn} and
vλ in H01(Ω) such that
vn* vλ weakly in H01(Ω) and
vn→ vλ strongly in Lq(Ω). (2.9)
First, we claim thatRΩf |vλ|qdx > 0. If not,
Kλ(vn) ≥ 1 2kvλk 2 H1− λ q Z Ω f |vλ|qdx + o(1) ≥ 1 2kvλk 2 H1+ o(1),
this contradicts Kλ(vn) → βλ(Ω) < 0 as n → ∞. Thus, R
Ωf |vλ|qdx > 0. In particular, vλ 6≡ 0. Now, we prove that vn → vλ strongly in H01(Ω). Suppose otherwise, then kvλkH1 < lim infn→∞kvnkH1 and so
kvλk2H1− λ Z Ω f |vλ|qdx < lim inf n→∞ ³ kvnk2H1− λ Z Ω f |vn|qdx ´ = 0. SinceRΩf |vλ|qdx > 0, there is a unique t06= 1 such that t0vλ∈ Nλ. Thus,
t0vn* t0vλ weakly in H01(Ω). Moreover,
Kλ(t0vλ) < Kλ(vλ) < lim
n→∞Kλ(vn) = βλ,
which is a contradiction. Hence vn → vλ strongly in H01(Ω). This implies vλ∈ Nλ and
Kλ(vn) → Kλ(vλ) = βλ as n → ∞.
Since Kλ(vλ) = Kλ(kvλk) and kvλk ∈ Nλ, without loss of generality, we may assume that vλ is a nontrivial nonnegative solution of equation (2.7). ¤
Then we have the following results. Lemma 2.6. (i) αλ≤ α+λ ≤ βλ< 0;
(ii) Jλ is coercive and bounded below on Mλ for all λ ∈ (0,p−2p−q].
Proof. (i) Let vλ be a positive solution of equation (2.7) such that K(vλ) = βλ. Since vλ∈ C2(Ω). Then we have
R
∂Ωg|vλ|pds = 0 and vλ∈ M+λ. This implies
Jλ(vλ) =1 2kvλk 2 H1− λ q Z Ω f |vλ|qdx = βλ< 0 and so αλ≤ α+λ ≤ βλ< 0.
(ii) For u ∈ Mλ, we have kuk2H1 = λ
R
Ωf |u|qdx + R
∂Ωg|u|pds. Then by the H¨older and Young inequalities,
Jλ(u) = p − 2 2p kuk 2 H1− λ ¡ p − q pq ¢Z Ω f |u|qdx ≥ p − 2 2p kuk 2 H1− λ ¡ p − q pq ¢ kf kLp∗SpqkukqH1 ≥£ p − 2 2p − λ ¡ p − q 2p ¢¤ kuk2H1− λ ¡ (p − q)(2 − q) 2pq ¢¡ kf kLp∗Spq ¢ 2 2−q = 1 2p £ (p − 2) − λ(p − q)¤kuk2 H1− λ ¡ (p − q)(2 − q) 2pq ¢¡ kf kLp∗Spq ¢ 2 2−q.
Thus, Jλ is coercive on Mλ and
Jλ(u) ≥ −λ¡ (p − q)(2 − q) 2pq ¢¡ kf kLp∗Spq ¢ 2 2−q for all λ ∈ (0,p−2p−q]. ¤
3. Proof of Theorem 1.1
First, we will use the idea of Ni-Takagi [12] to get the following results.
Lemma 3.1. For each u ∈ Mλ, there exist ² > 0 and a differentiable function
ξ : B(0; ²) ⊂ H1(Ω) → R+ such that ξ(0) = 1, the function ξ(v)(u − v) ∈ M λ and hξ0(0), vi = 2 R Ω∇u∇vdx − λq R Ωf |u|q−2uvdx − p R ∂Ωg|u|p−2uvds (2 − q)kuk2 H1− (p − q) R ∂Ωg|u|pds (3.1) for all v ∈ H1(Ω).
Proof. For u ∈ Mλ, define a function F : R × H1(Ω) → R by
Fu(ξ, w) = hJλ0(ξ(u − w)), ξ(u − w)i = ξ2 Z Ω |∇(u − w)|2+ (u − w)2dx − ξqλ Z Ω f |u − w|qdx − ξp Z ∂Ω g|u − w|pds. Then Fu(1, 0) = hJλ0(u), ui = 0 and
d dξFu(1, 0) = 2kuk 2 H1− λq Z ∂Ω f |u|qdx − p Z ∂Ω g|u|pds = (2 − q)kuk2 H1− (p − q) Z ∂Ω g|u|pds 6= 0.
According to the implicit function theorem, there exist ² > 0 and a differentiable function ξ : B(0; ²) ⊂ H1(Ω) → R such that ξ(0) = 1,
hξ0(0), vi = 2 R Ω∇u∇vdx − λq R Ωf |u|q−2uvdx − p R ∂Ωg|u|p−2uvds (2 − q)kuk2 H1− (p − q) R ∂Ωg|u|pds and Fu(ξ(v), v) = 0 for all v ∈ B(0; ²) which is equivalent to hJ0
λ(ξ(v)(u − v)), ξ(v)(u − v)i = 0 for all v ∈ B(0; ²),
that is ξ(v)(u − v) ∈ Mλ. ¤
Lemma 3.2. For each u ∈ M−λ, there exist ² > 0 and a differentiable function ξ− : B(0; ²) ⊂ H1(Ω) → R+ such that ξ−(0) = 1, the function ξ−(v)(u − v) ∈ M− λ and h(ξ−)0(0), vi = 2 R Ω∇u∇vdx − λq R Ωf |u|q−2uvdx − p R ∂Ωg|u|p−2uvds (2 − q)kuk2 H1− (p − q) R ∂Ωg|u|pds (3.2) for all v ∈ H1(Ω).
Proof. Similar to the argument in Lemma 3.1, there exist ² > 0 and a differentiable
function ξ− : B(0; ²) ⊂ H1(Ω) → R such that ξ−(0) = 1 and ξ−(v)(u − v) ∈ M λ for all v ∈ B(0; ²). Since
hψ0
λ(u), ui = (2 − q)kuk2H1− (p − q)
Z ∂Ω
Thus, by the continuity of the function ξ−, we have
hψ0
λ(ξ−(v)(u − v)), ξ−(v)(u − v)i = (2 − q)kξ−(v)(u − v)k2H1− (p − q)
Z ∂Ω
g|ξ−(v)(u − v)|pds < 0
if ² sufficiently small, this implies that ξ−(v)(u − v) ∈ M−
λ. ¤
Proposition 3.3. Let λ0= min{λ1, λ2,p−1p−q}, Then for λ ∈ (0, λ0): (i) There exists a minimizing sequence {un} ⊂ Mλ such that
Jλ(un) = αλ+ o(1),
Jλ0(un) = o(1) in H∗(Ω);
(ii) there exists a minimizing sequence {un} ⊂ M−λ such that
Jλ(un) = α−λ + o(1),
J0
λ(un) = o(1) in H∗(Ω).
Proof. (i) By Lemma 2.6 (ii) and the Ekeland variational principle [7], there exists
a minimizing sequence {un} ⊂ Mλ such that
Jλ(un) < αλ+1
n, (3.3)
Jλ(un) < Jλ(w) +1
nkw − unkH1 for each w ∈ Mλ. (3.4)
By taking n large, from Lemma 2.6 (i), we have
Jλ(un) = (1 2 − 1 p)kunk 2 H1− ( 1 q− 1 p)λ Z Ω f |un|qdx < αλ+ 1 n < βλ 2 . (3.5) This implies kf kLp∗SqpkunkqH1 ≥ Z Ω f |un|qdx > −pq 2λ(p − q)βλ> 0. (3.6) Consequently, un 6= 0 and putting together (3.5), (3.6) and the H¨older inequality, we obtain kunkH1 > h −pq 2λ(p − q)βλS −q p kf k−1Lp∗ i1/q (3.7) kunkH1 <h 2(p − q) (p − 2)qkf kLp∗S q p i1/(2−q) (3.8) Now, we show that
kJλ0(un)kH−1 → 0 as n → ∞.
Applying Lemma 3.1 with un to obtain the functions ξn : B(0; ²n) → R+ for some
²n > 0, such that ξn(w)(un− w) ∈ Mλ. Choose 0 < ρ < ²n. Let u ∈ H1(Ω) with
u 6≡ 0 and let wρ = kukρu
H1. We set ηρ = ξn(wρ)(un− wρ). Since ηρ ∈ Mλ, we
deduce from (3.4) that
Jλ(ηρ) − Jλ(un) ≥ −1
and by the mean value theorem, we have hJλ0(un), ηρ− uni + o(kηρ− unkH1) ≥ −1 nkηρ− unkH1. Thus, hJ0 λ(un), −wρi + (ξn(wρ) − 1)hJλ0(un), (un− wρ)i ≥ −1 nkηρ− unkH1+ o(kηρ− unkH1). (3.9) Since ξn(wρ)(un− wρ) ∈ Mλ and (3.9) it follows that
− ρhJ0 λ(un), u kukH1i + (ξn(wρ) − 1)hJ 0 λ(un) − Jλ0(ηρ), (un− wρ)i ≥ −1 nkηρ− unkH1+ o(kηρ− unkH1). Thus, hJ0 λ(un), u kukH1i ≤ kηρ− unkH1 nρ + o(kηρ− unkH1) ρ +(ξn(wρ) − 1) ρ hJ 0 λ(un) − Jλ0(ηρ), (un− wρ)i. (3.10)
Since kηρ− unkH1 ≤ ρkξn(wρ)k + kξn(wρ) − 1kkunkH1 and
lim ρ→0 kξn(wρ) − 1k ρ ≤ kξ 0 n(0)k,
if we let ρ → 0 in (3.10) for a fixed n, then by (3.8) we can find a constant C > 0, independent of ρ, such that
hJλ0(un), u kukH1i ≤ C n(1 + kξ 0 n(0)k). The proof will be complete once we show that kξ0
n(0)k is uniformly bounded in n. By (3.1), (3.8) and the H¨older inequality, we have
hξ0 n(0), vi ≤ bkvkH1 |(2 − q)kunkH1− (p − q) R ∂Ωg|un|pds| for some b > 0. We only need to show that
|(2 − q)kunkH1− (p − q)
Z ∂Ω
g|un|pds| > c (3.11) for some c > 0 and n large enough. We argue by contradiction. Assume that there exists a subsequence {un}, we have
(2 − q)kunkH1− (p − q)
Z ∂Ω
g|un|pds = o(1). (3.12) Combining (3.12) with (3.7), we can find a suitable constant d > 0 such that
Z ∂Ω
g|un|pds ≥ d for n sufficiently large. (3.13) In addition (3.12), and the fact that un∈ Mλ also give
λ Z Ω f |un|qdx = kunk2H1− Z ∂Ω g|un|pds = p − 2 2 − q Z ∂Ω g|un|pds + o(1)
and kunkH1 ≤ h λ(p − q p − 2)kf kLp∗S q p i 1 2−q + o(1). (3.14) This implies Iλ(un) = K(p, q) ³ kunk2(p−1)H1 R ∂Ωg|un|pds ´1/(p−2) − λ Z Ω f |un|qdx = o(1). (3.15) However, by (3.13), (3.14) and λ ∈ (0, λ0), Iλ(un) ≥ K(p, q) ³ kunk2(p−1)H1 R ∂Ωg|un|pds ´1/(p−2) − λSq pkf kLp∗kunkqH1 ≥ kunkqH1 ³ K(p, q)³ kunk 2(p−1) H1 Cppkunkp+q(p−2)H1 ´1/(p−2) − λSq pkf kLp∗ ´ ≥ kunkqH1 n K(p, q)C2−pp p λ 1−q 2−q£(p − q p − 2)kf kLp∗Spq ¤1−q 2−q − λkf k Lp∗ o .
this contradicts (3.15). We get
hJ0
λ(un), u
kukH1 i ≤C
n.
This completes the proof of (i).
(ii) Similarly, by using Lemma 3.2, we can prove (ii). We will omit detailed proof
here. ¤
Now, we establish the existence of a local minimum for Jλ on M+λ.
Theorem 3.4. Let λ0> 0 as in Proposition 3.3, then for λ ∈ (0, λ0) the functional
Jλ has a minimizer u+0 in M+λ and it satisfies (i) Jλ(u+0) = αλ= α+λ;
(ii) u+
0 is a nontrivial nonnegative solution of equation (1.1); (iii) Jλ(u+0) → 0 as λ → 0.
Proof. Let {un} ⊂ Mλ be a minimizing sequence for Jλ on Mλ such that
Jλ(un) = αλ+ o(1) and Jλ0(un) = o(1) in H∗(Ω).
Then by Lemma 2.6 and the compact imbedding theorem, there exist a subsequence
{un} and u+0 ∈ H1(Ω) such that
un * u+0 weakly in H1(Ω), un→ u+0 strongly in Lp(∂Ω) and
un→ u+0 strongly in Lq(Ω). (3.16)
First, we claim that RΩf (x)ku+
0kqdx 6= 0. Suppose otherwise, by (3.16) we can conclude that Z Ω f |un|qdx → Z Ω f |u+0|qdx = 0 as n → ∞ and so kunk2H1= Z ∂Ω g|un|pds + o(1).
Thus, Jλ(un) =1 2kunk 2 H1− λ q Z Ω f |un|qdx − 1 p Z ∂Ω g|un|pds = (1 2 − 1 p) Z ∂Ω g|un|pds + o(1) = (1 2 − 1 p) Z ∂Ω g|u+ 0|pds as n → ∞, this contradicts Jλ(un) → αλ< 0 as n → ∞. Moreover,
o(1) = hJλ0(un), φi = hJλ0(u0), φi + o(1) for all φ ∈ H1(Ω). Thus, u+
0 ∈ Mλ is a nonzero solution of equation (1.1) and Jλ(u+0) ≥ αλ. Now we prove that un → u+0 strongly in H1(Ω). Suppose otherwise, then ku+0kH1 <
lim infn→∞kunkH1 and so ku+0k2H1− λ Z Ω f |u+0|qdx − Z ∂Ω g|u+0|pds < lim inf n→∞ ³ kunk2H1− λ Z Ω f |un|qdx − Z ∂Ω g|un|pds ´ = 0, this contradicts u+0 ∈ Mλ. Hence un→ u+0 strongly in H1(Ω) and
Jλ(un) → Jλ(u+0) = αλ as n → ∞.
Moreover, we have u+0 ∈ M+λ. If not, then u+0 ∈ M−λ and by Lemma 2.4, there are unique t+
0 and t−0 such that t+0u+0 ∈ M+λ and t−0u+0 ∈ M−λ. In particular, we have
t+ 0 < t−0 = 1. Since d dtJλ(t + 0u+0) = 0 and d2 dt2Jλ(t + 0u+0) > 0, there exists t+
0 < ¯t ≤ t−0 such that Jλ(t+0u+0) < Jλ(¯tu+0). By Lemma 2.4,
Jλ(t+0u+0) < Jλ(¯tu+0) ≤ Jλ(t−0u+0) = Jλ(u+0),
which is a contradiction. Since Jλ(u+0) = Jλ(|u+0|) and |u+0| ∈ M+λ, by Lemma 2.2 we may assume that u+0 is a nontrivial nonnegative solution of equation (1.1). From Lemma 2.6 it follows that
0 > Jλ(u+0) ≥ −λ ³ (p − q)(2 − q) 2pq ´ (kf kLp∗Spq) 2 2−q and so Jλ(u+0) → 0 as λ → 0. ¤
Next, we establish the existence of a local minimum for Jλ on M−λ.
Theorem 3.5. Let λ0> 0 as in Proposition 3.3. Then for λ ∈ (0, λ0) the functional
Jλ has a minimizer u−0 in M−λ and satisfies (i) Jλ(u−0) = α−λ;
(ii) u−0 is a nontrivial nonnegative solution of equation (1.1).
Proof. By Proposition 3.3 (ii), there exists a minimizing sequence {un} for Jλ on M−λ such that
By Lemma 2.6 and the compact imbedding theorem, there exist a subsequence {un} and u− 0 ∈ H1(Ω) such that un* u−0 weakly in H1(Ω), un→ u−0 strongly in Lp(∂Ω), un→ u−0 strongly in Lq(Ω). Since (2 − q)kunk2H1 − (p − q) R
∂Ωg|un|pds < 0, by the Sobolev trace inequality there exists C > 0 such thatR∂Ωg|un|pds > C. Moreover,
o(1) = hJλ0(un), φi = hJλ0(u0), φi + o(1) for all φ ∈ H1(Ω) and (2 − q)ku0k2H1− (p − q) Z ∂Ω g|u0|pds ≤ lim inf n→∞ ³ (2 − q)kunk2H1− (p − q) Z ∂Ω g|un|pds ´ ≤ 0.
Thus, u−0 ∈ M−λ is a nonzero solution of equation (1.1). Now we prove that un→ u−0 strongly in H1(Ω). Suppose otherwise, then ku−
0kH1 < lim infn→∞kunkH1 and so ku−0k2 H1− λ Z Ω f |u−0|qdx − Z ∂Ω g|u−0|pds < lim inf n→∞ ³ kunk2H1− λ Z Ω f |un|qdx − Z ∂Ω g|un|pds ´ = 0, this contradicts u−
0 ∈ M−λ. Hence un→ u−0 strongly in H1(Ω). This implies
Jλ(un) → Jλ(u−0) = α−λ as n → ∞.
Since Jλ(u−0) = Jλ(|u0−|) and |u−0| ∈ M−λ, by Lemma 2.2 we may assume that u−0
is a nontrivial nonnegative solution of equation (1.1). ¤
Now, we complete the proof of Theorem 1.1. By Theorems 3.4, 3.5, we obtain equation (1.1) has two nontrivial nonnegative solutions u+0 and u−0 such that u+0 ∈
M+
λ and u−0 ∈ M−λ. Since M+λ∩M−λ = φ, this implies that u+0 and u−0 are different. References
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Tsung-fang Wu
Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
