Afr. Mat. DOI 10.1007/s13370-014-0254-y
On the existence of positive weak solutions for a class of (p,q)-Laplacian nonlinear elliptic system with sign-changing weights S. H. Rasouli · M. Choubin · G. A. Afrouzi · M. B. Ghaemi
Received: 6 November 2013 / Accepted: 23 April 2014 © African Mathematical Union and Springer-Verlag Berlin Heidelberg 2014
Abstract In this paper, we prove the existence of positive weak solution for the nonlinear elliptic system ⎧ ⎪ ⎨− p u = λ1 a(x) f (v) + μ1 α(x)h(u), x ∈ , −q v = λ2 b(x)g(u) + μ2 β(x)γ (v), x ∈ , ⎪ ⎩ u = 0 = v, x ∈ ∂,
where s z = div(|∇z|s−2 ∇z), s > 1,λ1 , λ2 , μ1 and μ2 are positive parameters, and is a bounded domain in R N . Here a(x), b(x), α(x) and β(x) are sign-changing functions that maybe negative near the boundary. We discuss the existence of positive solution via sub-super-solutions without assuming sign conditions on f (0), h(0), g(0) and γ (0). Keywords
Positive weak solution · Sign-changing weight · Sub- and super solutions
Mathematics Subject Classification (2000)
35J55, 35J65
S. H. Rasouli (B) Department of Mathematics, Faculty of Basic Sciences, Babol University of Technology, Babol, Iran e-mail:
[email protected] M. Choubin Department of Mathematics, Velayat University, Iranshahr, Iran e-mail:
[email protected] G. A. Afrouzi Department of Mathematics, Faculty of Basic Sciences, Mazandaran University, Babolsar, Iran e-mail:
[email protected] M. B. Ghaemi Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran e-mail:
[email protected]
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1 Introduction Consider the system ⎧ ⎪ ⎨− p u = λ1 a(x) f (v) + μ1 α(x)h(u), x ∈ , −q v = λ2 b(x)g(u) + μ2 β(x)γ (v), x ∈ , ⎪ ⎩ u = 0 = v, x ∈ ∂,
(1.1)
where s z = div(|∇z|s−2 ∇z), s > 1, λ1 , λ2 , μ1 and μ2 are nonnegative parameters, and is a bounded domain in R N with smooth boundary ∂. Hai and Shivaji in [8] stablished the existence of positive weak solutions for the system (1.1) when p = q, a ≡ 1, b ≡ 1, λ1 = λ2 , μ1 = μ2 = 0 and Ali and Shivaji in [3] studied the case a ≡ 1, b ≡ 1, α ≡ 1, β ≡ 1 for λ1 + μ1 and λ2 + μ2 large when lim
x→0 h(x) p−1 x→0 x
for every M > 0, lim
f (M[g(x)]1/q−1 ) =0 x p−1 γ (x) q−1 x→0 x
= 0 and lim
= 0. Also those in [2] studied the existence of
positive solutions of the case p = q = 2, a ≡ 1, β ≡ 1, λ1 = λ2 = μ1 = μ2 . See [10] for the case μ1 = μ2 = 0 and f (s), g(s) > 0 for all s > 0. Many results have been obtained on this kind of problems; see for example [1,4,5,9]. Here we focus on further extending the study in [3] to the system (1.1). In fact, we study the existence of positive solution to the system (1.1) with sign-changing weight functions a(x), b(x), α(x) and β(x). Due to this weight functions, the extensions are challenging and nontrivial. These problems arise in some physical models and are interesting in applications at combustion, mathematical biology, chemical reactions. Our approach is based on the method of sub- and supersolutions (see [6,7]). We make the following assumptions: (H1) f, h, g, γ ∈ C 1 ([0, ∞)) are nondecreasing functions such that lim
s→+∞
f (s) = lim h(s) = lim g(s) = lim γ (s) = +∞. s→+∞
s→+∞
s→+∞
1
f (M(g(s)) q−1 ) = 0, ∀M > 0 . (H2) lim s→+∞ s p−1 h(s) γ (s) (H3) lim p−1 = 0, lim q−1 = 0 . s→+∞ s s→+∞ s Let σr the first eigenvalue of −r with Dirichlet boundary conditions and φr the corresponding eigenfunction with φr > 0; and φr = 1 for r = p, q. Let m, η, δ > 0 be such that |∇φr |r − σr φrr ≥ m on δ = {x ∈ |d(x, ∂) ≤ δ} and φr ≥ η on \δ for r = p, q. (This is possible since |∇φr | = 0 on ∂ while φr = 0 on ∂ for r = p, q). Here we assume that the weights a(x), b(x), α(x) and β(x) take negative values in δ , but require a(x), b(x), α(x) and β(x) to be strictly positive in \δ . To be precise we assume that there exist positive constants a0 , a1 , α0 , α1 , b0 , b1 , β0 and β1 such that a(x) ≥ −a0 , α(x) ≥ −α0 , b(x) ≥ −b0 , β(x) ≥ −β0 , x ∈ δ , a(x) ≥ a1 , α(x) ≥ α1 , b(x) ≥ b1 , β(x) ≥ β1 ,
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x ∈ \δ .
On the existence of positive weak solutions
Also let s0 ≥ 0 be such that f (s0 ), h(s0 ), g(s0 ), γ (s0 ) > 0 and K 1 :=
1 1 p q p−1 q −1 s0 p−1 s0 q−1 p−1 q−1 and θ0 := max , η , K 2 := η . p q K1 K2
For θ > θ0 we define θσp , d a1 f (θ 1/q−1 K 2 ) θσp , μ1 ∗ (θ ) := d α1 h(θ 1/ p−1 K 1 ) θ σq , λ2∗ (θ ) := d b1 g(θ 1/ p−1 K 1 ) θ σq μ2 ∗ (θ ) := , d β1 γ (θ 1/q−1 K 2 ) λ1∗ (θ ) :=
θm d a0 f (θ 1/q−1 ) θm μ1 ∗ (θ ) := d α0 h(θ 1/ p−1 ) θm λ2 ∗ (θ ) := d b0 g(θ 1/ p−1 ) θm μ2 ∗ (θ ) := d β0 γ (θ 1/q−1 ) λ1 ∗ (θ ) :=
d , also assume where d > 1 and d = d−1
:= θ > θ0 : λ1∗ (θ ) < λ1 ∗ (θ ), μ1 ∗ (θ ) < μ1 ∗ (θ ), λ2∗ (θ ) < λ2 ∗ (θ ), μ2 ∗ (θ ) < μ2 ∗ (θ ) .
We shall establish the following result. Theorem 1.1 Let (H1)–(H3) hold, a(x), b(x), α(x), β(x) are in L ∞ () and = ∅. Let I := ∪θ ∈ [λ1∗ (θ ), λ1 ∗ (θ )] × [μ1 ∗ (θ ), μ1 ∗ (θ )] × [λ2∗ (θ ), λ2 ∗ (θ )] × [μ2 ∗ (θ ), μ2 ∗ (θ )] Then problem (1.1) has a positive weak solution for each (λ1 , μ1 , λ2 , μ2 ) ∈ I. Example 1.2 (see [3]) Let f (x) =
n
Ai x pi − c1 , g(x) =
m
i=1
h(x) =
s k=1
B j x q j − c2 ,
j=1
Ck x rk − c3 , γ (x) =
t
Dl x dl − c4 ,
l=1
where Ai , B j , Ck , Dl , pi , q j , rk , dl , c1 , c2 , c3 , c4 ≥ 0, pi q j < ( p−1)×(q−1), rk < ( p−1) and dl < (q − 1). Then f, g, h, and γ satisfy the hypotheses of Theorem 1.1.
2 Proof of Theorem 1.1 Proof We shall establish Theorem 1.1 by constructing a positive weak subsolution (ψ1 , ψ2 ) ∈ W 1, p () ∩ C() × W 1,q () ∩ C() and a supersolution (z 1 , z 2 ) ∈ W 1, p () ∩ C() × W 1,q () ∩ C() of (1.1) such that ψi ≤ z i for i = 1, 2. That is, ψi , z i satisfies (ψ1 , ψ2 ) = (0, 0) = (z 1 , z 2 ) on ∂,
p−2
|∇ψ1 |q−2 ∇ψ1 .∇ξ1 d x ≤ [λ1 a(x) f (ψ2 ) + μ1 α(x)h(ψ1 )]ξ1 d x,
|∇ψ2 |p−2 ∇ψ2 .∇ξ2 d x ≤ [λ2 b(x)g(ψ1 ) + μ2 β(x)γ (ψ2 )]ξ2 d x,
|∇z 1 |q−2 ∇z 1 .∇ξ1 d x ≥ [λ1 a(x) f (z 2 ) + μ1 α(x)h(z 1 )]ξ1 d x, ∇z 2 .∇ξ2 d x ≥ [λ2 b(x)g(z 1 ) + μ2 β(x)γ (z 2 )]ξ2 d x, |∇z 2 |
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for all test functions ξ1 ∈ W0 1, p and ξ2 ∈ W0 1,q with ξ1 , ξ2 ≥ 0. Let (λ1 , μ1 , λ2 , μ2 ) ∈ I and θ > θ0 be such that (λ1 , μ1 ,λ2 , μ2 )∈[λ1∗ (θ ), λ1 ∗ (θ )]×[μ1 ∗ (θ ), μ1 ∗ (θ )]×[λ2∗ (θ ), λ2 ∗ (θ )]×[μ2 ∗ (θ ), μ2 ∗ (θ )]. We shall verify that p − 1 p/ p−1 1/q−1 q − 1 q/q−1 , ,θ φp φq (ψ1 , ψ2 ) := θ 1/ p−1 p q is a subsolution of (1.1). Let the test function ξ1 (x) ∈ W0 1, p with ξ1 (x) ≥ 0. We have
|∇ψ1 | p−2 ∇ψ1 .∇ξ1 d x = θ
p−2 φ p ∇φ p ∇φ p .∇ξ1 d x
=θ
∇φ p p−2 ∇φ p .∇(φ p ξ1 ) − ∇φ p p ξ1 d x
=θ
p (σ p φ p p − ∇φ p )ξ1 d x.
Similarly,
|∇ψ2 |q−2 ∇ψ2 .∇ξ2 d x = θ
q (σq φq q − ∇φq )ξ2 d x,
for all ξ2 (x) ∈ W0 1,q with ξ2 (x) ≥ 0. Now on δ we have θ
p (σ p φ p p − ∇φ p )ξ1 d x ≤ −θ m
δ
ξ1 d x δ
= −θ m
1 1 + d d
ξ1 d x δ
1 1 ≤ [−λ1 a0 f θ q−1 − μ1 α0 h θ p−1 ] ξ1 d x
≤
δ
δ
1 q − 1 q − λ1 a0 f θ q−1 φq q−1 q
1 p − 1 p φ p p−1 ξ1 d x − μ1 α0 h θ p−1 p ≤ [λ1 a(x) f (ψ2 ) + μ1 α(x)h(ψ1 )]ξ1 d x, δ
and
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On the existence of positive weak solutions
θ
(σq φq
q
q − ∇φq )ξ2 d x ≤ −θ m
δ
δ
ξ2 d x
θm θm − = − d d
ξ2 d x δ
≤ [−λ2 b0 g(θ
≤
1 p−1
1
) − μ2 β0 γ (θ q−1 )]
ξ2 d x
δ
1 p − 1 p − λ2 b0 g θ p−1 φ p p−1 p
δ
1 q − 1 q φq q−1 ξ2 d x − μ2 β0 γ θ q−1 q ≤ [λ2 b(x)g(ψ1 ) + μ2 β(x)γ (ψ2 )]ξ2 d x. δ
On the other hand, on \δ we have p p (σ p φ p − ∇φ p )ξ1 d x ≤ θ σ p θ \δ
ξ1 d x
\δ
= θσp
1 1 + d d
ξ1 d x \δ
1 1 q−1 p−1 K 2 + μ1 α0 h θ K1 ] ≤ [λ1 a1 f θ
ξ1 d x
\δ
[λ1 a(x) f (ψ2 ) + μ1 α(x)h(ψ1 )]ξ1 d x,
≤ \δ
and similarly, q (σq φq q − ∇φq )ξ2 d x ≤ θ \δ
[λ2 b(x)g(ψ1 ) + μ2 β(x)γ (ψ2 )]ξ2 d x.
\δ
Therefore (ψ1 , ψ2 ) is subsolution. Next, we construct a supersolution of (1.1). Let er be a unique positive solution of −r er = 1 in , er = 0 on ∂ for r = p, q. We denote (z 1 , z 2 ) := Ce p , (λ2 b∞ + μ2 β∞ )1/q−1 [g(Ce p ∞ )]1/q−1 eq and we shall verify that (z 1 , z 2 ) is a supersolution of (1.1) such that (z 1 , z 2 ) ≥ (ψ1 , ψ2 ). By (H2)–(H3) we can choose C large enough so that C p−1 ≥ λ1 a∞ f (λ2 b∞ + μ2 β∞ )1/q−1 [g(Ce p ∞ )]1/q−1 eq + μ1 α∞ h(Ce p ∞ ).
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Hence |∇z 1 | p−2 ∇z 1 .∇ξ1 d x = C p−1 ξ1 d x
≥ λ1 a∞ f (λ2 b∞ +μ2 β∞ )1/q−1 [g(Ce p ∞ )]1/q−1 eq
+ μ1 α∞ h(Ce p ∞ ) ξ1 d x ≥ [λ1 a(x) f (z 2 ) + μ1 α(x)h(z 1 )]ξ1 d x.
(2.1)
Next
|∇z 2 |
q−2
∇z 2 .∇ξ2 d x = (λ2 b∞ + μ2 β∞ )
g(Ce p ∞ )ξ2 d x
[λ2 b(x)g(Ce p ∞ ) + μ2 β(x)g(Ce p ∞ )]ξ2 d x
≥
[λ2 b(x)g(z 1 ) + μ2 β(x)γ (z 2 )]ξ2 d x.
≥
(2.2)
By (H3) choose C large so that g(Ce p ∞ ) ≥ γ (λ2 b∞ + μ2 β∞ )1/q−1 [g(Ce p ∞ )]1/q−1 eq ∞ . Then from (2.2) we have [λ2 b(x)g(Ce p ∞ ) + μ2 β(x)g(Ce p ∞ )]ξ2 d x
λ2 b(x)g(Ce p ∞ ) ≥
+ μ2 β(x)γ (λ2 b∞ + μ2 β∞ )1/q−1 [g(Ce p ∞ )]1/q−1 eq ∞ ξ2 d x (2.3) ≥ [λ2 b(x)g(z 1 ) + μ2 β(x)γ (z 2 )]ξ2 d x.
According to (2.1) and (2.3), we can conclude that (z 1 , z 2 ) is a supersolution of (1.1). Further z i ≥ ψi for C large, i = 1, 2. Thus, there exists a solution (u, v) of (1.1) with ψ1 ≤ u ≤ z 1 , ψ2 ≤ v ≤ z 2 . This completes the proof of Theorem 1.1.
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