# Existence of Positive Weak Solutions for a New Class ofp,q Laplacian Nonlinear Elliptic System with Sign-Changing Weights

Authors: Rafik Guefaifia; Salah Mahmoud Boulaaras; Sultan Alodhaibi; Salem Alkhalaf
Journal: Complexity
 (2020)

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## Abstract

In this paper, by using subsuper solutions method, we study the existence of weak positive solutions for a new class ofp,q Laplacian nonlinear elliptic system in bounded domains, when ax, bx,αx, and βx are sign-changing functions that maybe negative near the boundary, without assuming sign conditions on f0,g0,h0, and γ0.


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## Body

## 1. Introduction

The study of differential equations and variational problems with nonstandardpx−growth conditions is a new and interesting topic. It arises from nonlinear elasticity theory, electrorheological fluids, etc. (see [1–15]). Many existence results have been obtained on this kind of problems, see, for example [10, 12, 16–18] and [19–30]. In [31, 32], Fan et al. studied the regularity of solutions for differential equations with nonstandard px–-growth conditions.In this article, we consider the following system:(1)−△pu−up−2u=λ1axfv+μ1αxhuinΩ,−△qv−vq−2v=λ2bxgu+μ2βxγvinΩ,u=v=0on∂Ω,where Δsz=div∇zs−2∇z, s>1,Ω⊂ℝN N≥3 is a bounded domain with smooth boundary ∂Ω,ax,bx, αx,βx∈CΩ¯, and λ1,λ2,μ1, and μ2 are nonnegative parameters.In fact, we study the existence of positive solutions to system (1) with sign-changing weight functions ax, bx, αx, and βx. Due to these weight functions, the extensions are challenging and nontrivial.These problems arise in some physical models and are interesting in applications at combustion, mathematical biology, and chemical reactions. Our approach is based on the method of sub- and supersolutions (see [6, 23–25, 27, 30]).We make the following assumptions:H1f, g, h, γ∈C10,∞ are nondecreasing functions such that lims⟶+∞fs=lims⟶+∞gs=lims⟶+∞hs=lims⟶+∞γs=+∞H2lims⟶+∞fMgs1/q−1/sp−1=0,∀M>0H3lims⟶+∞hs/sp−1=lims⟶+∞γs/sq−1=0Letσr be the first eigenvalue of −Δr with Dirichlet boundary conditions and ϕr be the corresponding eigenfunction with ϕr>0 in Ω and ϕr=1 for r=p,q. Let m,η,δ>0 be such that ∇ϕrr−σrϕr≥m on Ωδ¯=x∈Ω,dx,∂Ω≤δ and ϕr≥η on Ω/Ω¯δ for r=p,q.Here, we assume that the weightsax, bx, αx, and βx take negative values in Ωδ¯ but require ax, bx, αx, and βx to be strictly positive in Ω/Ω¯δ. To be precise, we assume that there exist positive constants a0,a1, b0, b1, α0, α1,β0, and β1 such that(2)ax≥−a0,bx≥−b0,αx≥−α0,βx≥−β0,x∈Ωδ¯,ax≥a1,bx≥b1,αx≥α1,βx≥β1,x∈Ω/Ω¯δ.Also, lets0≥0 be such that fs0, gs0, hs0, γs0>0, and(3)K1=p−1pηp/p−1,K2=q−1qηq/q−1,θ0=maxs0K11/p−1,s0K2q/q−1.Forθ>θ0, we define(4)λ1∗θ≔θσpda1fθ1/q−1K2,λ1∗θ≔θmda0fθ1/q−1,μ1∗θ≔θσpd′α1hθ1/p−1K1,μ1∗θ≔θmd′α0hθ1/q−1,λ2∗θ≔θσqdb1gθ1/p−1K1,λ2∗θ≔θmdb0gθ1/p−1,μ2∗θ≔θσqdβ′1γθ1/q−1K2,μ2∗θ≔θmdβ′0γθ1/q−1,where d>1 and 1/d+1/d′=1; also assume(5)Λ=θ>θ0:λ1∗θ<λ1∗θ,μ1∗θ<μ1∗θ,λ2∗θ<λ2∗θ,μ2∗θ<μ2∗θ.


## 2. Existence Result

We give the following two definitions before we give our main result.Definition 1.
Letu,v∈W1,pΩ∩CΩ¯×W1,qΩ∩CΩ¯, u,v be said a weak solution of (1) if it satisfies(6)∫Ω∇up−2∇u⋅∇ξdx−∫Ωup−2u⋅ξdx=λ1∫Ωaxfvξdx+μ1∫ΩαxhuξdxinΩ,∫Ω∇vq−2∇v⋅∇ζdx−∫Ωvq−2v⋅ζdx=λ2∫Ωbxguζdx+μ2∫ΩβxγvζdxinΩ,for all ξ,ζ∈W01,pΩ×W01,qΩ.Definition 2.
A pair of nonnegative functionsu¯,v¯,u¯,v¯ in W1,pΩ∩CΩ¯×W1,qΩ∩CΩ¯ is called a weak subsolution and supersolution of (1) if they satisfy u¯,v¯,u¯,v¯=0,0 on ∂Ω:(7)∫Ω∇u¯p−2∇u¯⋅∇ξdx−∫Ωu¯p−2u¯⋅ξdx≤λ1∫Ωaxfv¯ξdx+μ1∫Ωαxhu¯ξdxinΩ,∫Ω∇v¯q−2∇v¯⋅∇ζdx−∫Ωv¯q−2v¯⋅ζdx≤λ2∫Ωbxgu¯ζdx+μ2∫Ωβxγv¯ζdxinΩ,∫Ω∇u¯p−2∇u¯⋅∇ξdx−∫Ωu¯p−2u¯⋅ξdx≥=λ1∫Ωaxfv¯ξdx+μ1∫Ωαxhu¯ξdxinΩ,∫Ω∇v¯q−2∇v¯⋅∇ζdx−∫Ωv¯q−2v¯⋅ζdx≥=λ2∫Ωbxgu¯ζdx+μ2∫Ωβxγv¯ζdxinΩ,for all ξ,ζ∈W01,pΩ×W01,qΩ.
We shall establish the following result.Theorem 1.
Assume that the conditionsH1−H3 hold, ax, bx, αx, and βx are in L∞Ω and Λ≠∅. Let(8)I=∪θ∈Λλ1∗θ,λ1∗θ×μ1∗θ,μ1∗θ×λ2∗θ,λ2∗θ×μ2∗θ,μ2∗θ.
Then, problem (1) has a positive weak solution for each λ1,μ1,λ2,μ2∈I.Proof of Theorem 1.
Letλ1,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(9)u¯=θ1/p−1p−1pϕpp/p−1,v¯=θ1/q−1q−1qϕqq/q−1,is a subsolution of (1). Let the test function ξx∈W01,pΩ with ξx≥0. We have(10)∫Ω∇u¯p−2∇u¯⋅∇ξdx−∫Ωu¯p−2u¯⋅ξdx≤∫Ω∇u¯p−2∇u¯⋅∇ξdx=θ∫Ωϕp∇ϕpp−2∇ϕp⋅∇ξdx=θ∫Ω∇ϕpp−2∇ϕp⋅∇ϕpξ−∇ϕppξdx=θ∫Ωσpϕpp−∇ϕppξdx.
Similarly,(11)∫Ω∇v¯q−2∇v¯⋅∇ζdx−∫Ωv¯q−2v¯⋅ζdx≤θ∫Ωσqϕqq−∇ϕqqζdx.
For allζx∈W01,qΩ with ζx≥0.
Now, onΩδ¯ we have(12)θ∫Ωδ¯σpϕpp−∇ϕppξdx≤−θm∫Ωδ¯ξdx=−θm1d+1dd′∫Ωδ¯ξdx≤−λ1a0fθ1/q−1−μ1α0hθ1/p−1∫Ωδ¯ξdx≤∫Ωδ¯−λ1a0fθ1/q−1q−1qϕqq/q−1−μ1α0hθ1/p−1p−1pϕpp/p−1ξdx≤∫Ωδ¯λ1axfv¯+μ1αxhu¯ξdxθ∫Ωδ¯σqϕqq−∇ϕqqζdx≤−θm∫Ω¯δζdx,=−θmd−θmd′∫Ωδ¯ζdx≤−λ2b0gθ1/p−1−μ2β0γθ1/q−1∫Ωδ¯ζdx≤∫Ωδ¯−λ2b0gθ1/p−1p−1pϕpp/p−1−μ2β0γθ1/q−1q−1qϕqq/q−1ζdx≤∫Ωδ¯λ2bxgu¯+μ2βxγv¯ζdx.
On the contrary, onΩ/Ω¯δ we have(13)θ∫Ω/Ω¯δσpϕpp−∇ϕppξdx≤θσp∫Ω/Ω¯δξdx,=θσp1d+1d′∫Ω/Ω¯δξdx≤λ1a1fθ1/q−1K2+μ1α0hθ1/p−1K1∫Ω/Ω¯δξdx≤∫Ω/Ω¯δλ1axfv¯+μ1αxhu¯ξdx,and similarly,(14)θ∫Ω/Ω¯δσqϕqq−∇ϕqqζdx≤∫Ω/Ω¯δλ2bxgu¯+μ2βxγv¯ζdx.
Therefore,u¯,v¯ is subsolution of problem (1).
Next, we construct a supersolution of (1). Let ωr be a unique positive solution of(15)−△rωr=1inΩ,ωr=0on∂Ω,for r=p,q. We denote(16)u¯=Cνpλ1a∞+μ1α∞1−νpp−11/p−1ωp,v¯=λ2b∞+μ2β∞1−νqq−1gCλ1a∞+μ1α∞1−νpp−11/p−11/q−1ωq,where νr=ωr∞, r=p,q, and C>0 is a large number to be chosen later. We shall verify that u¯,v¯ is a supersolution of (1) such that u¯,v¯≥u¯,v¯. By H2−H3, we can choose C large enough so that(17)Cνpp−1≥fλ2b∞+μ2β∞1−νqq−1gCλ1a∞+μ1α∞1−νpp−11/p−11/q−1ωq+μ1hλ1a∞+μ1α∞1−νpp−11/p−1ωp.
Hence,(18)∫Ω∇u¯p−2∇u¯⋅∇ξdx−∫Ωu¯p−2u¯⋅ξdx=Cνpp−1λ1a∞+μ1α∞∫Ωξdx.
Using (17),(19)∫Ω∇u¯p−2∇u¯⋅∇ξdx−∫Ωu¯p−2u¯⋅ξdx≥λ1a∞fλ2b∞+μ2β∞1−νqq−1gCλ1a∞+μ1α∞1−νpp−11/p−11/q−1ωq∫Ωξdx+μ1α∞∫ΩhCλ1a∞+μ1α∞1−νpp−11/p−1ξdx≥∫Ωλ1axfv¯+μ1αxhu¯ξdx.
Next(20)∫Ω∇v¯q−2∇v¯∇∇ζdx−∫Ωv¯q−2v¯.ζdx=λ2b∞+μ2β∞gCλ1a∞+μ1α∞1−νpp−11/p−1ωq∫Ωξdx≥λ2b∞gCλ1a∞+μ1α∞1−νpp−11/p−1+μ2β∞gCλ1a∞+μ1α∞1−νpp−11/p−1∫Ωξdx.
ByH3, choose C large so that(21)gCλ1a∞+μ1α∞1−νpp−11/p−1≥γλ2b∞+μ2β∞1−νqq−1gCλ1a∞+μ1α∞1−νpp−11/p−11/q−1ωq∞.
Then, from (19), we have(22)∫Ω∇v¯q−2∇v¯⋅∇ζdx−∫Ωv¯q−2v¯⋅ζdx≥λ2b∞gCλ1a∞+μ1α∞1−νpp−11/p−1+μ2β∞γλ2b∞+μ2β∞1−νqq−1gCλ1a∞+μ1α∞1−νpp−11/p−11/q−1ωq∞≥∫Ωbxgu¯+μ2βxγv¯ζdx.
According to (19) and (20), we can conclude that u¯,v¯ is a supersolution of (1). Furthermore, u¯≥u¯ and v¯≥v for C large, Thus, there exists a solution u,v∈W1,pΩ∩CΩ¯×W1,qΩ∩CΩ¯ of (1) with u¯≤u≤u¯ and v¯≤v≤v¯. This completes the Proof of Theorem 1.



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*Source: 5302513-2020-05-04.xml*