# Existence and Nonexistence for Boundary Problem Involving the p-Biharmonic Operator and Singular Nonlinearities

Authors: Mohammed El Mokhtar Ould El Mokhtar
Journal: Journal of Function Spaces
 (2023)

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## Abstract

This article concerns the existence and the nonexistence of solution for the following boundary problem involving the p-biharmonic operator and singular nonlinearities,Δp2u=uγ−1u+μu−1−α/xβuinΩ and u=∂u/∂n=0on∂Ω, where 4<2p<N,0∈Ω,−∞<μ<μ∗,=N−2p1−α/pN,p<γ<p∗=pN/N−2p is the critical Sobolev exponent, 0≤β<Nγ+α/γ+1,0<α<1. Under some sufficient conditions on coefficients, we prove the existence of at least one nontrivial solutions in E by using variational methods. By using the Pohozaev identity type, we show the nonexistence of positive solution when Ω⊂ℝN be a bounded, smoothandstrictlystar-shapeddomain, β=0andγ≥γ∗,=pN1−α/N−2p1−α−μNp>p∗=pN/N−2p.


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## Body

## 1. Introduction and Main Results

The main purpose of this article is to investigate the existence and nonexistence of nontrivial solutions of the following problem:(1)Δp2u=uγ−1u+μu−1−αxβu,inΩ,u=∂u∂n=0,on∂Ω,where n is the exterior normal vector to ∂Ω, Ω is a smooth-bounded domain in ℝN,4<2p<N, 0∈Ω,−∞<μ<μ∗=N−2p1−α/Np, p<γ<p∗ with p∗≔pN/N−2p is the critical Sobolev exponent, 0≤β<Nγ+α/γ+1, 0<α<1. Δp2u≔ΔΔup−2Δu is the operator of the fourth order, so-called the p-biharmonic (or p-bilaplacian) operator. For p=2, the linear operator Δ22=Δ2=Δ.Δ is the iterated Laplacian that to a multiplicative positive constant appears often in the equations of Navier-Stokes as being a viscosity coefficient, and its reciprocal operator noted Δ2−1 is the celebrated Green’s operator see [1].The fourth-order differential equations with nonlinearity arise in the study of deflections of elastic beams on nonlinear elastic foundations. It furnishes a model to study travelling waves in suspension bridges. Lazer and McKenna [2] gave a survey of results in this direction. This fourth-order semilinear elliptic problem can be considered as an analogue of a class of second-order problems which have been studied by many authors (see [3] and references therein). Thus, they become very significant in engineering and physics and many authors considered this type of equation in recent years, and we refer to [4–11] and the references therein. For this reason, the existence of solutions of p-biharmonic equations has been studied by several authors; see [12–15]. Li [16] establish the existence of at least two distinct weak solutions for the following singular elliptic problems involving a p-biharmonic operator, subject to Navier boundary conditions in a smooth-bounded domain in ℝN.
(2)Δp2u+uq−1ux2q=λfx,u,inΩ,u=Δu=0,on∂Ω,with f being a Carathéodory function. Wang [17] studied the existence and multiplicity to p-biharmonic equation with the Sobolev–Hardy term under the Dirichlet boundary conditions and the Navier boundary conditions, respectively.
(3)Δp2u=λur−1uxs+fx,u,inΩ,u=∂u∂n=0or,u=Δu=0,on∂Ω.Before giving our main results, we state here some definitions, notations, and known results.


### 1.1. Notations

SetE=W02,pΩ∩L∞Ω a Banach space, with the norm u=Δup for 1≤p≤∞.We consider the following approximation equation:(4)Δp2u=uγ−1u+μxβu+θα,inΩ,u=∂u∂n=0,on∂Ω,for any θ>0 such that u+θ>0. The energy functional of (4) Jθ is defined by
(5)Jθu≔1pup−1γ+1∫Ωuγ+1dx−μ1−α∫Ωu+θ1−α−θ1−αxβdx.for all u∈E.We note thatJθ is a C1−function on E=W02,pΩ∩L∞Ω.A pointu∈E is a weak solution of the Equation (1) if it satisfies
(6)Jθ′u,φ≔∫ΩFpΔuΔφdx−∫Ωuγ−1uφdx−μ∫Ωφu+θαxβdx=0,forallφ∈E,where Fps=sp−2s with p>2.Here,.,. denotes the product in the duality E′,EE′thedualofE.In our work, we research the critical points as the minimizers of the energy functional associated to problem (1).Let(7)S≔infu∈E\0up∫Ωuγ+1dxp/γ+1.From [18], S is achieved.Let(8)A=2πN/2q+βNΓN/2q+β−αq+1q+β/q+1R0N/q+1q+β−α>0,with
(9)0≤α<Nq+1q+β.Now, we can state our main results.Theorem 1.
Assume that4<2p<N, 0∈Ω,p<γ<p∗ with p∗≔pN/N−2p is the critical Sobolev exponent, 0≤β<Nγ+α/γ+1, 0<α<1 and μ≤0; then, problem (1) has at least a nontrivial solution in E.Theorem 2.
LetΩ⊂ℝN be a bounded, smooth, and strictly star-shaped domain, 0≤μ<μ∗=N−2p1−α/Np and β=0, with γ≥γ∗=Np1−α/N−2p1−α−μNp>p∗=Np/N−2p. Then, problem (1) has no positive solutions in E.This paper is organized as follows. In Section2, we give some preliminaries. Sections 3 and 4 are devoted to the proofs of Theorems 1 and 2.


## 1.1. Notations

SetE=W02,pΩ∩L∞Ω a Banach space, with the norm u=Δup for 1≤p≤∞.We consider the following approximation equation:(4)Δp2u=uγ−1u+μxβu+θα,inΩ,u=∂u∂n=0,on∂Ω,for any θ>0 such that u+θ>0. The energy functional of (4) Jθ is defined by
(5)Jθu≔1pup−1γ+1∫Ωuγ+1dx−μ1−α∫Ωu+θ1−α−θ1−αxβdx.for all u∈E.We note thatJθ is a C1−function on E=W02,pΩ∩L∞Ω.A pointu∈E is a weak solution of the Equation (1) if it satisfies
(6)Jθ′u,φ≔∫ΩFpΔuΔφdx−∫Ωuγ−1uφdx−μ∫Ωφu+θαxβdx=0,forallφ∈E,where Fps=sp−2s with p>2.Here,.,. denotes the product in the duality E′,EE′thedualofE.In our work, we research the critical points as the minimizers of the energy functional associated to problem (1).Let(7)S≔infu∈E\0up∫Ωuγ+1dxp/γ+1.From [18], S is achieved.Let(8)A=2πN/2q+βNΓN/2q+β−αq+1q+β/q+1R0N/q+1q+β−α>0,with
(9)0≤α<Nq+1q+β.Now, we can state our main results.Theorem 1.
Assume that4<2p<N, 0∈Ω,p<γ<p∗ with p∗≔pN/N−2p is the critical Sobolev exponent, 0≤β<Nγ+α/γ+1, 0<α<1 and μ≤0; then, problem (1) has at least a nontrivial solution in E.Theorem 2.
LetΩ⊂ℝN be a bounded, smooth, and strictly star-shaped domain, 0≤μ<μ∗=N−2p1−α/Np and β=0, with γ≥γ∗=Np1−α/N−2p1−α−μNp>p∗=Np/N−2p. Then, problem (1) has no positive solutions in E.This paper is organized as follows. In Section2, we give some preliminaries. Sections 3 and 4 are devoted to the proofs of Theorems 1 and 2.


## 2. Some Preliminary Results

Definition 3 [1].
An operatorA: E⟶E′ is hemicontinuous if
(10)λ↦Au1+λu2,u3,is continuous from ℝ to ℝ, for all u1,u2,u3∈E.Definition 4 [1].
An operatorA: E⟶E′ is “ calculus of variations,” if it is bounded and if it can be represented by
d1Au=A~u,u where
(11)A~:E×E⟶E′,u,u^↦A~u,u^,is an operator satisfying the following properties.
d2(12)∀u∈E;u^↦A~u,u^hemicontinuous,boundedofE⟶E′A~u,u−A~u,u^,u−u^≥0.
d3 For all: u^∈E; u↦A~u,u^ is bounded, hemicontinuous of E⟶E′.
d4 If unn∈ℕ converges weakly to u in E and if
(13)A~un,un−A~un,u,un−u⟶0asn⟶∞,then, for all u^∈E, the sequence definite by A~un,u^ converges to A~u,u^ in E′.
d5 If unn∈N converges weakly to u in E, and if A~un,u^ converges weakly to φ in E′, then
(14)A~un,u^,unE′,E⟶φ,uE′,Easn⟶∞.


## 3. Existence Results

The energy functionalJθ associated to Equation (4) is defined on E by
(15)Jθu≔1pup−1γ+1∫Ωuγ+1dx−μ1−α∫Ωu+θ1−α−θ1−αxβdx.To prove Theorem1, we use the three following lemmas.SetJθu≔Au−Bu with Au=1/p∫ΩΔupdx=1/pup and Bu=1/γ+1∫Ωuγ+1dx+μ/1−α∫Ωu+θ1−α−θ1−α/xβdx.Firstly, we need the following Lemmas.Lemma 5.
JθisC1E,ℝ.Proof.
LetR0>0 such that Ω⊂B0,R0=x∈ℝN:x<R0. If u∈E and by the Hölder inequality, we obtain
(16)∫Ωu1−αxβdx≤C1u1−α,with C1=AS1−1−α/p,
where(17)S1≔infu∈E\0up∫Ωu1−α/xβdxp/1−α.
It is enough to show thatA and B are C1 on E.
According to an algebraic relation of Li and Tang [3] and the Hölder inequality, one has
(18)A′u1−A′u2E′≤FpΔu1−FpΔu2p′≤CΔu1−Δu2pΔu1p+Δu2pp−2,whereFpt=tp−2t.
ThereforeA is C1 on E and B is C1, since gu=uγ−1u is subcritical and by (16). Thus, Jθ is C1 on E.Lemma 6.
Jθ satisfies the Palais-Smale condition.Proof.
Letun be a Palais-Smale sequence in E (i.e., Jθun is bounded and Jθ′un⟶0). un is bounded in E. In fact,
(19)∫ΩΔunp−unγ+1dx⟶0,asn⟶∞,(20)∫Ω1pΔunp−μγ+1unγ+1isbounded.
Combining (19), (20), and (16), we obtain that un is bounded when p<γ<p∗,0≤β<Nγ+α/γ+1,0<α<1 and μ≤0.
The embeddingE↪LγΩ is compact for p<γ<p∗; then, there exists a subsequence, still denoted by un, which converges in LγΩ. Show that un converges in E.
Set(21)In,m=∫ΩFpΔun−FpΔumΔun−Δum.
Then, we can writeIn,m in the form
(22)In,m=Jθ′un−Jθ′um,un−um+Bx,un−Bx,umun−umdx.
In addition,Fp verifies an algebraic relation [1], from where
(23)Δun−Δump≤CFpΔun−FpΔumΔun−Δum,thus,
(24)un−ump≤CIn,m,and therefore
(25)limn,m⟶+∞un−um=0.
Then,un converges strongly in E.Lemma 7.
Suppose4<2p<N, 0∈Ω,p<γ<p∗ with p∗≔pN/N−2pis the critical Sobolev exponent, 0≤β<Nγ+α/γ+1, 0<α<1 and μ≤0; then, there exist ε and η positive constants such that
(i)
We have(26)Jθu≥η>0whenε=usmall.(ii)
There existsv∈Mρ when v>ε, with ε=u, such that Jθv≤0.Proof.
(i)
We have(27)∫Ωuγ+1dx≤uγ+1S−γ+1/p,∫Ωu1−αxβdx≤u1−αAS1−1−α/p.Then, we get
(28)Jθu≥1pup−S−γ+1/pγ+1uγ+1−μAS1−1−α/p1−αu1−α.
Jθu≥η>0 when ε=u small.
(ii)
Jθ0=0 and Jθw>0 for an element w≥0;w≢0; then,(29)limt⟶+∞Jtw=−∞,
Indeed
Lett be such that t>0,tsufficientlylarge(30)Jθtw=tpp∫ΩΔwpdx−tγ+1γ+1∫Ωwγ+1dx+−μt1−α1−α∫Ωw+θ1−α−θ1−αxβdx.
Howeverγ>p, we obtain
(31)limt⟶+∞Jθtw=−∞.
Owing to the fact thatJθ is continuous, then there exists v0≠0 such that Jθv0<0.Proof of Theorem 8.
From Lemmas5–7, we deduce that there exists u∈E−0 such that
(32)∫ΩFpΔu.Δv−∫Ωuγ−1uvdx−μ∫Ωvu+θαxβdx=0,∀v∈E.Finally, for everyθ∈0,1, problem (4) has solution uθ∈H such that Jθuθ=0. Thus, there exist θn⊂0,1 with θn⟶0 as n⟶∞. Then, we get u=limn⟶∞uθn.


## 4. Nonexistence Result

By a Pohozaev-type identity, one shows the nonexistence of positive solutions of (1) when 0≤μ<μ∗=N−2p1−α/Np and β=0, with γ≥γ∗=Np1−α/N−2p1−α−μNp>p∗=Np/N−2p.The Proof Theorem10 uses an identity of the Pohozaev type which we state in the following Lemma.Lemma 9.
Letu∈E be a positive solution of (1).
Then, the following identity is checked(33)N−2pp∫ΩΔupdx−Nγ∫Ωuγdx−Nμ1−α∫Ωu1−αdx=−p−1p∫∂ΩΔupx·n⟶dσ,where n⟶isthenormalexteriorat∂Ω.Proof.
Multiplying Equation (1) by x·∇u and integrating on Ω, we obtain
(34)∫Ωgux·∇udx=∫ΩΔwx·∇udxwherew=Δup−2Δuandgu=uγ+u1−α.
Set(35)A1u=∫Ωgux·∇udxandA2u=∫ΩΔwx·∇udx.
Calculation ofA1u(36)∫Ωgux.∇udx=∑j=1N∫Ωguxj∂u∂xjdσ,=∑j=1N∫Ω∂∂xiGuxjdσ.
According to the divergence theorem,(37)∑j=1N∫Ω∂∂xiGuxjdσ=−∑j=1N∫ΩGudx+∑j=1N∫ΩGuxjn⟶jdσ,=−N∫ΩGudx+∑j=1N∫∂ΩGux.n⟶dσ.
However,(38)u=0on∂Ω⇒Gu=0on∂Ω,from where one deduces from (36) and (37) that
(39)A1u=−N∫ΩGudx=−Nγ∫Ωuγdx−Nμ1−α∫Ωu1−αdx.
Calculation ofA2u
In [19], Mitidieri established the following relation:
(40)∫ΩΔvx·∇u+Δux·∇vdx=∫∂Ω∂v∂n⟶x·∇u+∂u∂n⟶x·∇v−∇u∇vx·n⟶dσ+N−2∫Ω∇u∇vdx.
Let us apply to our case withv=w=Δup−2Δu(41)A2u=∫Ω∂w∂n⟶x·∇u+∂u∂n⟶x·∇wdσ+−∫∂Ω∇u∇wx·n⟶dσ−∫ΩΔux·∇w+N−2∫Ω∇u∇wdx.
Set(42)H1=∫∂Ω∂w∂n⟶x·∇u+∂u∂n⟶x·∇wdσ,H2=∫∂Ω∇u∇wx·n⟶dσ,H3=∫ΩΔux·∇w,andH4=∫Ω∇u∇wdx.
Owing to the fact thatu=∇u=0 on ∂Ω,
we have(43)H1=∫∂Ω∂w∂n⟶x·∇u+∂u∂n⟶x·∇wdσ=0,(44)H2=∫∂Ω∇u∇wx·n⟶dσ=0.
Calculations ofH3
We have(45)H3=∫ΩΔux·∇wdx,=∫Ωx.∇p−1pΔupdx,by applying the divergence theorem, we obtain
(46)H3=p−1p∫∂ΩΔupx·n⟶dσ−Np−1p∫ΩΔupdx.
Calculations ofH4
Applying Green’s formula generalized, we obtain(47)H4=∫Ω∇u∇wdx,=−∫ΩwΔudx,therefore,
(48)H4=−∫ΩΔupdx.
From (43), (44), (46), and (48), we obtain
(49)A2u=−p−1p∫∂ΩΔupx·n⟶dσ+Np−1p∫ΩΔupdx+−N−2∫ΩΔupdx.
Thus,(50)A2u=−N−2pp∫ΩΔupdx−p−1p∫∂ΩΔupx·n⟶dσ.
From (39) and (50), we deduces that
(51)N−2pp∫ΩΔupdx−Nγ∫Ωuγdx−Nμ1−α∫Ωu1−αdx=−p−1p∫∂ΩΔupx·n⟶dσ,Proof of Theorem 10 (By the absurdity).
(52)N−2pp∫ΩΔupdx−Nγ∫Ωuγdx−Nμ1−α∫Ωu1−αdx=−p−1p∫∂ΩΔupx·n⟶dσ,thus, we obtain
(53)N−2pp−Nγ−Nμ1−α∫ΩΔupdx=−p−1p∫∂ΩΔupx·n⟶dσ.
Knowing thatΩ is strictly star-shaped then,
(54)N−2pp−Nγ−Nμ1−α<0,what contradicts the fact that γ≥γ∗=Np1−α/N−2p1−α−μNp>p∗=Np/N−2p when 0≤μ<μ∗=N−2p1−α/Np.


## 5. Conclusion

In our work, we have searched the critical points as the minimizers of the energy functional associated to the problem. Under some sufficient conditions on coefficients of equation of (1) such that 4<2p<N, 0∈Ω,p<γ<p∗ with p∗≔pN/N−2p is the critical Sobolev exponent, 0≤β<Nγ+α/γ+1, 0<α<1 and μ≤0. In Section 3, we have shown the existence of at least one nontrivial solution in E. In the Section 4, the nonexistence of positive solution at (1) has proved when Ω⊂ℝN can be a bounded, smooth, and stricly star-shaped domain, 0≤μ<μ∗=N−2p1−α/Np and β=0, with γ≥γ∗=Np1−α/N−2p1−α−μNp>p∗=Np/N−2p.



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