Multiplicity of Solutions of the Weighted p-Laplacian with Isolated Singularity and Diffusion Suppressed by Convection*

Liting YOU Huijuan SONG Jingxue YIN

1School of Mathematical Sciences,South China Normal University,Guangzhou 510631,China.E-mail:yoult@m.scnu.edu.cn yjx@scnu.edu.cn

2Corresponding Author.School of Mathematics and Statistics,Jiangxi Normal University,Nanchang 330022,China.E-mail:songhj@jxnu.edu.cn

Abstract In this paper,the authors study the multiplicity of solutions to the weighted p-Laplacian with isolated singularity and diffusion suppressed by convection

subject to nonlinear Robin boundary value condition

where λ ＞0,B ?RN(N ≥2)is the unit ball centered at the origin,α ＞0,p ＞1,β ∈R,γ ＞-N,g ∈C([0,1])with g(0)＞0,A ∈R,ρ ＞0 and →n is the unit outward normal.The same problem with diffusion promoted by convection,namely λ ≤0,has already been discussed by the last two authors(Song-Yin(2012)),where the existence,nonexistence and classification of singularities for solutions are presented.Completely different from[Song,H.J.and Yin,J.X.,Removable isolated singularities of solutions to the weighted p-Laplacian with singular convection,Math.Meth.Appl.Sci.,35,2012,1089–1100],in the present case λ ＞0,namely the diffusion is suppressed by the convection,non-singular solutions are not only existent but also may be infinite which vary according only to the values of solutions at the isolated singular point.At the same time,the singular solutions may exist only if the diffusion dominates the convection.

Keywords Weighted p-Laplacian,Multiplicity of solutions,Isolated singularities,Convection

1 Introduction

This is a continuation of the work[1]on the study of isolated singularities of solutions to the following nonlinear Robin boundary value problem:

where B ?RN(N ≥2)is the unit ball centered at the origin,α ＞0,p ＞1,γ ＞-N,λ,β,A ∈R,ρ ＞0,g(r)is a continuous function defined on[0,1]with g(0)＞0,and →n denotes the unit outward normal to the boundary ?B.We have classified the isolated singularities for the problem when λ ≤0 in[1].The essential point to set λ ≤0 in[1]is due to the fact that the convection may exhibit(sometimes very strong)degeneracy at the isolated singular point,being consistent with the diffusion,namely the convection promotes the diffusion,which ensures the uniqueness of solutions.When λ ＞0,the convection is with(sometimes very strong)singularity,being opposite from that of the diffusion,namely the convection suppresses the diffusion;this causes many difficulties for the classification of the isolated singularities and the proofs of the nonexistence of solutions.

The study of isolated singularities for quasilinear elliptic equations was initiated by Serrin in[2–3],where the growth of lower-order terms is at most that of the principal part.Since then,great attention has been paid to the study of isolated singularities of various equations;see[4–5]for the fractional Laplacian,[6]for nonhomogeneous divergence-form operators,[7]for nonlinear equations with singular potentials,[8–9]for equations with nonlinearities depending on the gradient,[10–11]for the weighted p-Laplacian and so on.However,as far as we know,there are only a few papers concerning isolated singularities for equations involving convection explicitly.An elaborate and rarely known result was obtained in 1995 by Guedda and Kirane[12]for positive solutions to the equation

Now we state the main results of this paper.We consider the balanced case β =2-α first.Recall that when λ ≤0,there may exist infinitely many singular solutions but one non-singular solution at most.However,when λ ＞0,not only singular solution does not exist,but also there may exist infinitely many non-singular solutions.

Theorem 1.1Assume β =2-α.

(i)If

α ≥p+γ,

then problem(1.1)–(1.2)has no solution.

(ii)If

(iii)If

then for any θ ∈R,(1.1),(1.2)and(1.5)has a unique solution,which also satisfies the equation(1.1)in the whole ball B.That is to say,problem(1.1)–(1.2)admits infinitely many non-singular solutions.

Remark 1.2When p,γ,N have different values,the figures are slightly different,but we prefer to show the case p ＞N.Being same as the previous remark,we show it for λ ∈R:

From the figure,we see that when 0 ＜λ ＜N+γ,the first critical exponent for α:min{p-N+λ,p+γ}=p-N+λ,does not change(see[1]),and as the value of λ increases,the length of the interval in which(1.1)–(1.2)has infinitely many non-singular solutions is lengthened,attains the maximum at λ = N +γ and keeps unchanged when λ ＞N +γ;correspondingly,the length of the interval in which(1.1)–(1.2)has a unique non-singular solution is shortened and becomes zero when λ ≥N +γ.Besides,for 0 ＜α ＜min{p-N +λ,p+γ},infinitely many solutions to(1.1)–(1.2)all have removable singularities at the origin here,but only one solution possesses removable singularities in the case λ ≤0.

In fact,if p ≤N,the singular solution in the case λ ≤0 does not exist,and if p ≤-γ,there is no solution with λ ∈R.

Next,we consider the case β ＞2-α.We find that when β ＞2-α and β ＞2-p-γ,the result that there are infinitely many non-singular solutions is different from that in the case λ ≤0,where problem(1.1)–(1.2)has a unique non-singular solution.

Theorem 1.2Suppose β ＞2-α.

(i)If β ≤2-p-γ,then problem(1.1)–(1.2)has no solution.

(ii)If β ＞2-p-γ,then for any θ ∈R,(1.1),(1.2)and(1.5)has a unique solution,which further satisfies(1.1)in the whole ball B.In other words,problem(1.1)–(1.2)has infinitely many non-singular solutions.

Remark 1.3The figure as α ＞p+γ is slightly different from the figure as α ≤p+γ.Again,we show the case α ＞p+γ for λ ∈R here:

From the figure,we see that even if p and γ have different values,there exist infinitely many non-singular solutions if β ＞max{2-p-γ,2-α},which is different from the case β =2-α where there is no solution if p ≤-γ.

Finally,we consider the case β ＜2-α,the result is exactly the same as that in the case λ ≤0.

Theorem 1.3Let β ＜2-α.

(i)If α ≥p+γ,then problem(1.1)–(1.2)has no solution.

(ii)If p-N ≤α ＜p+γ,then(1.1)–(1.2)has a unique solution,which is non-singular.

(iii)If 0 ＜α ＜p-N,then for any θ ∈R,(1.1)–(1.2)has a unique solution such that(1.5)holds,among which one and only one is non-singular.

Remark 1.4Although the result is the same as that in the case λ ≤0,we show it for λ ∈R:

In fact,the figure as p ≤N is slightly different from that as p ＞N we have shown.If p ≤N,the singular solution will not exist.

The rest of this paper is organized as follows.In Section 2,we introduce several related notations and present some auxiliary lemmas.Subsequently,we carry out the proofs of Theorem 1.1 in Section 3 and Theorem 1.2 in Section 4;since Theorem 1.3 can be proven by using a procedure quite similar to that in[1],for brevity we omit the details.

2 Preliminaries

3 Proof of Theorem 1.1

4 Proof of Theorem 1.2

and according to(3.14),

Since(4.10)has a unique root,the uniqueness of generalized solutions is derived.Next,using(4.6)and β ＞max{2-α,2-p-γ},one can see that(4.9)with U satisfying(4.10)solves(1.1),(1.2)and(1.5).Finally,the removability of singularities follows by a similar argument as that in the proof of(ii)of Theorem 1.1.The proof is complete.