Radially Symmetric Positive Solutions of the Dirichlet Problem for the p-Laplace Equation

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper studies the boundary value problem (3) with the Dirichlet boundary condition (4). Equation (3) is a differential equation involving the p-Laplacian in R^d.

By restricting to radially symmetric positive solutions, equations (3) and (4) reduce to (6) and (7).

This paper then focuses on solving (6) and (7). 

 

Theorems 1 and 2 provide a new lower estimate for positive solutions to (6) and (7).

Theorems 4 and 5 show the existence of positive solutions for (6) and (7) under certain conditions.

Theorem 6 and 7 presents a condition under which (6) and (7) do not have any positive solutions.

These results are illustrated by Example 1, where the existence of radially symmetric positive solutions rely on the size of lambda.

 

Overall, the results look interesting and meaningful, so I recommend its publication in Mathematics after some minor revision.

 

 

Comments:

 

- Lines 25-26: The parameter p is hidden in the definition of \Phi in (5), so it is not easy to recognize the dependency of (6) and (7) on p, even though p is an important parameter as can be seen in Theorems 4 and 5. It seems better to write Phi_p instead of Phi, in order to highlight its dependency on p.

 

- Lines 106-108: Theorem 3 is the fixed point theorem originally due to Krasnosel’skii. The negation of ≥ and ≤ was a bit confusing when I first read the conditions (K1) and (K2). Lu \not \geq u means Lu(t) < u(t) for some t. Maybe the authors can add a sentence describing the conditions (K1) and (K2). Also, are there any physical interpretations for the conditions (K1) and (K2)? 

 

- Line 113: “Krasnosel’skii’s fixed has become” → “Krasnosel’skii’s fixed point theorem has become” 

 

- The conclusion section is missing. I suggest the authors to add a conclusion section at the end of the paper. 

Author Response

Comments 1:  Lines 25-26: The parameter p is hidden in the definition of \Phi in (5), so it is not easy to recognize the dependency of (6) and (7) on p, even though p is an important parameter as can be seen in Theorems 4 and 5. It seems better to write Phi_p instead of Phi, in order to highlight its dependency on p.
Respond 1: I agree. I have replaced $\Phi$ by $\Phi_p$ in the revised version.
 

Comments 2:  Lines 106-108: Theorem 3 is the fixed point theorem originally due to Krasnosel’skii. The negation of ≥ and ≤ was a bit confusing when I first read the conditions (K1) and (K2). Lu \not \geq u means Lu(t) < u(t) for some t. Maybe the authors can add a sentence describing the conditions (K1) and (K2). Also, are there any physical interpretations for the conditions (K1) and (K2)? 
Respond 2: Thank you for the suggestions. I added a remark right after the statement of the fixed point theorem.
 

Comments 3:  Line 113: “Krasnosel’skii’s fixed has become” → “Krasnosel’skii’s fixed point theorem has become” 
Respond 3: Thanks! I made the changes in the revised version.
 

Comments 4:  The conclusion section is missing. I suggest the authors to add a conclusion section at the end of the paper. 
Respond 4: A conclusion section is added in the revised version of the manuscript.

Thank you very much for your time and the valuable suggestions!

 

Reviewer 2 Report

Comments and Suggestions for Authors

In this manuscript the author studies the existence of radially symmetric positive solutions for p-Laplacian BVP with Dirichlet conditions. The author reduces the considered problem to second order ODE and then he shows the existence of a fixed point. The arguments are based on Krasosel'skii's fixed point theorem in cones. The calculations seem to be correct and the paper is clearly written. However, there are a few issues that need to be fixed before publishing:

- References [3], [6] and [9] are not cited in the manuscript. Please, check if this is the case and cite them or remove them form the References;

- Also, there are a lot of numbered equations that are not cited in the text. Such as: 1,2,5,8,9,12-15,18,20. Please, check and remove the numbering or cite them in the text;

- on page 1, condition (H): remove "," after n in the beginning;

- on page 6, Proof of Theorem 6: It is better to be "Assume on contrary..." instead of "Assume to the contrary...";

- Section 2: In Section 2, the author presents "a new lower estimate" for positive solutions to the problem (6)-(7). However, it is not clear at all what is the "old one" and how his result improves is, which is maybe the main purpose of this paper.

Author Response

• Comments 1: References [3], [6] and [9] are not cited in the manuscript. Please, check if this is the case and cite them or remove them form the References;
  Respond 1: Thank you for pointing this out. I totally agree with you. References [3], [6] and [9] are now cited in the revised version.
• Comments 2: Also, there are a lot of numbered equations that are not cited in the text. Such as: 1,2,5,8,9,12-15,18,20. Please, check and remove the numbering or cite them in the text;
  Respond 3: Thank you for pointing this out. I have removed the redundant numberings.
• Comments 3: on page 1, condition (H): remove "," after n in the beginning;
  Respond 3: Agree. Changes are made in the revised version. 
• Comments 4: on page 6, Proof of Theorem 6: It is better to be "Assume on contrary..." instead of "Assume to the contrary...";
  Respond 4: Agree. I have made the changes in the revised version.
• Comments 5: In Section 2, the author presents "a new lower estimate" for positive solutions to the problem (6)-(7). However, it is not clear at all what is the "old one" and how his result improves is, which is maybe the main purpose of this paper.
  Respond 5: Agree. In the revised version, I added some explanation in the beginning of Section 2.
• Thank you very much for the valuable suggestions!