Radial solutions of equations and inequalities involving the <inline-formula><graphic file="1029-242X-1997-409301-i1.gif"/></inline-formula>-Laplacian

• Main Authors: Reichel Wolfgang, Walter Wolfgang
• Format: Article
• Language: English
• Published: SpringerOpen 1997-01-01
• Series: Journal of Inequalities and Applications
• Subjects: <it>p</it>-Laplacian; radial solutions; uniqueness; comparison principle; blow-up solutions
• Online Access: http://www.journalofinequalitiesandapplications.com/content/1/409301
• ISSN: 1025-5834; 1029-242X

<p/> <p>Several problems for the differential equation <inline-formula><graphic file="1029-242X-1997-409301-i2.gif"/></inline-formula> are considered. For <inline-formula><graphic file="1029-242X-1997-409301-i3.gif"/></inline-formula>, the operator <inline-formula><graphic file="1029-242X-1997-409301-i4.gif"/></inline-formula> is the radially symmetric <inline-formula><graphic file="1029-242X-1997-409301-i5.gif"/></inline-formula>-Laplacian in <inline-formula><graphic file="1029-242X-1997-409301-i6.gif"/></inline-formula>. For the initial value problem with given data <inline-formula><graphic file="1029-242X-1997-409301-i7.gif"/></inline-formula> various uniqueness conditions and counterexamples to uniqueness are given. For the case where <inline-formula><graphic file="1029-242X-1997-409301-i8.gif"/></inline-formula> is increasing in <inline-formula><graphic file="1029-242X-1997-409301-i9.gif"/></inline-formula>, a sharp comparison theorem is established; it leads to maximal solutions, nonuniqueness and uniqueness results, among others. Using these results, a strong comparison principle for the boundary value problem and a number of properties of blow-up solutions are proved under weak assumptions on the nonlinearity <inline-formula><graphic file="1029-242X-1997-409301-i10.gif"/></inline-formula>.</p>