Qualitative properties of solutions to elliptic singular problems

• Main Authors: Gladiali F, Porru G, Berhanu S
• Format: Article
• Language: English
• Published: SpringerOpen 1999-01-01
• Series: Journal of Inequalities and Applications
• Subjects: Singular equations; Boundary behaviour; Convexity
• Online Access: http://www.journalofinequalitiesandapplications.com/content/3/197624

<p/> <p>We investigate the singular boundary value problem <inline-formula><graphic file="1029-242X-1999-197624-i1.gif"/></inline-formula> in <inline-formula><graphic file="1029-242X-1999-197624-i2.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-1999-197624-i3.gif"/></inline-formula> on <inline-formula><graphic file="1029-242X-1999-197624-i4.gif"/></inline-formula>, where <inline-formula><graphic file="1029-242X-1999-197624-i5.gif"/></inline-formula>. For <inline-formula><graphic file="1029-242X-1999-197624-i6.gif"/></inline-formula>, we find the estimate <inline-formula><graphic file="1029-242X-1999-197624-i7.gif"/></inline-formula> where <inline-formula><graphic file="1029-242X-1999-197624-i8.gif"/></inline-formula> depends on <inline-formula><graphic file="1029-242X-1999-197624-i9.gif"/></inline-formula> only, <inline-formula><graphic file="1029-242X-1999-197624-i10.gif"/></inline-formula> denotes the distance from <inline-formula><graphic file="1029-242X-1999-197624-i11.gif"/></inline-formula> to <inline-formula><graphic file="1029-242X-1999-197624-i12.gif"/></inline-formula> and is <inline-formula><graphic file="1029-242X-1999-197624-i13.gif"/></inline-formula> suitable constant. For <inline-formula><graphic file="1029-242X-1999-197624-i14.gif"/></inline-formula>, we prove that the function <inline-formula><graphic file="1029-242X-1999-197624-i15.gif"/></inline-formula> is concave whenever <inline-formula><graphic file="1029-242X-1999-197624-i16.gif"/></inline-formula> is convex. A similar result is well known for the equation <inline-formula><graphic file="1029-242X-1999-197624-i17.gif"/></inline-formula>, with <inline-formula><graphic file="1029-242X-1999-197624-i18.gif"/></inline-formula>. For <inline-formula><graphic file="1029-242X-1999-197624-i19.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-1999-197624-i20.gif"/></inline-formula> and <inline-formula><graphic file="1029-242X-1999-197624-i21.gif"/></inline-formula> we prove convexity sharpness results.</p>