Asymptotically Linear Problems and Antimaximum Principle for the Square Root of the Laplacian Articles uri icon

publication date

  • November 2012

start page

  • 683

end page

  • 701

issue

  • 4

volume

  • 12

International Standard Serial Number (ISSN)

  • 1536-1365

abstract

  • This work deals with bifurcation of positive solutions for some asymptotically linear problems, involving the square root of the Laplacian (-Delta)(1/2). A simplified model problem is the following: {(-Delta)(1/2)u = lambda m(x)u + g(u) in Omega, u = 0 on partial derivative Omega, with Omega subset of R-N a smooth bounded domain, N >= 2, lambda > 0, m is an element of L-infinity(Omega), m(+) not equivalent to 0 and g is a continuous function which is super-linear at 0 and sub-linear at infinity. As a consequence of our bifurcation theory approach we prove some existence and multiplicity results. Finally, we also show an anti-maximum principle in the corresponding functional setting.