Estimates at infinity for positive solutions to problems involving the p-Laplacian
Ralph Thomas · Digital Scholarship - UNLV (University of Nevada Reno) · 2019
There has been much study of finding positive solutions to various logistic problems involving the Laplacian and the p-Laplacian; problems which, loosely speaking, contain a nonlinear term that behaves like lambdau p-1(1 - u gamma); In this paper we will first look for positive solutions to -Dpu=ax lup-2u-a xgu x∈RN 0.3 where g(s) behaves like sgamma-1, gamma > p, for s large. We will employ many of the same methods as Costa, Drabek and Tehrani, and in doing so will not only prove the existence of positive (weak) solutions, but will also have estimates for the behavior of these solutions at infinity. Namely we will show that a solution u0 satisfies u0x≥C x-N-pp-1 forx large. In addition, by an appropriate modification of our assumptions on g(u) and a( x), we prove that u0 is the unique positive solution to (0.3), and that the above estimate at infinity is sharp; Second, we will generalize our first scalar result to a system result, finding positive solutions (u0, v 0) to -Dp1u=a1 xm1 up1-2u-g1 u+F ux,u,v -Dp2v=a2 xm2 vp2-2v-g2 v+F vx,u,v satisfying u0x≥ CxN-p 1p1-1 forx large, and v0x≥ CxN-p 2p2-1 forx large, where the interaction term behaves like F( x,s,t) = bxsp1 mtp2 m-1m with m ≤ p1 and mm-1 ≤ p2; Finally, we will add harvesting terms to our system equations, finding solutions (u0, v0) to -Dp1u=a1 xm1 up1-2u-g1 u+F ux,u,v-n 1h1x -Dp2v=a2 xm2 vp2-2v-g2 v+F vx,u,v-n 2h2x In this case, we must have p1 = p2 = 2 to prove that the solution is positive and satisfies the same behaviors at infinity as above. (Abstract shortened by UMI.).