Existence, uniqueness and qualitative properties of positive solutions of quasilinear elliptic equations

• Authors: NGUYEN Phuoc-Tai; VO Hoang-Hung
• Year of publication: 2015
• Type: Article in Periodical
• Magazine / Source: Journal of Functional Analysis
• MU Faculty or unit: Faculty of Science
• web: https://www.sciencedirect.com/science/article/pii/S0022123615003626?via%3Dihub
• Doi: http://dx.doi.org/10.1016/j.jfa.2015.09.003
• Keywords: Quasilinear elliptic equations; Hardy potentials; Polynomial decay; Exponential decay
• Description: We study the following quasilinear elliptic equation -Delta(p)u (beta Phi(x) - a(x))u(p-1) + b(x)g(u) = 0 in R-N, (P-beta) where p > 1, a, b is an element of L-infinity(R-N), beta, b, g >= 0, b not equivalent to 0 and Phi is an element of L-loc(infinity)(R-N), inf(R)N, Phi > -infinity. We provide a sharp criterion in term of generalized principal eigenvalues for existence/non-existence of positive solution of (P-beta) in suitable classes of functions. Uniqueness result for (P-beta) in those classes is also derived. Under additional conditions on Phi, we further show that: i) either for every beta >= 0 nonexistence phenomenon occurs, ii) or there exists a threshold value beta* > 0 in the sense that for every beta is an element of [0, beta*) existence and uniqueness phenomenon occurs and for every beta >= beta* nonexistence phenomenon occurs. In the latter case, we study the limits, as beta -> 0 and beta -> beta*, of the sequence of positive solutions of (P-beta). Our results are new even in the case p = 2.