Bifurcation for a class of singular elliptic problems with quadratic convection term
Marius Ghergu, Vicenţiu D. Rădulescu · Comptes Rendus Mathématique · 2004
We study the bifurcation problem −Δ u = g ( u )+ λ |∇ u | 2 + μ in Ω , u = 0 on ∂ Ω , where λ , μ ⩾0 and Ω is a smooth bounded domain in ℝ N . The singular character of the problem is given by the nonlinearity g which is assumed to be decreasing and unbounded around the origin. In this Note we prove that the above problem has a positive classical solution (which is unique) if and only if λ ( a + μ )< λ 1 , where a =lim t →+∞ g ( t ) and λ 1 is the first eigenvalue of the Laplace operator in H 0 1 ( Ω ) . We also describe the decay rate of this solution, as well as a blow-up result around the bifurcation parameter.