Positive solutions for singular nonlinear elliptic equations

Abstract

This paper deals with singular semilinear elliptic equations in bounded domains with Dirichlet boundary data. The elliptic operator is a second-order operator not necessarily in divergence form. We consider existence, uniqueness and linearized stability of positive solutions for a series of nonlinear eigenvalue problems.

Periodic parabolic problems with nonlinearities indefinite in sign

“…In order to relate our results to others in the literature, let us mention that similar elliptic problems have been studied for example in [20] for L = −∆ and a, b ∈ C Ω using a variational approach, and recently for a, b ∈ C 1 (Ω) and allowing these functions to have a singularity in the boundary in [14,Section 3] (see also the references therein). The particular case Lu = mu p in Ω, with 0 < p < 1 and m changing sign, was treated in detail in [2] for L = −∆ and m ∈ C θ Ω , θ ∈ (0, 1), using sub and supersolutions to construct nonnegative solutions, and there is also a result for the associated parabolic initial boundary value problem there.…”

Periodic parabolic problems with concave and convex nonlinearities

“…Thus, if we had available linear theory for "singular" weights (as it is the case in the elliptic problem, see e.g. [12]), H1 could be removed.…”

“…On the other hand, for a non-selfadjoint operator, some existence results have been proved recently for 0 ≤ a, b ∈ C 1 (Ω) and allowing these functions to have a singularity in the boundary in [12], Section 3, and also in [4] assuming that a ≡ 1 and b ∈ C α Ω , α ∈ (0, 1). In the parabolic case, when b ≤ 0 (logistic-type equation), existence of positive periodic solutions was shown in [9], but no results seem to be known for (1.3) when b > 0 in a set of positive measure.…”

“…In both [1] and [6], existence of at least two (positive) solutions for (1.3) is proved under some growth restriction on q (namely, q ≤ (N + 2) / (N − 2)) using variational arguments, which of course are not eligible in our case. On the other hand, also under some restrictions on b and q, in [4] a second (positive) solution is found in a different way assuming that L is selfadjoint and making use of some spectral theory with singular potential 446 T. Godoy and U. Kaufmann NoDEA (see [12]) which is not available for the periodic parabolic problem. We believe that at least some of the above mentioned multiplicity results should still be true in the parabolic case, but we are not able to give a proof.…”

Global existence and blow-up to a class of degenerate parabolic equation with time dependent coefficients

“…We would like to refer some results on blowup solutions to the degenerate parabolic equations and system in [2,6,7] and references therein.…”