Quasilinear elliptic problems with singular and homogeneous lower order terms

A singularity as a break point for the multiplicity of solutions to quasilinear elliptic problems

“…In contrast, we also derive existence and uniqueness results for λ > 0 and q − 1 < α ≤ 1. We thus complement the results in [15,16], which are concerned with α = q − 1, and show that the value α = q − 1 plays the role of a break point for the multiplicity/uniqueness of solution.…”

“…The proof basically follows the steps of a similar result in [3]. However, up to our knowledge this is the first time that a comparison result has been proved including a general positive singular lower order term on the right hand side of the equation (see the comparison results in [16], where a specific 1-homogeneous singular term is considered). Theorem 2.1.…”

“…By taking ϕ 1 as test function in the weak formulation of (P λ ) one easily deduces that, if u is a solution to (P λ ), then λ < λ 1 . Furthermore, since α ∈ [0, 1], it can be proved as in [16,Appendix], which follows the ideas in [28], that every solution u to (P λ ), for any λ < λ 1 , satisfies that u ∈ C 0,η (Ω) for some η ∈ (0, 1). Finally, since the solutions to (P λ ) are positive in compact subsets of Ω, then it can be seen again as in the mentioned appendix that u ∈ W 1,N loc (Ω) for every solution to (P λ ) for any λ < λ 1 .…”

“…Indeed, we state and prove in Section 2 two comparison principles valid for continuous lower and upper solutions to singular equations. As far as we know, these two results are new, and they are interesting by themselves as only few uniqueness results for singular equations are known (see [2,5,6,14,16]). We follow in their proofs the arguments in [3] and [16].…”

“…This suggests that the nonlinear term does not play an essential role in this case, since the situation is analogous to the linear problem (µ ≡ 0). Recall that this is not the case when α = q − 1, for which one has existence if and only if λ < λ * , where λ * < λ 1 provided µ > 0 (see [16,Remark 6.3]).…”