Existence and Multiplicity of Positive Solutions for Singular $p$-Laplacian Equations

Abstract

Abstract. Positive solutions are obtained for the boundary value problemwhereN −p are two constants, λ > 0 is a real parameter. We obtain that Problem ( * ) has two positive weakly solutions if λ is small enough.

A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

“…Moreover, we show that for every λ ∈ L := (0, λ * ] problem (P λ ) admits a minimal positive solution u * λ , and establish the monotonicity and continuity properties of the map λ → u * λ . Our work here extends that of Lü-Xie [25], who considered equations driven by the p-Laplacian only and f (z, x) = x r −1 with p < r < p * (recall…”

“…the critical Sobolev exponent corresponding to p). In [25], the authors did not prove the precise dependence of the set of positive solutions on the parameter λ > 0, that is, they did not prove a bifurcation-type theorem as described above and they did not produce the minimum positive solution.…”