In this work we establish existence results for a class of nonhomogeneous and singular quasilinear elliptic equations involving a convection term. The gradient term makes the problem non variational, and in addition to this difficulty we have to handle the singular term with a sign changing nonlinearity. The proof of the results are made combining the sub‐super solution method, fixed point theorem, Leray–Schauder degree theory and comparison theorems.
We establish the existence of connected components of positive solutions for the equation \( (-\Delta_p)^s u = \lambda f(u)\), under Dirichlet boundary conditions, where the domain is a bounded in \(\mathbb{R}^N\) and has smooth boundary, \((-\Delta_p)^s\) is the fractional p-Laplacian operator, and \(f:(0,\infty) \to \mathbb{R}\) is a continuous function which may blow up to \(\pm \infty\) at the origin.
In a recent paper D. D. Hai showed that the equation −∆ p u = λf (u) in Ω, under Dirichlet boundary condition, where Ω ⊂ R N is a bounded domain with smooth boundary ∂Ω, ∆ p is the p-Laplacian, f : (0, ∞) → R is a continuous function which may blow up to ±∞ at the origin, admits a solution if λ > λ 0 and has no solution if 0 < λ < λ 0 . In this paper we show that the solution set S of the equation above, which is not empty by Hai's results, actually admits a continuum of positive solutions.
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