Abstract
We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian with 1<q<p. The reaction is parametric and exhibits the competing effects of a singular term and of concave and convex nonlinearities. We are looking for positive solutions and prove a bifurcation-type theorem describing in a precise way the set of positive solutions as the parameter varies. Moreover, we show the existence of a minimal positive solution and we study it as a function of the parameter.
Highlights
Let ⊆ RN be a bounded domain with a C2-boundary ∂
For r ∈ (1, ∞) we denote by r the r -Laplace differential operator defined by r u = div |∇u|r−2∇u for all u ∈ W01,r ( )
We have in problem (Pλ) the combined effects of singular terms, of sublinear terms and of superlinear terms (the function s → f (x, s))
Summary
Let ⊆ RN be a bounded domain with a C2-boundary ∂. The study of elliptic problems with combined nonlinearities was initiated with the seminal paper of Ambrosetti–Brezis–Cerami [1] who studied semilinear Dirichlet equations driven by the Laplacian without any singular term. Their work has been extended to nonlinear problems driven by the p-Laplacian by García Azorero–Peral Alonso–Manfredi [5] and Guo–Zhang [11] In both works there is no singular term and the reaction has the special form x → λsτ−1 + sr−1 for all s ≥ 0 with 1 < τ < p < r < p∗, where p∗ is the critical Sobolev exponent to p given by p∗ =. We refer to the works of Papageorgiou–Radulescu–Repovš [23] for Robin problems and Papageorgiou–Winkert [19], Leonardi–Papageorgiou [14] and Marano–Marino–Papageorgiou [16] for Dirichlet problems None of these works involves a singular term. We refer to Papageorgiou– Radulescu–Repovš [26,28], Papageorgiou–Zhang [22] and Ragusa–Tachikawa [30]
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