Nonlinear singular problems with indefinite potential term

Abstract

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearities. In this paper the concave term is parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter λ varies. This work continues our research published in arXiv:2004.12583, where ξ ≡ 0 and in the reaction the parametric term is the singular one. Rabinowitz condition (the AR-condition for short). So in problem (P λ ) we have the competing effects of singular, concave and convex terms.

Positive solutions for singular double phase problems

“…The function µ u * (•) is decreasing on (0, t 1 ). Hence we have (12)).…”

“…A nice survey of the recent works on such equations can be found in Rǎdulescu [16]. We also mention the recent works on (p, q)-equations (equations driven by the sum of a p-Laplacian and of a q-Laplacian) with singular terms of Papageorgiou-Rǎdulescu-Repovš [12] and Papageorgiou-Vetro-Vetro [13]. For such differential operators, the integrand of the energy functional is k(t) = 1 p t p + 1 q t q for all t > 0 (that is, ξ(z) = 1) and so the use of the global regularity theory of Lieberman [9] and the nonlinear maximum principle of Pucci-Serrin [15] is possible.…”

Positive solutions for anisotropic singular $$\varvec{(p,q)}$$-equations

“…By varying and restricting the parameter, we are able to satisfy the geometry of the minimax theorems of critical point theory and then use them to produce a positive solution. Indicatively, we mention the works of Bai-Motreanu-Zeng [3], Candito-Gasiński-Livrea [5], Gasiński-Papageorgiou [13], Ghergu-Rȃdulescu [17,18], Giacomoni-Schindler-Takáč [19], Haitao [21], Kyritsi-Papageorgiou [22], Papageorgiou-Rȃdulescu-Repovš [26][27][28], Papageorgiou-Repovš-Vetro [31], Papageorgiou-Smyrlis [32], Papageorgiou-Vetro-Vetro [35], Sun-Wu-Long [40]. All the aforementioned works consider parametric isotropic singular semilinear or nonlinear problems.…”