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

Nikolaos S. Papageorgiou, Andrea Scapellato
2020 Zeitschrift für Angewandte Mathematik und Physik  
We consider a nonlinear elliptic Dirichlet problem driven by the anisotropic (p, q)-Laplacian and with a reaction which is nonparametric and has the combined effects of a singular and of a superlinear terms. Using variational tools together with truncation and comparison techniques, we show that the problem has at least two positive smooth solutions. Mathematics Subject Classification. 35J75, 35J20, 35J60. ZAMP superlinear perturbation. In contrast, the study of anisotropic singular problems is
more » ... ingular problems is lagging behind. To the best of our knowledge, there is only the recent work of Byun-Ko [4], who study an equation driven by the p(z)-Laplacian and with a reaction of the form λu −η(z) + u r(z) , where λ > 0 is a parameter and r ∈ C(Ω), p(z) < r(z) + 1 for all z ∈ Ω. We also mention the works of 14] , 16] , Papageorgiou-Rȃdulescu-Repovš [29] and Papageorgiou-Vetro [33] , which also deal with anisotropic equations with a superlinear reaction, but no singular term. We mention that partial differential equations with variable exponents arise in several models of electrorheological fluids (see Qian [37], Ruzicka [39] ) and in image processing and image restoration (see Chen-Levine-Rao [6]). Further applications can be found in the book of Rȃdulescu-Repovš [38] .
doi:10.1007/s00033-020-01385-7 fatcat:dlolmawgzrch3imktncahfed7y