Positive solutions for anisotropic singular $$\varvec{(p,q)}$$ ( p , q ) -equations
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Positive solutions for anisotropic singular (p, q)-equations Nikolaos S. Papageorgiou and Andrea Scapellato Abstract. 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. Keywords. Anisotropic (p, q)-Laplacian, Regularity theory, Truncation, Comparison principle, Pairs of positive solutions.
1. Introduction Let Ω ⊆ RN be a bounded domain with a C 2 -boundary ∂Ω. In this paper, we study the following anisotropic singular (p, q)-equation (double phase problem) −η(z) + f (z, u(z)) in Ω −Δ p(z) u(z) − Δq(z) u(z) = u(z) , (1.1) u = 0, u > 0 ∂Ω
Given r ∈ C(Ω) with 1 < min r, by Δr(z) we denote the r(z)-Laplace differential operator defined by Ω
1,r(z) (Ω). Δr(z) = div |Du|r(z)−2 Du for all u ∈ W0 In Problem (1.1), we have the sum of two such operators (double phase problem). In the reaction (right-hand side of (1.1)), we have the competing effects of two different terms of different nature. One is the singular term u−η(z) , and the other term is a Carath´eodory perturbation f (z, x) (that is, for all x ∈ R, z → f (z, x) is measurable and for a.a. z ∈ Ω, x → f (z, x) is continuous) which exhibits (p+ − 1)superlinear growth as x → +∞ (here p+ = max p). We point out that problem (1.1) is nonparametric. Ω
Our aim is to prove the existence and the multiplicity of positive solutions for problem (1.1). Usually, singular problems are studied with a parameter involved in the reaction. 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 BaiMotreanu-Zeng [3], Candito-Gasi´ nski-Livrea [5], Gasi´ nski-Papageorgiou [13], Ghergu-R˘ adulescu [17,18], Giacomoni-Schindler-Tak´ aˇc [19], Haitao [21], Kyritsi-Papageorgiou [22], Papageorgiou-R˘ adulescu-Repovˇs [26–28], Papageorgiou-Repovˇs-Vetro [31], Papageorgiou-Smyrlis [32], Papageorgiou-Vetro-Vetro [35], SunWu-Long [40]. All the aforementioned works consider parametric isotropic singular semilinear or nonlinear problems. Nonparametric isotropic singular problems were considered by Bai-Gasi´ nski-Papageorgiou [2], Papageorgiou-R˘ adulescu-Repovˇs [25] and Papageorgiou-Vetro-Vetro [34]. Papers [2,25] deal with equations driven by the p-Laplacian and in [2] the perturbation f (z, ·) is (p − 1)-superlinear, while in [25] the perturbation f (z, ·) is (p − 1)-linear and resonant. In [34], the authors consider a (p, 2)-equation with 0123456789().: V,-vol
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N. S. Papageorgiou and A. Scapellato
ZAMP
superlinear perturbation. In contrast, the study of anisotropic singular problems is lagging behind
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