Singular Double Phase Problems with Convection
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Singular Double Phase Problems with Convection Nikolaos S. Papageorgiou1 · Calogero Vetro2 · Francesca Vetro3,4
Received: 14 February 2020 / Accepted: 24 September 2020 © Springer Nature B.V. 2020
Abstract We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplacian and of a q-Laplacian (double phase equation). In the reaction we have the combined effects of a singular term and of a gradient dependent term (convection) which is locally defined. Using a mixture of variational and topological methods, together with suitable truncation and comparison techniques, we prove the existence of a positive smooth solution. Mathematics Subject Classification (2010) 35B50 · 35J75 · 35J92 · 47H10 Keywords (p, q)-Laplacian · Nonlinear regularity · Nonlinear maximum principle · Fixed point · Positive solution
1 Introduction Let ⊆ RN be a bounded domain with a C 2 -boundary ∂. We study the following Dirichlet (p, q)-equation with a singular term and a gradient dependent perturbation (convection): −p u(z) − q u(z) = u(z)−η + f (z, u(z), ∇u(z)) in , u = 0, u > 0. (1) ∂
B F. Vetro
[email protected] N.S. Papageorgiou [email protected] C. Vetro [email protected]
1
Department of Mathematics, National Technical University, Zografou campus, 15780, Athens, Greece
2
Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123, Palermo, Italy
3
Nonlinear Analysis Research Group, Ton Duc Thang University, Ho Chi Minh City, Vietnam
4
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
N.S. Papageorgiou et al.
In this problem 1 < q < p < +∞, 0 < η < 1. For every r ∈ (1, +∞) by r we denote the r-Laplace differential operator defined by r u = div(|∇u|r−2 ∇u) for all u ∈ W01,r (). The differential operator in problem (1) is the sum of two such operators with different indices (double phase equation) and so it is not homogeneous. This is a source of difficulties in the analysis of problem (1). In the reaction of problem (1) we have the combined effects of two terms of different nature. One is the singular term u−η and the other is a perturbation f (z, u, ∇u) which is a Carathéodory function (that is, for all (x, y) ∈ R × RN , z → f (z, x, y) is measurable and for a.a. z ∈ , (x, y) → f (z, x, y) is continuous). There are two special features of this perturbation. The first is that it is gradient dependent (convection) and this make the problem nonvariational. The other special feature is that our conditions on f (z, ·, y) are only local (near zero). No restrictions are imposed on x → f (z, x, y) for large x ≥ 0. Since the problem is nonvariational (due to the convection), our approach is necessarily topological, based on the fixed point theory. The idea is to freeze the gradient term in the perturbation. This way we have a variational problem which we hope to solve using tools from the critical point theory. However, the presence of the singular term creates problems in this direction since the energy functional is
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