Anisotropic double-phase problems with indefinite potential: multiplicity of solutions
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Anisotropic double-phase problems with indefinite potential: multiplicity of solutions 3,4,5 Nikolaos S. Papageorgiou1 · Dongdong Qin2 · Vicen¸tiu D. Radulescu ˘
Received: 20 June 2020 / Revised: 1 September 2020 / Accepted: 15 October 2020 © The Author(s) 2020
Abstract We consider an anisotropic double-phase problem plus an indefinite potential. The reaction is superlinear. Using variational tools together with truncation, perturbation and comparison techniques and critical groups, we prove a multiplicity theorem producing five nontrivial smooth solutions, all with sign information and ordered. In this process we also prove two results of independent interest, namely a maximum principle for anisotropic double-phase problems and a strong comparison principle for such solutions. Keywords Anisotropic regularity · Anisotropic maximum principle · Strong comparison · Constant sign and nodal solutions · Critical groups · Double phase Mathematics Subject Classification Primary 35J20; Secondary 35J70
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Vicen¸tiu D. R˘adulescu [email protected] Nikolaos S. Papageorgiou [email protected] Dongdong Qin [email protected]
1
Department of Mathematics, National Technical University, Zografou Campus, 15780 Athens, Greece
2
School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, People’s Republic of China
3
Faculty of Applied Mathematics, AGH University of Science and Technology, al. Mickiewicza 30, 30-059 Kraków, Poland
4
Department of Mathematics, University of Craiova, 200585 Craiova, Romania
5
“Simion Stoilow” Institute of Mathematics of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania 0123456789().: V,-vol
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N. S. Papageorgiou et al.
1 Introduction and origin of double-phase problems Let ⊆ R N be a bounded domain with a C 2 -boundary ∂. In this paper we deal with the following anisotropic double phase Dirichlet problem
− p(z) u(z) − q(z) u(z) + ξ(z)|u(z)| p(z)−2 u(z) = f (z, u(z)) in , u|∂ = 0.
(1)
In this problem, we assume that p, q ∈ C 1 () and 1 < q− ≤ q(z) ≤ q+ < p− ≤ p(z) ≤ p+ < p ∗ (z), where p ∗ (z) = NN−p(z) p(z) if p+ < N and +∞ otherwise. ∞ The potential function ξ ∈ L () is sign-changing and so the differential operator (left-hand side) of problem (1) is not coercive. The reaction f (z, x) is a Carathéodory function (that is, for all x ∈ R the mapping z → f (z, x) is measurable and for a.a. z ∈ the function x → f (z, x) is continuous) which exhibits ( p+ − 1)-superlinear growth near ±∞, but without satisfying the Ambrosetti-Rabinowitz condition (the A R-condition). Using variational tools from the critical point theory, together with truncation, perturbation and comparison techniques and critical groups, we show that the problem has at least five nontrivial smooth solutions, all with sign information and ordered. The energy functional associated to problem (1) is a double-phase variational integral, according to the terminology of Marcellini and Mingione. Problems with unbalanced growth have been studied fo
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