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.
Highlights
Let ⊆ RN be a bounded domain with a C2-boundary ∂
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
Problems with unbalanced growth have been studied for the first time by Ball [4,5] in relationship with patterns arising in nonlinear elasticity
Summary
Let ⊆ RN be a bounded domain with a C2-boundary ∂. Marcellini considered continuous functions f = f (x, u) with unbalanced growth that satisfy c1 |u|q ≤ | f (x, u)| ≤ c2 (1 + |u|p) for all (x, u) ∈ × R, where c1, c2 are positive constants and 1 ≤ q ≤ p These contributions are in relationship with the works of Zhikov [37,38], in order to describe the behavior of phenomena arising in nonlinear elasticity. We mention the work of Papageorgiou & Vetro [28], who deal with anisotropic double phase problems with no potential term (that is, ξ ≡ 0) and with a superlinear reaction that has a different geometry near zero. They prove a multiplicity theorem producing three nontrivial solutions.
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