Existence and multiplicity of solutions to concave-convex-type double-phase problems with variable exponent. (English) Zbl 1492.35124
Summary: This paper is devoted to the study of the \(L^\infty\)-bound of solutions to the double-phase nonlinear problem with variable exponent by the case of a combined effect of concave-convex nonlinearities. The main tools are the De Giorgi iteration method and a truncated energy technique. Applying this and a variant of Ekeland’s variational principle, we give the existence of at least two distinct nontrivial solutions belonging to \(L^\infty\)-space when the condition on a nonlinear convex term does not assume the Ambrosetti-Rabinowitz condition in general. In addition, our problem admits a sequence of small energy solutions whose converge to zero in \(L^\infty\) space. To achieve this result, we apply the modified functional method and global variational formulation as the main tools.
MSC:
| 35J62 | Quasilinear elliptic equations |
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
Keywords:
double phase problem; concave-convex nonlinearities; De Giorgi iteration; Ekeland’s variational principleReferences:
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