| Abstract: |
| This talk 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 variational methods, we give the existence of nontrivial solutions belonging to $L^{\infty}$-space when the condition on a nonlinear convex term does not assume the Ambrosetti-Rabinowitz condition in general. Also we introduce the recent works related to the double-phase nonlinear problems. In particular, on a new class of nonlinear terms we give the existence result of small energy solutions via applying the dual fountain theorem. |
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