November 8, 2025

Quasilinear problems with superlinear or asymptotically linear nonlinearities

Key Points

Ground state solutions were proven to exist for quasilinear problems involving generalized Orlicz functions.
The method addresses the major challenge posed by the double phase problem within the analysis.
Utilization of the associated Laplacian operator allowed for effectively establishing the foundational groundwork in this context.
Such findings suggest potential advancement in the understanding of nonlinear differential equations involving complex nonlinearity.

Abstract

We establish the existence of a nontrivial solution for the quasilinear problem Formula: see text where Formula: see text is a smooth bounded domain, Formula: see text is a generalized Orlicz function, and Formula: see text denotes the associated Formula: see text-Laplacian operator. The nonlinearity Formula: see text is assumed to be either superlinear or asymptotically linear at both infinity and the origin. We prove the existence of bound and ground state solutions. The main difficulty here comes from the fact Formula: see text can be a generalized Orlicz function which covers the double phase problem.

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