Multiple solutions of double phase variational problems with variable exponent

• Published in: Advances in calculus of variations Vol. 13; no. 4; pp. 385 - 401
• Main Authors: Shi, Xiayang, Rădulescu, Vicenţiu D., Repovš, Dušan D., Zhang, Qihu
• Format: Journal Article
• Language: English
• Published: Berlin De Gruyter 01.10.2020; Walter de Gruyter GmbH
• Subjects: 35J20; 35J62; 35J70; Critical point; Divergence; Elliptic functions; integral functionals; Mathematical analysis; Operators (mathematics); Partial differential equations; Schrodinger equation; Sobolev space; Standing waves; Variable exponent elliptic operator; variable exponent Orlicz–Sobolev spaces; Variational methods; Weighting methods
• ISSN: 1864-8258; 1864-8266
• DOI: 10.1515/acv-2018-0003

This paper deals with the existence of multiple solutions for the quasilinear equation which involves a general variable exponent elliptic operator in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like for small and like for large , where . Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz–Sobolev spaces with variable exponent. Our results extend the previous works [A. Azzollini, P. d’Avenia and A. Pomponio, Quasilinear elliptic equations in via variational methods and Orlicz–Sobolev embeddings, Calc. Var. Partial Differential Equations 49 2014, 1–2, 197–213] and [N. Chorfi and V. D. Rădulescu, Standing wave solutions of a quasilinear degenerate Schrödinger equation with unbounded potential, Electron. J. Qual. Theory Differ. Equ. 2016 2016, Paper No. 37] from cases where the exponents and are constant, to the case where and are functions. We also substantially weaken some of the hypotheses in these papers and we overcome the lack of compactness by using the weighting method.