Multiple solutions of double phase variational problems with variable exponent

Shi, Xiayang (author), Rǎdulescu, Vicenţiu (author), Repovš, Dušan (author), Zhang, Qihu (author)

This paper deals with the existence of multiple solutions for the quasilinear equation ▫$$-\text{div} \mathbf{A} (x,\nabla u)+|u|^{\alpha(x)-2}u = f(x,u) \quad \text{in} \; \mathbb{R}^N,$$▫ which involves a general variable exponent elliptic operator ▫$\mathbf{A}$▫ in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has various types of behavior like ▫$|\xi|^{q(x)-2} \xi$▫ for small ▫$|\xi|$▫ and like ▫$|\xi|^{p(x)-2} \xi$▫ for large ▫$|\xi|$▫, where ▫$1 < \alpha(\cdot) \le p(\cdot) < q(\cdot) < N$▫. 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 ▫$\mathbb{R}^N$▫ via variational methods and Orlicz-Sobolev embeddings, Calc. Var. Partial Differential Equations 49 2014, 1-2, 197-213] and [N. Chorfi and V. D. Rǎdulescu, 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 ▫$p$▫ and ▫$q$▫ are constant, to the case where ▫$p(\cdot)$▫ and ▫$q(\cdot)$▫ 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.

• Language: English
• Keywords: variable exponent elliptic operator, integral functionals, variable exponent Orlicz-Sobolev spaces, critical point
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: PEF - Faculty of Education; FMF - Faculty of Mathematics and Physics
• Year: 2020
• Number of pages: Str. 385-401
• Numbering: Vol. 13, iss. 4
• PID: 20.500.12556/RUL-121505
• UDC: 517.956
• ISSN on article: 1864-8258
• DOI: 10.1515/acv-2018-0003
• COBISS.SI-ID: 18383193
• Publication date in RUL: 13.10.2020

Record is a part of a journal

• Title: Advances in calculus of variations
• Shortened title: Adv. calc. var.
• Publisher: de Gruyter
• ISSN: 1864-8258
• COBISS.SI-ID: 517905177