Multiple solutions of double phase variational problems with variable exponent

Author:

Shi Xiayang1,Rădulescu Vicenţiu D.2,Repovš Dušan D.3,Zhang Qihu4

Affiliation:

1. Software Engineering College, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, P. R. China

2. Institute of Mathematics, Physics and Mechanics, 1000Ljubljana, Slovenia; and Faculty of Applied Mathematics, AGH University of Science and Technology, 30-059 Kraków, Poland; and Department of Mathematics, University of Craiova, 200585 Craiova, Romania

3. Faculty of Education and Faculty of Mathematics and Physics, University of Ljubljana; and Institute of Mathematics, Physics and Mechanics, 1000Ljubljana, Slovenia

4. Department of Statistics, University of Georgia, Athens, GA 30602, USA

Abstract

AbstractThis paper deals with the existence of multiple solutions for the quasilinear equation{-\operatorname{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\,)\leq 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. 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 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.

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Analysis

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