Year: 2018
Type: article
Abstract: This paper deals with the existence of multiple solutions for the quasilinear equation -div𝐀(x,∇u)+|u|α(x)-2u=f(x,u) in ℝN, {-\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 𝐀 {\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 |ξ|q(x)-2ξ {|\xi|^{q(x)-2}\xi} for small |ξ| {|\xi|} and like |ξ|p(x)-2ξ {|\xi|^{p(x)-2}\xi} for large |ξ| {|\xi|} , where 1<α(⋅)≤p(⋅)<q(⋅)<N {1<\alpha(\,\cdot\,)\leq p(\,\cdot\,) . 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 ℝN \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 exponentspandqare constant, to the case where p(⋅) {p(\,\cdot\,)} and q(⋅) {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.
Authors: Xiayang Shi, Vicenţiu D. Rădulescu, Dušan D. Repovš, Qihu Zhang
Language: Portuguese
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2809094757