TY - JOUR

T1 - High perturbations of quasilinear problems with double criticality

AU - Alves, Claudianor O.

AU - Garain, Prashanta

AU - Rădulescu, Vicenţiu D.

N1 - Funding Information:
Claudianor O. Alves was partially supported by CNPq/Brazil 304804/2017-7. The work of Vicenţiu D. Rădulescu was supported by a grant of the Romanian Ministry of Education and Research, CNCS-UEFISCDI, Project number PN-III-P4-ID-PCE-2020-0068, within PNCDI III. Vicenţiu D. Rădulescu was also supported by the Slovenian Research Agency program P1-0292.
Publisher Copyright:
© 2021, The Author(s).

PY - 2021/12/1

Y1 - 2021/12/1

N2 - This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems {-ΔΦu=f(x,u)inΩ,u=0on∂Ω,where ΔΦu=div(φ(x,|∇u|)∇u) and Φ(x,t)=∫0|t|φ(x,s)sds is a generalized N-function. We assume that Ω ⊂ RN is a smooth bounded domain that contains two open regions Ω N, Ω p with Ω ¯ N∩ Ω ¯ p= ∅. The features of this paper are that - Δ Φu behaves like - Δ Nu on Ω N and - Δ pu on Ω p, and that the growth of f: Ω × R→ R is like that of eα|t|NN-1 on Ω N and as |t|p∗-2t on Ω p when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.

AB - This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems {-ΔΦu=f(x,u)inΩ,u=0on∂Ω,where ΔΦu=div(φ(x,|∇u|)∇u) and Φ(x,t)=∫0|t|φ(x,s)sds is a generalized N-function. We assume that Ω ⊂ RN is a smooth bounded domain that contains two open regions Ω N, Ω p with Ω ¯ N∩ Ω ¯ p= ∅. The features of this paper are that - Δ Φu behaves like - Δ Nu on Ω N and - Δ pu on Ω p, and that the growth of f: Ω × R→ R is like that of eα|t|NN-1 on Ω N and as |t|p∗-2t on Ω p when |t| is large enough. The main result establishes the existence of solutions in a suitable Musielak–Sobolev space in the case of high perturbations with respect to the values of a positive parameter.

KW - Musielak–Sobolev space

KW - Quasilinear problems

KW - Variational methods

UR - http://www.scopus.com/inward/record.url?scp=85104969690&partnerID=8YFLogxK

U2 - 10.1007/s00209-021-02757-z

DO - 10.1007/s00209-021-02757-z

M3 - Article

AN - SCOPUS:85104969690

VL - 299

SP - 1875

EP - 1895

JO - Mathematische Zeitschrift

JF - Mathematische Zeitschrift

SN - 0025-5874

IS - 3-4

ER -