Radially Symmetric Solutions for Quasilinear Elliptic Equations Involving Nonhomogeneous Operators in an Orlicz-Sobolev
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
RADIALLY SYMMETRIC SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING NONHOMOGENEOUS OPERATORS IN AN ORLICZ-SOBOLEV SPACE SETTING∗ Jae-Myoung KIM Department of Mathematics Education, Andong National University, Andong 36729, Republic of Korea E-mail :[email protected]
Yun-Ho KIM† Department of Mathematics Education, Sangmyung University, Seoul 03016, Republic of Korea E-mail :[email protected]
Jongrak LEE Department of Mathematics, Jeju National University, Jeju 63243, Republic of Korea E-mail :[email protected] Abstract
We investigate the following elliptic equations:
Z −M
RN
u(x) → 0,
φ(|∇u|2 )dx div(φ′ (|∇u|2 )∇u) + |u|α−2 u = λh(x, u),
in
RN ,
as |x| → ∞,
Np , φ(t) behaves where N ≥ 2, 1 < p < q < N , α < q, 1 < α ≤ p∗ q ′ /p′ with p∗ = N−p q/2 p/2 ′ ′ like t for small t and t for large t, and p and q are the conjugate exponents of p and q, respectively. We study the existence of nontrivial radially symmetric solutions for the problem above by applying the mountain pass theorem and the fountain theorem. Moreover, taking into account the dual fountain theorem, we show that the problem admits a sequence of small-energy, radially symmetric solutions.
Key words
radial solution; quasilinear elliptic equations; variational methods, OrliczSobolev spaces
2010 MR Subject Classification
35J50; 35J62; 46E30; 46E35
∗ Received Sepbember 10, 2019; revised July 27, 2020. Jae-Myoung Kim’s work was supported by a Research Grant of Andong National University. Yun-Ho Kim’s work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2019R1F1A1057775). Jongrak Lee’s work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2018R1D1A1B07048620). † Corresponding author: Yun-Ho KIM.
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ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
Introduction
In recent years, great attention has been paid to the study of certain nonlinear equations, including non-homogeneous operators of the type −div(φ′ (|∇u|2 )∇u),
where φ ∈ C 1 (R+ , R+ ).
The interest in such operators has consistently increased in light of the relevence of the pure or applied mathematical theory to some concrete phenomena, such as nonlinear elasticity, fluid mechanics, plasticity theory, biophysics problems, and plasma physics; see [22–25, 28] and the references therein. In the case of nonlinear quasilinear elliptic equations, a functional setting is achieved by using Sobolev spaces to treat the problem variationally. In contrast, the study of nonhomogeneous differential operators is based on the theory of Orlicz-Sobolev spaces. In this direction, variational problems for elliptic equations of this type have been widely studied in recent years; for example, see [1, 2, 10, 13–15, 18–20, 26, 27, 37, 43, 44, 46], and see [8, 11, 47, 56] for problems with variable exponents. We also refer the rea
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