On Some Elliptic Equation Involving the p ( x )-Laplacian in $$\mathbb {R}^N$$ R N with a Possibly Discontinuous Non
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On Some Elliptic Equation Involving the p(x)-Laplacian in RN with a Possibly Discontinuous Nonlinearities Sami Aouaoui1,2 · Ala Eddine Bahrouni2,3 Received: 29 July 2020 / Revised: 21 August 2020 / Accepted: 26 August 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, we prove some existence and uniqueness results for some elliptic quasilinear equation defined on the whole Euclidean space R N , N ≥ 2, involving the p(x)-Laplacian operator and whose nonlinearities can be discontinuous. Some new ideas and tools are used to reach our main results. Keywords Variable exponents · p(x)-Laplacian · Degree theory · (S+ ) operator · Banach semilattice · Fixed point theorem · Discontinuity Mathematics Subject Classification 35D30 · 35J15 · 35J62 · 03G10
1 Introduction and Statement of Main Results In this paper, we are concerned with the following equation N ∂u p(x)−2 p(x)−2 ∇u + |u| u+ u. P − div |∇u| ∂x j j=1
|u|
α(xi )−2
i= j
= λ f (x, u) + h(x) in R N ,
Communicated by Maria Alessandra Ragusa.
B
Sami Aouaoui [email protected] Ala Eddine Bahrouni [email protected]
1
Department of Mathematics, High Institute of Applied Mathematics and Informatics of Kairouan, University of Kairouan, Avenue Assad Iben Fourat, 3100 Kairouan, Tunisia
2
Laboratory of Algebra, Number Theory and Nonlinear Analysis LR18ES15, Faculty of Sciences of Monastir, University of Monastir, Rue de l’environnement, 5019 Monastir, Tunisia
3
High School of Sciences and Technology of Hammam Sousse, University of Sousse, Rue Lamine ElAbbessi, 4011 Hammam Sousse, Tunisia
123
S. Aouaoui, A. E. Bahrouni
N ≥ 3, where p is some Lipschitz continuous function such that 2 < p − = inf p(x) ≤ p + = sup p(x) < N . x∈R N
x∈R N
Clearly, the Lipschitz-continuity of the function p implies | p(x) − p(y)| ≤
C 1 , ∀ |x − y| ≤ . − log |x − y| 2
where C is some positive constant. That last property guarantees the density of C0∞ (R N ) in W 1,p(x) (R N ). The problem P is taken under the following assumptions: (H1 ) α ∈ C(R) ∩ L ∞ (R) is such that p + − 1 ≤ α − = inf α(x) ≤ α + = sup α(x) ≤ x∈R
x∈R
N ( p − − 1) . N − p+
(H2 ) f : R N × R −→ R is a measurable function such that there exist g ∈ L ∞ (R N ), g(x) ≥ 0 a.e. x ∈ R N and β ∈ C(R N ) ∩ L ∞ (R N ), β(x) ≤ p ∗ (x) = N p(x) N − p(x) satisfying | f (x, s)| ≤ g(x)|s|β(x)−1 a.e. x ∈ R N , ∀ s ∈ R. We also assume that f (x, s) = 0, a.e. x ∈ R N , ∀ s ≤ 0, and f (x, s) ≥ 0, a.e. x ∈ R N , ∀ s ≥ 0. (H3 ) There exist two open nonempty sets 1 and 2 of R N such that inf x∈1
p(x) β(x)
> 1, inf
x∈2
β(x) p(x)
c > 1, and g(x) = 0 a.e. x ∈ 1 ∪ 2 .
∗ (H4 ) h ∈ W 1, p(x) (R N ) \ {0}, h ≥ 0 ( i.e. h, u ≥ 0, ∀u ∈ W 1, p(x) (R N ), u ≥ 0). p(x) g(x) p(x)−β(x) dx < +∞, 2 is bounded. (H5 ) 1
Nowadays, the importance of the study of quasilinear problems with variable exponents is becoming a confirmed fact. The same can be said about the motivation of this attention in such types of problem
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