On a class of p ( x )-Choquard equations with sign-changing potential and upper critical growth
- PDF / 2,599,558 Bytes
- 25 Pages / 439.37 x 666.142 pts Page_size
- 99 Downloads / 159 Views
On a class of p(x)‑Choquard equations with sign‑changing potential and upper critical growth B. B. V. Maia1 Received: 27 June 2020 / Accepted: 12 August 2020 © Springer-Verlag Italia S.r.l., part of Springer Nature 2020
Abstract Motived by the Hardy–Littlewood–Sobolev inequality for variable exponents, in this paper we use variational methods to prove the existence of a weak solution for a class of p(x)Choquard equations with upper critical growth. Using truncation arguments and Krasnoselskii’s genus, we also show a multiplicity of solutions for a class of p(x)-Choquard equations with a nonlocal and non-degenerate Kirchhoff term. Also we show that the solutions obtained belong to L∞ (ℝN ) and have polynomial decay. Keywords p(x)-Laplacian · Choquard equation · Kirchhoff equation · Variational methods · Critical growth · Krasnoselskii’s genus · Polynomial decay Mathematics Subject Classification 35A15 · 35J60 · 58E05
1 Introduction This paper focuses on a study of the existence and multiplicity of a class of elliptic equations of the p(x)-Choquard type. Specifically, in the first part of our paper, we are interested in determining the existence of solution for the problem:
� � ⎧ �u(x)�r(x) ⎪ −Δp(x) u + V(x)�u�p(x)−2 u = h(x) + �u(y)�r(y)−2 u(y), in ℝN , (P1 )⎨ ∫ℝN �x − y�𝛼(x,y) ⎪ u ∈ W 1,p(x) (ℝN ), ⎩ where p, V, r ∶ ℝN → ℝ , h ∶ ℝN → ℝ and 𝛼 ∶ ℝN × ℝN → ℝ are continuous functions specified later. The operator −Δp(x) denotes the p(x)-Laplacian given by
Δp(x) u = div(|∇u|p(x)−2 ∇u). In the second part, we prove the existence of infinitely many solutions for the following class of p(x)-Kirchhoff–Choquard equation: * B. B. V. Maia [email protected]; [email protected] 1
Universidade Federal Rural da Amazônia, campus de Capitão‑Poço, Capitão‑Poço, PA, Brazil
13
Vol.:(0123456789)
B. B. V. Maia
� � ⎧ � � �u(x)�r(x) ⎪ M(��u��2 ) −Δp(x) u + V(x)�u�p(x)−2 u = 𝜆�u�s(x)−2 u + �u(y)�r(y)−2 u(y), (P2 )⎨ ∫ℝN �x − y�𝛼(x,y) ⎪ u ∈ W 1,p(x) (ℝN ), ⎩
where M ∶ [0, ∞) → (0, ∞) , s ∶ ℝN → ℝ are continuous functions and 𝜆 is a positive parameter. The study of differential equations involving p(x)-growth has increased in recent decades, this interest has been created by their presence in many models from real-world applications such as image restoration [14, 15] and electrorheological fluids [1, 2, 8, 44]. See also the monograph [42] for a comprehensive treatment of nonlinear partial differential equations with variable exponent. When p ≡ 2 , the problem (P1 ) is related with the equation ( ) 1 r(x) −Δu + V(x)u = h(x) + ∗ |u| |u|r(x)−2 u, in ℝN , (1) |x|𝛼(x,y) which is known in the literature as a Choquard equation. Equations of the type (1) have a strong physical motivation. However, despite being named the “Choquard equation”, they were first studied by Fröhlich and Pekar in the quantum theory of a polaron at rest [23, 24, 40]. Also, according to [30], the equation ( ) A 2 −Δu + u = ∗ |u| u, in ℝ3 (2) |x|𝜇 (A is a constant) was introduced by Ph. Choquard in 1976 as an approximation to Hartree–Fock theory for a o
Data Loading...
