Existence of renomalized solution for nonlinear elliptic boundary value problem without $$\Delta _{2}$$
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Existence of renomalized solution for nonlinear elliptic boundary value problem without 12 -condition Nourdine El Amarty1 · Badr El Haji2
· Mostafa EL Moumni1
Received: 4 April 2020 / Accepted: 26 May 2020 © Sociedad Española de Matemática Aplicada 2020
Abstract In this paper we will prove in Musielak–Orlicz spaces, the existence of renomalized solution for nonlinear elliptic equations of Leray-Lions type, in the case where the Musielak–Orlicz function ϕ doesn’t satisfy the 2 condition while the right hand side f belongs to W −1 E ψ (). Keywords Musielak–Orlicz–Sobolev spaces · Elliptic equation · Renormalized solutions · Truncations Mathematics Subject Classification 35J25 · 35J60 · 46E30
1 Introduction and basic assumptions This work deals with existence of solutions for strongly nonlinear boundary value problem whose model is: A(u) − div (u) + g(x, u, ∇u) = f in (1.1) u ≡ 0, on ∂ where be a bounded domain of R N , N ≥ 2, A(u) = − div a(x, u, ∇u) be a Leray-Lions operator defined from the space W01 L ϕ () into its dual W −1 L ψ (), and ∈ C0 R, R N . where a is a function satisfying the following conditions : a(x, s, ξ ) : × R × R N −→ R N is a Carath´eodory function.
B
(1.2)
Badr El Haji [email protected] Nourdine El Amarty [email protected] Mostafa EL Moumni [email protected]
1
Department of Mathematics, Faculty of Sciences El Jadida, University Chouaib Doukkali, P. O. Box 20, 24000 El Jadida, Morocco
2
Laboratory LAMA, Department of Mathematics, Faculty of Sciences Fez, University Sidi Mohamed Ben Abdellah, P. O. Box 1796, Atlas Fez, Morocco
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N. E. Amarty et al.
There exist two Musielak–Orlicz functions ϕ and γ such that γ ≺≺ ϕ, a positive function d(·) ∈ E ψ () and positive constants k1 , k2 and k3 such that for a.e. x ∈ and for all s ∈ R, ξ ∈ R N |a(x, s, ξ )| ≤ k1 d(x) + ψx−1 γ (x, k2 |s|) + ψx−1 ϕ(x, k3 |ξ |); (1.3) a(x, s, ξ ) − a x, s, ξ ξ − ξ > 0; (1.4) a(x, s, ξ ).ξ ≥ αϕ(x, |ξ |).
(1.5)
Furthermore, let g(x, s, ξ ) : × R × R N −→ R be a Carathéodory function such that for a.e. x ∈ and for all s ∈ R, ξ ∈ R N , satisfying the following conditions |g(x, s, ξ )| ≤ c(x) + b(|s|)ϕ(x, |ξ |); g(x, s, ξ )s ≥ 0;
(1.6)
(1.7) where b : R+ −→ R+ is a continuous positive function which belongs to L R+ and c(·) ∈ L 1 () The right-hand side of (1.1) and : R → R N are assumed to satisfy f ∈ W −1 E ψ (); ∈ C 0 R, R N .
1
(1.8) (1.9)
Note that no growth hypothesis is assumed on the function , which implies that the term −div (u) may be meaningless, even as a distribution. Several researches deals with the existence solutions of elliptic and parabolic problems under various assumptions and in different contexts (see [1–10,13–20,24–28,35,37,39,40] for more details), indeed we can’t recite all examples; we will just choose some of them, So we mention that: the problem (1.1) was treated by Boccardo (see [23]) in the case g ≡ 0 and for p such that 2 − 1/N < p < N where he proved the existence and regularity of an entropy sol
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