Existence of solutions of anisotropic elliptic equations with variable exponents of nonlinearity in unbounded domains

For anisotropic quasilinear second order elliptic equations in divergence form with a non-standard growth conditions Σ︁𝑖=1 (𝑎𝑖(x, 𝑢,∇𝑢))𝑥𝑖 - 𝑎0(x, 𝑢,∇𝑢) = 0, x ∈ Ω; (1) in domain Ω of the space R𝑛, Ω ( R𝑛, ≥ 2, the Dirichlet problem is considered with homogeneous boundary condition 𝑢⃒⃒⃒ 𝜕Ω = 0. (2) It is assumed that the functions 𝑎𝑖(x, 𝑠0, 𝑠1,..., 𝑠𝑛) have a polinomial growth on variable with powers 𝑝𝑖(x) ∈ (1,∞), = 0, 1,..., 𝑛. As example we can use the equation Σ︁𝑖=1 (|𝑢𝑥𝑖 |𝑝𝑖(x)-2𝑢𝑥𝑖)𝑥𝑖 - |𝑢|𝑝0(x)-2𝑢 = Σ︁𝑖=1 ( 𝑖(x))𝑥𝑖 - 0(x). In the paper by M.B. Benboubker, E. Azroul, A. Barbara (Quasilinear elliptic problems with nonstandard growths, Electronic Journal of Differential Equations, 2011) the existence of solutions of the Dirichlet problem in a bounded domain was proved for an isotropic elliptic equations with variable nonlinearities. For isotropic equations with constant power of nonlinearity the existence of solutions of the Dirichlet problem in an arbitrary domain was established by F.E. Browder (Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains, Proc. Nati. Acad. Sci. USA, 1977). The proof is based on an abstract theorem for pseudomonotone operators. In this paper we prove the existence of solutions of the problem (1), (2) without the assumption of boundedness of Ω and the smoothness of its boundary. Note by 𝐿𝑝(·)(Ω) Lebesgue spaces with variable exponent 𝑝(x) and the Luxemburg norm ‖·‖𝑝(·). Let the -→p (x) = (𝑝0(x), 𝑝1(x),..., 𝑝𝑛(x)) ∈ (𝐿+ ∞(Ω))𝑛+1∩ ∩ (𝐶+(Ω))𝑛+1. The Sobolev - Orlicz space with variable exponents ˚ 1- →p (·)(Ω) is defined as the completion of the space 𝐶∞0 (Ω) in the norm ‖𝑣‖˚ 1- →p (·) (Ω) = ‖𝑣‖𝑝0(·) + Σ︁𝑖=1 ‖𝑣𝑥𝑖‖𝑝𝑖(·). It is assumed that 𝑝+(x) ≤ 𝑝0(x) 𝑛, +∞, 𝑝(x) ≤ 𝑛, 𝑝(x) = 𝑛(︃ Σ︁𝑖=1 1/𝑝𝑖(x))︃-1. And it is also assumed that 𝑎𝑖(x, 𝑠0, s), = 0,..., 𝑛, x ∈ Ω, s = (𝑠0, s) = = (𝑠0, 𝑠1,..., 𝑠𝑛) ∈ R𝑛+1, are the Caratheodory functions, and there exist positive numbers ̂︀𝑎, and measurable non-negative function (x) ∈ 𝐿1(Ω), Φ𝑖(x) ∈ ∈ 𝐿𝑝′𝑖(·)(Ω), 𝑝′𝑖(x) = 𝑝𝑖(x)/(𝑝𝑖(x) - 1), = 0, 1,..., 𝑛, such that for almost all x ∈ Ω and any s = (𝑠0, s) ∈ R𝑛+1 the inequalities hold: |𝑎𝑖(x, 𝑠0, s)| ≤ ̂︀𝑎(|𝑠𝑖|𝑝𝑖(x)-1 + |𝑠0|𝑝0(x)/𝑝′𝑖(x)) + Φ𝑖(x), = 0, 1,..., 𝑛; (4) Σ︁𝑖=1 (𝑎𝑖(x, 𝑠0, s) - 𝑎𝑖(x, 𝑠0, t))(𝑠𝑖 - 𝑡𝑖) > 0, s ̸= t; (5) Σ︁𝑖=0 𝑎𝑖(x, 𝑠0, s)𝑠𝑖 ≥ 𝑛 Σ︁𝑖=0 |𝑠𝑖|𝑝𝑖(x) - (x). (6) Elliptic operators A : ˚𝑊 1- →p (·)(Ω) → (︁˚ 1- →p (·)(Ω))︁′, corresponding to the problem (1), (2), defined by the equation: ⟨A(𝑢), = w Ω Σ︁𝑖=0 𝑎𝑖(x, 𝑢,∇𝑢)𝑣𝑥𝑖𝑑x, 𝑢(x), 𝑣(x) ∈ ˚ 1- →p (·)(Ω). It is proved that operator A is pseudomonotone, bounded and coercitive. On the basis of these properties we prove the theorem. Theorem. If the conditions (3)-(6), there is a generalized solution of the problem (1), (2).