We consider the Poisson equation with the Dirichlet and the Neumann boundary conditions in weighted Sobolev spaces. The weight is a positive power of the distance to a distinguished plane. We prove the existence of solutions in a suitably defined weighted space.
The existence of solutions to the elliptic problem rot v = w, div v = 0 in a bounded domain Ω ⊂ ℝ³, , S = ∂Ω in weighted -Sobolev spaces is proved. It is assumed that an axis L crosses Ω and the weight is a negative power function of the distance to the axis. The main part of the proof is devoted to examining solutions of the problem in a neighbourhood of L. The existence in Ω follows from the technique of regularization.
We consider the Cauchy problem for nonlinear parabolic equations with functional dependence. We prove Schauder-type existence results for unbounded solutions. We also prove existence of maximal solutions for a wide class of differential functional equations.
This work is devoted to the study of the initial boundary value problem for a general non isothermal model of capillary fluids derived by J. E Dunn and J. Serrin (1985) in [9, 16], which can be used as a phase transition model.We distinguish two cases, when the physical coefficients depend only on the density, and the general case. In the first case we can work in critical scaling spaces, and we prove global existence of solution and uniqueness for data close to a stable equilibrium. For general...
We establish the existence of at least three weak solutions for the (p1,…,pₙ)-biharmonic system ⎧ in Ω, ⎨ ⎩ on ∂Ω, for 1 ≤ i ≤ n. The proof is based on a recent three critical points theorem.
The existence of at least three weak solutions is established for a class of quasilinear elliptic equations involving the p(x)-biharmonic operator with Navier boundary value conditions. The proof is mainly based on a three critical points theorem due to B. Ricceri [Nonlinear Anal. 70 (2009), 3084-3089].
Using a three critical points theorem and variational methods, we study the existence of at least three weak solutions of the Navier problem ⎧ in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω, where (N ≥ 1) is a non-empty bounded open set with a sufficiently smooth boundary ∂Ω, λ > 0, μ > 0 and f,g: Ω × ℝ → ℝ are two L¹-Carathéodory functions.