We examine the Poisson equation with boundary conditions on a cylinder in a weighted space of $L_p$, p≥ 3, type. The weight is a positive power of the distance from a distinguished plane. To prove the existence of solutions we use our result on existence in a weighted L₂ space.

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, ${v·n̅|}_S=0$, S = ∂Ω in weighted $L_p$-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 establish the existence of at least three weak solutions for the (p1,…,pₙ)-biharmonic system ⎧$\Delta \left(\right|\Delta u_i{|}^{p-2}\Delta u_i)=\lambda F_{u_i}(x,u₁,\dots ,uₙ)$ in Ω, ⎨ ⎩$u_i=\Delta u_i=0$ 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 ⎧${\Delta \left(\right|\Delta u|}^{p-2}{\Delta u\left)-div\right(\left|\nabla u\right|}^{p-2}\nabla u)=\lambda f(x,u)+\mu g(x,u)$ in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ (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.

We study the following singular elliptic equation with critical exponent ⎧$-\Delta u=Q\left(x\right)u^{2*-1}+\lambda u^{-\gamma }$ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where $\Omega \subset {ℝ}^N$ (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.

This paper presents several sufficient conditions for the existence of weak solutions to general nonlinear elliptic problems of the type $$\left\{\begin{array}{cc}-\mathrm{div}a(x,u,\nabla u)+b(x,u,\nabla u)=0\hfill & \text{in}\Omega ,\hfill \\ u=0\hfill & \text{on}\partial \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is a bounded domain of ${ℝ}^n$, $n\ge 2$. In particular, we do not require strict monotonicity of the principal part $a(x,z,·)$, while the approach is based on the variational method and results of the variable exponent function spaces.