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We consider Kirchhoff type problems of the form
⎧ -M(ρ(u))(div(a(|∇u|)∇u) - a(|u|)u) = K(x)f(u) in Ω
⎨
⎩ ∂u/∂ν = 0 on ∂Ω
where , N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, , M: [0,∞) → ℝ is a continuous function, , and f: ℝ → ℝ is a continuous function not satisfying the Ambrosetti-Rabinowitz type condition. Using variational methods, we obtain some existence and multiplicity results.
We consider a model of generalized Cahn-Hilliard equations with a logarithmic free energy and a degenerate mobility, and obtain a result on the existence of solutions.
We study the existence of solutions to a nonlinear parabolic equation describing the temporal evolution of a cloud of self-gravitating particles with a given external potential. The initial data are in spaces of (generalized) pseudomeasures. We prove existence of local and global-in-time solutions, and also a kind of stability of global solutions.
The existence of solutions for boundary value problems for a nonlinear discrete system involving the -Laplacian is investigated. The approach is based on critical point theory.
This paper deals with the existence of solutions to the following system:−Δu+u=αα+βa(x)|v|β|u|α−2u inRN−Δv+v=βα+βa(x)|u|α|v|β−2v inRN. With the help of the Nehari manifold and the linking theorem, we prove the existence of at least two nontrivial solutions. One of them is positive. Our main tools are the concentration-compactness principle and the Ekeland’s variational principle.
In this paper we study the existence of classical solutions for a class of abstract neutral integro-differential equation with unbounded delay. A concrete application to partial neutral integro-differential equations is considered.
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