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Existence of solutions for a class of Kirchhoff type problems in Orlicz-Sobolev spaces

Nguyen Thanh Chung (2015)

Annales Polonici Mathematici

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 Ω ⊂ ℝ N , N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, ρ ( u ) = ∫ Ω ( Φ ( | ∇ u | ) + Φ ( | u | ) ) d x , M: [0,∞) → ℝ is a continuous function, K ∈ L ∞ ( Ω ) , 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.

Existence of solutions for a semilinear elliptic system

Mohamed Benrhouma (2013)

ESAIM: Control, Optimisation and Calculus of Variations

This paper deals with the existence of solutions to the following system: - Δ u + u = α α + β a ( x ) | v | β | u | α - 2 u in ℝ N - Δ v + v = β α + β a ( x ) | u | α | v | β - 2 v in ℝ N . −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.

existence of solutions for an elliptic-algebraic system describing heat explosion in a two-phase medium

Cristelle Barillon, Georgy M. Makhviladze, Vitaly A. Volpert (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The paper is devoted to analysis of an elliptic-algebraic system of equations describing heat explosion in a two phase medium filling a star-shaped domain. Three types of solutions are found: classical, critical and multivalued. Regularity of solutions is studied as well as their behavior depending on the size of the domain and on the coefficient of heat exchange between the two phases. Critical conditions of existence of solutions are found for arbitrary positive source function.

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