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 $\Omega \subset {ℝ}^N$, N ≥ 3, is a smooth bounded domain, ν is the outward unit normal to ∂Ω, $\rho \left(u\right)={\int }_{\Omega }\left(\Phi \right(\left|\nabla u\right|)+\Phi \left(\right|u\left|\right))dx$, M: [0,∞) → ℝ is a continuous function, $K\in L^{\infty }\left(\Omega \right)$, 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.

This paper deals with the existence of solutions to the following system:$$\left\{\begin{array}{c}-\Delta u+u=\frac{\alpha }{\alpha +\beta }{a\left(x\right)\left|v\right|}^{\beta }{\left|u\right|}^{\alpha -2}u\text{in}{ℝ}^N\hfill \\ -\Delta v+v=\frac{\beta }{\alpha +\beta }{a\left(x\right)\left|u\right|}^{\alpha }{\left|v\right|}^{\beta -2}v\text{in}{ℝ}^N.\hfill \end{array}\right.$$−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.

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.

In this paper we are interested in the existence and uniqueness of solutions for the Navier problem associated to the degenerate nonlinear elliptic equations $${\Delta \left(v\left(x\right)\right|\Delta u|}^{q-2}\Delta u)-\sum _{j=1}^n{D}_j\left[\omega \left(x\right){𝒜}_j(x,u,\nabla u)\right]=f_0\left(x\right)-\sum _{j=1}^n{D}_j{f}_j\left(x\right),\text{in}\Omega$$ in the setting of the weighted Sobolev spaces.