Existence and multiplicity results for a fourth order mean field equation

Existence and multiplicity of solutions for a fourth-order elliptic equation

where Ω ⊂ R is a bounded smooth domain, f : Ω × R ® R and M : R ® R are continuous functions. The existence and multiplicity results for Equation (1) are considered in [1-3] by using variational methods and fixed point theorems in cones of ordered Banach space with space dimension is one. On the other hand, The four-order semilinear elliptic problem { 2u + c u = f (x, u), in , u = u = 0, on ∂ ,...

Existence results for mean field equation with turbulence

on Σ; ( where ∆g is the Laplace-Beltrami, ρ a real parameter) which arises in the study of limit of point vortices of Euler flows, spherical Onsager vortex theory and condensates in some Chern-Simons-Higgs models, see for example the papers [3], [4], [8], [9], [10], [11], [14], [25], [40] and the references therein. An other example is what we refer as mean field equation with turbulence on a c...

Multiplicity Results for Perturbed Fourth-order Kirchhoff-type Problems

Existence Results for Mean Field Equations

Let Ω be an annulus. We prove that the mean field equation −∆ψ = e ∫ Ω e−βψ in Ω

Asymptotic Behavior of a Fourth Order Mean Field Equation with Dirichlet Boundary Condition

where Σ is a smooth bounded domain in R, has been extensively studied by many authors. Let (uk, ρk) be a unbounded sequence of solutions to (1.2) with ρk ≤ C,maxx∈Σ uk(x) → +∞. Then it has been proved that (P1) (no boundary bubbles) uk is uniformly bounded near a neighborhood of ∂Σ (Nagasaki-Suzuki [33], Ma-Wei [29]); (P2) (bubbles are simple) ρk → 8mπ for some m ≥ 1 and uk(x) → 8π ∑m j=1 G(·, ...