Uniform estimates and blow–up behavior for solutions of −δ(u)=v(x)e^uin two dimensions

Singular Limits of a Two-Dimensional Boundary Value Problem Arising in Corrosion Modelling

“…It is easily checked that the solutions (1.2) correspond exactly to this description in the case Ω = D. It is interesting to mention the analogy existing between this result and the problem −∆u = λe u under Dirichlet boundary conditions, whose solutions with λ Ω e u uniformly bounded have become well understood after the works [10,2,8]. It follows from those results that concentration occurs in the form λe u 8π δ ξj .…”

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

“…follows from equation (4) multiplied by |ξ| 2 and from the integrability of u given by (6) and (8). Mimicking a standard computation for the parabolic-elliptic Keller-Segel system by writing v = − 1 2 π log(·) * u +ṽ, the above identity reads [5,8] for more details.…”

Convergence for a Liouville equation

“…However, as pointed out in [5], there are regions, for example, the rectangles when the ratio between the sides is large enough, for which the concentration does not occur, so there is at least one solution for (1.1-2) with λ = 8π . The problem (1.1-2) also has other interesting features, which are different from other nonlinear problems like Nirenberg's problem (see [15]) and the scalar curvature problem (see [2]); in particular the blow-up analysis of H.Brezis and F.Merle [3]. In fact, for (1.1-2) we do have multiple bubbles [12] and when K = 1 , S.Baraket and F.Pacard [26] construct them by using the fixed point theorem.…”

“…Then there is a subsequence of {ξ n } , still denoted by {ξ n } satisfying either (i) We remark that the original statement of Theorem 3 in [3] is different from the form above, but their argument can be modified to cover this case. Now we recall some well-known facts including the Pohozaev identities.…”

“…One may see [5] and [11] for discussions about physical background. Motivated by the interesting work of H.Brezis and F.Merle [3], we study the location of possible bubble points of a solution sequence when the Dirichlet boundary condition is imposed.…”