Using some nonlinear domain decomposition method, we prove the existence of branches of solutions having singular limits for some 4-dimensional semilinear elliptic problem with exponential nonlinearity. Résumé: En utilisant une variante non linéaire de la méthode de décomposition de domaines, nous démontrons l'existence de branches de solutions ayant une limite singulière, pour une équation semilinéaire elliptique avec nonlinéarité exponentielle, en dimension 4.

We study two classes of nonhomogeneous elliptic problems with Dirichlet boundary condition and involving a fourth-order differential operator with variable exponent and power-type nonlinearities. The first result of this paper establishes the existence of a nontrivial weak solution in the case of a small perturbation of the right-hand side. The proof combines variational methods, including the Ekeland variational principle and the mountain pass theorem of Ambrosetti and Rabinowitz. Next we consider a very related eigenvalue problem and we prove the existence of nontrivial weak solutions for large values of the parameter. The direct method of the calculus of variations, estimates of the levels of the associated energy functional and basic properties of the Lebesgue and Sobolev spaces with variable exponent have an important role in our arguments.

In this work, we study the weighted Kirchhoff problem {g(?B(?(x)|?u|2 + V(x)u2)dx) [?div(?(x)?u) + V(x)u] = f (x, u) in B u > 0 in B u = 0 on ?B, where B is the unit ball in R2, ?(x) = log e |x| , the singular logarithm weight in the Trudinger-Moser embedding, g is a continuous positive function on R+ and the potential V is a continuous positve function. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities. We prove the existence of non-trivial solutions via the critical point theory. In the critical case, the associated energy function does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check the min-max compactness level.