We study the asymptotic behavior of the solution of linear and nonlinear elliptic problems in cylindrical domains becoming unbounded in one or several directions. In particular we show that this solution converges in H 1 norm toward the solution of problems set on the cross section of the domains.
Abstract. The aim of this work is to analyze the asymptotic behaviour of the eigenmodes of some elliptic eigenvalue problems set on domains becoming unbounded in one or several directions. In particular, in the case of a linear elliptic operator in divergence form, we prove that the sequence of the k-th eigenvalues convergences to the first eigenvalue of an elliptic problems set on the section of the domain. Moreover, an optimal rate of convergence of this sequence is given.
Abstract. We consider quasilinear elliptic equations where the diffusion at each point depends on all the values of the solution in a neighborhood of this point. The size of this neighborhood is parameterized by some non negative number which represents the range of nonlocal interactions. For fixed values of the parameter, the issue of the existence and local uniqueness of the solution is addressed. In a radial symmetric setting, we give pointwise estimates of the solutions and prove the existence of multiple solutions. Regarding bifurcation theory, we show that many local branches of solutions may exist while, among them, only one is global and has no bifurcation point.
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