1,721,263 research outputs found
Correction to: Eigenvalue bifurcation in doubly nonlinear problems with an application to surface plasmon polaritons (Nonlinear Differential Equations and Applications NoDEA, (2021), 28, 1, (9), 10.1007/s00030-020-00668-2)
No full textThis erratum corrects some errors which appear in [Dohnal T., Romani G., Eigenvalue bifurcation in doubly nonlinear problems with an application to surface plasmon polaritons, Nonlinear Differ. Equ. Appl. 28, 9 (2021).]. In particular, it replaces the (unsuitable) definition of isolatedness of the linear eigenvalue, from which the bifurcation occurs
Schrödinger–Poisson systems with zero mass in the Sobolev limiting case
No full textWe study the existence of positive solutions for a class of systems which strongly couple a quasilinear Schröodinger equation driven by a weighted N-Laplace operator and without the mass term, and a higher-order fractional Poisson equation. Since the system is set in RN\mathbb {R}<^>N, the limiting case for the Sobolev embedding, we consider nonlinearities with exponential growth. Existence is proved relying on the study of a corresponding Choquard equation in which the Riesz kernel is a logarithm, hence sign-changing and unbounded from above and below. This is in turn solved by means of a variational approximating procedure for an auxiliary Choquard equation, where the logarithm is uniformly approximated by polynomial kernels. Our results are new even in the planar case N=2
Choquard equations with critical exponential nonlinearities in the zero mass case
No full textWe investigate Choquard equations in RN driven by a weighted N-Laplace operator with polynomial kernel and zero mass. Since the setting is limiting for the Sobolev embedding, we work with nonlinearities which may grow up to the critical exponential. We establish the existence of a positive solution by variational methods, complementing the analysis in [32], where the case of a logarithmic kernel was considered
Uniform bounds for higher-order semilinear problems in conformal dimension
No full textWe establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem (−Δ)mu=h(x,u) in Ω, u=∂nu=⋯=∂nm−1u=0 on ∂Ω,where h is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger–Moser–Adams inequality, either when Ω is a ball or, provided an energy control on solutions is prescribed, when Ω is a smooth bounded domain. Our results are sharp within the class of distributional solutions. The analogous problem with Navier boundary conditions is also studied. Finally, as a consequence of our results, existence of a positive solution is shown by degree theory
A-priori bounds for quasilinear problems in critical dimension
No full textWe establish uniform a-priori bounds for solutions of the quasilinear problem
Perturbed eigenvalues of polyharmonic operators in domains with small holes
Full text linkWe study singular perturbations of eigenvalues of the polyharmonic operator on bounded domains under removal of small interior compact sets. We consider both homogeneous Dirichlet and Navier conditions on the external boundary, while we impose homogeneous Dirichlet conditions on the boundary of the removed set. To this aim, we develop a notion of capacity which is suitable for our higher-order context, and which permits to obtain a description of the asymptotic behaviour of perturbed simple eigenvalues in terms of a capacity of the removed set, in dependence of the respective normalized eigenfunction. Then, in the particular case of a subset which is scaling to a point, we apply a blow-up analysis to detect the precise convergence rate, which turns out to depend on the order of vanishing of the eigenfunction. In this respect, an important role is played by Hardy-Rellich inequalities in order to identify the appropriate functional space containing the limiting profile. Remarkably, for the biharmonic operator this turns out to be the same, regardless of the boundary conditions prescribed on the exterior boundary
Eigenvalue bifurcation in doubly nonlinear problems with an application to surface plasmon polaritons
Full text linkWe consider a class of generally non-self-adjoint eigenvalue problems which are nonlinear in the solution as well as in the eigenvalue parameter (“doubly” nonlinear). We prove a bifurcation result from simple isolated eigenvalues of the linear problem using a Lyapunov–Schmidt reduction and provide an expansion of both the nonlinear eigenvalue and the solution. We further prove that if the linear eigenvalue is real and the nonlinear problem PT-symmetric, then the bifurcating nonlinear eigenvalue remains real. These general results are then applied in the context of surface plasmon polaritons (SPPs), i.e. localized solutions for the nonlinear Maxwell’s equations in the presence of one or more interfaces between dielectric and metal layers. We obtain the existence of transverse electric SPPs in certain PT-symmetric configurations
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