Existence and multiplicity results for resonant fractional boundary value problems

We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory

Variational methods in relativistic quantum mechanics

This review is devoted to the study of stationary solutions of linear and nonlinear equations from relativistic quantum mechanics, involving the Dirac operator. The solutions are found as critical points of an energy functional. Contrary to the Laplacian appearing in the equations of nonrelativistic quantum mechanics, the Dirac operator has a negative continuous spectrum which is not bounded from below. This has two main consequences. First, the energy functional is strongly indefinite. Second, the Euler-Lagrange equations are linear or nonlinear eigenvalue problems with eigenvalues lying in a spectral gap (between the negative and positive continuous spectra). Moreover, since we work in the space domain R^3, the Palais-Smale condition is not satisfied. For these reasons, the problems discussed in this review pose a challenge in the Calculus of Variations. The existence proofs involve sophisticated tools from nonlinear analysis and have required new variational methods which are now applied to other problems

Asymmetric (p, 2)-equations with double resonance

We consider a nonlinear Dirichlet elliptic problem driven by the sum of a p-Laplacian and a Laplacian [a (p, 2)-equation] and with a reaction term, which is superlinear in the positive direction (without satisfying the Ambrosetti-Rabinowitz condition) and sublinear resonant in the negative direction. Resonance can also occur asymptotically at zero. So, we have a double resonance situation. Using variational methods based on the critical point theory and Morse theory (critical groups), we establish the existence of at least three nontrivial smooth solutions

Robin problems with a general potential and a superlinear reaction

We consider semilinear Robin problems driven by the negative Laplacian plus an indefinite potential and with a superlinear reaction term which need not satisfy the Ambrosetti-Rabinowitz condition. We prove existence and multiplicity theorems (producing also an infinity of smooth solutions) using variational tools, truncation and perturbation techniques and Morse theory (critical groups)

Nonlinear, nonhomogeneous Robin problems with indefinite potential and general reaction

We consider a nonlinear elliptic equation driven by a nonhomogeneous differential operator plus an indefinite potential. On the reaction term we impose conditions only near zero. Using variational methods, together with truncation and perturbation techniques and critical groups, we produce three nontrivial solutions with sign information. In the semilinear case we improve this result by obtaining a second nodal solution for a total of four nontrivial solutions. Finally, under a symmetry condition on the reaction term, we generate a whole sequence of distinct nodal solutions.Comment: arXiv admin note: text overlap with arXiv:1907.04999, arXiv:1706.0356

Symmetry breaking in the periodic Thomas--Fermi--Dirac--von Weizs{\"a}cker model

We consider the Thomas--Fermi--Dirac--von~Weizs{\"a}cker model for a system composed of infinitely many nuclei placed on a periodic lattice and electrons with a periodic density. We prove that if the Dirac constant is small enough, the electrons have the same periodicity as the nuclei. On the other hand if the Dirac constant is large enough, the 2-periodic electronic minimizer is not 1-periodic, hence symmetry breaking occurs. We analyze in detail the behavior of the electrons when the Dirac constant tends to infinity and show that the electrons all concentrate around exactly one of the 8 nuclei of the unit cell of size 2, which is the explanation of the breaking of symmetry. Zooming at this point, the electronic density solves an effective nonlinear Schr\"odinger equation in the whole space with nonlinearity $u^{7/3}-u^{4/3}$. Our results rely on the analysis of this nonlinear equation, in particular on the uniqueness and non-degeneracy of positive solutions

Constant sign and nodal solutions for nonlinear Robin equations with locally defined source term

We consider a parametric Robin problem driven by a nonlinear, nonhomogeneous differential operator which includes as special cases the p-Laplacian and the (p,q)-Laplacian. The source term is parametric and only locally defined (that is, in a neighborhood of zero). Using suitable cut-off techniques together with variational tools and comparison principles, we show that for all big values of the parameter, the problem has at least three nontrivial smooth solutions, all with sign information (positive, negative and nodal)