Three nontrivial solutions for nonlocal anisotropic inclusions under nonresonance

In this article, we study a pseudo-differential inclusion driven by a nonlocal anisotropic operator and a Clarke generalized subdifferential of a nonsmooth potential, which satisfies nonresonance conditions both at the origin and at infinity. We prove the existence of three nontrivial solutions: one positive, one negative and one of unknown sign, using variational methods based on nosmooth critical point theory, more precisely applying the second deformation theorem and spectral theory. Here, a nosmooth anisotropic version of the Hölder versus Sobolev minimizers relation play an important role

Dirichlet problems for several nonlocal operators via variational and topological methods

The main topic of the thesis is the study of elliptic differential equations with fractional order driven by nonlocal operators, as the fractional p-Laplacian, the fractional Laplacian for p=2, the general nonlocal operator and its anisotropic version. Recently, great attention has been focused on the study of fractional and nonlocal operators of elliptic type, both for pure mathematical research and in view of concrete real-world applications. This type of operators arises in a quite natural way in many different contexts, such as, among others, game theory, image processing, optimization, phase transition, anomalous diffusion, crystal dislocation, water waves, population dynamics and geophysical fluid dynamics. The main reason is that nonlocal operators are the infinitesimal generators of Lévy-type stochastic processes. Such processes extend the concept of Brownian motion, where the infinitesimal generator is the Laplace operator, and may contain jump discontinuities. Our aim is to show existence and multiplicity results for nonlinear elliptic Dirichlet problems, driven by a nonlocal operator, by applying variational and topological methods. Such methods usually exploit the special form of the nonlinearities entering the problem, for instance its symmetries, and offer complementary information. They are powerful tools to show the existence of multiple solutions and establish qualitative results on these solutions, for instance information regarding their location. The topological and variational approach provides not just existence of a solution, usually several solutions, but allow to achieve relevant knowledge about the behavior and properties of the solutions, which is extremely useful because generally the problems cannot be effectively solved, so the precise expression of the solutions is unknown. As a specific example of property of a solution that we look for is the sign of the solution, for example to be able to determine whether it is positive, or negative, or nodal (i.e., sign changing)

A sufficient and necessary condition of existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator

In this paper, we establish the results of nonexistence and existence of blow-up radial solutions for a k-Hessian equation with a nonlinear operator. Under some suitable growth conditions for nonlinearity, the result of nonexistence of blow-up solutions is established, a sufficient and necessary condition on existence of blow-up solutions is given, and some further results are obtained.&nbsp

Bloch Waves in Crystals and Periodic High Contrast Media

Analytic representation formulas and power series are developed describing the band structure inside periodic photonic and acoustic crystals made from high contrast inclusions. Central to this approach is the identification and utilization of a resonance spectrum for quasi-periodic source free modes. These modes are used to represent solution operators associated with electromagnetic and acoustic waves inside periodic high contrast media. Convergent power series for the Bloch wave spectrum is recovered from the representation formulas. Explicit conditions on the contrast are found that provide lower bounds on the convergence radius. These conditions are sufficient for the separation of spectral branches of the dispersion relation

Three nontrivial solutions for nonlocal anisotropic inclusions under nonresonance

In this article, we study a pseudo-differential inclusion driven by a nonlocal anisotropic operator and a Clarke generalized subdifferential of a nonsmooth potential, which satisfies nonresonance conditions both at the origin and at infinity. We prove the existence of three nontrivial solutions: one positive, one negative and one of unknown sign, using variational methods based on nosmooth critical point theory, more precisely applying the second deformation theorem and spectral theory. Here, a nosmooth anisotropic version of the Holder versus Sobolev minimizers relation play an important role.publishe