Keywords: Nonlinear PDEs, Variational methods, Existence, multiplicity and symmetry of solutions, Higher-order PDEs, Optimization eigenvalue problems, Non-standard growth functionals, Supercritical problems, Kirchhoff problems

OPTIMIZATION EIGENVALUE PROBLEMS

We consider a fourth-order optimization eigenvalue problem, governed by the biharmonic operator, with Dirichlet or Navier boundary conditions. In the two-dimensional case, this problem has a physical interpretation in Continuum Mechanics for inhomogeneous linear elastic plates: build a plate of prescribed shape and mass, out of different materials (with different densities), with the property that the principal frequency of the resulting body is the lowest possible. We prove the existence of optimal configurations and their qualitative properties, such as regularity and symmetry. Symmetry issues in this setting are complicated by the lack of a general maximum principle for biharmonic equations and by the fact that higher-order Sobolev spaces are not invariant under symmetric rearrangements. Under Navier boundary conditions, it is possible to reduce the fourth-order equation to a second-order elliptic system, which inherits the classical comparison and maximum principles for second-order equations. These, together with moving planes techniques, allow to prove symmetry properties of the solutions. For Dirichlet boundary conditions, symmetry issues are far more delicate. In this case it is possible to prove symmetry properties thanks to the combination of the polarization technique -which plays the role of symmetric rearrangements- with the properties of the Green function of the Dirichlet biharmonic operator, whose explicit representation is known when the domain is a ball of R^n.

SUPERCRITICAL QUASILINEAR PROBLEMS

We consider quasilinear equations governed by the p-Laplacian operator, set in the unit ball of R^n, coupled with Neumann boundary conditions. These equations involve a nonlinearity which is possibly supercritical in the sense of Sobolev embeddings. In the case p>2, we overcome the lack of compactness by working in the cone of non-negative, radial, non-decreasing functions, where we are able to prove a priori estimates on the solutions of the problem. This allows us to use variational techniques to prove the existence of a non-constant solution of the problem. The problem for 1<p<2 is more delicate, since the energy functional is less regular than in the case p>2. Recently, we attacked the problem by using a different technique, i.e. the shooting method, that allows us to obtain existence and multiplicity results for every p>1.

PROBLEMS GOVERNED BY INHOMOGENEOUS OPERATORS

We study some properities of the variational spectrum of two inhomogeneous operators, i.e. the p(x)-Laplacian, and a non-autonomous sum of a p-Laplacian and a q-Laplacian operator (the so called double-phase operator). The energy functionals associated to these operators arise in some models for strongly anisotropic materials. For both operators we prove the stability of the variational spectrum, via Gamma-convergence, and a Weyl-type law describing the asymptotic growth of the eigenvalues.

We study also qualitative properties of solutions to a problem governed by an inhomogeneous operator which is the sum of different 2m-Laplacians. These problems naturally arise as approximations of the Born-Infeld equation in the nonlinear theory of electromagnetism.

KIRCHHOFF PROBLEMS

We study some Kirchhoff-type problems in bounded domains of R^n, governed by the polyharmonic operator, with homogeneous Dirichlet boundary conditions. These problems are non-local due to the presence of the so-called Kirchhoff function (which is a function of the norm of u) in front of the leading differential operator. Furthermore, they involve both a nonlinear damping term and a nonlinear subcritical source force term. When the source force term is "stronger" than the damping one (this condition is related to the growth of the two terms), we prove that the problem does not admit any global (in time) solutions and we provide an a priori estimate of the lifespan of maximal solutions. For strongly damped problems (i.e. problems involving also an interior damping term), we prove that the solutions blow-up at infinity.We study also the stationary version of this kind of problems. In this case, we show existence and multiplicity results via variational methods, also for problems governed by more general operators.