We study the Schrodinger-Newton equations proposed by Penrose to describe his view that quantum state reduction occurs due to some gravitational e?ffect. They consist of a system of equations obtained by coupling together the linear Schrodinger equation of quantum mechanics with the Poisson equation from Newtonian mechanics. We assume that the magnetic vector potential A and the electric potential V in the linear Schrodinger equation are compatible with the action of a group G of linear isometries of R^3. Then, for any given homomorphism ? $ au$ on G into the unit complex numbers, we show that there is a combined e?ffect of the symmetries and the potential V on the number of semiclassical solutions u which satisfy $u(gx) = au?(g)u(x)$ for all $g in G, x in R3$. We also study the concentration behavior of these solutions in the semiclassical limit.

We consider a compact, connected, orientable, boundaryless Riemannian manifold $(M,g)$ of class $C^infty$ where $g$ denotes the metric tensor. Let $n= dim M geq 3$. Using Morse techniques, we prove the existence of $2{mathcal P}_1(M) -1$ non-costant solutions $uin H^{1,p}(M)$ to the quasilinear problem [ (P_epsilon) left{ begin{array}{l} -epsilon^p , Delta_{p,g} u +u^{p-1}=u^{q-1} \ u>0 end{array} ight. label{eqab} ] for $varepsilon>0$ small enough, where $2 leq p<n$, $p < q <p^*$, $p^* = np/(n-p)$ and $Delta_{p,g} u = extrm{div}_g (|nabla u|_g^{p-2}nabla u)$ is the $p$-laplacian associated to $g$ of $u$ (note that $Delta_{2,g} = Delta_g$) and ${mathcal P}_t(M)$ denotes the Poincar'e Polynomial of $M$. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem $(P_varepsilon)$.

Regularity results and critical group estimates are studied for critical (p,r)-systems. Multiplicity results of solutions for a critical potential quasilinear system are also proved using Morse theory.

This work deals with Morse index estimates for a solution $ uin H_1^p(M)$ of the quasilinear elliptic equation $ - extrm{div}_g left( left(alpha +|nabla u|_g^2 ight)^{(p-2)/2}nabla u ight)=h(x,u) $, where (M, g) is a compact, Riemannian manifold, 0 < alpha, 2 leq p < n. The nonlinear right-hand side h(x,s) is allowed to be subcritical or critical.

We consider a compact, connected, orientable, boundaryless Riemannian manifold $(M,g)$ of class $C^\infty$ where $g$ denotes the metric tensor. Let $n= \dim M \geq 3$. Using Morse techniques, we prove the existence of $2{\mathcal P}_1(M) -1$ non-costant solutions $u\in H^{1,p}(M)$ to the quasilinear problem \[ (P_\epsilon) \left\{ \begin{array}{l} -\epsilon^p \, \Delta_{p,g} u +u^{p-1}=u^{q-1} \\ u>0 \end{array} ight. \label{eqab} \] for $\varepsilon>0$ small enough, where $2 \leq p<n$, $p < q <p^*$, $p^* = np/(n-p)$ and $\Delta_{p,g} u = extrm{div}_g (|\nabla u|_g^{p-2}\nabla u)$ is the $p$-laplacian associated to $g$ of $u$ (note that $\Delta_{2,g} = \Delta_g$) and ${\mathcal P}_t(M)$ denotes the Poincar\'e Polynomial of $M$. We also establish results of genericity of nondegenerate solutions for the quasilinear elliptic problem $(P_\varepsilon)$.

We consider singularly perturbed nonlinear Schrödinger equations − ε2u + V(x)u = f (u), u > 0, v ∈ H1(RN ) (0.1) where V ∈ C(RN ,R) and f is a nonlinear term which satisfies the so-called Berestycki–Lions conditions. We assume that there exists a bounded domain Omega ⊂ RN such that m0 ≡ inf inf{V(x) |x∈ Omega} < inf{V(x) |x∈∂ Omega } and we set K = {x ∈ Omega | V(x) = m0}. For ε > 0 small we prove the existence of at least cupl(K) + 1 solutions to (0.1) concentrating, as ε → 0 around K. We remark that, under our assumptions of f , the search of solutions to (0.1) cannot be reduced to the study of the critical points of a functional restricted to a Nehari manifold.

We consider a class of quasilinear elliptic equations whose principal part includes the p-area (for 1 < p < ∞)) and the p-Laplace (for 1 < p ≤ 2) operator. For the critical points of the associated functional, we provide estimates of the corresponding critical polynomial.

The semi-classical regime of standing wave solutions of a Schrödinger equation in the presence of non-constant electric and magnetic potentials is studied in the case of non-local nonlinearities of Hartree type. It is shown that there exists a family of solutions having multiple concentration regions which are located around the minimum points of the electric potential.

We consider the magnetic NLS equation (εi∇ + A(x)) 2 u + V (x)u = K(x) |u| p-2 u, x ∈ ℝ N, where N ≥ 3, 2 < p < 2* := 2N/(N - 2), A : ℝ N → ℝ N is a magnetic potential and V : ℝ N → ℝ, K : ℝ N → ℝ are bounded positive potentials. We consider a group G of orthogonal transformations of ℝ N and we assume that A is G-equivariant and V, K are G-invariant. Given a group homomorphism τ : G → S 1 into the unit complex numbers we look for semiclassical solutions u ε: ℝ N → ℂ to the above equation which satisfy u ε (gx) = τ(g)u ε (x) for all g ∈ G, x ∈ ℝ N. Using equivariant Morse theory we obtain a lower bound for the number of solutions of this type.

In this paper we illustrate the lineguides of our research group. We describe some recent results concerning the study of some nonlinear differential equations and systems having a variational nature and arising from physics, geometry and applied sciences. In particular we report existence, multiplicity and regularity results for the solutions of these nonlinear problems. We point out that, in treating the above problems, the used methods for finding solutions are variational and topological, indeed the existence of solutions of the considered equations is obtained searching for critical points of suitable functionals defined on manifolds embedded into infinite dimensional functional spaces, while the regularity of the solutions is studied by means of geometric and harmonic analysis tools.