Existence and Qualitative Properties of Solutions to Nonlinear Schrodinger-Poisson Systems

Our main equation of study is the nonlinear Schr¨odinger-Poisson system⇢−Du+u+r(x)fu = |u|p−1u, x 2 R3,−Df = r(x)u2, x 2 R3,with p 2 (2,5) and r : R3 ! R a nonnegative measurable function. In the spirit ofthe classical work of P. H. Rabinowitz [55] on nonlinear Schr¨odinger equations, wefirst prove existence of positive mountain-pass solutions and least energy solutions tothis system under different assumptions on r at infinity. Our results cover the rangep 2 (2,3) where the lack of compactness phenomena may be due to the combinedeffect of the invariance by translations of a ‘limiting problem’ at infinity and of thepossible unboundedness of the Palais-Smale sequences. In the case of a coercive r,namely r(x)!+• as |x|!+•, we then prove the existence of infinitely many distinctpairs of solutions. For p 2 (3,5) we exploit the symmetry of the problem bythe action of Z2 as well as some well-known properties of the Krasnoselskii-genus,whereas for p 2 (2,3] we use an appropriate abstract min-max scheme, which requiressome additional assumptions on r.After establishing these existence and multiplicity results, we are then interested inthe qualitative properties of solutions the singularly perturbed problem⇢−e2Du+lu+r(x)fu = |u|p−1u, x 2 R3−Df = r(x)u2, x 2 R3,with r : R3 ! R a nonnegative measurable function, l 2 R, and l > 0, taking advantageof a shrinking parameter e ⌧ 1. In particular, we seek to understand theconcentration phenomena purely driven by r. To this end, we first find necessaryconditions for concentration at points to occur for solutions in various functionalsettings which are suitable for both variational and perturbation methods. We thendiscuss a variational/penalisation method, which has been exploited in the case ofnonlinear Schr¨odinger equations, and discuss its applications to the present nonlinearSchr¨odinger-Poisson context, in the attempt of showing that the necessary conditionsare, in fact, sufficient conditions on r for point concentration of solutions. Finally,we present some preliminary results in this direction that elicit interesting standalonequalitative properties of the solutions.