We study the Choquard equation with a local perturbation −Δu=λu+(Iα∗|u|p)|u|p−2u+μ|u|q−2u,x∈RN having prescribed mass ∫RN|u|2dx=a2. For L2-critical or L2-supercritical μ|u|q−2u, we prove nonexistence, existence and symmetry of normalized ground states, by using mountain pass lemma, Pohožaev constraint method, Schwartz symmetrization rearrangements some theories polarizations. In particular, our...

In this paper, we consider a neutral molecule that possesses two distinct stable positions for its nuclei, and look for a mountain pass point between the two minima in the non-relativistic Schrödinger framework. We first prove some properties concerning the spectrum and the eigenstates of a molecule that splits into pieces, a behaviour which is observed when the Palais-Smale sequences obtained ...

We provide a max-min characterization of the mountain pass energy level for a family of variational problems. As a consequence we deduce the mountain pass structure of solutions to suitable PDEs, whose existence follows from classical minimization argument. In the literature the existence of solutions for variational PDE is often reduced to the existence of critical points of functionals F havi...

We prove a saddle point theorem for locally Lipschitz functionals with arguments based on a version of the mountain pass theorem for such kind of functionals. This abstract result is applied to solve two diierent types of multivalued semilinear elliptic boundary value problems with a Laplace{Beltrami operator on a smooth compact Riemannian manifold. The mountain pass theorem of Ambrosetti and R...

This paper is concerned with the existence of two non-trivial weak solutions for a p(x)-Kirchho type problem by using the mountain pass theorem of Ambrosetti and Rabinowitz and Ekeland's variational principle and the theory of the variable exponent Sobolev spaces.

In the last decade or so, variational gluing methods have been widely used to construct homoclinic and heteroclinic type solutions of nonlinear elliptic equations and Hamiltonian systems. This note is concerned with the procedure of gluing mountain-pass type solutions. The rst procedure to glue mountain-pass type solutions was developed through the work of S er e, and Coti Zelati-Rabinowitz. Th...

In this paper, a system of elliptic equations is investigated, which involves multiple critical Sobolev exponents and singular points. The best Sobelev constant related to the system is studied, which is verified to be independent of the location of singular points. By a variant of the concentration compactness principle and the mountain-pass argument, the existence of positive solutions to the...