Critical Gagliardo–Nirenberg, Trudinger, Brezis–Gallouet–Wainger inequalities on graded groups and ground states

In this paper, we investigate critical Gagliardo–Nirenberg, Trudinger-type and Brezis–Gallouet–Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which include the cases of [Formula: see text], Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo–Nirenberg inequality, the existence of least energy solutions of nonlinear Schrödinger type equations is obtained. We also express the best constant in the critical Gagliardo–Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The obtained results are already new in the setting of general stratified Lie groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland’s analysis of Hölder spaces from stratified Lie groups to general homogeneous Lie groups.

Ground state solution of semilinear Schrödinger system with local super-quadratic conditions

In this paper, we dedicate to studying the following semilinear Schrödinger system equation*-Δu+V1(x)u=Fu(x,u,v)amp;mboxin~RN,r-Δv+V2(x)v=Fv(x,u,v)amp;mboxin~RN,ru,v∈H1(RN),endequation* where the potential Vi are periodic in x,i=1,2, the nonlinearity F is allowed super-quadratic at some x ∈ R N and asymptotically quadratic at the other x ∈ R N . Under a local super-quadratic condition of F, an approximation argument and variational method are used to prove the existence of Nehari–Pankov type ground state solutions and the least energy solutions.

Critical polyharmonic systems and optimal partitions

<p style='text-indent:20px;'>We establish the existence of solutions to a weakly-coupled competitive system of polyharmonic equations in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^N $\end{document}</tex-math></inline-formula> which are invariant under a group of conformal diffeomorphisms, and study the behavior of least energy solutions as the coupling parameters tend to <inline-formula><tex-math id="M2">\begin{document}$ -\infty $\end{document}</tex-math></inline-formula>. We show that the supports of the limiting profiles of their components are pairwise disjoint smooth domains and solve a nonlinear optimal partition problem of <inline-formula><tex-math id="M3">\begin{document}$ \mathbb R^N $\end{document}</tex-math></inline-formula>. We give a detailed description of the shape of these domains.</p>

The asymptotically linear Hénon problem

In this paper, we consider the Hénon problem in the ball with Dirichlet boundary conditions. We study the asymptotic profile of radial solutions and then deduce the exact computation of their Morse index when the exponent [Formula: see text] is close to [Formula: see text]. Next we focus on the planar case and describe the asymptotic profile of some solutions which minimize the energy among functions which are invariant for reflection and rotations of a given angle [Formula: see text]. By considerations based on the Morse index we see that, depending on the values of [Formula: see text] and [Formula: see text], such least energy solutions can be radial, or nonradial and different one from another.