Schrödinger–Poisson system with zero mass and convolution nonlinearity in R 2

In this paper, we prove the existence of nontrivial solutions and ground state solutions for the following planar Schrödinger–Poisson system with zero mass − Δ u + ϕ u = ( I α ∗ F ( u ) ) f ( u ) , x ∈ R 2 , Δ ϕ = u 2 , x ∈ R 2 , where α ∈ ( 0 , 2 ), I α : R 2 → R is the Riesz potential, f ∈ C ( R , R ) is of subcritical exponential growth in the sense of Trudinger–Moser. In particular, some new ideas and analytic technique are used to overcome the double difficulties caused by the zero mass case and logarithmic convolution potential.

The cone Moser–Trudinger inequalities and applications

In this paper, we first study the cone Moser–Trudinger inequalities and their best exponents on both bounded and unbounded domains R + 2 . Then, using the cone Moser–Trudinger inequalities, we study the asymptotic behavior of Cerami sequences and the existence of weak solutions to the nonlinear equation − Δ B u = f ( x , u ) , in x ∈ int ( B ) , u = 0 , on ∂ B , where Δ B is an elliptic operator with conical degeneration on the boundary x 1 = 0, and the nonlinear term f has the subcritical exponential growth or the critical exponential growth.

On a nonlocal asymmetric Kirchhoff problem

This work is devoted to study the existence of nontrivial solutions to nonlocal asymmetric problems involving the [Formula: see text]-Laplacian. [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text] is a Kirchhoff function, [Formula: see text] and [Formula: see text] is of subcritical polynomial or subcritical exponential growth. Moreover, the existence of nontrivial solutions for the above problem is obtained by using variational methods combined with the Moser–Trudinger inequality. Our interest then is to study [Formula: see text] without the analogue of Ambrosetti–Rabinowitz superquadratic condition ([Formula: see text] condition for short) in the positive semi-axis. To the best of our best knowledge, our results are new even in the asymmetric Kirchhoff Laplacian and [Formula: see text]-Laplacian cases.

A Biharmonic Equation in ℜ4 Involving Nonlinearities with Subcritical Exponential Growth

In this paper we consider a biharmonic equation of the form Δ