Existence of a ground state solution for a nonlinear scalar field equation with critical growth

Abstract

In the present paper, we establish the existence of Ground State Solutions for some class of Elliptic problems with Critical Growth in R N for N ≥ 2. Our results complete the study made in Berestycki and Lions (Arch Rat Mech Anal 82: 1983) and Berestycki, Gallouët and Kavian (C R Acad Sci Paris Ser I Math 297: [307][308][309][310] 1984), in the sense that, in those papers only the subcritical growth was considered.

Fractional Kirchhoff equation with a general critical nonlinearity

“…When a = 1 and b = 0, (1.1) becomes the fractional Schrödinger equations which have been studied by many authors. We refer the readers to [2,[5][6][7] and the references therein for the details. When s = 1, the problem (1.1) reduces to the well-known Kirchhoff equation…”

Ground states of nonlocal scalar field equations with Trudinger-Moser critical nonlinearity

“…In [11] the Trudinger-Moser inequality was extended to the whole R 2 and the authors gave some applications to study equations like (1.3) when the nonlinear term has critical growth of Trudinger-Moser type. For further results and applications, we would like to mention also [2,3,17,29] and references therein. When the potential V is a positive constant and f (x, s) = f (s) for (x, s) ∈ R N × R, that is the autonomous case, the existence of ground states for subcritical nonlinearities was established in [6] for N ≥ 3 and [7] for N = 2 respectively, while in [3] the critical case for N ≥ 3 and N = 2 was treated.…”

Ground States for Elliptic Equations in $\pmb{\mathbb{R}}^{2}$ with Exponential Critical Growth

“…Recently Alves, Montenegro and Souto [3] improved the arguments in [8], assuming g(s) = f (s) − s and considering nonlinearities with critical exponential growth.…”

“…Theorem 3 [3] Assume that (f 0 ) f : R → R is continuous and has critical exponential growth with α 0 = 4π ;…”