Nguyen Lam

N-laplacian equations in RN with subcritical and critical growthwithout the ambrosetti-rabinowitz condition

Abstract Let Ω be a bounded domain in ℝN. In this paper, we consider the following nonlinear elliptic equation of N-Laplacian type: when f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N−Laplacian in ℝN when the nonlinear term f has the exponential growth only deal with the case when f satisfies the (AR) condition. Our approach is based on a suitable version of the Mountain Pass Theorem introduced by G. Cerami [9, 10, 21]. Examples of f and comparison with earlier assumptions in the literature are given.

On nonuniformly subelliptic equations of Q-sub-laplacian type with critical growth in the heisenberg group

Abstract Let ℍn = ℝ2n × ℝ be the n−dimensional Heisenberg group, be its subelliptic gradient operator, and ρ (ξ) = ( |z|4 + t2 )1/4 for ξ = (z, t) ∈ ℍn be the distance function in ℍn. Denote ℍ = ℍn, Q = 2n + 2 and Q′ = Q/(Q − 1). Let Ω be a bounded domain with smooth boundary in ℍ. Motivated by the Moser-Trudinger inequalities on the Heisenberg group, we study the existence of solution to a nonuniformly subelliptic equation of the form where f : Ω × ℝ → ℝ behaves like exp ( α |u|Q′ ) when |u| → ∞. In the case of Q−sub- Laplacian we will apply minimax methods to obtain multiplicity of weak solutions.

Sharp constants and optimizers for a class of the Caffarelli-Kohn-Nirenberg inequalities

Abstract In this paper, we use a suitable transform of quasi-conformal mapping type to investigate the sharp constants and optimizers for the following Caffarelli–Kohn–Nirenberg inequalities for a large class of parameters ( r , p , q , s , μ , σ ) {(r,p,q,s,\mu,\sigma)} and 0 ≤ a ≤ 1 {0\leq a\leq 1} : ( ∫ | u | r d ⁢ x | x | s ) 1 r ≤ C ( ∫ | ∇ u | p d ⁢ x | x | μ ) a p ( ∫ | u | q d ⁢ x | x | σ ) 1 - a q . \bigg{(}\int\lvert u|^{r}\frac{dx}{|x|^{s}}\bigg{)}^{\frac{1}{r}}\leq C\bigg{(% }\int\lvert\nabla u|^{p}\frac{dx}{\lvert x|^{\mu}}\bigg{)}^{\frac{a}{p}}\bigg{% (}\int\lvert u|^{q}\frac{dx}{\lvert x|^{\sigma}}\bigg{)}^{\frac{1-a}{q}}. We compute the best constants and the explicit forms of the extremal functions in numerous cases. When 0 < a < 1 {0<a<1} , we can deduce the existence and symmetry of optimizers for a wide range of parameters. Moreover, in the particular cases r = p ⁢ q - 1 p - 1 {r=p\frac{q-1}{p-1}} and q = p ⁢ r - 1 p - 1 {q=p\frac{r-1}{p-1}} , the forms of maximizers will also be provided in the spirit of Del Pino and Dolbeault [14, 15]. In the case a = 1 {a=1} , that is, the Caffarelli–Kohn–Nirenberg inequality without the interpolation term, we will provide the exact maximizers for all the range of μ ≥ 0 {\mu\geq 0} . The Caffarelli–Kohn–Nirenberg inequalities with arbitrary norms on Euclidean spaces will also be considered in the spirit of Cordero-Erausquin, Nazaret and Villani [13]. Due to the absence of the classical Polyá–Szegö inequality in the weighted case, we establish a symmetrization inequality with power weights which is of independent interest.

Sharp Singular Trudinger-Moser-Adams Type Inequalities with Exact Growth

The main purpose of this paper is two fold. On the one hand, we review some recent progress on best constants for various sharp Moser-Trudinger and Adams inequalities in Euclidean spaces \(\mathbb{R}^{N}\), hyperbolic spaces and other settings, and such sharp inequalities of Lions type. On the other hand, we present and prove some new results on sharp singular Moser-Trudinger and Adams type inequalities with exact growth condition and their affine analogues of such inequalities (Theorems 1.1, 1.2 and 1.3). We also establish a sharpened version of the classical Moser-Trudinger inequality on finite balls (Theorem 1.4).