The classical Trudinger–Moser inequality says that for functions with Dirichlet norm smaller or equal to 1 in the Sobolev space H 0 1 ( Ω ) (with Ω ⊂ R 2 a bounded domain), the integral ∫ Ω e 4 π u 2 dx is uniformly bounded by a constant depending only on Ω . If the volume | Ω | becomes unbounded then this bound tends to infinity, and hence the Trudinger–Moser inequality is not available for such domains (and in particular for R 2 ). In this paper, we show that if the Dirichlet norm is replaced by the standard Sobolev norm, then the supremum of ∫ Ω e 4 π u 2 dx over all such functions is uniformly bounded, independently of the domain Ω . Furthermore, a sharp upper bound for the limits of Sobolev normalized concentrating sequences is proved for Ω = B R , the ball or radius R, and for Ω = R 2 . Finally, the explicit construction of optimal concentrating sequences allows to prove that the above supremum is attained on balls B R ⊂ R 2 and on R 2 .

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A Sharp Trudinger-Moser Type Inequality for Unbounded Domains in Rn

We study a mountain pass characterization of least energy solutions of the following nonlinear scalar field equation in R N : -Δu=g(u), u ∈ H 1 (R N ), where N > 2. Without the assumption of the monotonicity of t → g(t)/t, we show that the mountain pass value gives the least energy level.

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A remark on least energy solutions in RN

In this paper we study the existence and concentration behaviors of positive solutions to the Kirchhoff type equations 

$$- \varepsilon^2 M \left(\varepsilon^{2-N}\!\!\int_{\mathbf{R}^N}|\nabla u|^2\,\mathrm{d} x \right) \Delta u \!+\! V(x) u \!=\! f(u) \quad{\rm in}\ \mathbf{R}^N, \quad u \!\in\! H^1(\mathbf{R}^N), \ N \!\geqq\!1,$$

where M and V are continuous functions. Under suitable conditions on M and general conditions on f, we construct a family of positive solutions ${(u_\varepsilon)_{\varepsilon \in (0,\tilde{\varepsilon}]}}$ which concentrates at a local minimum of V after extracting a subsequence (ek).

Existence and Concentration Result for the Kirchhoff Type Equations with General Nonlinearities

We study the behavior of solitary-wave solutions of some generalized nonlinear Schrodinger equations with an external potential. The equations have the feature that in the absence of the external potential, they have solutions describing inertial motions of stable solitary waves. We consider solutions of the equations with a non-vanishing external potential corresponding to initial conditions close to one of these solitary wave solutions and show that, over a large interval of time, they describe a solitary wave whose center of mass motion is a solution of Newton's equations of motion for a point particle in the given external potential, up to small corrections corresponding to radiation damping.

Solitary Wave Dynamics in an External Potential

We establish an interpolation of Hardy inequality and Trudinger–Moser inequality in R N (N ≥ 2). Denote u u
1,τ ≤1 1,τ RN = R N (|∇u| 1 |x|β eα|u| N+ τ |u| N )dx N/(N−1)N−2−m=0 1/N for any τ > 0. There holds

α m|u|mN/(N−1) m!dx < ∞ 1/(N−1)βα, ω N−1 α N + N ≤ 1, where 0 ≤ β < N, α N = Nω N−1 N−1 . The above interpolation can be used to establish if and only if is the volume of the
unit sphere sufficient conditions under which the quasilinear nonhomogeneous partial differential equation 

−Nu+V(x)|u| N−2 u =f(x,u)|x|β + h(x) in R N

has weak solutions, where −uous potential, f behaves like
N−2 ∇u) is the N-Laplacian, V isN u = −div(|∇u|N/(N−1)
when |u| → ∞, h ∈ (W1,N (R N ))∗ , 0 < β eα|u|
> 0.

An Interpolation of Hardy Inequality and Trudinger–Moser Inequality in ℝN and Its Applications

We study Trudinger type inequalities in RN and their best exponents αN . We show for α ∈ (0, αN ), αN = Nω N−1 (ωN−1 is the surface area of the unit sphere in RN ), there exists a constant Cα > 0 such that ∫ RN ΦN α( |u(x)| ‖∇u‖LN (RN ) ) N N−1  dx ≤ Cα ‖u‖NLN (RN ) ‖∇u‖N LN (RN ) (∗) for all u ∈W 1,N (RN ) \ {0}. Here ΦN (ξ) is defined by ΦN (ξ) = exp(ξ) − N−2 ∑ j=0 1 j! ξ . It is also shown that (∗) with α ≥ αN is false, which is different from the usual Trudinger’s inequalities in bounded domains. 0. Introduction In this note, we study the limit case of Sobolev’s inequalities; suppose N ≥ 2 and let D ⊂ R be an open set. We denote by W 1,N 0 (D) the usual Sobolev space, that is, the completion of C∞ 0 (D) with the norm ‖u‖W 1,p 0 (D) = ‖∇u‖p+‖u‖p. Here ‖u‖p = (∫

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Trudinger type inequalities in ^{} and their best exponents

Trudinger type inequalities in R^N and their best exponents

We consider the existence of positive solutions of the following semilinear elliptic problem in ${\mathbb R}^N$:

Four positive solutions for the semilinear elliptic equation: $-\Delta u+u=a(x)u^p+f(x)$ in ${\mathbb R}^N$

We are concerned with the uniqueness result of positive solutions for a class of quasilinear elliptic equation arising from plasma physics. We convert a quasilinear elliptic equation into a semilinear one and show the unique existence of positive radial solution for original equation under the suitable conditions on the power of nonlinearity and quasilinearity. We also investigate the non-degeneracy of a positive radial solution for a converted semilinear elliptic equation.

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Uniqueness of the ground state solutions of quasilinear Schrödinger equations

We are concerned with a quasilinear elliptic equation of the form \begin{equation*} -\Delta u+a(x) u-\Delta(|u|^\alpha)|u|^{\alpha-2}u=h(u)\quad \hbox{in }\mathbb R^N, \end{equation*} where $\alpha> 1$ and $N\geq 1$. By using variational approaches, we prove the existence of at least one positive solution of the above equation under suitable conditions on $a(x)$ and $h$. In particular, we are interested in the situation that $a(x)$ is invariant under the finite group action $G$.

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Shinji Adachi

Papers

Trudinger type inequalities in ^{} and their best exponents

Trudinger type inequalities in R^N and their best exponents

Four positive solutions for the semilinear elliptic equation: $-\Delta u+u=a(x)u^p+f(x)$ in ${\mathbb R}^N$

Uniqueness of the ground state solutions of quasilinear Schrödinger equations

$G$-invariant positive solutions for a quasilinear Schrödinger equation