Ferrari, Fausto Vitolo, Antonio
Published in
Advanced Nonlinear Studies

We consider the elliptic differential operator defined as the sum of the minimum and the maximum eigenvalue of the Hessian matrix, which can be viewed as a degenerate elliptic Isaacs operator, in dimension larger than two. Despite of nonlinearity, degeneracy, non-concavity and non-convexity, such an operator generally enjoys the qualitative propert...

Gheraibia, Billel Wang, Chunhua
Published in
Advanced Nonlinear Studies

In this paper, we study the following nonlinear Schrödinger–Newton type system: { - ϵ 2 Δ u + u - Φ ( x ) u = Q ( x ) | u | u , x ∈ ℝ 3 , - ϵ 2 Δ Φ = u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} &\displaystyle{-}\epsilon^{2}\Delta u+u-\Phi(x)u=Q(x)|u% |u,&&\displaystyle x\in\mathbb{R}^{3},\\ &\displaystyle{-}\epsilon^{2}\Delta\Phi=u^{2}...

Guo, Yuxia Liu, Ting Nie, Jianjun
Published in
Advanced Nonlinear Studies

We consider the following fractional Schrödinger equation involving critical exponent: { ( - Δ ) s u + V ( y ) u = u 2 s * - 1 in ℝ N , u > 0 , y ∈ ℝ N , \left\{\begin{aligned} &\displaystyle(-\Delta)^{s}u+V(y)u=u^{2^{*}_{s}-1}&&% \displaystyle\text{in }\mathbb{R}^{N},\\ &\displaystyle u>0,&&\displaystyle y\in\mathbb{R}^{N},\end{aligned}\ri...

Cazenave, Thierry Martel, Yvan Zhao, Lifeng
Published in
Advanced Nonlinear Studies

We prove that any sufficiently differentiable space-like hypersurface of ℝ 1 + N {{\mathbb{R}}^{1+N}} coincides locally around any of its points with the blow-up surface of a finite-energy solution of the focusing nonlinear wave equation ∂ t t u - Δ u = | u | p - 1 u {\partial_{tt}u-\Delta u=|u|^{p-1}u} on ℝ × ℝ N {{\mathbb{R}}\times{\mathb...

Flynn, Joshua
Published in
Advanced Nonlinear Studies

The main purpose of this paper is to establish several general Caffarelli–Kohn–Nirenberg (CKN) inequalities on Carnot groups G (also known as stratified groups). These CKN inequalities are sharp for certain parameter values. In case G is an Iwasawa group, it is shown here that the L 2 {L^{2}} -CKN inequalities are sharp for all parameter values exc...

Goel, Divya Rădulescu, Vicenţiu D. Sreenadh, K.
Published in
Advanced Nonlinear Studies

We consider the following Choquard equation: { - Δ u = ( ∫ Ω | u ( y ) | 2 μ * | x - y | μ 𝑑 y ) | u | 2 μ * - 2 u in Ω , u = 0 on ∂ Ω , \left\{\begin{aligned} \displaystyle-\Delta u&\displaystyle=\Bigg{(}\int_{% \Omega}\frac{|u(y)|^{2^{*}_{\mu}}}{|x-y|^{\mu}}\,dy\Bigg{)}|u|^{2^{*}_{\mu}-2}% u&&\displaystyle\phantom{}\text{in }\Omeg...

Gui, Changfeng Guo, Hui
Published in
Advanced Nonlinear Studies

This paper deals with the general Choquard equation - Δ u + V ( | x | ) u = ( I α * | u | p ) | u | p - 2 u in ℝ N , -\Delta u+V(|x|)u=(I_{\alpha}*|u|^{p})|u|^{p-2}u\quad\text{in }\mathbb{R}^{N}, where V ∈ C ( [ 0 , ∞ ) , ℝ + ) {V\in C([0,\infty),\mathbb{R}^{+})} is bounded below by a positive constant, and I α {I_{\alpha}} denotes th...

Mugnai, Dimitri Proietti Lippi, Edoardo
Published in
Advanced Nonlinear Studies

In this paper, we consider fractional Choquard equations with confining potentials. First, we show that they admit a positive ground state and infinitely many bound states. Then we prove the existence of two signed solutions when a superlinear and subcritical perturbation is added; in this case, the main feature is that such a perturbation does not...

Aguirre, Natham
Published in
Advanced Nonlinear Studies

We study a concept of renormalized solution to the problem { - Δ p u = 0 in ℝ + N , | ∇ u | p - 2 u ν + g ( u ) = μ on ∂ ℝ + N , \begin{cases}-\Delta_{p}u=0&\mbox{in }{\mathbb{R}}^{N}_{+},\\ \lvert\nabla u\rvert^{p-2}u_{\nu}+g(u)=\mu&\mbox{on }\partial{\mathbb{R}}^{N}_% {+},\end{cases} where 1 0 } {{\mathbb{R}}^{N}_{+}=\{(x^{\prime},...

Luo, Haijun Zhang, Zhitao
Published in
Advanced Nonlinear Studies

We study a Schrödinger system of four equations with linear coupling functions and nonlinear couplings, including the case that the corresponding elliptic operators are indefinite. For any given nonlinear coupling β > 0 {\beta>0} , we first use minimizing sequences on a normalized set to obtain a minimizer, which implies the existence of positive s...