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Vicentiu D. Radulescu

Publications and source records attributed to Vicentiu D. Radulescu.

14 recordsLinked to original sources
arXiv2025

Another look at quasilinear Schrödinger equations with prescribed mass via dual method

In this paper, we aim to study the existence of ground state normalized solutions for the following quasilinear Schrödinger equation $-Δu-Δ(u^2)u=h(u)+λu,\,\, x\in\R^N$, under the mass constraint $\int_{\R^N}|u|^2\text{d}x=a,$ where $N\geq2$, $a>0$ is a given mass, $λ$ is a Lagrange multiplier and $h$ is a nonlinear reaction term with some suitable conditions. By employing a suitable transformation $u=f(v)$, we reformulate the original problem into the equivalent form $-Δv =h(f(v))f'(v)+λf(v)f'(v),\,\, x\in\R^N,$ with prescribed mass $ \int_{\R^N}|f(v)|^2\text{d}x=a. $ To address the challenge posed by the $L^2$-norm $\|f(v)\|^2_2$ not necessarily equaling $a$, we introduce a novel stretching mapping: $ v_t(x):=f^{-1}(t^{N/2}f(v(tx))). $ This construction, combined with a dual method and detailed analytical techniques, enables us to establish the following existence results: (1)Existence of solutions via constrained minimization using dual methods; (2) Existence of ground state normalized solutions under general $L^2$-supercritical growth conditions, along with nonexistence results, analyzed via dual methods; (3)Existence of normalized solutions under critical growth conditions, treated via dual methods. Additionally, we analyze the asymptotic behavior of the ground state energy obtained in {\bf(P2)}. Our results extend and refine those of Colin-Jeanjean-Squassina [Nonlinearity 20: 1353-1385, 2010], of Jeanjean-Luo-Wang [J. Differ. Equ. 259: 3894-3928, 2015], of Li-Zou [Pacific J. Math. 322: 99-138, 2023], of Zhang-Li-Wang [Topol. Math. Nonl. Anal. 61: 465-489, 2023] and so on. We believe that the methodology developed here can be adapted to study related problems concerning the existence of normalized solutions for quasilinear Schrödinger equations via the dual method.

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arXiv2025

Fractional Morrey-Sobolev type embeddings and nonlocal subelliptic problems with oscillating nonlinearities on stratified Lie groups

In this paper, we establish the fractional Morrey-Sobolev type embeddings on stratified Lie groups. This extends and complements the Sobolev type embeddings derived in \cite{GKR}. As an application of the results, we study the following nonlocal subelliptic problem, \begin{equation} \begin{cases} (-Δ_{\mathbb{G}, p})^s u= λβ(x) g(u) & \text{in} \quad Ω, \\ u=0\quad & \text{in}\quad \mathbb{G}\backslash Ω, \end{cases} \end{equation} where $0 0})$ and $g \in C(\mathbb{R}, \R) $ oscillates near the origin or at infinity. By using the variational principle of Ricceri, we prove the existence and asymptotic behaviors of infinitely many solutions to the problem under consideration. We emphasize that the results obtained here are also novel for $\mathbb{G}$ being the Heisenberg group and $p=2$.

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arXiv2024

Regularity for a class of degenerate fully nonlinear nonlocal elliptic equations

We consider a wide class of fully nonlinear integro-differential equations that degenerate when the gradient of the solution vanishes. By using compactness and perturbation arguments, we give a complete characterization of the regularity of viscosity solutions according to different diffusion orders. More precisely, when the order of the fractional diffusion is sufficiently close to 2, we obtain Hölder continuity for the gradient of any viscosity solutions and further derive an improved gradient regularity estimate at the origin. For the order of the fractional diffusion in the interval $(1, 2)$, we prove that there is at least one solution of class $C^{1, α}_{\rm loc}$. Additionally, if the order of the fractional diffusion is in the interval $(0,1]$, the local Hölder continuity of solutions is inferred.

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arXiv2024

Uniqueness of positive solutions to fractional nonlinear elliptic equations with harmonic potential

In this paper, we establish the uniqueness of positive solutions to the following fractional nonlinear elliptic equation with harmonic potential \begin{align*} (-Δ)^s u+ \left(ω+|x|^2\right) u=|u|^{p-2}u \quad \mbox{in}\,\, \R^n, \end{align*} where $n \geq 1$, $0 -λ_{1,s}$, $2 0$ is the lowest eigenvalue of the operator $(-Δ)^s + |x|^2$. This solves an open question raised in \cite{SS} concerning the uniqueness of solutions to the equation.

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arXiv2024

Blow-up of cylindrically symmetric solutions for Fractional NLS

In this paper, we consider blow-up of solutions to the Cauchy problem for the following fractional NLS, $$ \textnormal{i} \, \partial_t u=(-Δ)^s u-|u|^{2 σ} u \quad \text{in} \,\, \R \times \R^N, $$ where $N \geq 2$, $1/2 <s<1$ and $0<σ<2s/(N-2s)$. In the mass critical and supercritical cases, we establish a criterion for blow-up of solutions to the problem for cylindrically symmetric data. The results extend the known ones with respect to blow-up of solutions to the problem for radially symmetric data in \cite{BHL}.

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arXiv2024

Non-autonomous double phase eigenvalue problems with indefinite weight and lack of compactness

In this paper, we consider eigenvalues to the following double phase problem with unbalanced growth and indefinite weight, $$ -Δ_p^a u-Δ_q u =λm(x) |u|^{q-2}u \quad \mbox{in} \,\, \R^N, $$ where {$N \geq 2$}, {$1<p, q<N$, $p \neq q$}, ${a \in C^{0, 1}(\R^N, [0, +\infty))}$, $a \not\equiv 0$ and $m: \R^N \to \R$ is {an indefinite sign weight which may admit nontrivial positive and negative parts}. Here $Δ_q$ is the $q$-Laplacian operator and $Δ_p^a$ is the weighted $p$-Laplace operator defined by $Δ_p^a u:=\textnormal{div}(a(x) |\nabla u|^{p-2} \nabla u)$. The problem can be degenerate, in the sense that the infimum of $a$ in $\R^N$ may be zero. Our main results distinguish between the cases $p<q$ and $q<p$. In the first case, we establish the existence of a {\it continuous} family of eigenvalues, starting from the principal frequency of a suitable single phase eigenvalue problem. In the latter case, we prove the existence of a {\it discrete} family of positive eigenvalues, which diverges to infinity.

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arXiv2022

A large class of nonlocal elliptic equations with singular nonlinearities

In this work, we address the questions of existence, uniqueness, and boundary behavior of the positive weak-dual solution of equation $\mathbb{L}_γ^s u = \mathcal{F}(u)$, posed in a $C^2$ bounded domain $Ω\subset \mathbb{R}^N$, with appropriate homogeneous boundary or exterior Dirichlet conditions. The operator $\mathbb{L}_γ^s$ belongs to a general class of nonlocal operators including typical fractional Laplacians such as restricted fractional Laplacian, censored fractional Laplacian and spectral fractional Laplacian. The nonlinear term $\mathcal{F}(u)$ covers three different amalgamation of nonlinearities: a purely singular nonlinearity $\mathcal{F}(u) = u^{-q}$ ($q>0$), a singular nonlinearity with a source term $\mathcal{F}(u) = u^{-q} + f(u)$, and a singular nonlinearity with an absorption term $\mathcal{F}(u) = u^{-q}-g(u)$. Based on a delicate analysis of the Green kernel associated to $\mathbb{L}_γ^s$, we develop a new unifying approach that empowered us to construct a theory for equation $\mathbb{L}_γ^s u = \mathcal{F}(u)$. In particular, we show the existence of two critical exponents $q^{\ast}_{s, γ}$ and $q^{\ast \ast}_{s, γ}$ which provides a fairly complete classification of the weak-dual solutions via their boundary behavior. Various types of nonlocal operators are discussed to exemplify the wide applicability of our theory.

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arXiv2022

Equivalence of weak and viscosity solutions for the nonhomogeneous double phase equation

We establish the equivalence between weak and viscosity solutions to the nonhomogeneous double phase equation with lower-order term $$ -{\rm div}(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du)=f(x,u,Du),\quad 1<p\le q<\infty, a(x)\ge0. $$ We find some appropriate hypotheses on the coefficient $a(x)$, the exponents $p, q$ and the nonlinear term $f$ to show that the viscosity solutions with {\em a priori} Lipschitz continuity are weak solutions of such equation by virtue of the $\inf$($\sup$)-convolution techniques. The reverse implication can be concluded through comparison principles. Moreover, we verify that the bounded viscosity solutions are exactly Lipschitz continuous, which is also of independent interest.

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arXiv2022

Nonhomogeneous quasilinear elliptic problems: linear and sublinear cases

We are concerned with a class of second order quasilinear elliptic equations driven by a nonhomogeneous differential operator introduced by C.A. Stuart and whose study is motivated by models in Nonlinear Optics. We establish sufficient conditions for the existence of at least one or two non-negative solutions. Our analysis considers the cases when the reaction has either a sublinear or a linear growth. In the sublinear case, we also prove a nonexistence property. The proofs combine energy estimates and variational methods. In particular, the monotonicity trick is applied in order to overcome the lack of a priori bounds on the Palais-Smale sequences.

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arXiv2021

Combined effects in mixed local-nonlocal stationary problems

In this work, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects, and in either the presence or the absence of a singular nonlinearity. We investigate existence or non-existence properties, power and exponential type Sobolev regularity results, and the boundary behavior of the weak solution, in the light of the interplay between the summability of the datum and the power exponent in singular nonlinearities.

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arXiv2021

Regularity of solutions to degenerate fully nonlinear elliptic equations with variable exponent

We consider the fully nonlinear equation with variable-exponent double phase type degeneracies $$ \big[|Du|^{p(x)}+a(x)|Du|^{q(x)}\big]F(D^2u)=f(x). $$ Under some appropriate assumptions, by making use of geometric tangential methods and combing a refined improvement-of-flatness approach with compactness and scaling techniques we obtain the sharp local $C^{1,α}$ regularity of viscosity solutions to such equations.

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arXiv2020

Positive supersolutions for the Lane-Emden system with inverse-square potentials

In this paper, we study the nonexistence of positive supersolutions for the following Lane-Emden system with inverse-square potentials \begin{equation}\label{0} \left\{ \begin{array}{lll} -Δu+\frac{μ_1}{|x|^2} u= v^p \quad {\rm in}\ \, Ω\setminus\{0\},\\[2mm] -Δv+\frac{μ_2}{|x|^2} v= u^q \quad {\rm in}\ \, Ω\setminus\{0\} \end{array} \right. \end{equation} for suitable $p,q>0$, $μ_1,μ_2\geq -(N-2)^2/4$, where $Ω$ is a smooth bounded domain containing the origin in $\mathbb{R}^N$ with $N\geq 3$. Precisely, we provide sharp supercritical regions of $(p,q)$ for the nonexistence of positive supersolutions to system (\ref{0}) in the cases $-(N-2)^2/4\leq μ_1,μ_2<0$ and $-(N-2)^2/4\leq μ_1<0\leq μ_2$. Due to the negative coefficients $μ_1,μ_2$ of the inverse-square potentials, an initial blowing-up at the origin could be derived and an iteration procedure could be applied in the supercritical case to improve the blowing-up rate until the nonlinearities are not admissible in some weighted $L^1$ spaces. In the subcritical case, we prove the existence of positive supersolutions for system (\ref{s 1.1}) by specific radially symmetric functions.

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arXiv2019

Coron problem for nonlocal equations invloving Choquard nonlinearity

We study the problem \[ -\De u = \left(\int_{\Om}\frac{|u(y)|^{2^*_μ}}{|x-y|^μ}dy\right)|u|^{2^*_μ-2}u, \; \text{in}\; \Om,\quad u = 0 \; \text{ on } \pa \Om , \] where $\Om$ is a smooth bounded domain in $\mathbb{R}^N( N\geq 3)$, $2^*_μ=\frac{2N-μ}{N-2}$. we prove the existence of a positive solution of the above problem in an annular type domain when the inner hole is sufficiently small.

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