Vicentiu D. Radulescu

• PhD, 1995 (Université Pierre et Marie Curie - Paris VI); Habilitation, 2003 (Univ. Pierre et Marie Curie - Paris VI)
• Professor AGH at AGH University of Krakow

We investigate normalized solutions of the following Choquard equation perturbed by saturable nonlinearity where \(2_{\alpha }:=\frac{N+\alpha }{N}\le p\le 2_{\alpha }^{*}:=\frac{N+\alpha }{N-2}\), \(\mu \in \mathbb {R}\backslash \{0\},\) and g(x) is a bounded intensity function on \(\mathbb {R}^{N}\). Under different assumptions on \(p,\mu \) and...

In this work, we introduce two novel classes of quasilinear elliptic equations, each driven by the double phase operator with variable exponents. The first class features a new double phase equation where exponents depend on the gradient of the solution. We delve into proving various properties of the corresponding Musielak-Orlicz Sobolev spaces, i...

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...

In this paper we consider a non-linear Robin problem driven by the Orlicz g-Laplacian operator. Using variational technique combined with a suitable truncation and Morse theory (critical groups), we prove two multiplicity theorems with sign information for all the solutions. In the first theorem, we establish the existence of at least two non-trivi...

In this paper, we establish the Harnack inequality of nonnegative weak solutions to the doubly nonlinear mixed local and nonlocal parabolic equations. This result is obtained by combining a related comparison principle, a local boundedness estimate, and an integral Harnack-type inequality. Our proof is based on the expansion of positivity together...

This paper is concerned with the existence and multiplicity of solutions for a class of problems involving the Φ‐Laplacian operator with general assumptions on the nonlinearities, which include both semipositone cases and critical concave convex problems. The research is based on the subsupersolution technique combined with a truncation argument an...

In this paper, we study an inverse problem of estimating three discontinuous parameters in a double phase implicit obstacle problem with multivalued terms and mixed boundary conditions which is formulated by a regularized optimal control problem. Under very general assumptions, we introduce a multivalued function called a parameter-to-solution map...

This paper focuses on the study of multiplicity and concentration phenomena of positive solutions for the singularly perturbed double phase problem with nonlocal Choquard reaction{−ϵpΔpu−ϵqΔqu+V(x)(|u|p−2u+|u|q−2u)=ϵμ−N(1|x|μ⁎G(u))g(u),inRN,u∈W1,p(RN)∩W1,q(RN),u>0,inRN, where 1<p<q<N, 0<μ<N, ϵ is a small positive parameter and V is the absorption p...

In this paper, we consider the Schrödinger equation involving the fractional (p,p1,⋯,pm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p,p_1,\dots ,p_m)$$\end{documen...

This paper is devoted to studying a complicated implicit obstacle problem involving a nonhomogenous differential operator, called double phase operator, a nonlinear convection term (i.e. a reaction term depending on the gradient), and a multivalued term which is described by Clarke’s generalized gradient. We develop a general framework to deliver a...

We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplacian and of a q-Laplacian, \(1<p<q\), (a (p, q)-equation). The reaction is parametric (eigenvalue problem) and exhibits the competing effects of a strongly singular term and of \((p-1)\)-superlinear Carathéodory perturbation. We show that when the parameter (eigenvalue) is smal...

In this work, we address the questions of existence, uniqueness, and boundary behavior of the positive weak-dual solution of equation $\mathbb{L}_\gamma^s u = \mathcal{F}(u)$, posed in a $C^2$ bounded domain $\Omega \subset \mathbb{R}^N$, with appropriate homogeneous boundary or exterior Dirichlet conditions. The operator $\mathbb{L}_\gamma^s$ belo...

In this paper, we study three-dimensional Kirchhoff equations with critical growth and singular nonlinearity. We are concerned with the qualitative analysis of solutions to the following nonlocal problem -a+b∫Ω|∇u|2dxΔu=λu-γ+u5,inΩ,u>0,inΩ,u=0,on∂Ω,where Ω⊂R3 is a bounded domain with smooth boundary, 0<γ<1, and a,b,λ are positive constants. By comb...

In this paper we study an anisotropic implicit obstacle problem driven by the (p(⋅),q(⋅))-Laplacian and an isotropic implicit obstacle problem involving a nonlinear convection term (a reaction term depending on the gradient) which contain several interesting and challenging untreated problems. These two implicit obstacle problems have both highly n...

In this paper, we investigate the following fractional Sobolev critical Nonlinear Schrödinger coupled systems: $$\begin{aligned} \left\{ \begin{array}{lll} (-\Delta )^{s} u=\mu _{1} u+|u|^{2^{*}_{s}-2}u+\eta _{1}|u|^{p-2}u+\gamma \alpha |u|^{\alpha -2}u|v|^{\beta } ~ \text {in}~ {\mathbb {R}}^{N},\\ (-\Delta )^{s} v=\mu _{2} v+|v|^{2^{*}_{s}-2}v+\e...

We establish the equivalence between weak and viscosity solutions to the nonhomogeneous double phase equation with lower-order term − div(|Du| p−2 Du + a(x)|Du| q−2 Du) = f (x, u, Du), 1 < p ≤ q < ∞, a(x) ≥ 0. We find some appropriate hypotheses on the coefficient a(x), the exponents p, q and the nonlinear term f to show that the viscosity solution...

The classical Ambrosetti–Prodi problem considers perturbations of the linear Dirichlet Laplace operator by a nonlinear reaction whose derivative jumps over the principal eigenvalue of the operator. In this paper, we develop a related analysis for parametric problems driven by the nonlinear Robin (p,q)-Laplace operator (sum of a p-Laplacian and a q-...

In this paper, we study the critical fractional Choquard equation with a local perturbation \((-\Delta )^su=\lambda u+\mu |u|^{q-2}u+(I_{\alpha }*|u|^{2^*_{\alpha ,s}})|u|^{2^*_{\alpha ,s}-2}u,~~x\in {\mathbb {R}}^N,\) having prescribed mass \(\int _{{\mathbb {R}}^N}u^2\text {d}x=a^2,\) where \(I_{\alpha }(x)\) is the Riesz potential, \(s\in (0,1),...

For studying the evolution of the transverse deflection of an extensible beam derived from the connection mechanics, we investigate the initial boundary value problem of nonlinear extensible beam equation with linear strong damping term, nonlinear weak damping term, and nonlinear source term. The key idea of our analysis is to describe the invarian...

In this paper, we consider the following fractional Kirchhoff equation with discontinuous nonlinearity $$\begin{aligned} \left\{ \begin{array}{ll} \left( \varepsilon ^{2\alpha }a+\varepsilon ^{4\alpha -3}b\int _{{\mathbb {R}}^3}|(-\Delta )^{\frac{\alpha }{2}} u|^2{{\mathrm{d}}}x\right) (-\Delta )^\alpha {u}+V(x)u = H(u-\beta )f(u) &{} \quad \text{...

We are concerned with the following Kirchhoff equation: -a+b∫R2|∇u|2dxΔu+V(x)u=f(u),inR2,u∈H1(R2),where a,b are positive constants, V∈C(R2,(0,∞)) is a trapping potential, and f has critical exponential growth of Trudinger–Moser type. By developing some new analytical approaches and techniques, we prove the existence of nontrivial solutions and leas...

We are concerned with a class of second order quasilinear elliptic equations driven by a nonhomogeneous differential operator introduced by C.A. Stuart [22] 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...

In this paper, we focus on the existence of positive solutions to the following planar Schrödinger-Newton system with general subcritical growth{−Δu+u+ϕu=f(u)inR2,Δϕ=u2inR2, where f is a smooth reaction. We introduce a new variational approach, which enables us to study the above problem in the Sobolev space H1(R2). The analysis developed in this p...

We study the following class of stationary Schrödinger equations of Choquard type−Δu+V(x)u=[|x|−μ⁎(Q(x)F(u))]Q(x)f(u),x∈R2, where the potential V and the weight Q decay to zero at infinity like (1+|x|γ)−1 and (1+|x|β)−1 for some (γ,β) in variously different ranges, ⁎ denotes the convolution operator with μ∈(0,2), and F is the primitive of f that fu...

In this paper, we establish concentration and multiplicity properties of ground state solutions to the following perturbed double phase problem with competing potentials: $$\begin{aligned} \left\{ \begin{array}{ll} -\epsilon ^{p}\Delta _{p} u-\epsilon ^{q}\Delta _{q} u +V(x)(|u|^{p-2}u+|u|^{q-2}u)=K(x)f(u),&{} \quad \hbox {in}~\mathbb {R}^{N},\\ u\...

We consider the Navier problem −Δk,p2u(x)=f(x,u(x),∇u(x),Δu(x))inΩ,u∂Ω=Δu∂Ω=0,$$ -{\Delta}_{k,p}^2u(x)=f\left(x,u(x),\nabla u(x),\Delta u(x)\right)\kern0.30em \mathrm{in}\kern0.5em \Omega, \kern0.30em u{\left|{}_{\mathrm{\partial \Omega }}=\Delta u\right|}_{\mathrm{\partial \Omega }}=0, $$ driven by the sign‐changing (degenerate) Kirchhoff type p(x...

In this paper, we investigate the following fractional Sobolev critical nonlinear Schr\"{o}dinger (NLS) coupled systems: \begin{equation*} \left\{\begin{array}{lll} (-\Delta)^{s} u=\mu_{1} u+|u|^{2^{*}_{s}-2}u+\eta_{1}|u|^{p-2}u+\gamma\alpha|u|^{\alpha-2}u|v|^{\beta} ~ \text{in}~ \mathbb{R}^{N},\\ (-\Delta)^{s} v=\mu_{2} v+|v|^{2^{*}_{s}-2}v+\eta_{...

In this paper, we study the following weighted nonlocal system with critical exponents related to the Stein–Weiss inequality -Δu=1|x|α∫RNvp(y)|x-y|μ|y|αdyuq,-Δv=1|x|α∫RNuq(y)|x-y|μ|y|αdyvp,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \use...

We consider a nonlinear parametric Neumann problem driven by the anisotropic $(p,q)$-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive solutions. Using a combination of topological and variational tools together with suitable truncation and compa...

In the present paper, we investigate the existence and multiplicity properties of the normalized solutions to the following Kirchhoff-type equation with Sobolev critical growth ( P ) { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + λ u = μ | u | p − 2 u + | u | 4 u , a m p ; in R 3 , u > 0 , ∫ R 3 | u | 2 d x = c 2 , a m p ; in R 3 , \begin{equation*} \begi...

In this paper, we investigate a nonlinear and nonsmooth dynamics system (NNDS, for short) involving two multi-valued maps which are a convex subdifferential operator and a generalized subdifferential operator in the sense of Clarke, respectively. Under general assumptions, by using a surjectivity theorem for multi-valued mappings combined with the...

This paper is concerned with the existence and multiplicity of solutions for singular Kirchhoff-type problems involving the fractional p-Laplacian operator. More precisely, we study the following nonlocal problem $$\left\{ {\matrix{{M\left( {\int\!\!\!\int_{{^{2N}}} {{{{{\left| x \right|}^{{\alpha _1}p}}{{\left| y \right|}^{{\alpha _2}p}}{{\left| {...

This paper is dedicated to show the existence of ground state solution for a magnetic Choquard equation with critical exponential growth. By introducing a Moser type function involving magnetic potential and applying analytical techniques, we surmount the obstacles brought from the magnetic potential which makes it a complex-valued problem and the...

In this paper, we are concerned with the following magnetic Schrödinger–Poisson system \begin{align*} \begin{cases} -(\nabla+i A(x))^{2}u+(\lambda V(x)+1)u+\phi u=\alpha f(\left | u\right |^{2})u+\vert u\vert^{4}u,& \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi = u^{2}, & \text{ in } \mathbb{R}^{3}, \end{cases} \end{align*} where $\lambda>0$ is a para...

The aim of this article is to derive some Lewy-Stampacchia estimates and the existence of solutions for equations driven by a nonlocal integro-differential operator on the Heisenberg group.

We consider a double phase Dirichlet equation with a reaction which is asymptotically as x→±∞, resonant with respect to the first eigenvalue of a related eigenvalue problem. Using variational tools together with Morse theoretic arguments, we prove the existence of at least two bounded nontrivial solutions for the problem.

We consider a perturbed version of the Robin eigenvalue problem for the p-Laplacian. The perturbation is (p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p - 1)$$\...

We study double phase singular problems with strong singularity and unbounded coefficient (that is, in the singular term u↦g(z)u(z)η, where η⩾ 1 and g(·) is not bounded). First we deal with the purely singular problem. We consider two distinct cases. In the first one, we assume that η= 1 and the double phase operator ((p, q)-Laplacian with weight)...

The aim of this paper is to study the following time-space fractional diffusion problem $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t^\beta u+(-\Delta )^\alpha u+(-\Delta )^\alpha \partial _t^\beta u=\lambda f(x,u) +g(x,t) &{}\text{ in } \Omega \times {\mathbb {R}}^{+},\\ u(x,t)=0\ \ &{}\text{ in } ({\mathbb {R}}^N{\setminu...

We consider a nonlinear elliptic Dirichlet equation driven by a double phase operator and a Carathéodory (p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p-1)$$\en...

In this paper, we first establish the uniqueness and non-degeneracy of positive solutions to the fractional Kirchhoff problem \begin{equation*} \Big(a+b{\int_{\mathbb{R}^{N}}}|(-\Delta)^{\frac{s}{2}}u|^2dx\Big)(-\Delta)^su+mu=|u|^{p-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b,m>0$, $0<\frac{N}{4}<s<1$, $2<p<2^*_s=\frac{2N}{N-2s}...

In this paper, we consider the following singularly perturbed fractional Kirchhoff problem \begin{equation*} \Big(\varepsilon^{2s}a+\varepsilon^{4s-N} b{\int_{\mathbb{R}^{N}}}|(-\Delta)^{\frac{s}{2}}u|^2dx\Big)(-\Delta)^su+V(x)u=|u|^{p-2}u,\quad \text{in}\ \mathbb{R}^{N}, \end{equation*} where $a,b>0$, $2s<N<4s$ with $s\in(0,1)$, $2<p<2^*_s=\frac{2...

This paper deals with the qualitative analysis of solutions to the following $(p,q)$-fractional equation: \begin{equation*} (-\Delta)^{s_1}_{p}u+(-\Delta)^{s_2}_{q}u+V(x) \big(|u|^{p-2}u+|u|^{q-2}u\big) = K(x)\frac{f(u)}{|x|^\ba} \quad \text{in } \mathbb R^N, \end{equation*} where $1< q< p$, $0< s_2\leq s_1< 1$, $ps_1=N$, $\ba\in[0,N)$, and $V,K\co...

The variational methods are adopted for establishing the existence of at least two nontrivial solutions for a Dirichlet problem driven by a non-homogeneous differential operator of p -Laplacian type. A large class of nonlinear terms is considered, covering the concave-convex case. In particular, two positive solutions to the problem are obtained un...

In this paper, we consider the following non-autonomous Schrödinger–Bopp–Podolsky system $$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u + V(x) u + q^2\phi u = f(u)\\ -\Delta \phi + a^2 \Delta ^2 \phi = 4\pi u^2 \end{array}\right. } \hbox { in }{\mathbb {R}}^3. \end{aligned}$$ - Δ u + V ( x ) u + q 2 ϕ u = f ( u ) - Δ ϕ + a 2 Δ 2 ϕ = 4 π u 2...

In this paper, using variational methods, we study multiplicity of multi-bump solutions for the following nonlinear magnetic Choquard equation{−(∇+iA(x))2u+(λV(x)+1)u=(1|x|μ⁎|u|p)|u|p−2ux∈RN,u∈H1(RN,C), where N≥2, λ>0 is a real parameter, 0<μ<2, i is the imaginary unit, p∈(2,2⁎(2(N−μ)2N)), where 2⁎=2NN−2 if N≥3, 2⁎=+∞, if N=2. The magnetic potentia...

We study a double phase Neumann problem with a superlinear reaction which need not satisfy the Ambrosetti-Rabinowitz condition. Using the Nehari manifold method, we show that the problem has at least three nontrivial bounded ground state solutions, all with sign information (positive, negative and nodal).

We consider the existence and concentration properties of standing waves for a fourth-order Schrödinger equation with mixed dispersion, which was introduced to regularize and stabilize solutions to the classical time-dependent Schrödinger equation. This leads to study multi-peak solutions to the following singularly perturbed fourth-order nonlinear...

In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomono...

In this paper, we investigate the following nonlinear magnetic Schrödinger equation with exponential growth: \begin{document}$ (-i\nabla+A(x))^2 u+V(x)u = f(x, |u|^2)u\ \mbox{in}\; \mathbb{R}^2 , $\end{document} where \begin{document}$ V $\end{document} is the electric potential and \begin{document}$ A $\end{document} is the magnetic potential. We...

The prime goal of this paper is to introduce and study a highly nonlinear inverse problem of identification discontinuous parameters (in the domain) and boundary data in a nonlinear variable exponent elliptic obstacle problem involving a nonhomogeneous, nonlinear partial differential operator, which is formulated the sum of a weighted anisotropic \...

We consider a Dirichlet elliptic equation driven by a weighted \begin{document}$ (p,q) $\end{document}-Laplace differential operator. The weights are in general different. When the reaction is "superlinear", using the fountain theorem, we show the existence of a sequence of distinct smooth solutions with energies diverging to \begin{document}$ +\in...

This paper is concerned with the qualitative analysis of solutions to the following class of quasilinear problems $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta _{\Phi }u=f(x,u) &{}\quad \text {in } \Omega ,\\ u=0 &{}\quad \text {on }\partial \Omega , \end{array} \right. \end{aligned}$$ (P) where \(\Delta _{\Phi }u=\mathrm{div}\,(\varphi (x,|\...

In this paper, we study the following nonlinear magnetic Kirchhoff equation with critical growth \begin{document}$ \begin{align*} \left\{ \begin{aligned} &-\Big(a\epsilon^{2}+b\epsilon\, [u]_{A/\epsilon}^{2}\Big)\Delta_{A/\epsilon} u+V(x)u = f(|u|^{2})u+\vert u\vert^{4}u \quad \hbox{in }\mathbb{R}^3, \\ &u\in H^{1}(\mathbb{R}^{3}, \mathbb{C}), \end...

We consider an anisotropic double phase problem with a reaction in which we have the competing effects of a parametric singular term and a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies on \begin{document}$ \mathring{\mathbb{R}}_+ = (0, +\infty) $\end{docu...

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 th...

This paper is concerned with concentration and multiplicity properties of solutions to the following fractional problem with unbalanced growth and critical or supercritical reaction:{(−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=h(u)+|u|r−2u in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0, in RN,} where ε is a positive parameter, 0<s<1, 2⩽p<q<N/s, (−Δ)ts (t∈{p,q}) is the f...

In this paper, we are concerned with a Kirchhoff problem in the presence of a strongly-singular term perturbed by a discontinuous nonlinearity of the Heaviside type in the setting of Orlicz-Sobolev space. The presence of both strongly-singular and non-continuous terms brings up difficulties in associating a differentiable functional to the problem...

Weighted inequality theory for fractional integrals is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Basic weighted inequalities are often associated to Hardy, Littlewood and Sobolev [6, 11], Caffarelli, Kohn and Nirenberg [4], respectively to Stein and Weiss [12]. A key att...

We consider a thermoelastic theory in which the equations that govern the evolution are linear with respect to the thermal displacement and nonlinear with regards to gradients of displacements and temperature. Our results refer to the non-existence of solutions for some mixed problems, considered in this context. We also address the instability of...

In this paper, we consider positive supersolutions of the semilinear fourth-order problem{(−Δ)2u=ρ(x)f(u)inΩ,−Δu>0inΩ, where Ω is a domain in RN (bounded or not), f:Df=[0,af)→[0,∞) (0<af⩽+∞) is a non-decreasing continuous function with f(u)>0 for u>0 and ρ:Ω→R is a positive function. Using a maximum principle-based argument, we give explicit estima...

In this paper we introduce a new double phase Baouendi-Grushin type operator with variable coefficients. We give basic properties of the corresponding functions space and prove a compactness result. In the second part, using topological argument, we prove the existence of weak solutions of some nonvariational problems in which this new operator is...

In this paper we introduce a new double phase Baouendi-Grushin type operator with variable coefficients. We give basic properties of the corresponding functions space and prove a compactness result. In the second part, using topological argument, we prove the existence of weak solutions of some nonvariational problems in which this new operator is...

We study the following class of pseudo-relativistic Hartree equations−ε2Δ+m2u+V(x)u=εμ−N(|x|−μ⁎F(u))f(u)inRN, where the nonlinearity satisfies general hypotheses of Berestycki-Lions type. By using the method of penalization arguments, we prove the existence of a family of localized positive solutions that concentrate at the local minimum points of...

In this paper, we investigate the following fractional p-Kirchhoff type problem $$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^{p(\theta -1)}\right) (-\Delta )^s_pu = \Big ({\mathcal {I}}_\mu *|u|^q\Big )|u|^{q-2}u+\frac{|u|^{p_{\alpha }^*-2}u}{|x|^\alpha },\ u>0, &{}\text{ in }\ \Omega ,\\ u=0, \ &{} \mathrm{in}\ {\mathbb {R}}^N\b...

In this paper we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution s...

In this paper, we establish the existence of normalized solutions to the following Kirchhoff-type equation $$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{{\mathbb {R}}^3}|\nabla u|^2{\mathrm {d}}x\right) \Delta u-\lambda u=K(x)f(u), &{} x\in {\mathbb {R}}^3; \\ u\in H^1({\mathbb {R}}^3), \end{array} \right. \end{aligned}$$where \(a,...

In this paper, we first develop the fractional Trudinger–Moser inequality in singular case and then we use it to study the existence and multiplicity of solutions for a class of perturbed fractional Kirchhoff type problems with singular exponential nonlinearity. Under some suitable assumptions, the existence of two nontrivial and nonnegative soluti...

We consider a parametric (p,q)-equations with sign-changing reaction and Robin boundary condition. We show that for all values of the parameter λ bigger than a certain value which we determine precisely, the problem has at least three nontrivial solutions all with sign information and ordered. For the particular case of (p,2)-equations we produce a...

We consider a Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and a reaction with gradient dependence (convection). The presence of the gradient in the source term excludes from consideration a variational approach in dealing with the qualitative analysis of this problem with unbalanced growth. Using the frozen variable method and eve...

In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomono...

We consider a Dirichlet problem driven by the anisotropic (p, q)-Laplacian and a reaction with gradient dependence (convection). The presence of the gradient in the source term excludes from consideration a variational approach in dealing with the qualitative analysis of this problem with unbalanced growth. Using the frozen variable method and even...

We consider Robin problems driven by the anisotropic p-Laplace operator and with a logistic reaction. Our analysis covers superdiffusive, subdiffusive and equidiffusive equations. We examine all three cases, and we prove multiplicity properties of positive solutions (superdiffusive case) and uniqueness (subdiffusive and equidiffusive cases). The eq...

In this paper, we study the following nonlinear magnetic Kirchhoff equation: { - ( a ⁢ ϵ 2 + b ⁢ ϵ ⁢ [ u ] A / ϵ 2 ) ⁢ Δ A / ϵ ⁢ u + V ⁢ ( x ) ⁢ u = f ⁢ ( | u | 2 ) ⁢ u in ⁢ ℝ 3 , u ∈ H 1 ⁢ ( ℝ 3 , ℂ ) , \left\{\begin{aligned} &\displaystyle{-}(a\epsilon^{2}+b\epsilon[u]_{A/% \epsilon}^{2})\Delta_{A/\epsilon}u+V(x)u=f(\lvert u\rvert^{2})u&&\display...

This paper is concerned with the zero-viscosity limit of the three-dimensional (3D) incompressible stationary magnetohydrodynamics (MHD) equations in the 3D unbounded domain [Formula: see text]. The main result of this paper establishes that the solution of 3D incompressible stationary MHD equations converges to the solution of the 3D incompressibl...

We are concerned with the qualitative analysis of positive solutions to the fractional Choquard equation{(−Δ)su+a(x)u=(Iα⁎|u|2α,s⁎)|u|2α,s⁎−2u,x∈RN,u∈Ds,2(RN),u(x)>0,x∈RN, where Iα(x) is the Riesz potential, s∈(0,1), N>2s, 0<α<min⁡{N,4s}, and 2α,s⁎=2N−αN−2s is the fractional critical Hardy-Littlewood-Sobolev exponent. We first establish a nonlocal...

We consider a nonlinear Robin problem driven by the sum of p-Laplacian and q-Laplacian (i.e. the (p,q)-equation). In the reaction there are competing effects of a singular term and a parametric perturbation λf(z,x), which is Carathéodory and (p−1)-superlinear at x∈R, without satisfying the Ambrosetti–Rabinowitz condition. Using variational tools, t...

We are concerned with the qualitative and asymptotic analysis of solutions to the nonlocal equation $$ (-\Delta)^su+V(|z|)u=Q(|z|)u^p\quad \text{in} \ \mathbb{R}^{N},$$ where $N\geq 3,\ 0

We provide an overview of recent results concerning elliptic variational problems with nonstandard growth conditions and related to different kinds of nonuniformly elliptic operators. Regularity theory is at the center of this paper.

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 l...

In this paper, using variational methods, we establish the existence and multiplicity of multi-bump solutions for the following nonlinear magnetic Schrödinger equation $$\begin{aligned} -(\nabla +\mathrm{i} A(x))^2 u+(\lambda V(x)+Z(x))u=f(\vert u\vert ^{2})u\quad \text {in}\, \,{\mathbb {R}}^{2}, \end{aligned}$$where \(\lambda >0\), f(t) is a cont...

In this paper, we prove the global well-posedness for the incompressible magnetohydrodynamics (MHD) equations in the three-dimensional unbounded domain Ω:=R+×R2. More precisely, we construct global small Sobolev regularity solutions with the initial data near 0 for the three-dimensional MHD equations in Ω. The key point of the proof is to find the...

We consider a nonlinear parametric Neumann problem driven by the anisotropic (p, q)-Laplacian and a reaction which exhibits the combined effects of a singular term and of a parametric superlinear perturbation. We are looking for positive solutions. Using a combination of topological and variational tools together with suitable truncation and compar...

In this paper, we are concerned with the problem -div∇u1+|∇u|2=f(u)inΩ,u=0on∂Ω,where Ω⊂R2 is a bounded smooth domain and f:R→R is a superlinear continuous function with critical exponential growth. We first make a truncation on the prescribed mean curvature operator and obtain an auxiliary problem. Next, we show the existence of positive solutions...

In this paper, we are concerned with the following nonlinear magnetic Schrödinger equation with critical growth: \(\left\{ {\matrix{ {{{\left( {{\varepsilon \over i}\nabla - A\left( x \right)} \right)}^2}u + V\left( x \right)u = f\left( {{{\left| u \right|}^2}} \right)u + {{\left| u \right|}^{{2^*} - 2}}u\,\,\,\,\,\,{\rm{in}}\,{\mathbb{R}^N},} \hfi...

In this paper we study a class of singular systems with double-phase energy. The main feature is that the associated Euler equation is driven by the Baouendi-Grushin operator with variable coefficient. In such a way, we continue the analysis introduced in [6] to the case of lack of compactness corre-sponding to the whole Euclidean space. After esta...

This paper deals with the mathematical analysis of solutions for a new class of Choquard equations. The main features of the problem studied in this paper are the following: (i) the equation is driven by a differential operator with variable exponent; (ii) the Choquard term contains a nonstandard potential with double variable growth; and (iii) the...

This paper deals with the study of combined effects of logarithmic and critical nonlinearities for the following class of fractional p–Kirchhoff equations: M([u]s,pp)(−Δ)psu=λ|u|q−2uln|u|2+|u|ps∗−2uinΩ,u=0inRN∖Ω,where Ω⊂RN is a bounded domain with Lipschitz boundary, N>sp with s∈(0,1), p≥2, ps∗ = Np/(N - ps) is the fractional critical Sobolev expon...

We are concerned with the existence of ground state solutions to the nonhomogeneous perturbed Choquard equation \begin{document}$ - \Delta_{p(x)} u + V(x)|u|^{p(x) - 2} u $\end{document} \begin{document}$ = \left( \int_{\mathbb R^N} r(y)^{-1}|u(y)|^{r(y)}|x-y|^{-\lambda(x,y)} dy\right) |u|^{r(x)-2} u+g(x,u)\ \mbox{in}\ \mathbb R^N, $\end{document}...

The aim of this article is to derive some Lewy-Stampacchia estimates and existence of solutions for equations driven by a nonlocal integro-differential operator on the Heisenberg group.

In this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: a+b[u]s,pp(-Δ)psu=λ|u|q-2uln|u|2+|u|ps∗-2uinΩ,u=0inRN\Ω,where N>sp with s∈(0,1), p>1, and [u]s,pp=∬R2N|u(x)-u(y)|p|x-y|N+psdxdy,ps∗=Np/(N-ps) is the fractional critical...

In this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi-Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic...