We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where either $f(u)=-λ|u|^{p-2}u+|u|^{r-2}u$ or $f(u)=λ|u|^{p-2}u-|u|^{r-2}u$, with $p$, $q>1$, $p+q<\min\{N,r\}$, and $r<(Np-N+p)/(N-p)$. In the first case we show the existence of infinitely many weak solutions for any $λ>0$. In the second case we prove the existence of a nontrivial weak solution if $λ$ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.
We study a nonlinear Neumann boundary value problem associated to a nonhomogeneous differential operator. Taking into account the competition between the nonlinearity and the bifurcation parameter, we establish sufficient conditions for the existence of nontrivial solutions in a related Orlicz--Sobolev space.
We study the nonlinear eigenvalue problem $-{\rm div}(a(|\nabla u|)\nabla u)=λ|u|^{q(x)-2}u$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded open set in $\RR^N$ with smooth boundary, $q$ is a continuous function, and $a$ is a nonhomogeneous potential. We establish sufficient conditions on $a$ and $q$ such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases $a(t)=t^{p-2}\log (1+t^r)$ and $a(t)= t^{p-2} [\log (1+t)]^{-1}$.
We prove Harnack's inequality for bounded weak solutions to quasilinear second order elliptic equations with generalized Orlicz growth conditions. Our approach covers new cases of variable exponent and (p,q) growth conditions.
In this paper we prove the existence of a nontrivial non-negative radial solution for a quasilinear elliptic problem. Our aim is to approach the problem variationally by using the tools of critical points theory in an Orlicz-Sobolev space. A multiplicity result is also given.
We consider the Dirichlet problem for quasilinear elliptic equations with Musielak-Orlicz (p,q)-growth and non-logarithmic conditions on the coefficients. A sufficient Wiener-type condition for the regularity of a boundary point is established.
In this paper, we prove the existence and multiplicity of solutions for a large class of quasilinear problems on a nonreflexive Orlicz-Sobolev space. Here, we use the variational methods developed by Szulkin combined with some properties of the weak$^*$ topology.
In this work we estend a recent result of Kristály, Marzantowicz and Varga concerning the existence of three critical points certain non-smooth functionals. Using this result, we guarantee the existence of three solutions to a inclusion quasilinear problem, with non-homogeneous boundary condition through Orlicz Sobolev spaces.
In this paper, we prove existence of \emph{very weak solutions} to nonhomogeneous quasilinear parabolic equations beyond the duality pairing. The main ingredients are a priori esitmates in suitable weighted spaces combined with the compactness argument developed in \cite{bulicek2018well}. In order to obtain the a priori estimates, we make use of the full Calderón-Zygmund machinery developed in the past few years and combine it with some sharp bounds for the subclass of Muckenhoupt weights considered in this paper.
We consider the nonlinear eigenvalue problem $-{\rm div}(|\nabla u|^{p(x)-2}\nabla u)=λ|u|^{q(x)-2}u$ in $Ω$, $u=0$ on $\partialΩ$, where $Ω$ is a bounded open set in $\RR^N$ with smooth boundary and $p$, $q$ are continuous functions on $\barΩ$ such that $1<\inf\_Ωq< \inf\_Ωp<\sup\_Ωq$, $\sup\_Ωp<N$, and $q(x)<Np(x)/(N-p(x))$ for all $x\in\barΩ$. The main result of this paper establishes that any $λ>0$ sufficiently small is an eigenvalue of the above nonhomogeneous quasilinear problem. The proof relies on simple variational arguments based on Ekeland's variational principle.
Surface-to-Volume Homogenization of NonLinear Steklov Eigencouples with NonHomogeneous Orlicz Growth in Perforated Domains
University of Yaoundé I（雅温得大学）；Higher Teachers Training College, University of Yaoundé I（雅温得第一师范学院）；Higher Teachers Training College, University of Bamenda（班达师范学院）；University of Dschang（德尚大学）
It is established some existence and multiplicity of solution results for a quasilinear elliptic problem driven by $Φ$-Laplacian operator. One of these solutions is built as a ground state solution. In order to prove our main results we apply the Nehari method combined with the concentration compactness theorem in an Orlicz-Sobolev framework. One of the difficulties in dealing with this kind of operator is the lost of homogeneity properties.
The aim of this work is to establish the existence of multi-peak solutions for the following class of quasilinear problems \[ - \mbox{div}\big(ε^{2}ϕ(ε|\nabla u|)\nabla u\big) + V(x)ϕ(| u|)u = f(u)\quad \mbox{in} \quad \mathbb{R}^{N}, \] where $ε$ is a positive parameter, $ N\geq 2$, $V, f$ are continuous functions satisfying some technical conditions and $ϕ$ is a $C^{1}$-function.
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 t…
In this work, we study the existence and multiplicity of solutions for a class of problems involving the $ϕ$-Laplacian operator in a bounded domain, where the nonlinearity has a critical growth. The main tool used is the variational method combined with the genus theory for even functionals.
We study the following boundary value problem (P)\ \ \ \ \ {-\mathrm{div}(a(|\nabla u|)\nabla u)=f(x,u),\ & in $Ω$, u=0, & on $\partialΩ$} with nonhomogeneous principal part. By assuming the nonlinearity $f(x, t)$ being subcritical growth, some abstract results of problem (P) are obtained: (1) Regularity; (2) Orlicz-Sobolev versus Hölder local minimizer; (3) Strong comparison principle. Applying these abstract results and critical point theory, we prove the existence of multiple solutions of problem (P) in an Orlicz-Sobolev space.
In this paper, we study existence, multiplicity and concentration of positive solutions for the following class of quasilinear problems \[ - Δ_Φu + V(εx)ϕ(\vert u\vert)u = f(u)\quad \mbox{in} \quad \mathbb{R}^{N} \,\,\, ( N\geq 2 ), \] where $Φ(t) = \int_{0}^{\vert t\vert}ϕ(s)sds$ is a N-function, $ Δ_Φ$ is the $Φ$-Laplacian operator, $ε$ is a positive parameter, $V : \mathbb{R}^{N} \rightarrow \mathbb{R} $ is a continuous function and $f : \mathbb{R} \rightarrow \mathbb{R} $ is a $C^{1}$-function.
Existence of positive solution for a class of quasilinear Schrödinger equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev spaces
This work is concerned with the existence and multiplicity of solutions for the following class of quasilinear problems $$ -Δ_Φu+ϕ(|u|)u=f(u)~\text{in} ~Ω_λ, u(x)>0 ~\text{in}~Ω_λ, u=0~ \mbox{on} ~\partialΩ_λ, $$ where $Φ(t)=\int_0^{|t|} ϕ(s) s \, ds $ is an $N-$function, $Δ_Φ$ is the $Φ-$Laplacian operator, \linebreak $Ω_λ=λΩ,$ $Ω$ is a smooth bounded domain in $\mathbb{R}^N,$ $N \geq 2$, $λ$ is a positive parameter and $f: \mathbb{R}\rightarrow \mathbb{R}$ is a continuous function. Here, we use variational methods to get multiplicity of solutions by using of Lusternik-Schnirelmann category o…
We construct an efficient approach to deal with the global regularity estimates for a class of elliptic double-obstacle problems in Lorentz and Orlicz spaces. The motivation of this paper comes from the study on an abstract result in the viewpoint of the fractional maximal distributions and this work also extends some regularity results proved in \cite{PN_dist} by using the weighted fractional maximal distributions (WFMDs). We further investigate a pointwise estimates of the gradient of weak solutions via fractional maximal operators and Riesz potential of data. Moreover, in the setting of the…
We prove the continuity of bounded solutions for a wide class of parabolic equations with $(p,q)$-growth $$ u_{t}-{\rm div}\left(g(x,t,|\nabla u|)\,\frac{\nabla u}{|\nabla u|}\right)=0, $$ under the generalized non-logarithmic Zhikov's condition $$ g(x,t,{\rm v}/r)\leqslant c(K)\,g(y,τ,{\rm v}/r), \quad (x,t), (y,τ)\in Q_{r,r}(x_{0},t_{0}), \quad 0<{\rm v}\leqslant Kλ(r), $$ $$ \quad \lim\limits_{r\rightarrow0}λ(r)=0, \quad \lim\limits_{r\rightarrow0} \frac{λ(r)}{r}=+\infty, \quad \int_{0} λ(r)\,\frac{dr}{r}=+\infty. $$ In particular, our results cover new cases of double-phase parabolic equat…
In this paper, we mainly investigate the critical points associated to solutions $u$ of a quasilinear elliptic equation with nonhomogeneous Dirichlet boundary conditions in a connected domain $Ω$ in $\mathbb{R}^2$. Based on the fine analysis about the distribution of connected components of a super-level set $\{x\in Ω: u(x)>t\}$ for any $\mathop {\min}_{\partialΩ}u(x)<t<\mathop {\max}_{\partialΩ}u(x)$, we obtain the geometric structure of interior critical points of $u$. Precisely, when $Ω$ is simply connected, we develop a new method to prove $Σ_{i = 1}^k {m_i}+1=N$, where $m_1,\cdots,m_k$ ar…
\noindent In this paper we study existence of solution for a class of problem of the type $$ \left\{ \begin{array}{ll} -Δ_Φ{u}=f(u), \quad \mbox{in} \quad Ωu=0, \quad \mbox{on} \quad \partial Ω, \end{array} \right. $$ where $Ω\subset \mathbb{R}^N$, $N \geq 2$, is a smooth bounded domain, $f:\mathbb{R} \to \mathbb{R}$ is a continuous function verifying some conditions, and $Φ:\mathbb{R} \to \mathbb{R}$ is a N-function which is not assumed to satisfy the well known $Δ_2$-condition, then the Orlicz-Sobolev space $W^{1,Φ}_0(Ω)$ can be non reflexive. As main model we have the function $Φ(t)=(e^{t^{…
This article deals with the study of the following singular quasilinear equation: \begin{equation*} (P) \left\{ \ -Δ_{p}u -Δ_{q}u = f(x) u^{-δ},\; u>0 \text{ in }\; \Om; \; u=0 \text{ on } \pa\Om, \right. \end{equation*} where $\Om$ is a bounded domain in $\mathbb{R}^N$ with $C^2$ boundary $\pa\Om$, $1< q< p<\infty$, $\de>0$ and $f\in L^\infty_{loc}(\Om)$ is a non-negative function which behaves like $\textnormal{dist}(x,\pa\Om)^{-\ba},$ $\ba\ge 0$ near the boundary of $\Om$. We prove the existence of a weak solution in $W^{1,p}_{loc}(\Om)$ and its behaviour near the boundary for $\ba<p$. Cons…
In this paper, we establish pointwise estimates for supersolutions of quasilinear elliptic equations with structural conditions involving a generalized Orlicz growth in terms of a Wolff type potential. As a consequence, under the extra assumption, we obtain that the supersolutions satisfy a Harnack inequality and local Hölder continuity.
In this paper, we study the interplay between Orlicz-Sobolev spaces $L^{M}$ and $W^{1,M}$ and fractional Sobolev spaces $W^{s,p}$. More precisely, we give some qualitative properties of the new fractional Orlicz-Sobolev space $W^{s,M}$, where $s\in (0,1)$ and $M$ is an $N-$function. We also study a related non-local operator, which is a fractional version of the nonhomogeneous $M$-Laplace operator. As an application, we prove existence of weak solution for a non-local problem involving the new fractional $M-$Laplacian operator.
In this paper, we prove the existence of solutions to quasilinear elliptic equations on a bounded domain of $\R^N$ under subcritical Musielak-Orlicz-Sobolev growth. Our proofs rely essentially on Mountain Pass Theorem with corresponding variational techniques. Furthermore, we establish the sharpness of our central assumptions. As far as we know, our approach is new, even for the Orlicz case.
The main goal this work is to prove two results like Strauss and Lions for Orlicz-Sobolev spaces. After, we use these results for study the existence of solutions for a class of quasilinear problems in $\mathbb{R}^{N}$.
This paper deals with existence of positive solutions for a class of quasilinear elliptic systems involving the $Φ$-Laplacian operator and convex-concave singular terms. Our approach is based on the generalized Galerkin Method along with perturbartion techniques and comparison arguments in the setting of Orlicz-Sobolev spaces
We establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in the Orlicz Sobolev spaces and under general growth conditions on the convection term. The sub and supersolutions method is a key tool in the proof of the existence results.
Boundary value problems for a class of quasilinear elliptic equations, with an Orlicz type growth and L^1 right-hand side are considered. Both Dirichlet and Neumann problems are contemplated. Existence and uniqueness of generalized solutions, as well as their regularity, are established. The case of measure right-hand sides is also analyzed.
Our purpose is to show the existence of weak solutions for unsteady flow of non-Newtonian incompressible nonhomogeneous, heat-conducting fluids with generalised form of the stress tensor without any restriction on its upper growth. Motivated by fluids of nonstandard rheology we focus on the general form of growth conditions for the stress tensor which makes anisotropic Musielak-Orlicz spaces a suitable function space for the considered problem. We do not assume any smallness condition on initial data in order to obtain long-time existence. Within the proof we use monotonicity methods, integrat…
We study a nonlinear parametric Neumann problem driven by a nonhomogeneous quasilinear elliptic differential operator $\operatorname{div}(a(x,\nabla u))$, a special case of which is the $p$-Laplacian. The reaction term is a nonlinearity function $f$ which exhibits $(p-1)$-subcritical growth. By using variational methods, we prove a multiplicity result on the existence of weak solutions for such problems. An explicit example of an application is also presented.
Mountain pass in a suitable Orlicz space is employed to prove the existence of soliton solutions for a quasilinear Schrödinger equation involving critical exponent in ${\BR}^N$. These equations contain strongly singular nonlinearities which include derivatives of the second order. Such equations have been studied as models of several physical phenomena. The nonlinearity here corresponds to the superfluid film equation in plasma physics.
In this paper we prove a Lions type result for a large class of Orlicz-Sobolev space that can be nonreflexive and use this result to show the existence of solution for a large class of quasilinear problem on a nonreflexive Orlicz-Sobolev space.
We are concerned with the existence and regularity of the solutions to the Dirichlet problem, for a class of quasilinear elliptic equations driven by a general differential operator, depending on $(x,u,\nabla u)$, and with a convective term $f$. The assumptions on the members of the equation are formulated in terms of Young's functions, therefore we work in the Orlicz-Sobolev spaces. After establishing some auxiliary properties, that seem new in our context, we present two existence and two regularity results. We conclude with several examples.
We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak-Orlicz space avoiding growth restrictions. Namely, we consider \begin{equation*} -{\rm div} A(x,\nabla u)= f\in L^1(Ω), \end{equation*} on a Lipschitz bounded domain in $\mathbb{R}^N$. The growth of the monotone vector field $A$ is controlled by a generalized nonhomogeneous and anisotropic $N$-function $M $. The approach does not require any particular type of growth condition of $M$ or its conjugate $M^*$ (neither $Δ_2$, nor $\nabla_2$). The condition we impose is log-Holder continuity of $M$, whic…
We study global regularity for solutions of quasilinear elliptic equations of the form $÷\A(x,u,\nabla u) = ÷\F $ in rough domains $Ω$ in $\R^n$ with nonhomogeneous Dirichlet boundary condition. The vector field $\A$ is assumed to be continuous in $u$, and its growth in $\nabla u$ is like that of the $p$-Laplace operator. We establish global gradient estimates in weighted Morrey spaces for weak solutions $u$ to the equation under the Reifenberg flat condition for $Ω$, a small BMO condition in $x$ for $\A$, and an optimal condition for the Dirichlet boundary data.
We study the existence of positive solutions to the quasilinear elliptic problem -εΔu+V(x)u-εk(\Del(|u|^{2}))u=g(u), \quad u>0, x \in R^N, where g has superlinear growth at infinity without any restriction from above on its growth. Mountain pass in a suitable Orlicz space is employed to establish this result. These equations contain strongly singular nonlinearities which include derivatives of the second order which make the situation more complicated. Such equations arise when one seeks for standing wave solutions for the corresponding quasilinear Schrödinger equations. Schrödinger equations …
Three solutions for a quasilinear inclusion systems driven by a nonstandard Laplacian operator in $\mathbb{R}^N$
Kunming University of Science and Technology（昆明理工大学）；Research Center for Mathematics and Interdisciplinary Sciences, Kunming University of Science and Technology（昆明理工大学数学与交叉科学研究中心）
We investigate an initial-boundary value problem for a quasilinear nonhomogeneous, anisotropic Maxwell system subject to an absorbing boundary condition of Silver & Müller type in a smooth, bounded, strictly star-shaped domain of $\mathbb{R}^{3}$. Imposing usual smallness assumptions in addition to standard regularity and compatibility conditions, a nonlinear stabilizability inequality is obtained by showing nonlinear dissipativity and observability-like estimates enhanced by an intricate regularity analysis. With the stabilizability inequality at hand, the classic nonlinear barrier method is …
We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in \cite{DFMST}. We apply it to generalized $p$-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative solutions. Based on a generalized hidden convexity result, we show that uniqueness holds among strongly positive solutions and nonnegative global minimizers. Problems involving nonhomogeneous operators as the so-called $(p,r)$-Laplacian are also treated.
This paper is principally devoted to revisit the remarkable works of Keller and Osserman and generalize some previous results related to the those for the class of quasilinear elliptic problem $$ \left\{ \begin{array}{l} {\rm{div}} \left( ϕ(|\nabla u|)\nabla u\right) = a(x)f(u)\quad \mbox{in } Ω,\\ u\geq0\ \ \mbox{in}\ Ω,\ \ u=\infty\ \mbox{on}\ \partialΩ, \end{array} \right. $$ where either $Ω\subset \mathbb{R}^N$ with $N \geq 1$ is a smooth bounded domain or $Ω= \mathbb{R}^N$. The function $ϕ$ includes special cases appearing in mathematical models in nonlinear elasticity, plasticity, genera…
It is established $L^{p}$ estimates for the fractional $Φ$-Laplacian operator defined in bounded domains where the nonlinearity is subcritical or critical in a suitable sense. Furthermore, using some fine estimates together with the Moser's iteration, we prove that any weak solution for fractional $Φ$-Laplacian operator defined in bounded domains belongs to $L^\infty(Ω)$ under appropriate hypotheses on the $N$-function $Φ$. Using the Orlicz space and taking into account the fractional setting for our problem the main results are stated for a huge class of nonlinear operators and nonlinearities…
