Given $N\geq 3$, $1<p<N$, two measurable functions $V\left(r \right)\geq 0$, $K\left(r\right)> 0$ and a continuous function $A(r) >0$ ($r>0$), we study the quasilinear elliptic equation \[ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) u+V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}. \] We find existence of nonegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space $X$ into the sum of Lebesgue spaces $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. Our results do not require any compatibility between how the potentials $A$, $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of $V$ and $K$, not of the potentials separately. The nonlinearity $f$ has a double-power behavior, whose standard example is $f(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$, recovering the usual case of a single-power behavior when $q_1 = q_2$.
Method of compensated compactness is used to show that the almost everywhere limit of quasilinear viscous approximations is the unique entropy solution (in the sense of {\it Bardos et.al}\cite{MR542510}) of the corresponding scalar conservation laws in a bounded domain in $\mathbb{R}^{d}$, where the viscous term is of the form $\varepsilon div\left(B(u^{\varepsilon})\nabla u^{\varepsilon}\right)$.
We use the method of Compensated Compactness and Kinteic Formulation to show that the almost everywhere limit of quasilinear viscous approximations is the unique entropy solution (in the sense of {\it F. Otto}) of the corresponding scalar conservation laws on a bounded domain in $\mathbb{R}^{d}$, where the viscous term is of the form $\varepsilon\,div\left(B(u^{\varepsilon})\nabla u^{\varepsilon}\right)$.
One of the major difficulties in nonlinear elliptic problems involving critical nonlinearities is the compactness of Palais-Smale sequences. In their celebrated work \cite{BN}, Brézis and Nirenberg introduced the notion of critical level for these sequences in the case of a critical perturbation of the Laplacian homogeneous eigenvalue problem. In this paper, we give a natural and general formula of the critical level for a large class of nonlinear elliptic critical problems. The sharpness of our formula is established by the construction of suitable Palais-Smale sequences which are not relativ…
It is established existence, multiplicity and asymptotic behavior of positive solutions for a quasilinear elliptic problem driven by the $Φ$-Laplacian operator. One of these solutions is obtained as ground state solution by applying the well known Nehari method. The semilinear term in the quasilinear equation is a concave-convex function which presents a critical behavior at infinity. The concentration compactness principle is used in order to recover the compactness required in variational methods.
Via minimization arguments and Concentration Compactness Principle, we prove the orbital stability of standing wave solutions for a class of quasilinear Schrödinger equation arising from physics.
In this work we use variational methods to prove results on existence and concentration of solutions to a problem in $\mathbb{R}^N$ involving the $1-$Laplacian operator. A thorough analysis on the energy functional defined in the space of functions of bounded variation $BV(\mathbb{R}^N)$ is necessary, where the lack of compactness is overcome by using the Concentration of Compactness Principle of Lions.
Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the non-linearities.
The purpose of this paper is to provide tools for analyzing the compactness of sequences in Sobolev spaces, in particular if the sequence gets mapped onto a compact set by some nonlinear operator. Here, our focus lies on a very general class of nonlinear operators arising in quasilinear systems of partial differential equations of second order, in divergence form. Our approach, based on a suitable decomposition lemma, admits the discussion of problems with some inherent loss of compactness, for example due to a domain with infinite measure or a lower order term with critical growth. As an appl…
The existence of entire solutions to quasilinear elliptic systems exhibiting both singular and convective reaction terms is discussed. An auxiliary problem, obtained by `freezing' the convection terms and `shifting' the singular ones, is first solved. Then, a priori estimates, fixed point arguments, nonlinear regularity, compactness results concerning the gradient terms, besides a regularization-localization procedure, yield the existence of a weak solution to the problem.
The main purpose of this paper is to establish the existence of positive solutions to a class of quasilinear elliptic equations involving the (p-q)-Laplacian operator. We consider a nonlinearity that can be subcritical at infinity and supercritical at the origin; we also consider potential functions that can vanish at infinity. The approach is based on variational arguments dealing with the mountain-pass lemma and an adaptation of the penalization method. In order to overcome the lack of compactness we modify the original problem and the associated energy functional. Finally, to show that the …
In this note we complete a previous study, where we got existence results for the quasilinear elliptic equation \begin{equation*} -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \end{equation*} with singular or vanishing continuous radial potentials $V(r)$, $K(r)$. In our previuos study we assumed, for technical reasons, that $K(r)$ was vanishing as $r \rightarrow 0$, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables $w=f(u)$ and we find existence of non negative solutio…
Existence of positive solution for a class of quasilinear Schrödinger equations with potential vanishing at infinity on nonreflexive Orlicz-Sobolev spaces
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 general theory on exact boundary controllability for general first order quasilinear hyperbolic systems requires that the characteristic speeds of system do not vanish. This paper deals with exact boundary controllability, when this is not the case. Some important models are also shown as applications of the main result. The strategy uses the return method, which allows in certain situations to recover non zero characteristic speeds.
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 prove the existence of one positive, one negative, and one sign-changing solution of a $p$-Laplacian equation on $\mathbb{R}^N$, with a $p$-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of $\mathbb{R}^N$ have only been scarcely investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by…
Mixed local-nonlocal quasilinear problems with mixed interpolated Hardy potential
Ghent University（根特大学）；National Institute of Technology Calicut（卡利卡特国立技术学院）；Indian Institute of Technology (BHU)（印度理工学院（贝拿勒斯））；Queen Marry University of London（伦敦玛丽女王大学）
In this paper, we consider the existence and non-existence of non-trivial solution to a Brezis-Nirenberg type problem with singular weights. First, we obtain a compact imbedding theorem which is an extension of the classical Rellich-Kondrachov compact imbedding theorem, and consider the corresponding eigenvalue problem. Secondly, we deduce a Pohozaev type identity and obtained a non-existence result. Thirdly, based on a generalized concentration compactness principle, we will give some abstract conditions when the functional satisfies the (PS)$_c$ condition. Finally, based on the explicit form…
In this paper, by using the theory of compensated compactness coupled with the kinetic formulation by Lions, Perthame, Souganidis and Tadmor \cite{LPT,LPS}, we prove the existence and nonexistence of global generalized (nonnegative) solutions of the nonlinearly degenerate wave equations $v_{tt} =c (|v|^{s-1} v)_{xx}$ with the nonnegative initial data $v_{0}(x)$ and $ s > 1$. This result is an extension of the results in the second author's paper \cite{Su}, where the existence and the nonexistence of the unique global classical solution were studied with a threshold on $\int_{-\infty}^{\infty} …
In three spatial dimensions, we study the Cauchy problem for the model wave equation $- \partial_t^2 Ψ+ (1 + Ψ)^P ΔΨ= 0$ for $P \in \lbrace 1,2 \rbrace$. We exhibit a stable form of finite-time Tricomi-type degeneracy formation that has not previously been studied for quasilinear wave equations. Specifically, using only energy methods and ODE-type techniques, we exhibit an open (in an appropriate Sobolev topology) set of data such that $Ψ$ is initially near $0$ while $1 + Ψ$ vanishes in finite time. In fact, generic data profiles, when appropriately rescaled, lead to the vanishing of $1 + Ψ$ i…
For $0 < p-1 < q$ and $\ge=\pm 1$, we prove the existence of solutions of $-\Gd_pu=\ge u^q$ in a cone $C_S$, with vertex 0 and opening $S$, vanishing on $\prt C_S$, under the form $u(x)=|x|^\gb\gw(\frac{x}{|x|})$. The problem reduces to a quasilinear elliptic equation on $S$ and existence is based upon degree theory and homotopy methods. We also obtain a non-existence result in some critical case by an integral type identity.
The aim of this paper is to establish regularity for weak solutions to the nondiagonal quasilinear degenerate elliptic systems related to Hörmander's vector fields, where the coefficients are bounded with vanishing mean oscillation. We first prove $L^p$($p \ge 2$) estimates for gradients of weak solutions by using a priori estimates and a known reverse Hölder inequality, and consider regularity to the corresponding nondiagonal homogeneous degenerate elliptic systems. Then we get higher Morrey and Campanato estimates for gradients of weak solutions to original systems and Hölder estimates for w…
We consider planar solutions to certain quasilinear elliptic equations subject to the Dirichlet boundary conditions; the boundary data is assumed to have finite number of relative maximum and minimum values. We are interested in certain vanishing properties of sign changing solutions to such a Dirichlet problem. Our method is applicable in the plane.
We prove the global strong solvability of a quasilinear initial-boundary value problem with fractional time derivative of order less than one. Such problems arise in mathematical physics in the context of anomalous diffusion and the modelling of dynamic processes in materials with memory. The proof relies heavily on a regularity result about the interior Hölder continuity of weak solutions to time fractional diffusion equations, which has been proved recently by the author. We further establish a basic $L_2$ decay estimate for the special case with vanishing external source term and homogeneou…
We use blow up analysis for local integral equations to prove compactness of solutions to higher order critical elliptic equations provided the potentials only have non-degenerate zeros. Secondly, corresponding to Schoen's Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow up points of solutions.
In this paper we prove compact embedding of a subspace of the fractional Orlicz-Sobolev space $W^{s, G}\left(\mathbb{R}^{N}\right)$ consisting of radial functions, our target embedding spaces are of Orlicz type. Also, we prove a Lions and Lieb type results for $W^{s,G}\left(\mathbb{R}^{N}\right)$ that works together in a particular way to get a sequence whose the weak limit is nontrivial. As an application, we study the existence of solutions to Quasilinear elliptic problems in the whole space $\mathbb{R}^N$ involving the fractional $g-$Laplacian operator, where the conjugated function $\widet…
We show that all nonnegative solutions of the critical semilinear elliptic equation involving the regional fractional Laplacian are locally universally bounded. This strongly contrasts with the standard fractional Laplacian case. Second, we consider the fractional critical elliptic equations with nonnegative potentials. We prove compactness of solutions provided the potentials only have non-degenerate zeros. Corresponding to Schoen's Weyl tensor vanishing conjecture for the Yamabe equation on manifolds, we establish a Laplacian vanishing rate of the potentials at blow-up points of solutions.
In this article, we establish the existence of a family of hypersurfaces $(Γ(t))_{0< t \leq T}$ which evolve by the vanishing mean curvature flow in Minkowski space and which as $t$ tends to~$0$ blow up towards a hypersurface which behaves like the Simons cone at infinity. This issue amounts to investigate the singularity formation for a second order quasilinear wave equation. Our constructive approach consists in proving the existence of finite time blow up solutions of this hyperbolic equation under the form $u(t,x) \sim t^ {ν+1} Q\Big(\frac {x} {t^ {ν+1}} \Big) $, where~$Q$ is a stationary …
Fitzpatrick's variational representation of maximal monotone operators is here extended to a class of pseudo-monotone operators in Banach spaces. On this basis, the initial-value problem associated with the first-order flow of such an operator is here reformulated as a minimization principle, extending a method that was pioneered by Brezis, Ekeland and Nayroles for gradient flows. This formulation is used to prove that the problem is stable w.r.t.\ arbitrary perturbations not only of data but also of operators. This is achieved by using the notion of evolutionary $Γ$-convergence w.r.t.\ a nonl…
The authors of this paper deal with the existence and regularities of weak solutions to the homogenous $\hbox{Dirichlet}$ boundary value problem for the equation $-\hbox{div}(|\nabla u|^{p-2}\nabla u)+|u|^{p-2}u=\frac{f(x)}{u^α}$. The authors apply the method of regularization and $\hbox{Leray-Schauder}$ fixed point theorem as well as a necessary compactness argument to prove the existence of solutions and then obtain some maximum norm estimates by constructing three suitable iterative sequences. Furthermore, we find that the critical exponent of $m$ in $\|f\|_{L^{m}(Ω)}$. That is, when $m$ li…
Weighted good-$λ$ type inequalities and Muckenhoupt-Wheeden type bounds are obtained for gradients of solutions to a class of quasilinear elliptic equations with measure data. Such results are obtained globally over sufficiently flat domains in $\mathbb{R}^n$ in the sense of Reifenberg. The principal operator here is modeled after the $p$-Laplacian, where for the first time singular case $\frac{3n-2}{2n-1}<p\leq 2-\frac{1}{n}$ is considered. Those bounds lead to useful compactness criteria for solution sets of quasilinear elliptic equations with measure data. As an application, sharp existence…
In this paper we continue the work that we began in arXiv:1912.07537. Given $1<p<N$, two measurable functions $V\left(r \right)\geq 0$ and $K\left(r\right)> 0$, and a continuous function $A(r) >0\ (r>0)$, we consider the quasilinear elliptic equation \[ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) +V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}, \] where all the potentials $A,V,K$ may be singular or vanishing, at the origin or at infinity. We find existence of nonnegative solutions by the application of variational methods, for which we nee…
This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps. Therefore we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling, and approximate the model by its viscous regularization. In fact, the analysis of the latter PDE system presents remarkable difficulties, due to its highl…
We study the quasilinear equation \[(P)\qquad - {\rm div} (A(x,u) |\nabla u|^{p-2} \nabla u) + \frac1p\ A_t(x,u) |\nabla u|^p + |u|^{p-2}u\ =\ g(x,u) \qquad \hbox{in ${\mathbb R}^N$,} \] with $N\ge 3$, $p > 1$, where $A(x,t)$, $A_t(x,t) = \frac{\partial A}{\partial t}(x,t)$ and $g(x,t)$ are Carathéodory functions on ${\mathbb R}^N \times {\mathbb R}$. Suitable assumptions on $A(x,t)$ and $g(x,t)$ set off the variational structure of $(P)$ and its related functional ${\cal J}$ is $C^1$ on the Banach space $X = W^{1,p}({\mathbb R}^N) \cap L^\infty({\mathbb R}^N)$. In order to overcome the lack o…
This paper concerns the existence of positive ground state solutions for generalized quasilinear Schrödinger equations in $\mathbb{R}^N$ with critical growth which arise from plasma physics, as well as high-power ultrashort laser in matter. By applying a variable replacement, the quasilinear problem reduces to a semilinear problem which the associated functional is well defined in the Sobolev space $H^1(\mathbb{R}^N)$. We use the method of Nehari manifold for the modified equation, establish the minimax characterization, then obtain each Palais-Smale sequence of the associated energy functiona…
Given two continuous functions $V\left(r \right)\geq 0$ and $K\left(r\right)> 0$ ($r>0$), which may be singular or vanishing at zero as well as at infinity, we study the quasilinear elliptic equation \[ -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \] where $N\geq3$. To study this problem we apply a change of variables $w=f(u)$, already used by several authors, and find existence results for nonnegative solutions by the application of variational methods. The main features of our results are that they do not require any compatibilit…
In this paper we study the Hausdorff dimension of a elliptic measure $μ_{f}$ in space associated to a positive weak solution to a certain quasilinear elliptic PDE in an open subset and vanishing on a portion of the boundary of that open set. We show that this measure is concentrated on a set of $σ-$finite $n-1$ dimensional Hausdorff measure for $p>n$ and the same result holds for $p=n$ with an assumption on the boundary. We also construct an example of a domain in space for which the corresponding measure has Hausdorff dimension $\leq n-1-δ$ for $p\geq n$ for some $δ$ which depends on various …
This paper is concerned with the nodal set of weak solutions to a broken quasilinear partial differential equation, \begin{equation*} \mbox{div} (a_+ \nabla u^+ - a_- \nabla u^-) = \mbox{div} f, \end{equation*} where $a_+$ and $a_-$ are uniformly elliptic, Dini continuous coefficient matrices, subject to a strong correlation that $a_+$ and $a_-$ are a multiple of some scalar function to each other. Under such a structural condition, we develop an iteration argument to achieve higher-order approximation of solutions at a singular point, which is also new for standard elliptic PDEs below Hölder …
We study the problem of conformal deformation of Riemannian structure to constant scalar curvature with zero mean curvature on the boundary. We prove compactness for the full set of solutions when the boundary is umbilic and the dimension $n \leq 24$. The Weyl Vanishing Theorem is also established under these hypotheses, and we provide counter-examples to compactness when $n \geq 25$. Lastly, our methods point towards a vanishing theorem for the umbilicity tensor, which is anticipated to be fundamental for a study of the nonumbilic case.
We investigate a class of nonlinear nonautonomous scalar field equations with fractional diffusion, critical power nonlinearity and a subcritical term. The involved potentials are allowed for vanishing behavior at infinity. The problem is set on the whole space and compactness issues have to be tackled.
In this work it is studied a quasilinear elliptic problem in the whole space $\mathbb{R}^N$ involving the $1-$Laplacian operator, with potentials which can vanish at infinity. The Euler-Lagrange functional is defined in a space whose definition resembles $BV(\mathbb{R}^N)$ and, in order to avoid working with extensions of it to some Lebesgue space, we state and prove a version of the Mountain Pass Theorem without the Palais-Smale condition to Lipschitz continuous functionals.
We study the perturbation by a critical term and a $(p-1)$-superlinear subcritical nonlinearity of a quasilinear elliptic equation containing a singular potential. By means of variational arguments and a version of the concentration-compactness principle in the singular case, we prove the existence of solutions for positive values of the parameter under the principal eigenvalue of the associated singular eigenvalue problem.
In this paper, we reformulate a mathematical model for the dynamics of an idealized electrostatically actuated MEMS device with two elastic membranes as an initial value problem for an abstract quasilinear evolution equation. Applying the Contraction Mapping Theorem, it is shown that the model is locally well-posed in time for any value of the source voltage of the device. In addition it is proven that the MEMS model considered here possesses global solutions for small source voltages whereas for large source voltages solutions of the model have a finite maximal existence time. Furthermore, we…
The aim of this paper is to extend previous results regarding the multiplicity of solutions for quasilinear elliptic problems with critical growth to the variable exponent case. We prove, in the spirit of \cite{DPFBS}, the existence of at least three nontrivial solutions to the following quasilinear elliptic equation $-Δ_{p(x)} u = |u|^{q(x)-2}u +λf(x,u)$ in a smooth bounded domain $Ω$ of $\R^N$ with homogeneous Dirichlet boundary conditions on $\partialΩ$. We assume that $\{q(x)=p^*(x)\}\not=\emptyset$, where $p^*(x)=Np(x)/(N-p(x))$ is the critical Sobolev exponent for variable exponents and …
We consider a scalar, possibly degenerate parabolic equation with a source term, in several space dimensions. For initial data with bounded variation we prove the existence of solutions to the initial-value problem. Then we show that these solutions converge, in the vanishing-viscosity limit, to the Kruzhkov entropy solution of the corresponding hyperbolic equation. The proof exploits the H-measure compactness in several space dimensions.
This paper deals with thermoelectric problems including the Peltier and Seebeck effects. The coupled elliptic and doubly quasilinear parabolic equations for the electric and heat currents are stated, respectively, accomplished with power-type boundary conditions that describe the thermal radiative effects. To verify the existence of weak solutions to this coupled problem (Theorem 1), analytical investigations for abstract multi-quasilinear elliptic-parabolic systems with nonsmooth data are presented (Theorem 2 and 3). They are essentially approximated solutions based on the Rothe method. It co…
In this work we study the existence of ground-state solutions of Dirac equations with potentials which are allowed to vanish at infinity. The approach is based on minimization of the energy functional over a generalized Nehari set. Some conditions on the potentials are given in order to overcome the lack of compactness.
The paper studies continutity of Moser nonlinearity in two dimensions with respect to weak convergence. Unlike the critical nonlinearity in the Sobolev inequality, which lacks weak continuity at any point, Moser functional fails to be weakly continuous only in an exceptional case of a concentrating sequence of functions from the Moser family (up to translations and the remainder vanishing in Sobolev norm). The argument is based on a structural description of the defect of weak converegence, analogous to the profile decomposition established by Solimini for Sobolev inequalities, but involving g…
We obtain nontrivial solutions of a critical fractional $p$-Laplacian equation in the whole space and with possibly vanishing potentials. In addition to the usual difficulty of the lack of compactness associated with problems involving critical Sobolev exponents, the problem is further complicated by the absence of a direct sum decomposition suitable for applying classical linking arguments. We overcome this difficulty using a generalized linking construction based on the ${\mathbb Z}_2$-cohomological index.
