Luigi Montoro, Berardino Sciunzi, Marco Squassina
By virtue of a weak comparison principle in small domains we prove axial symmetry in convex and symmetric smooth bounded domains as well as radial symmetry in balls for regular solutions of a class of quasi-linear elliptic systems in non-variational form. Moreover, in the two dimensional case, we study the system when set in a half-space.
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In this work we develop a systematic geometric approach to study fully nonlinear elliptic equations with singular absorption terms as well as their related free boundary problems. The magnitude of the singularity is measured by a negative parameter $(γ-1)$, for $0 < γ< 1$, which reflects on lack of smoothness for an existing solution along the singular interface between its positive and zero phases. We establish existence as well sharp regularity properties of solutions. We further prove that minimal solutions are non-degenerate and obtain fine geometric-measure properties of the free boundary…
For a linear nonvariational operator structured on smooth Hörmander's vector fields, with Hölder continuous coefficients, we prove a regularity result in the spaces of Hölder functions. We deduce an analogous regularity result for nonvariational degenerate quasilinear equations.
We establish the existence of multiple solutions for a nonvariational elliptic systems involving $p(x)$-Laplacian operator. The approach combines the methods of sub-supersolution and Leray--Schauder topological degree.
Established in the 30's, Schauder {\it a priori} estimates are among the most classical and powerful tools in the analysis of problems ruled by 2nd order elliptic PDEs. Since then, a central problem in regularity theory has been to understand Schauder type estimates fashioning particular borderline scenarios. In such context, it has been a common accepted aphorism that the continuity of the Hessian of a solution could never be superior than the continuity of the medium. Notwithstanding, in this article we show that solutions to uniformly elliptic, linear equations with $C^{0,\barε}$ coefficien…
In this work, we study regularity properties for nonvariational singular elliptic equations ruled by the infinity Laplacian. We obtain optimal $C^{1,α}$ regularity along the free boundary. We also show existence of solutions, nondegeneracy properties and fine geometric estimates for the free boundary.
We prove an existence result for a quasilinear elliptic equation satisfying natural growth conditions. As a consequence, we deduce an existence result for a quasilinear elliptic equation containing a singular drift. A key tool, in the proof, is the study of an auxiliary variational inequality playing the role of "natural constraint"
We consider quasicomplexes of pseudodifferential operators on a smooth compact manifold without boundary. To each quasicomplex we associate a complex of symbols. The quasicomplex is elliptic if this symbol complex is exact away from the zero section. We prove that elliptic quasicomplexes are Fredholm. Moreover, we introduce the Euler characteristic for elliptic quasicomplexes and prove a generalization of the Atiyah-Singer index theorem.
In a smooth bounded domain we obtain existence, uniqueness, regularity and boundary behavior for a class of singular quasi-linear elliptic equations.
We prove the uniqueness of positive radial solutions for a class of quasi-linear elliptic problems containing, in particular, the quasi-linear Schrodinger equation.
We prove a global compactness result for Palais-Smale sequences associated with a class of quasi-linear elliptic equations on exterior domains.
In this paper, we prove existence and regularity of positive solutions for singular quasilinear elliptic systems involving gradient terms. Our approach is based on comparison properties, a priori estimates and Schauder's fixed point theorem.
We study spectral properties of divergence form elliptic operators $-\textrm{div} [A(z) \nabla f(z)]$ with the Neumann boundary condition in planar domains (including some fractal type domains), that satisfy to the quasihyperbolic boundary conditions. Our method is based on an interplay between quasiconformal mappings, elliptic operators and composition operators on Sobolev spaces.
We prove nonexistence of nontrivial, possibly sign changing, stable solutions to a class of quasilinear elliptic equations with a potential on Riemannian manifolds, under suitable weighted volume growth conditions on geodesic balls.
For quasi-linear elliptic equations we detect relevant properties which remain invariant under the action of a suitable class of diffeomorphisms. This yields a connection between existence theories for equations with degenerate and non-degenerate coerciveness.
A system of quasilinear elliptic equations on an unbounded domain is considered. The existence of a sequence of radially symmetric weak solutions is proved via variational methods.
The existence of positive weak solutions to a singular quasilinear elliptic system in the whole space is established via suitable a priori estimates and Schauder's fixed point theorem.
The present paper is concerned a class of quasi-linear elliptic degenerate equations. The degenerate operator comes from the analysis of manifolds with corner singularity. Variational methods are applied to verify the existence of infinity many solutions for the problems.
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.
We study existence, uniqueness, multiplicity and symmetry of large solutions for a class of quasi-linear elliptic equations. Furthermore, we characterize the boundary blow-up rate of solutions, including the case where the contribution of boundary curvature appears.
We prove radial symmetry of singular solutions to an overdetermined boundary value problem for a class of degenerate quasilinear elliptic equations.
We prove the existence of multiple solutions for a quasilinear elliptic equation containing a term with natural growth, under assumptions that are invariant by diffeomorphism. To this purpose we develop an adaptation of degree theory.
We prove the following theorem: if $w$ is a quasiconformal mapping of the unit disk onto itself satisfying elliptic partial differential inequality $|L[w]|\le \mathcal{B}|\nabla w|^2+Γ$, then $w$ is Lipschitz continuous. This {result} extends some recent results, where instead of an elliptic differential operator is {only} considered {the} Laplace operator.
Liouville-type results for some quasilinear anisotropic elliptic equations
本文证明了拟线性各向异性椭圆方程稳定解的Liouville型定理，并涵盖纯Finsler p-Laplacian情形。
We generalize Polya-Szego inequality to integrands depending on $u$ and its gradient. Under minimal additional assumptions, we establish equality cases in this generalized inequality. We also give relevant applications of our study to a class of quasi-linear elliptic equations and systems.
The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity results are then employed to show that the obtained solution is actually strong.
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.
Multiple solutions for quasilinear elliptic problems with concave and convex nonlinearities
本文研究具有凹凸非线性的拟线性椭圆问题，通过在交集Banach空间中应用变分方法与截断技术，证明了多个有界符号解的存在性。
We consider a system of quasilinear elliptic equations, with indefinite super-linear nonlinearity, depending on two real parameters $λ,μ$. By using the Nehari manifold and the notion of extremal parameter, we extend some results concerning existence of positive solutions.
We consider a class of quasi-linear anisotropic elliptic equations, possibly degenerate or singular, which are of interest in several applications such as computer vision and continuum mechanics. We prove a Hopf Lemma as well as local and global regularity estimates for positive solutions, generalizing previous results known in the context of p-Laplacian equations.
We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi's counterexample. Here we assume a condition on the support of off-diagonal coefficients that "keeps away" the counterexample and allows us to prove local boundedness of weak solutions.
In this paper we consider maximal regularity for the vector-valued quasi-steady linear elliptic problems. The equations are the elliptic equation in the domain and the evolution equations on its boundary. We prove the maximal $L_p$-$L_q$ regularity for these problems and give examples that our results are applicable. The Lopatinskii--Shapiro and the asymptotic Lopatinskii--Shapiro conditions are important to get boundedness of solution operators.
Quasi-linear elliptic equations with superlinear convection
本文研究了涉及Leray-Lions算子和超线性对流项的拟线性椭圆方程解的存在性与正则性，涵盖p-Laplacian方程，并推广了Boccardo等人（2024年）的部分结果。
For a general class of autonomous quasi-linear elliptic equations on R^n we prove the existence of a least energy solution and show that all least energy solutions do not change sign and are radially symmetric up to a translation in R^n.
In this paper we study quasilinear elliptic systems with nonlinear boundary condition with fully coupled perturbations even on the boundary. Under very general assumptions our main result says that each weak solution of such systems belongs to $L^\infty(\close)\times L^\infty(\close)$. The proof is based on Moser's iteration scheme. The results presented here can also be applied to elliptic systems with homogeneous Dirichlet boundary condition.
We consider the existence and multiplicity of solutions for a class of quasi-linear Schrödinger equations which include the modified nonlinear Schrödinger equations. A new perturbation approach is used to treat the sub-cubic nonlinearity.
We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to $x$.
Two opposite constant-sign solutions to a non-variational p-Laplacian system with Robin boundary conditions are obtained via sub-super-solution techniques. A third nontrivial one comes out by means of topological degree arguments.
Using a standard linearization technique and previously obtained microlocal properties for pseudodifferential operators with smooth coefficients, the authors state results of microlocal regularity in generalized Besov spaces for solutions to non linear PDE.
In this paper, we prove $L^q$-estimates for gradients of solutions to singular quasilinear elliptic equations with measure data $$-\operatorname{div}(A(x,\nabla u))=μ,$$ in a bounded domain $Ω\subset\mathbb{R}^{N}$, where $A(x,\nabla u)\nabla u \asymp |\nabla u|^p$, $p\in (1,2-\frac{1}{n}]$ and $μ$ is a Radon measure in $Ω$
In this article we establish sharp $C^{1,α}$ estimates for weak solutions of singular and degenerate quasilinear elliptic equation $$-\,div\, a(x, \nabla u) = f,$$ which includes the standard $p$-laplacean equation with varying coefficients as a special case. The sharp exponent $α$ is asymptotically optimal and is determined by the Hölder regularity of the coefficients, the exponent $p$ and the $q$-integrability of the source term $f$.
Optimal estimates on the asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \[-Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u=Q(x)|u|^{\frac{Np}{N-p}-2}u,\quad\,x\in \mathbb{R}^{N},\] where $1<p<N, 0\leqμ<\left({(N-p)}/{p}\right)^{p}$ and $Q\in L^{\infty}(\mathbb{R}^{N})$.
Existence results for quasimonotone elliptic systems with growth up to critical exponents
本文研究了具有临界指数增长的拟单调椭圆系统弱解的存在性，采用非线性偏微分方程的分析方法。
Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), & & x\in\R^{N}, \end{eqnarray*} where $1<p<N$, $0\leqμ<\left((N-p)/p\right)^{p}$, $m>0$ and $f$ is a continuous function.
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 establish a gradient estimate for a very weak solution to a quasilinear elliptic equation with a nonstandard growth condition, which is a natural generalization of the $p$-Laplace equation. We investigate the maximum extent for the gradient estimate to hold without imposing any regularity assumption on the nonlinearity other than basic structure assumptions. Our results also include a higher integrability result of the gradient and the existence for the very weak solutions to such nonlinear problems.
Periodic operators over a component domain and homogenization of some class of quasi-linear elliptic problems in two-component domain with interfacial resistance
本文提出周期扩展算子用于双组分域中拟线性椭圆问题的周期均质化，处理具有界面电阻的$ L^2 $和$ L^1 $数据问题。
We prove Hölder continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This includes the $p$-Laplace operator for all $p \in (1,\infty)$, but also operators with unbounded coefficients. Based on the elliptic result we show that the corresponding parabolic problem is well-posed in the space $C(\barΩ)$ provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in…
This work is dedicated to the study of quasi-linear elliptic problems with $L^1$ data, the simple model will be the next equation on $ (M,g) $ a compact Riemannian manifold. $$-Δ_{p} u=f$$ Where $f\in L^{1}(M) $ .Our goal is to develop the functional framework and tools that are necessary to prove the existence and the uniqueness of the solution for the previous problem. Notice that our argument can be used to deal with a more general class of quasi-linear equations.
In this paper we prove the existence of multiple solutions for a quasilinear elliptic boundary value problem, when the p-derivative at zero and the p-derivative at infinity of the nonlinearity are greater than the first eigenvalue of the p-Laplace operator. Our proof uses bifurcation from infinity and bifurcation from zero to prove the existence of unbounded branches of positive solutions (resp. of negative solutions). We show the existence of multiple solutions and we provide qualitative properties of these solutions.
We study a boundary value elliptic problem having a lower order nonlinear term with subquadratic growth in the gradient of the solution and possibly singular when the solution vanishes. If the singularity is mild enough (and even in the absence of the singularity), we prove an existence and multiplicity result. On the contrary, we prove an existence and uniqueness result for strong singularities.
This paper deals with an existence and uniqueness result of the weak solution for a quasilinear elliptic PDE with nonlinear Robin boundary conditions.This problem is defined on a domain whose boundary is the union of two disjoint surfaces.where a Dirichelet condition is prescribed on the exterior boundary and a nonlinear Robin condition on the interior one.To achieve that, we use the theory of monotone operators
Optimal estimates on asymptotic behaviors of weak solutions both at the origin and at the infinity are obtained to the following quasilinear elliptic equations \begin{eqnarray*} -Δ_{p}u-\fracμ{|x|^{p}}|u|^{p-2}u+m|u|^{p-2}u=f(u), & & x\in\R^{N}, \end{eqnarray*} where $1<p<N,0\leqμ<\left((N-p)/p\right)^{p}$, $m>0$ and $f$ is a nonlinear function.
We consider weak solutions to $$-Δ_pu+a(x,u)|\nabla u|^q=f(x,u),$$ with $p>1$, $q\geq\max\,\{p-1,1\}$. We exploit the Moser iteration technique to prove a Harnack comparison inequality for $C^1$ weak solutions. As a consequence we deduce a strong comparison principle.
Second-order boundary estimates for solutions to a class of quasilinear elliptic equations
本文针对一类拟线性椭圆方程，在齐次Dirichlet和Neumann边界条件下证明了全局二阶正则性，并利用边界第二基本形式的可积性条件及源项符号假设，进一步获得了梯度之逆的可积性结果。
Rigidity results via a Poincaré type formula for stable solutions of quasilinear anisotropic elliptic equations
本文通过推广Poincaré型公式，证明了二维和三维空间中拟线性各向异性椭圆方程稳定解的一维对称性结果。
We consider nonlinear elliptic equations that are naturally obtained from the elliptic Schrödinger equation $-Δu +Vu=0$ in the setting of the calculus of variations, and obtain $L^q$-estimates for the gradient of weak solutions. In particular, we generalize a result of Shen in [Ann. Inst. Fourier 45 (1995), no. 2, 513--546] in the nonlinear setting by using a different approach. This allows us to consider discontinuous coefficients with a small BMO semi-norm and non-smooth boundaries which might not be Lipschitz continuous.
We consider local weak large solutions with its blow-up rate near the boundary to certain class of degenerate and/or singular quasilinear elliptic equation\\ ${\rm div}(d^α(x,\partial{}B)Φ_p(\nabla u)) = b(x)f(u)$ in a ball B, where $f$ is normalized regularly varying at infinity with index $σ+1>p-1,\ p>1$. In particular, how the asymptotic behavior of the solution changes over the varying index and degeneracy and/ or singularity present in the equation. We also include the second order blow-up rate for the corresponding semilinear problem.
In this paper we prove the monotonicity of positive solutions to $ -Δ_p u = f(u) $ in half-spaces under zero Dirichlet boundary conditions, for $(2N+2)/(N+2) < p < 2$ and for a general class of regular changing-sign nonlinearities $f$. The techniques used in the proof of the main result are based on a fine use of comparison and maximum principles and on an adaptation of the celebrated moving plane method to quasilinear elliptic equations in unbounded domains.
We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove $C^{1,α}$ local estimates for solutions of a degenerate equation in non divergence form.
In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in $\mathbb{R}^N$ for $N=2,3$. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks though non-degenerate are anisotropic. We also show, within the framework of homogenization, that it is possible to get isotropic regular approximate cloaks. This work generalizes to quasi-linear settings previous work on cloaking i…
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