Dušan Repovš
We are concerned with the asymptotic analysis of positive blow-up boundary solutions for a class of quasilinear elliptic equations with absorption term. By means of the Karamata theory we establish the first two terms in the expansion of the singular solution near the boundary. Our analysis includes large classes of nonlinearities of Keller-Osserman type.
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This paper is devoted to the study, with variational technique, of (p,q)-Laplacian equations in presence of general nonlinearities. Especially we obtain the existence result for the zero mass case, which includes a large class of pure power nonlinearities. More general quasilinear problems of Born-Infeld type are also considered.
We study the existence and multiplicity of solutions to the elliptic system where RN is a bounded and smooth domain. Using fibering maps and extracting Palais-Smale sequences in the Nehari manifold, we prove the existence of at least two distinct nontrivial nonnegative solutions.
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"
In this paper, we consider existence of positive solutions for the Schrödinger quasilinear elliptic problem $$ \left\{ \begin{array}{l} Δ_pu+Δ_p(|u|^{2γ})|u|^{2γ-2}u = a(x)g(u)~ \mbox{on}~ \mathbb{R}^N,\\ u>0\ \mbox{in}~\mathbb{R}^N,\ u(x)\stackrel{\left|x\right|\rightarrow \infty}{\longrightarrow} \infty, \end{array} \right. $$ where $a(x), ~x\in \mathbb{R}^N$ and $g(s)~s>0$ are a nonnegative and continuous functions with $g$ being nonincreasing as well, $γ>{1}/{2}$, and $N \geq 1$. By a dual approach we establish sufficient conditions for existence and multiplicity of solutions for this prob…
We establish that the elliptic equation $Δu+f(x,u)+g(| x|)x\cdot \nabla u=0$, where $x\in\mathbb{R}^{n}$, $n\geq3$, and $| x|>R>0$, has a positive solution which decays to 0 as $| x|\to +\infty$ under mild restrictions on the functions $f,g$. The main theorem extends and complements the conclusions of the recent paper [M. Ehrnström, O.G. Mustafa, On positive solutions of a class of nonlinear elliptic equations, Nonlinear Anal. TMA 67 (2007), 1147--1154]. Its proof relies on a general result about the long-time behavior of the logarithmic derivatives of solutions for a class of nonlinear ordina…
This paper deals with bounded solutions of quasilinear elliptic equations on Riemannian manifolds satisfying special condition.
We show the existence of nodal solutions to perturbed quasilinear elliptic equations with critical Sobolev exponent on compact Riemannian manifolds. A nonexistence result is also given.
In this paper we establish existence of smooth positive solutions for a singular quasilinear elliptic system involving gradient terms. The approach combines sub-supersolutions method and Schauder's fixed point theorem.
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 find a solution of a quasilinear elliptic equation with Dirichlet's boundary condition on a smooth bounded domain and involving an unbounded continuous nonlinearity with oscillatory behavior near the origin.
We establish existence and regularity of positive solutions for a class of quasilinear elliptic systems with singular and superlinear terms. The approach is based on sub-supersolution methods for systems of quasilinear singular equations and the Schauder's fixed point Theorem.
In this paper we establish existence, nonexitence and regularity of positive solutions for a class of singular quasilinear elliptic systems subject to (super-) homogeneous condition. The approach is based on sub-supersolution methods for systems of quasilinear singular equations combined with perturbation arguments involving singular terms.
In this paper we establish the existence of two positive solutions for a class of quasilinear singular elliptic systems. The main tools are sub and supersolution method and Leray-Schauder Topological degree.
We obtain existence and multiplicity results for quasilinear fourth order elliptic equations on $\mathbb{R}^{N}$ with sign-changing potential. Our results generalize some recent results on this problem.
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.
In this paper, we investigate a general quasilinear elliptic and singular system. By monotonicity methods, we give some existence and uniqueness results. Next, we give some applications to biological models.
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.
We obtain estimates and blow-up conditions for solutions of quasilinear elliptic inequalities containing terms with lower-order derivatives
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 prove radial symmetry of singular solutions to an overdetermined boundary value problem for a class of degenerate quasilinear elliptic equations.
This article sets forth results on the existence, positivity and boundedness of solutions for quasilinear elliptic systems involving p-Laplacian and q-Laplacian operators. The approach combines Schaefer's fixed point, comparison principle as well as Moser's iteration procedure.
We consider a quasilinear elliptic equation with right hand side measure, in which the lower order term has a behavior of jumping type. By means of techniques of degree theory, we prove the existence of one or two entropy solutions.
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.
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 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.
We establish a Lipschitz stability inequality for the problem of determining the nonlinear term in a quasilinear elliptic equation by boundary measurements. We give a proof based on a linearization procedure together with special solutions constructed from the fundamental solution of the linearized problem.
A new proof of Hölder estimates for the gradient of quasilinear elliptic equations
本文受Evans-Krylov定理证明启发，采用覆盖方法给出拟线性椭圆方程梯度Hölder估计的新证明，并通过相同方法获得全非线性椭圆方程梯度的Hölder估计。
In this paper, existence and localization results of $C^1$-solutions to elliptic Dirichlet boundary value problems are established. The approach is based on the nonlinear alternative of Leray-Schauder.
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.
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
In this paper we consider two elliptic problems. The first one is a Dirichlet problem while the second is Neumann. We extend all the known results concerning Landesman-Laser conditions by using the Mountain-Pass theorem with the Cerami $(PS)$ condition.
It is established existence, uniqueness and multiplicity of solutions for a quasilinear elliptic problem problems driven by $Φ$-Laplacian operator. Here we consider the reflexive and nonreflexive cases using an auxiliary problem. In order to prove our main results we employ variational methods, regularity results and truncation techniques.
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 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.
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.
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.
In this article we are interested in addressing the question of existence and uniqueness of positive solution to a quasilinear elliptic equation involving p-laplacian in Heisenberg Group. The idea is to prove the uniqueness by using Diaz-Saa Inequality in Heisenberg Group which we obtain via a generalized version of Picone's Identity.
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.
We extend some classical results dealing with boundary Harnack inequatilities to a class of quasilinear elliptic equations and derive some new estimates for solutions of such equations with an isolated singularity on the boundary of a domain.
We study a boundary-value quasilinear elliptic problem on a generic time scale. Making use of the fixed-point index theory, sufficient conditions are given to obtain existence, multiplicity, and infinite solvability of positive solutions.
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.
In this paper we present some very recent results regarding existence, uniqueness, and multiplicity of solutions for quasilinear elliptic equations and systems, exhibiting both singular and convective reaction terms. The importance of boundary conditions (Dirichlet, Neumann, or Robin) is also discussed. Existence is achieved via sub-supersolution and truncation techniques, fixed point theory, nonlinear regularity, and set-valued analysis, while uniqueness and multiplicity are obtained by monotonicity arguments.
We consider a family of quasilinear second order elliptic differential operators which are not coercive and are defined by functions in Marcinkiewicz spaces. We prove the existence of a solution to the corresponding Dirichlet problem. The associated obstacle problem is also solved. Finally, we show higher integrability of a solution to the Dirichlet problem when the datum is more regular.
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 $x$.
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.
This paper is concerned with a class of quasilinear elliptic equations involving some potentials related to the Caffarelli-Korn-Nirenberg inequality. We prove the local boundedness and Hölder continuity of weak solutions by using the classical De Giorgi techniques. Our result extends the results of Serrin \cite{Serrin64} and Corolado and Peral \cite{CP04}.
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.
We prove general a priori estimates of solutions of a class of quasilinear elliptic system on Carnot groups. As a consequence, we obtain several non existence theorems. The results are new even in the Euclidean setting.
This article sets forth results on the existence, a priori estimates and boundedness of positive solutions of a singular quasilinear systems of elliptic equations involving variable exponents. The approach is based on Schauder's fixed point Theorem. A Moser iteration procedure is also obtained for singular cooperative systems involving variable exponents establishing a priori estimates and boundedness of solutions.
In this paper we establish a unilateral bifurcation result for a class of quasilinear elliptic system strongly coupled, extending a bifurcation theorem due to J. López-Gómez. To this aim, we use several results, such that Fredholm operator of index $0$ theory, eigenvalues of elliptic operators and the Krein-Rutman theorem. Lastly, we apply this result to some particular systems arising from population dynamics and determine a region of existence of coexistence states.
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.
Removable singularities in the boundary for quasilinear elliptic equations
该研究聚焦拟线性椭圆方程的边界奇异集可去性问题，提出利用解的无穷远渐近行为与沿奇异集方向的Sobolev范数条件两种途径开展分析。
Liouville-type theorems and existence of solutions for quasilinear elliptic problems
本文建立了上半空间中不定拟线性椭圆方程的Liouville型定理，并利用纤维化方法证明了解的存在性，关键工具是新发展的加权Sobolev嵌入。
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 $Q\in L^{\infty}(\mathbb{R}^{N})$.
In a very general quasilinear setting, we show that the regularizing effect of a first order term causes the existence of energy solutions for problems involving the Hardy potential and $L^1$ data. In the same setting we study sharp (local and global) integral estimates for the second derivatives of the solutions.
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.
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