Local boundedness of solutions to quasilinear elliptic systems

The mathematical analysis to achieve everywhere regularity in the interior of weak solutions to nonlinear elliptic systems usually starts from their local boundedness. Having in mind De Giorgi’s counterexamples, some structure conditions must be imposed to treat systems of partial differential equations. On the contrary, in the scalar case of a general elliptic single equation a well established theory of regularity exists. In this paper we propose a unified approach to local boundedness of weak solutions to a class of quasilinear elliptic systems, with a structure condition inspired by Ladyzhenskaya–Ural’tseva’s work for linear systems, as well as valid for the general scalar case. Our growth assumptions on the nonlinear quantities involved are new and general enough to include anisotropic systems with sharp exponents and the p, q-growth case.     

On positive solutions of quasilinear elliptic systems

In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems where –_p is the p-Laplace operator, p > 1 and is a C ^1,-domain in . We prove an analogue of [7, 16] for the eigenvalue problem with and obtain a non-existence result of positive solutions for the general systems.     

Existence and uniqueness of solutions for quasilinear elliptic systems

We obtain necessary and sufficient conditions for the existence of positive solutions for a class of sublinear Dirichlet quasilinear elliptic systems.

     

Multiplicity of solutions for quasilinear elliptic systems with singularity

In this paper, we study the existence of multiple solutions for the following quasilinear elliptic system:p*(t)|u-2β- △pu1-μ|-2u up1= α1u + β1-2|xp||xt|vβ2||u|u, x∈,|q*β- △qv-μ2 |v|q-2v αv(s)-2|2x|q=|x|sv + β2|uβ1||v2 |-2v, x∈,u(x) = v(x) = 0, x∈ .Multiplicity of solutions for the quasilinear problem is obtained via variational method.     

Existence of bounded solutions for some degenerated quasilinear elliptic equations

Summary We prove the existence of bounded solutions in L () of degenerate elliptic boundary value problems of second order in divergence form with natural growth in the gradient. For the Dirichlet problem our results cover also unbounded domains .Work performed under the auspicies of G.N.A.F.A. of the C.N.R., partially supported by M.P.I. of Italy (40%).     

Existence and multiplicity of nontrivial solutions for quasilinear elliptic systems

The existence and multiplicity of nontrivial solutions are obtained for the quasilinear elliptic systems by the linking argument, the cohomological index theory and the pseudo-index theory.     

Multiplicity of solutions for quasilinear elliptic systems with critical growth

We study the existence of multiple solutions for a quasilinear elliptic system of gradient type with critical growth and the possibility of coupling on the subcritical term. The solutions are obtained from a version of the Symmetric Mountain Pass Theorem. The Concentration-Compactness Principle allows to verify that the Palais-Smale condition is satisfied below a certain level. The authors were partially supported by CNPq/Brazil     

On positive solutions for a class of quasilinear elliptic systems

We investigate the existence and properties of solutions for a class of systems of Dirichlet problems involving the perturbed phi-Laplace operators. We apply variational methods associated with the Fenchel conjugate. Our results cover both sublinear and superlinear cases of nonlinearities.     

Regularity and weak regularity of solutions to quasilinear elliptic higher-order systems

The weak regularity of solutions to quasilinear elliptic higher-order systems is studied. Bibliography: 9 titles. Translated fromProblemy Matematicheskogo Analiza, No. 17, 1997, pp. 238–262.     

A counter-example to the boundary regularity of solutions to elliptic quasilinear systems

It is shown that solutions to the Dirichlet problem for quasilinear elliptic systems in a domain ofR ⁿ n3 with smooth boundary datum can be singular at the boundary.     

Multibump solutions for quasilinear elliptic equations

The current paper is concerned with constructing multibump type solutions for a class of quasilinear Schrödinger type equations including the Modified Nonlinear Schrödinger Equations. Our results extend the existence results on multibump type solutions in Coti Zelati and Rabinowitz (1992) [17] to the quasilinear case. Our work provides a theoretic framework for dealing with quasilinear problems, which lack both smoothness and compactness, by using more refined variational techniques such as gluing techniques, Morse theory, Lyapunov–Schmidt reduction, etc.     

Extremal solutions for nonvariational quasilinear elliptic systems via expanding trapping regions

We consider the Dirichlet boundary value problem for quasilinear elliptic systems in a bounded domain \(\Omega \subset \mathbb {R}^N\) with a diagonal \((p_1, p_2)\)-Laplacian as leading differential operator of the form

$$\begin{aligned} -\Delta _{p_i} u_i=f_i(x, u_1,u_2,\nabla u_1,\nabla u_2)\ \ \text {in }\Omega ,\ \ u_i=0\ \ \text {on }\partial \Omega , \end{aligned}$$

where the component functions \(f_i\) (\(i=1,2\)) of the lower order vector field may also depend on the gradient of the solution \(u=(u_1,u_2)\). The main goal of this paper is twofold. First, we establish an enclosure and existence result by means of the trapping region which is formed by pairs of appropriately defined sub-supersolutions. Second, by a suitable construction of sequences of expanding trapping regions we are able to prove the existence of extremal positive and negative solutions of the system. The theory of pseudomonotone operators, regularity results due to Cianchi-Maz’ya, as well as a strong maximum principle due to Pucci-Serrin are essential tools in the proofs.

     

On positive solutions for a class of singular quasilinear elliptic systems

We study through the lower and upper-solution method, the existence of positive weak solution to the quasilinear elliptic system with weights

     

Entire solutions of quasilinear elliptic systems on Carnot groups

We prove general a priori estimates of the solutions of a class of quasilinear elliptic systems on Carnot groups. As a consequence, we obtain several nonexistence theorems. The results are new even in the Euclidean setting.