Some remarks on bounded and unbounded weak solutions of elliptic systems

The aim of this chapter is to discuss some full (that is everywhere) regularity results for scalar-valued weak solutions to second order elliptic equations in divergence forms. Weak solutions only belong to some Sobolev space (consequently, neither are they necessarily continuous nor do derivatives a priori exist in the classical sense), and hence, their regularity needs to be investigated. We here prove local Holder regularity of weak solution and present two different (and classical) strategies of proof dating back to the late 1950s. First, we explain De Giorgi’s level set technique, in a unified approach that applies both to weak solutions of elliptic equations and to minimizers of variational integrals, via the study of Q-minimizers of suitable functionals. We then address, for the specific case of linear elliptic equations, an alternative proof of the everywhere regularity result of weak solutions via Moser’s iteration method.

Globally smooth solutions of quasilinear hyperbolic systems in diagonal form

The main objective in this paper is to obtain the existence results for bounded and unbounded solutions of some quasilinear elliptic systems. Related results as obtained here have been established recently in [C. O. Alves and A. R. F. de Holanda, Existence of blow-up solutions for a class of elliptic systems, Differ. Integral Eqs.26(1/2) (2013) 105–118]. Also, we present some references to give the connection between these types of problems with probability and stochastic processes, hoping that these are interesting for the audience of analysts likely to read this paper.

We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space R N {{\mathbb{R}}}^{N} . We assume that the nonlinear term satisfies the locally super- ( m 1 , m 2 ) \left({m}_{1},{m}_{2}) condition, that is, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ G x\in G , where G G is a domain in R N {{\mathbb{R}}}^{N} , which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim ∣ ( u , v ) ∣ → + ∞ F ( x , u , v ) ∣ u ∣ m 1 + ∣ v ∣ m 2 = + ∞ {\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x ∈ R N x\in {{\mathbb{R}}}^{N} . We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

In this paper we study regularity and partial regularity for the weak solution of a class of general quasi-linear elliptic equations and systems, which are of the quasi-linear main coefficients satisfying the VMO conditions in x uniformly with respect to u, and of the lower order items satisfying controllable growth.Contents 1. Introduction 87 2. Preliminaries 90 3. Proofs of Main Results 92 4. Conclusion 97

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 that off-diagonal coefficients have a “butterfly support”: this allows us to prove local boundedness of weak solutions.

For the weak solutions of second-order equations in a plane bounded domain, one obtains the asymptotics and estimates of the solutions near the interior angular points of the line of discontinuity of the coefficients. It is established that the number of terms of the asymptotics, not in the Sobolev classW 2 2 , can be arbitrarily large.

Multiple symmetric invariant non trivial solutions for a class of quasilinear elliptic variational systems

The aim of this chapter is to prove a Sobolev space regularity result for weak solutions of elliptic equations. This result will be used for the construction of \({L^p}\)-strong Feller processes in the next chapter. It will be also used for the construction of the boundary local time.

An existence result for a class of quasilinear singular competitive elliptic systems

Large solutions for a cooperative elliptic system of [formula omitted]-Laplacian equations

We consider, for a bounded open domain Ω in Rn and a function u : Ω → Rm, the quasilinear elliptic system: (1). We generalize the system (QES)(f,g) in considering a right hand side depending on the jacobian matrix Du. Here, the star in (QES)(f,g) indicates that f may depend on Du. In the right hand side, v belongs to the dual space W-1,P’(Ω, ω*, Rm), , f and g satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for σ, but with only very mild monotonicity assumptions.

It is well known that distributional solutions of an elliptic equation with constant coefficients behave asymptotically near an interior point as sums of polynomials and linear combinations of derivatives of a fundamental solution. We consider a class of quasilinear elliptic systems and give mild conditions ensuring the same asymptotic behaviour. The sharpness of our conditions is illustrated by examples. The results are obtained as corollaries of a general theorem on the asymptotics of solutions to nonlinear ordinary differential equations in Banach spaces.

We obtain a result contained in [10] in a different manner. An optimal sufficient condition is supplied in order to get the interior Holder re gularity of a weak H1-solution to a linear elliptic system with measurable bounded coefficients. Explicit estimate of the norm of the gradient of a solution in a suitable weighted Morrey space is given. The condition obtained is then applied to quasilinear and nonlinear elliptic systems.

We present a simple proof of the existence and uniqueness of a weak solution for a class of quasilinear elliptic reaction-diffusion systems. The proof is based on an iterative process and on some a priori estimates.