Existence results to elliptic systems with nonstandard growth conditions

Nonstandard growth conditions in partial differential equations have been the subject of recent developments in elastic mechanics and electrorheological fluid dynamics [Lecture Notes in Mathematics, vol. 1748, 2000; C. R. Acad. Sci. Paris Sér. I Math. 329 (1999) 393-398; Math. USSR Izv. 29 (1987) 33-66]. In this work, elliptic systems with nonstandard growth conditions are studied. Existence and multiplicity results, under growth conditions on the reaction terms, are established.     

Quasilinear elliptic equations with subquadratic growth

In this paper we consider nonlinear boundary value problems whose simplest model is the following:

(0.1)     

Singular solutions of elliptic equations with nonstandard growth

We prove a removability result for nonlinear elliptic equations withp (x)‐type nonstandard growth and estimate the growth of solutions near a nonremovable isolated singularity. To accomplish this, we employ a Harnack estimate for possibly unbounded solutions and the fact that solutions with nonremovable isolated singularities are p (x)‐superharmonic functions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multidimensional BSDEs with weak monotonicity and general growth generators

This paper aims at solving a multidimensional backward stochastic differential equation （BSDE） whose generator g satisfies a weak monotonicity condition and a general growth condition in y. We first establish an existence and uniqueness result of solutions for this kind of BSDEs by using systematically the technique of the priori estimation, the convolution approach, the iteration, the truncation and the Bihari inequality. Then, we overview some assumptions related closely to the monotonieity condition in the literature and compare them in an effective way, which yields that our existence and uniqueness result really and truly unifies the Mao condition in y and the monotonieity condition with the general growth condition in y, and it generalizes some known results. Finally, we prove a stability theorem and a comparison theorem for this kind of BSDEs, which also improves some known results.     

Quasilinear elliptic equations at critical growth

The existence of nontrivial solutions of quasilinear elliptic equations at critical growth is proved. The solutions are obtained by variational methods: as the corresponding functional is nonsmooth, the analysis of Palais-Smale sequences requires suitable generalizations of the techniques involved in the study of the corresponding semilinear problem with lack of compactness. Received September 20, 1996     

Boundary continuity of solutions to elliptic equations with nonstandard growth

We study regularity properties of weak solutions to elliptic equations involving variable growth exponents. We prove the sufficiency of a Wiener type criterion for the regularity of boundary points. This criterion is formulated in terms of the natural capacity involving the variable growth exponent. We also prove the Hölder continuity of weak solutions up to the boundary in domains with uniformly fat complements, provided that the boundary values are Hölder continuous.     

Deterministic homogenization of nonlinear degenerate elliptic operators with nonstandard growth

We study the deterministic homogenization of nonlinear degenerate elliptic equations with nonstandard growth.One fundamental in this topic is to extend the classical compactness results of theΣ-convergence method to the Orlicz spaces.We also show that one can homogenize nonlinear Dirichlet problems in a general way by leaning on a simple abstract hypothesis contrary to what has been done in the determinstic homogenization theory.     

Boundedness of solutions for anisotropic degenerate elliptic equations with nonstandard growth

Under nonstandard growth conditions, following Moser's iteration technique, we prove boundedness of solutions for fourth order nonlinear elliptic equation in divergence form.     

Study of weak solutions for parabolic equations with nonstandard growth conditions

The authors of this paper study the existence and uniqueness of weak solutions of the initial and boundary value problem for ut=div((uσ+d₀)|∇u|^p(x,t)−2∇u)+f(x,t). Localization property of weak solutions is also discussed.     

Quasilinear elliptic equations with boundary blow-up

Assume that Ω is a bounded domain in ℝ _N withN ≥2, which has aC ²-boundary. We show that forp ∃ (1, ∞) there exists a weak solutionu of the problem δ_p u(x) = f(u(x)), x ∃ Ω with boundary blow-up, wheref is a positive, increasing function which meets some natural conditions. The boundary blow-up ofu(x) is characterized in terms of the distance ofx from ∂Ω. For the Laplace operator, our results coincide with those of Bandle and Essén [1]. Finally, for a rather wide subclass of the class of the admissible functionsf, the solution is unique whenp ∃ (1, 2].     

Quasilinear elliptic problems with multivalued terms

We study the quasilinear elliptic problem with multivalued terms.We consider the Dirichlet problem with a multivalued term appearing in the equation and a problem of Neumann type with a multivalued term appearing in the boundary condition. Our approach is based on Szulkin's critical point theory for lower semicontinuous energy functionals.