We investigate which elliptic PDEs have the property that every viscosity supersolution belongs to a Sobolev space $$W^{1,q}_{loc}(\Omega )$$ , $$\Omega \subseteq \mathbb {R}^n$$ . The asymptotic cone of operator’s sublevel set seems be essential. It turns out much can said if we know how is related dominative p-Laplacian, with $$p = \frac{n-1}{n}q + 1$$ show that, in certain sense, this minima...

In the present paper we deal with a quasilinear elliptic equation involvinga critical Sobolev exponent on non-compact Finsler manifolds, i.e. Randers spaces. Under very general assumptions perturbation prove existence of non-trivial solution. The approach is based direct methods calculus variations. One key step to that energy functional associated problem weakly lower semicontinuous small ball...

In this work, we study the existence of non-trivial multiple solutions for a class of quasilinear elliptic systems equipped with concave-convex nonlinearities and critical growth terms in bounded domains. By using the variational method, especially Nehari manifold and Palais-Smale condition, we prove the existence and multiplicity results of positive solutions.

Abstract We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and doubling metric measure spaces. show that the strongly amv-harmonic are Hölder continuous for any exponent below one. More generally, we define class of finite amv-norm this belong a fractional Hajłasz–Sobolev space their blow-ups satisfy mean-value property. Furt...

Bilinear operators are investigated in the context of Sobolev spaces and various techniques useful in the study of their boundedness properties are developed. In particular, several classes of symbols for bilinear operators beyond the so called CoifmanMeyer class are considered. Some of the Sobolev space estimates obtained apply to both the bilinear Hilbert transform and its singular multiplier...

Weighted inequality theory for fractional integrals is a relatively less known branch of calculus that offers remarkable opportunities to simulate interdisciplinary processes. Basic weighted inequalities are often associated Hardy, Littlewood and Sobolev [6, 11], Caffarelli, Kohn Nirenberg [4], respectively Stein Weiss [12]. A key attempt in the present paper prove Stein–Weiss with lack symmetr...

We present a unified strategy to derive Hardy–Poincaré inequalities on bounded and unbounded domains. The approach allows proving general inequality from which the classical Poincaré Hardy immediately follow. extend idea more context of variable exponent Sobolev spaces. Surprisingly, despite well-known counterexamples Fan et al. (2005) [28] , we show that modular form is actually possible provi...