We obtain some nonlocal characterizations for a class of variable exponent Sobolev spaces arising in nonlinear elasticity theory and in the theory of electrorheological fluids. We also get a singular limit formula extending Nguyen results to the anisotropic case.
Mean value formulas are of great importance in the theory of partial differential equations: many very useful results are drawn, for instance, from the well known equivalence between harmonic functions and mean value properties. In the nonlocal setti
ng of fractional harmonic functions, such an equivalence still holds, and many applications are now-days available. The nonlinear case, corresponding to the $p$-Laplace operator, has also been recently investigated, whereas the validity of a nonlocal, nonlinear, counterpart remains an open problem. In this paper, we propose a formula for the emph{nonlocal, nonlinear mean value kernel}, by means of which we obtain an asymptotic representation formula for harmonic functions in the viscosity sense, with respect to the fractional (variational) $p$-Laplacian (for $pgeq 2$) and to other gradient dependent nonlocal operators.
We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-Delta)^s_p$ with $sin(0,1)$ and $pge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $Omega$. We prove that local minimizers of th
e associated energy functional in the fractional Sobolev space $W^{s,p}_0(Omega)$ and in the weighted Holder space $C^0_s(overlineOmega)$, respectively, do coincide.
We obtain approximate convexity principles for solutions to some classes of nonlinear elliptic partial differential equations in convex domains involving approximately concave nonlinearities. Furthermore, we provide some applications to some meaningful special cases.
We investigate the stability of ground states to a nonlinear focusing Schrodinger equation in presence of a Kirchhoff term. Through a spectral analysis of the linearized operator about ground states, we show a modulation stability estimate of ground
states in the spirit of one due to Weinstein [{it SIAM J. Math. Anal.}, 16(1985),472-491].
We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $pge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $Omega$. By means of barriers, a nonlocal superposition pri
nciple, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted Holder regularity up to the boundary, that is, $u/d^sin C^alpha(overlineOmega)$ for some $alphain(0,1)$, $d$ being the distance from the boundary.
We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole
Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.
Ultrafunctions are a particular class of generalized functions defined on a hyperreal field $mathbb{R}^{*}supsetmathbb{R}$ that allow to solve variational problems with no classical solutions. We recall the construction of ultrafunctions and we study
the relationships between these generalized solutions and classical minimizing sequences. Finally, we study some examples to highlight the potential of this approach.
We provide new characterizations of Sobolev ad BV spaces in doubling and Poincare metric spaces in the spirit of the Bourgain-Brezis-Mironescu and Nguyen limit formulas holding in domains of R^N.
We establish improv
