315 research outputs found
Existence, uniqueness, and decay results for singular -Laplacian systems in
Existence of solutions to a -Laplacian singular system is obtained via
shifting method and variational methods. A priori estimates are furnished
through De Giorgi's technique, Talenti's rearrangement argument, and exploiting
the weak Harnack inequality, while decay of solutions is obtained via
comparison with radial solutions to auxiliary problems. Finally, uniqueness is
investigated, and a Diaz-Saa type result is provided
Liouville rigidity and time-extrinsic Harnack estimates for an anisotropic slow diffusion
We prove that ancient non-negative solutions to a fully anisotropic prototype
evolution equation are constant if they satisfy a condition of finite speed of
propagation and if they are both one-sided bounded, and bounded in space at a
single time level. A similar statement is valid when the bound is given at a
single space point. As a general paradigm, H\"older estimates provide the
basics for rigidity. Finally, we show that recent intrinsic Harnack estimates
can be improved to a Harnack inequality valid for non-intrinsic times. Locally,
they are equivalent.Comment: 15 page
A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form
For solutions of we show that the quasiconformality
of is the key property leading to the Sobolev regularity of
the stress field , in relation with the summability of . This class
of nonlinearities encodes in a general way the notion of uniform ellipticity
and encompasses all known instances where the stress field is known to be
Sobolev regular. We provide examples showing the optimality of this assumption
and present three applications: the study of the strong locality of the
operator , a nonlinear Cordes condition for equations in
divergence form, and some partial results on the -conjecture.Comment: Amended version, applications on locality removed due to a flaw in
the previous proof of Lemma 3.
The sub-supersolution method for variable exponent double phase systems with nonlinear boundary conditions
In this paper we study quasilinear elliptic systems driven by variable
exponent double phase operators involving fully coupled right-hand sides and
nonlinear boundary conditions. The aim of our work is to establish an enclosure
and existence result for such systems by means of trapping regions formed by
pairs of sup- and supersolutions. Under very general assumptions on the data we
then apply our result to get infinitely many solutions. Moreover, we also
discuss the case when we have homogeneous Dirichlet boundary conditions and
present some existence results for this kind of problem
Some recent results on singular -Laplacian systems
Some recent existence, multiplicity, and uniqueness results for singular p-Laplacian systems either in bounded domains or in the whole space are presented, with a special attention to the case of convective reactions. A extensive bibliography is also provided
Some recent results on singular p-Laplacian equations
A short account of some recent existence, multiplicity, and uniqueness results for singular p-Laplacian problems either in bounded domains or in the whole space is performed, with a special attention to the case of convective reactions. An extensive bibliography is also provided
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