We establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by−Δu=h(u)finΩ, where f is an irregular ...datum, possibly a measure, and h is a continuous function that may blow up at zero. We also provide regularity results on both the solution and the lower order term depending on the regularity of the data, and we discuss their optimality.
In this paper we consider equations −|∇u|αM±(D2u)=|u|p−1u in an annulus. M± is one of the Pucci’s operators, α is some real >−1 and p>1+α. The solutions are intended in the sense of the definition ...given by Birindelli and Demengel (2006) We prove the existence of non trivial positive/negative radial solutions.
We consider a class of quasilinear Schrödinger equations which include the Modified Nonlinear Schrödinger Equations. A new perturbation approach is used to treat the critical exponent case giving new ...existence results.
We prove higher Hölder regularity for solutions of equations involving the fractional p-Laplacian of order s, when p≥2 and 0<s<1. In particular, we provide an explicit Hölder exponent for solutions ...of the non-homogeneous equation with data in Lq and q>N/(sp), which is almost sharp whenever sp≤(p−1)+N/q. The result is new already for the homogeneous equation.
We prove existence of solutions to problems whose model is{−Δpu+uq=fuγinΩ,u≥0inΩ,u=0on∂Ω, where Ω is an open bounded subset of RN (N≥2), Δpu is the p-laplacian operator for 1≤p<N, q>0, γ≥0 and f is a ...nonnegative function in Lm(Ω) for some m≥1. In particular we analyze the regularizing effect produced by the absorption term in order to infer the existence of finite energy solutions in case γ≤1. We also study uniqueness of these solutions as well as examples which show the optimality of the results. Finally, we find local W1,p-solutions in case γ>1.
Motivated by applications to congested traffic problems, we establish higher integrability results for the gradient of local weak solutions to the strongly degenerate or singular elliptic ...PDE−div((|∇u|−1)+q−1∇u|∇u|)=f,inΩ, where Ω is a bounded domain in Rn for n≥2, 1<q<∞ and (⋅)+ stands for the positive part. We assume that the datum f belongs to a suitable Sobolev or Besov space. The main novelty here is that we deal with the case of subquadratic growth, i.e. 1<q<2, which has so far been neglected. In the latter case, we also prove the higher fractional differentiability of the solution to a variational problem, which is characterized by the above equation. For the sake of completeness, we finally give a Besov regularity result also in the case q≥2.
We provide the classification of the positive solutions to −Δpu=up⁎−1 in D1,p(RN) in the case 2<p<N. Since the case 1<p≤2 is already known this provides the complete classification for 1<p<N.
This paper studies Sobolev regularity of weak solution of degenerate elliptic equations in divergence form divA(X)∇u=divF(X), where X=(x,y)∈Rn×R. The coefficient matrix A(X) is a symmetric, ...measurable (n+1)×(n+1) matrix, and it could be degenerate or singular in the one dimensional y-variable as a weight function in the A2 Muckenhoupt class. Our results give weighted Sobolev regularity estimates of Calderón–Zygmund type for weak solutions of this class of degenerate/singular equations. As an application of these estimates, we establish global fractional Sobolev regularity estimates for solutions of the spectral fractional elliptic equation with measurable coefficients. This result can be considered as the Sobolev counterpart of the recently established Schauder regularity theory of fractional elliptic equations.
We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by
−
Δ
p
u
=
f
u
γ
+
g
u
q
in
Ω
,
u
=
0
on
∂
Ω
,
where Ω is an open bounded subset of
ℝ
N
...where Ω is an open bounded subset of
ℝ
N
, Δ
p
u
:= ÷(|∇
u
|
p
− 2
∇
u
) is the usual
p
-Laplacian operator,
γ
≥ 0 and 0 ≤
q
≤
p
− 1;
f
and
g
are nonnegative functions belonging to suitable Lebesgue spaces.
We are concerned with positive maximal and minimal solutions for non-homogeneous elliptic equations of the form−div(a(|∇u|p)|∇u|p−2∇u)=f(x,u,∇u) in Ω, supplied with Dirichlet boundary conditions. ...First we localize maximal and minimal solutions between not necessarily bounded sub-super solutions. Then using a uniform gradient estimate, which seems of independent interest, we show the existence of positive maximal and minimal solutions in some situations. More precisely, we obtain positive maximal and minimal solution to some classes of non-homogeneous equations depending on the gradient which may be perturbed by unbounded, singular or logistic sources.
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