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The Dirichlet problem for possibly singular elliptic equations with degenerate coercivity
Full text linkWe deal with existence, uniqueness and regularity of nonnegative solutions to
a Dirichlet problem for equations as \begin{equation*} \displaystyle
-\operatorname{div}\left(\frac{|\nabla u|^{p-2}\nabla
u}{(1+u)^{\theta(p-1)}}\right) = h(u)f \quad \text{in }\Omega, \end{equation*}
where is an open bounded subset of (), ,
, belongs to a suitable Lebesgue space and is a
continuous, nonnegative function which may blow up at zero and it is bounded at
infinity.Comment: 37 page
Elliptic equations with general singular lower order term and measure data
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Abstract
In this paper we study a nonlinear elliptic boundary value problem with a general singular lower order term, whose model is {-Δu=H(u)μinΩ,u=0on∂Ω,u>0onΩ, where Ω is an open bounded subset of RN, μ is a nonnegative bounded Radon measure on Ω and H is a continuous positive function outside the origin such that lims→0+H(s)=+∞. We do not require any monotonicity property on the singular function H
Finite energy solutions for nonlinear elliptic equations with competing gradient, singular and terms
Full text linkIn this paper we deal with the following boundary value problem
\begin{equation*}
\begin{cases}
-Δ_{p}u + g(u) | \nabla u|^{p} = h(u)f & \text{in ,} \newline
u\geq 0 & \text{in ,} \newline
u=0 & \text{on ,} \
\end{cases}
\end{equation*}
in a domain , where , is a positive and continuous function on , and is a continuous function on (possibly blowing up at the origin). We show how the presence of regularizing terms and allows to prove existence of finite energy solutions for nonnegative data only belonging to
Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data
Full text linkWe study existence and uniqueness of solutions to a nonlinear elliptic
boundary value problem with a general, and possibly singular, lower order term,
whose model is Here is an open bounded
subset of (), () is the -laplacian operator, is a
nonnegative bounded Radon measure on and is a continuous,
positive and finite function outside the origin which grows at most as
, with , near zero
Finite and Infinite energy solutions of singular elliptic problems: existence and uniqueness
No full textWe establish existence and uniqueness of solution for the homogeneous Dirichlet problem associated to a fairly general class of elliptic equations modeled by
where is an irregular datum, possibly a measure, and 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
On a singular elliptic equation with a general measure source
Full text linkWe prove existence of solutions for a class of singular elliptic problems with a general measure as source term whose model is {-Δu = f(x)/uγ + μ in Ω, u = 0 on ∂Ω, u > 0 on Ω, where Ω is an open bounded subset of RN. Here γ > 0, f is a nonnegative function on Ω, and μ is a nonnegative bounded Radon measure on Ω
On the behaviour of the first eigenvalue of the -Laplacian with Robin boundary conditions as goes to
Full text linkIn this paper we study the -limit, as , of the functional
where is a smooth bounded open set in , and
is a real number. Among our results, for , we derive an
isoperimetric inequality for
which is the limit as of We show that among all
bounded and smooth open sets with given volume, the ball maximizes
when and minimizes
when
The Dirichlet problem for the -Laplacian with a general singular term and -data
No full textsee attached fil
Behaviour of solutions to -Laplacian with Robin boundary conditions as goes to
Full text linkWe study the asymptotic behaviour, as , of the solutions of the
following inhomogeneous Robin boundary value problem: \begin{equation}
\label{pbabstract} \tag{P}
\left\{\begin{array}{ll}
\displaystyle -\Delta_p u_p = f & \text{in }\Omega,
\displaystyle |\nabla u_p|^{p-2}\nabla u_p\cdot \nu +\lambda |u_p|^{p-2}u_p =
g& \text{on } \partial\Omega,
\end{array}\right. \end{equation} where is a bounded domain in
with sufficiently smooth boundary, is its unit outward
normal vector and is the -Laplacian operator with . The
data (which denotes the Marcinkiewicz space) and
are bounded functions defined on with
. We find the threshold below which the family of --solutions
goes to 0 and above which this family blows up. As a second interest we deal
with the -Laplacian problem formally arising by taking in
\eqref{pbabstract}
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