3,447 research outputs found
About curvature, conformal metrics and warped products
We consider the curvature of a family of warped products of two
pseduo-Riemannian manifolds and furnished with metrics of
the form and, in particular, of the type , where are smooth
functions and is a real parameter. We obtain suitable expressions for the
Ricci tensor and scalar curvature of such products that allow us to establish
results about the existence of Einstein or constant scalar curvature structures
in these categories. If is Riemannian, the latter question involves
nonlinear elliptic partial differential equations with concave-convex
nonlinearities and singular partial differential equations of the
Lichnerowicz-York type among others.Comment: 32 pages, 3 figure
A nonlinear parabolic problem with singular terms and nonregular data
We study existence of nonnegative solutions to a nonlinear parabolic boundary value problem with a general singular lower order term and a nonnegative measure as nonhomogeneous datum, of the form
\begin{cases}
\dys u_t - \Delta_p u = h(u)f+\mu & \text{in}\ \Omega \times (0,T),\\
u=0 &\text{on}\ \partial\Omega \times (0,T),\\
u=u_0 &\text{in}\ \Omega \times \{0\},
\end{cases}
where is an open bounded subset of (), is a nonnegative integrable function, is the -Laplace operator, is a nonnegative bounded Radon measure on and is a nonnegative function of . The term is a positive continuous function possibly blowing up at the origin. Furthermore, we show uniqueness of finite energy solutions in presence of a nonincreasing
Multiple positive solutions of a Sturm-Liouville boundary value problem with conflicting nonlinearities
We study the second order nonlinear differential equation \begin{equation*}
u"+ \sum_{i=1}^{m} \alpha_{i} a_{i}(x)g_{i}(u) - \sum_{j=0}^{m+1} \beta_{j}
b_{j}(x)k_{j}(u) = 0, \end{equation*} where ,
are non-negative Lebesgue integrable functions defined in
, and the nonlinearities are
continuous, positive and satisfy suitable growth conditions, as to cover the
classical superlinear equation , with . When the positive
parameters are sufficiently large, we prove the existence of at
least positive solutions for the Sturm-Liouville boundary value
problems associated with the equation. The proof is based on the Leray-Schauder
topological degree for locally compact operators on open and possibly unbounded
sets. Finally, we deal with radially symmetric positive solutions for the
Dirichlet problems associated with elliptic PDEs.Comment: 23 pages, 6 PNG figure
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