Uniqueness of very weak solutions for a fractional filtration equation

We prove existence and uniqueness of distributional, bounded, nonnegative solutions to a fractional filtration equation in ${\mathbb R}^d$. With regards to uniqueness, it was shown even for more general equations in [19] that if two bounded solutions $u,w$ of (1.1) satisfy $u-w\in L^1({\mathbb R}^d\times(0,T))$, then $u=w$. We obtain here that this extra assumption can in fact be removed and establish uniqueness in the class of merely bounded solutions, provided they are nonnegative. Indeed, we show that a minimal solution exists and that any other solution must coincide with it. As a consequence, distributional solutions have locally-finite energy.Comment: Final version. To appear on Adv. Mat

A curve of positive solutions for an indefinite sublinear Dirichlet problem

We investigate the existence of a curve $q\mapsto u_{q}$, with $q\in(0,1)$, of positive solutions for the problem $(P_{a,q})$: $-\Delta u=a(x)u^{q}$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded and smooth domain of $\mathbb{R}^{N}$ and $a:\Omega\rightarrow\mathbb{R}$ is a sign-changing function (in which case the strong maximum principle does not hold). In addition, we analyze the asymptotic behavior of $u_{q}$ as $q\rightarrow0^{+}$ and $q\rightarrow1^{-}$. We also show that in some cases $u_{q}$ is the ground state solution of $(P_{a,q})$. As a byproduct, we obtain existence results for a singular and indefinite Dirichlet problem. Our results are mainly based on bifurcation and sub-supersolutions methods

Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations

We study the fully nonlinear elliptic equation $F(D^2u,Du,u,x) = f$ in a smooth bounded domain $\Omega$, under the assumption the nonlinearity $F$ is uniformly elliptic and positively homogeneous. Recently, it has been shown that such operators have two principal "half" eigenvalues, and that the corresponding Dirichlet problem possesses solutions, if both of the principal eigenvalues are positive. In this paper, we prove the existence of solutions of the Dirichlet problem if both principal eigenvalues are negative, provided the "second" eigenvalue is positive, and generalize the anti-maximum principle of Cl\'{e}ment and Peletier to homogeneous, fully nonlinear operators.Comment: 32 page

Asymptotic behavior and existence of solutions for singular elliptic equations

We study the asymptotic behavior, as $\gamma$ tends to infinity, of solutions for the homogeneous Dirichlet problem associated to singular semilinear elliptic equations whose model is $$-\Delta u=\frac{f(x)}{u^\gamma}\,\text{ in }\Omega,$$ where $\Omega$ is an open, bounded subset of \RN and $f$ is a bounded function. We deal with the existence of a limit equation under two different assumptions on $f$: either strictly positive on every compactly contained subset of $\Omega$ or only nonnegative. Through this study we deduce optimal existence results of positive solutions for the homogeneous Dirichlet problem associated to $$-\Delta v + \frac{|\nabla v|^2}{v} = f\,\text{ in }\Omega.$$Comment: 31 page