Singular boundary behaviour and large solutions for fractional elliptic equations

We perform a unified analysis for the boundary behaviour of solutions to nonlocal fractional equations posed in bounded domains. Based on previous findings for some models of the fractional Laplacian operator, we show how it strongly differs from the boundary behaviour of solutions to elliptic problems modelled upon the Laplace–Poisson equation with zero boundary data. In the classical case it is known that, at least in a suitable weak sense, solutions of the homogeneous Dirichlet problem with a forcing term tend to zero at the boundary. Limits of these solutions then produce solutions of some non-homogeneous Dirichlet problem as the interior data concentrate suitably to the boundary. Here, we show that, for equations driven by a wide class of nonlocal fractional operators, different blow-up phenomena may occur at the boundary of the domain. We describe such explosive behaviours and obtain precise quantitative estimates depending on simple parameters of the nonlocal operators. Our unifying technique is based on a careful study of the inverse operator in terms of the corresponding Green function

On explosive solutions for a class of quasi-linear elliptic equations

We study existence, uniqueness, multiplicity and symmetry of large solutions for a class of quasi-linear elliptic equations. Furthermore, we characterize the boundary blow-up rate of solutions, including the case where the contribution of boundary curvature appears.Comment: 34 page

On Uniqueness of Boundary Blow-up Solutions of a Class of Nonlinear Elliptic Equations

We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.Comment: To appear in Comm. Partial Differential Equations; 10 page

Uniform bounds for higher-order semilinear problems in conformal dimension

We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem \begin{equation} \begin{cases} (-\Delta)^m u=h(x,u)\quad&\mbox{in }\Omega,\\ u=\partial_nu=\cdots=\partial_n^{m-1}u=0\quad&\mbox{on }\partial\Omega, \end{cases} \end{equation} where $h$ is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger-Moser-Adams inequality, either when $\Omega$ is a ball or, provided an energy control on solutions is prescribed, when $\Omega$ is a smooth bounded domain. The analogue problem with Navier boundary conditions is also studied. Finally, as a consequence of our results, existence of a positive solution is shown by degree theory.Comment: Minor correction

Uniform bounds for higher-order semilinear problems in conformal dimension

We establish uniform a-priori estimates for solutions of the semilinear Dirichlet problem \begin{equation} \begin{cases} (-\Delta)^m u=h(x,u)\quad&\mbox{in }\Omega,\\ u=\partial_nu=\cdots=\partial_n^{m-1}u=0\quad&\mbox{on }\partial\Omega, \end{cases} \end{equation} where $h$ is a positive superlinear and subcritical nonlinearity in the sense of the Trudinger-Moser-Adams inequality, either when $\Omega$ is a ball or, provided an energy control on solutions is prescribed, when $\Omega$ is a smooth bounded domain. The analogue problem with Navier boundary conditions is also studied. Finally, as a consequence of our results, existence of a positive solution is shown by degree theory.Comment: Minor correction