We study a Dirichlet problem for an elliptic equation defined by a degenerate coercive operator and a singular right-hand side. We will show that the right-hand side has some regularizing effects on the solutions, even if it is singular.
We solve the nonlinear Dirichlet problem (uniquely) for functions with prescribed asymptotic singularities at a finite number of points, and with arbitrary continuous boundary data, on a domain in euclidean space. The main results apply, in particular, to subequations with a Riesz characteristic $p geq 2$. In this case it is shown that, without requiring uniform ellipticity, the Dirichlet problem can be solved uniquely for arbitrary continuous boundary data with singularities asymptotic to the Riesz kernel: $Theta_j K_p(x - x_j)$, where $K_p(x) = - {1over|x|^{p-2}}$ for $p>2$ and $K_2(x) = log |x|$, at any prescribed finite set of points $x_1,...,x_k$ in the domain and any finite set of positive real numbers $Theta_1,..., Theta_k$. This sharpens a previous result of the authors concerning the discreteness of high-density sets of subsolutions. Uniqueness and existence results are also established for finite-type singularities such as $Theta_j |x - x_j|^{2-p}$ for $1leq p<2$. The main results apply similarly with prescribed singularities asymptotic to the fundamental solutions of Armstrong-Sirakov-Smart (in the uniformly elliptic case).
Given a smooth domain $OmegasubsetRR^N$ such that $0 in partialOmega$ and given a nonnegative smooth function $zeta$ on $partialOmega$, we study the behavior near 0 of positive solutions of $-Delta u=u^q$ in $Omega$ such that $u = zeta$ on $partialOmegasetminus{0}$. We prove that if $frac{N+1}{N-1} < q < frac{N+2}{N-2}$, then $u(x)leq C abs{x}^{-frac{2}{q-1}}$ and we compute the limit of $abs{x}^{frac{2}{q-1}} u(x)$ as $x to 0$. We also investigate the case $q= frac{N+1}{N-1}$. The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.
The existence and multiplicity of solutions for a class of non-local elliptic boundary value problems with superlinear source functions are investigated in this paper. Using variational methods, we examine the changes arise in the solution behaviours as a result of the non-local effect. Comparisons are made of the results here with those of the elliptic boundary value problem in the absence of the non-local term under the same prescribed conditions to highlight this effect of non-locality on the solution behaviours. Our results here demonstrate that the complexity of the solution structures is significantly increased in the presence of the non-local effect with the possibility ranging from no permissible positive solution to three positive solutions and, contrary to those obtained in the absence of the non-local term, the solution profiles also vary depending on the superlinearity of the source functions.
In this paper, we develop the Littman-Stampacchia-Weinberger duality approach to obtain global W^1,p estimates for a class of elliptic problems involving Leray-Hardy operators and measure sources in a distributional framework associated with a dual formulation with a specific weight function.