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Register a new userQualitative properties of singular solutions to semilinear elliptic problems
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We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
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This paper deals with existence and regularity of positive solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $Phi$-Laplacian operator. The proof of existence is based on a variant of the generalized Galerkin method that we developed inspired on ideas by Browder and a comparison principle. By using a kind of Moser iteration scheme we show $L^{infty}(Omega)$-regularity for positive solutions
We investigate qualitative properties of positive singular solutions of some elliptic systems in bounded and unbounded domains. We deduce symmetry and monotonicity properties via the moving plane procedure. Moreover, in the unbounded case, we study some cooperative elliptic systems involving critical nonlinearities in $mathbb{R}^n$.
Let $Omega subset mathbb{R}^N$ be a bounded domain and $delta(x)$ be the distance of a point $xin Omega$ to the boundary. We study the positive solutions of the problem $Delta u +frac{mu}{delta(x)^2}u=u^p$ in $Omega$, where $p>0, ,p e 1$ and $mu in mathbb{R},,mu e 0$ is smaller then the Hardy constant. The interplay between the singular potential and the nonlinearity leads to interesting structures of the solution sets. In this paper we first give the complete picture of the radial solutions in balls. In particular we establish for $p>1$ the existence of a unique large solution behaving like $delta^{- frac2{p-1}}$ at the boundary. In general domains we extend results of arXiv:arch-ive/1407.0288 and show that there exists a unique singular solutions $u$ such that $u/delta^{beta_-}to c$ on the boundary for an arbitrary positive function $c in C^{2+gamma}(partialOmega) , (gamma in (0,1)), c ge 0$. Here $beta_-$ is the smaller root of $beta(beta-1)+mu=0$.
We prove existence results of two solutions of the problem [ begin{cases} L(u)+u^{m-1}=lambda u^{p-1} & text{ in $Omega$}, quad u>0 &text{ in $Omega$}, quad u=0 & text{ on $partial Omega$}, end{cases} ] where $L(v)=-{rm div}(M(x) abla v)$ is a linear operator, $pin (2,2^{*}]$ and $lambda$ and $ m$ sufficiently large. Then their asymptotical limit as $mto +infty$ is investigated showing different behaviors.
In this paper we prove the following long-standing conjecture: stable solutions to semilinear elliptic equations are bounded (and thus smooth) in dimension $n leq 9$. This result, that was only known to be true for $nleq4$, is optimal: $log(1/|x|^2)$ is a $W^{1,2}$ singular stable solution for $ngeq10$. The proof of this conjecture is a consequence of a new universal estimate: we prove that, in dimension $n leq 9$, stable solutions are bounded in terms only of their $L^1$ norm, independently of the nonlinearity. In addition, in every dimension we establish a higher integrability result for the gradient and optimal integrability results for the solution in Morrey spaces. As one can see by a series of classical examples, all our results are sharp. Furthermore, as a corollary we obtain that extremal solutions of Gelfand problems are $W^{1,2}$ in every dimension and they are smooth in dimension $n leq 9$. This answers to two famous open problems posed by Brezis and Brezis-Vazquez.
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