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Register a new userPositive solutions of semilinear elliptic problems with a Hardy potential
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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 $beta(beta-1)+mu=0$.
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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 study the positive solutions of sub linear elliptic equations with a Hardy potential which is singular at the boundary. By means of ODE techniques a fairly complete picture of the class of radial solutions is given. Local solutions with a prescribed growth at the boundary are constructed by means of contraction operators. Some of those radial solutions are then used to construct ordered upper and lower solutions in general domains. By standard iteration arguments the existence of positive solutions is proved. An important tool is the Hardy constant.
Let $Omega subset {mathbb R}^N$ ($N geq 3$) be a $C^2$ bounded domain and $delta$ be the distance to $partial Omega$. We study positive solutions of equation (E) $-L_mu u+ g(| abla u|) = 0$ in $Omega$ where $L_mu=Delta + frac{mu}{delta^2} $, $mu in (0,frac{1}{4}]$ and $g$ is a continuous, nondecreasing function on ${mathbb R}_+$. We prove that if $g$ satisfies a singular integral condition then there exists a unique solution of (E) with a prescribed boundary datum $ u$. When $g(t)=t^q$ with $q in (1,2)$, we show that equation (E) admits a critical exponent $q_mu$ (depending only on $N$ and $mu$). In the subcritical case, namely $1<q<q_mu$, we establish some a priori estimates and provide a description of solutions with an isolated singularity on $partial Omega$. In the supercritical case, i.e. $q_muleq q<2$, we demonstrate a removability result in terms of Bessel capacities.
We study a nonlinear equation in the half-space ${x_1>0}$ with a Hardy potential, specifically [-Delta u -frac{mu}{x_1^2}u+u^p=0quadtext{in}quad mathbb R^n_+,] where $p>1$ and $-infty<mu<1/4$. The admissible boundary behavior of the positive solutions is either $O(x_1^{-2/(p-1)})$ as $x_1to 0$, or is determined by the solutions of the linear problem $-Delta h -frac{mu}{x_1^2}h=0$. In the first part we study in full detail the separable solutions of the linear equations for the whole range of $mu$. In the second part, by means of sub and supersolutions we construct separable solutions of the nonlinear problem which behave like $O(x_1^{-2/(p-1)})$ near the origin and which, away from the origin have exactly the same asymptotic behavior as the separable solutions of the linear problem. In the last part we construct solutions that behave like $O(x_1^{-2/(p-1)})$ at some prescribed parts of the boundary, while at the rest of the boundary the solutions decay or blowup at a slower rate determined by the linear part of the equation.
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
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