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Register a new userSublinear elliptic problems with a Hardy potential
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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.
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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 $beta(beta-1)+mu=0$.
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.
The paper is concerned with the slightly subcritical elliptic problem with Hardy term [ left{ begin{aligned} -Delta u-mufrac{u}{|x|^2} &= |u|^{2^{ast}-2-epsilon}u &&quad text{in } Omega, u &= 0&&quad text{on } partialOmega, end{aligned} right. ] in a bounded domain $Omegasubsetmathbb{R}^N$ with $0inOmega$, in dimensions $Nge7$. We prove the existence of multi-bubble nodal solutions that blow up positively at the origin and negatively at a different point as $epsilonto0$ and $mu=epsilon^alpha$ with $alpha>frac{N-4}{N-2}$. In the case of $Omega$ being a ball centered at the origin we can obtain solutions with up to $5$ bubbles of different signs. We also obtain nodal bubble tower solutions, i.e. superpositions of bubbles of different signs, all blowing up at the origin but with different blow-up order. The asymptotic shape of the solutions is determined in detail.
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.
By means of variational methods we establish existence and multiplicity of solutions for a class of nonlinear nonlocal problems involving the fractional p-Laplacian and a combined Sobolev and Hardy nonlinearity at subcritical and critical growth.
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