Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMulti-bubble nodal solutions to slightly subcritical elliptic problems with Hardy terms
62
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
In this paper, we consider the existence of nodal solutions with two bubbles to the slightly subcritical problem with the fractional Laplacian begin{equation*} left{aligned &(-Delta)^su=|u|^{p-1-varepsilon}u mbox{in} Omega &u=0 mbox{on} partialOmega, endaligned right. end{equation*} where $Omega$ is a smooth bounded domain in $mathbb R^N$, $N>2s$, $0<s<1$, $ p=frac{N+2s}{N-2s}$ and $varepsilon>0$ is a small parameter, which can be seen as a nonlocal analog of the results of Bartsch, Micheletti and Pistoia (2006) cite{Bartsch1}.
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 $beta(beta-1)+mu=0$.
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.
We study a nonlinear equation with an elliptic operator having degenerate coercivity. We prove the existence of a W^{1,1}_0 solution which is distributional or entropic, according to the growth assumptions on a lower order term in divergence form.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
