اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThree solutions for a fractional elliptic problems with critical and supercritical growth
91
0
0.0
(
0
)
تأليف
Jinguo Zhang
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
In this paper, we deal with the existence and multiplicity of solutions for the fractional elliptic problems involving critical and supercritical Sobolev exponent via variational arguments. By means of the truncation combining with the Moser iteration, we prove that the problems has at least three solutions.
قيم البحث
اقرأ أيضاً
We prove a global fractional differentiability result via the fractional Caccioppoli-type estimate for solutions to nonlinear elliptic problems with measure data. This work is in fact inspired by the recent paper [B. Avelin, T. Kuusi, G. Mingione, {e
m Nonlinear Calderon-Zygmund theory in the limiting case}, Arch. Rational. Mech. Anal. {bf 227}(2018), 663--714], that was devoted to the local fractional regularity for the solutions to nonlinear elliptic equations with right-hand side measure, of type $-mathrm{div}, mathcal{A}( abla u) = mu$ in the limiting case. Being a contribution to recent results of identifying function classes that solutions to such problems could be defined, our aim in this work is to establish a global regularity result in a setting of weighted fractional Sobolev spaces, where the weights are powers of the distance function to the boundary of the smooth domains.
We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases
where the solution for a corresponding linear equation is not known. By using a fractional order adaptation of this method, we show that the results of [LLLS20a, LLLS20b] remain valid for general power type nonlinearities.
The aim of this paper is to establish two results about multiplicity of solutions to problems involving the $1-$Laplacian operator, with nonlinearities with critical growth. To be more specific, we study the following problem $$ left{ begin{array}{l}
- Delta_1 u +xi frac{u}{|u|} =lambda |u|^{q-2}u+|u|^{1^*-2}u, quadtext{in }Omega, u=0, quadtext{on } partialOmega. end{array} right. $$ where $Omega$ is a smooth bounded domain in $mathbb{R}^N$, $N geq 2$ and $xi in{0,1}$. Moreover, $lambda > 0$, $q in (1,1^*)$ and $1^*=frac{N}{N-1}$. The first main result establishes the existence of many rotationally non-equivalent and nonradial solutions by assuming that $xi=1$, $Omega = {x in mathbb{R}^N,:,r < |x| < r+1}$, $Ngeq 2$, $N ot = 3$ and $r > 0$. In the second one, $Omega$ is a smooth bounded domain, $xi=0$, and the multiplicity of solutions is proved through an abstract result which involves genus theory for functionals which are sum of a $C^1$ functional with a convex lower semicontinuous functional.
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 $mu in m
athbb{R},,mu e 0$ 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 describe the asymptotic behavior of positive solutions $u_epsilon$ of the equation $-Delta u + au = 3,u^{5-epsilon}$ in $Omegasubsetmathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sen
se of Hebey and Vaugon and the functions $u_epsilon$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brezis and Peletier (1989). Similar results are also obtained for solutions of the equation $-Delta u + (a+epsilon V) u = 3,u^5$ in $Omega$.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
