Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Kirchhoff type equations in $mathbb{R}^{N}$
78
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Consider a nonlinear Kirchhoff type equation as follows begin{equation*} left{ begin{array}{ll} -left( aint_{mathbb{R}^{N}}| abla u|^{2}dx+bright) Delta u+u=f(x)leftvert urightvert ^{p-2}u & text{ in }mathbb{R}^{N}, uin H^{1}(mathbb{R}^{N}), & end{array}% right. end{equation*}% where $Ngeq 1,a,b>0,2<p<min left{ 4,2^{ast }right}$($2^{ast }=infty $ for $N=1,2$ and $2^{ast }=2N/(N-2)$ for $Ngeq 3)$ and the function $fin C(mathbb{R}^{N})cap L^{infty }(mathbb{R}^{N})$. Distinguishing from the existing results in the literature, we are more interested in the geometric properties of the energy functional related to the above problem. Furthermore, the nonexistence, existence, unique and multiplicity of positive solutions are proved dependent on the parameter $a$ and the dimension $N.$ In particular, we conclude that a unique positive solution exists for $1leq Nleq4$ while at least two positive solutions are permitted for $Ngeq5$.
rate research
Read More
We are concerned with a class of Kirchhoff type equations in $mathbb{R}^{N}$ as follows: begin{equation*} left{ begin{array}{ll} -Mleft( int_{mathbb{R}^{N}}| abla u|^{2}dxright) Delta u+lambda Vleft( xright) u=f(x,u) & text{in }mathbb{R}^{N}, uin H^{1}(mathbb{R}^{N}), & end{array}% right. end{equation*}% where $Ngeq 1,$ $lambda>0$ is a parameter, $M(t)=am(t)+b$ with $a,b>0$ and $min C(mathbb{R}^{+},mathbb{R}^{+})$, $Vin C(mathbb{R}^{N},mathbb{R}^{+})$ and $fin C(mathbb{R}^{N}times mathbb{R}, mathbb{R})$ satisfying $lim_{|u|rightarrow infty }f(x,u) /|u|^{k-1}=q(x)$ uniformly in $xin mathbb{R}^{N}$ for any $2<k<2^{ast}$($2^{ast}=infty$ for $N=1,2$ and $2^{ast}=2N/(N-2)$ for $Ngeq 3$). Unlike most other papers on this problem, we are more interested in the effects of the functions $m$ and $q$ on the number and behavior of solutions. By using minimax method as well as Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity of positive solutions for the above problem.
This paper is concerned with the existence of ground states for a class of Kirchhoff type equation with combined power nonlinearities begin{equation*} -left(a+bint_{mathbb{R}^{3}}| abla u(x)|^{2}right) Delta u =lambda u+|u|^{p-2}u+u^{5}quad text{for some} lambdainmathbb{R},quad xinmathbb{R}^{3}, end{equation*} with prescribed $L^{2}$-norm mass begin{equation*} int_{mathbb{R}^{3}}u^{2}=c^{2} end{equation*} in Sobolev critical case and proves that the equation has a couple of solutions $(u_{c},lambda_{c})in S(c)times mathbb{R}$ for any $c>0$, $a,b >0$ and $frac{14}{3}leq p< 6,$ where $S(c)={uin H^{1}(mathbb{R}^{3}):int_{mathbb{R}^{3}}u^{2}=c^{2}}.$ textbf{Keywords:} Kirchhoff type equation; Critical nonlinearity; Normalized ground states oindent{AMS Subject Classification:, 37L05; 35B40; 35B41.}
We establish the Trudinger-Moser inequality on weighted Sobolev spaces in the whole space, and for a class of quasilinear elliptic operators in radial form of the type $displaystyle Lu:=-r^{-theta}(r^{alpha}vert u(r)vert^{beta}u(r))$, where $theta, betageq 0$ and $alpha>0$, are constants satisfying some existence conditions. It worth emphasizing that these operators generalize the $p$- Laplacian and $k$-Hessian operators in the radial case. Our results involve fractional dimensions, a new weighted Polya-Szeg{o} principle, and a boundness value for the optimal constant in a Gagliardo-Nirenberg type inequality.
Let $W^{1,n} ( mathbb{R}^{n} $ be the standard Sobolev space and $leftVert cdotrightVert _{n}$ be the $L^{n}$ norm on $mathbb{R}^n$. We establish a sharp form of the following Trudinger-Moser inequality involving the $L^{n}$ norm [ underset{leftVert urightVert _{W^{1,n}left(mathbb{R} ^{n}right) }=1}{sup}int_{ mathbb{R}^{n}}Phileft( alpha_{n}leftvert urightvert ^{frac{n}{n-1}}left( 1+alphaleftVert urightVert _{n}^{n}right) ^{frac{1}{n-1}}right) dx<+infty ]in the entire space $mathbb{R}^n$ for any $0leqalpha<1$, where $Phileft( tright) =e^{t}-underset{j=0}{overset{n-2}{sum}}% frac{t^{j}}{j!}$, $alpha_{n}=nomega_{n-1}^{frac{1}{n-1}}$ and $omega_{n-1}$ is the $n-1$ dimensional surface measure of the unit ball in $mathbb{R}^n$. We also show that the above supremum is infinity for all $alphageq1$. Moreover, we prove the supremum is attained, namely, there exists a maximizer for the above supremum when $alpha>0$ is sufficiently small. The proof is based on the method of blow-up analysis of the nonlinear Euler-Lagrange equations of the Trudinger-Moser functionals. Our result sharpens the recent work cite{J. M. do1} in which they show that the above inequality holds in a weaker form when $Phi(t)$ is replaced by a strictly smaller $Phi^*(t)=e^{t}-underset{j=0}{overset{n-1}{sum}}% frac{t^{j}}{j!}$. (Note that $Phi(t)=Phi^*(t)+frac{t^{n-1}}{(n-1)!}$).
The existence of positive weak solutions to a singular quasilinear elliptic system in the whole space is established via suitable a priori estimates and Schauders fixed point theorem.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
