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Register a new userThe effect of nonlocal term on the superlinear Kirchhoff type equations in $mathbb{R}^{N}$
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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.
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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 $Ngeq5$.
We investigate a class of Kirchhoff type equations involving a combination of linear and superlinear terms as follows: begin{equation*} -left( aint_{mathbb{R}^{N}}| abla u|^{2}dx+1right) Delta u+mu V(x)u=lambda f(x)u+g(x)|u|^{p-2}uquad text{ in }mathbb{R}^{N}, end{equation*}% where $Ngeq 3,2<p<2^{ast }:=frac{2N}{N-2}$, $Vin C(mathbb{R}^{N})$ is a potential well with the bottom $Omega :=int{xin mathbb{R}^{N} | V(x)=0}$. When $N=3$ and $4<p<6$, for each $a>0$ and $mu $ sufficiently large, we obtain that at least one positive solution exists for $% 0<lambdaleqlambda _{1}(f_{Omega}) $ while at least two positive solutions exist for $lambda _{1}(f_{Omega })< lambda<lambda _{1}(f_{Omega})+delta_{a}$ without any assumption on the integral $% int_{Omega }g(x)phi _{1}^{p}dx$, where $lambda _{1}(f_{Omega })>0$ is the principal eigenvalue of $-Delta $ in $H_{0}^{1}(Omega )$ with weight function $f_{Omega }:=f|_{Omega }$, and $phi _{1}>0$ is the corresponding principal eigenfunction. When $Ngeq 3$ and $2<p<min {4,2^{ast }}$, for $% mu $ sufficiently large, we conclude that $(i)$ at least two positive solutions exist for $a>0$ small and $0<lambda <lambda _{1}(f_{Omega })$; $% (ii)$ under the classical assumption $int_{Omega }g(x)phi _{1}^{p}dx<0$, at least three positive solutions exist for $a>0$ small and $lambda _{1}(f_{Omega })leq lambda<lambda _{1}(f_{Omega})+overline{delta }% _{a} $; $(iii)$ under the assumption $int_{Omega }g(x)phi _{1}^{p}dx>0$, at least two positive solutions exist for $a>a_{0}(p)$ and $lambda^{+}_{a}< lambda<lambda _{1}(f_{Omega})$ for some $a_{0}(p)>0$ and $lambda^{+}_{a}geq0$.
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 carry out an analysis of the existence of solutions for a class of nonlinear partial differential equations of parabolic type. The equation is associated to a nonlocal initial condition, written in general form which includes, as particular cases, the Cauchy multipoint problem, the weighted mean value problem and the periodic problem. The dynamic is transformed into an abstract setting and by combining an approximation technique with the Leray-Schauder continuation principle, we prove global existence results. By the compactness of the semigroup generated by the linear operator, we do not assume any Lipschitzianity, nor compactness on the nonlinear term or on the nonlocal initial condition. In addition, the exploited approximation technique coupled to a Hartman-type inequality argument, allows to treat nonlinearities with superlinear growth. Moreover, regarding the periodic case we are able to prove the existence of at least one periodic solution on the half line.
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 $Phi(t)=Phi^*(t)+frac{t^{n-1}}{(n-1)!}$).
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