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Register a new userOn the solvability of an indefinite nonlinear Kirchhoff equation via associated eigenvalue problems
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We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% begin{equation*} left{ begin{array}{l} -Mleft( int_{mathbb{R}^{3}}leftvert abla urightvert ^{2}dxright) Delta u+mu Vleft( xright) u=Q(x)leftvert urightvert ^{p-2}u+lambda fleft( xright) utext{ in }mathbb{R}^{N}, uin H^{1}left( mathbb{R}^{N}right) ,% end{array}% right. end{equation*}% where $Ngeq 3,2<p<2^{ast }:=frac{2N}{N-2},Mleft( tright) =at+b$ $left( a,b>0right) ,$ the potential $V$ is a nonnegative function in $mathbb{R}% ^{N}$ and the weight function $Qin L^{infty }left( mathbb{R}^{N}right) $ with changes sign in $overline{Omega }:=left{ V=0right} .$ We mainly prove the existence of at least two positive solutions in the cases that $% left( iright) $ $2<p<min left{ 4,2^{ast }right} $ and $0<lambda <% left[ 1-2left[ left( 4-pright) /4right] ^{2/p}right] lambda _{1}left( f_{Omega }right) ;$ $left( iiright) $ $pgeq 4,lambda geq lambda _{1}left( f_{Omega }right) $ and near $lambda _{1}left( f_{Omega }right) $ for $mu >0$ sufficiently large, where $lambda _{1}left( f_{Omega }right) $ is the first eigenvalue of $-Delta $ in $% H_{0}^{1}left( Omega right) $ with weight function $f_{Omega }:=f|_{% overline{Omega }},$ whose corresponding positive principal eigenfunction is denoted by $phi _{1}.$ Furthermore, we also investigated the non-existence and existence of positive solutions if $a,lambda $ belongs to different intervals.
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We consider the global solvability to the Cauchy problem of Kirchhoff equation with generalized classes of Manfrins class. Manfrins class is a subclass of Sobolev space, but we shall extend this class as a subclass of the ultradifferentiable class, and we succeed to prove the global solvability of Kirchhoff equation with large data in wider classes from the previous works.
It is established existence and multiplicity of solutions for strongly nonlinear problems driven by the $Phi$-Laplacian operator on bounded domains. Our main results are stated without the so called $Delta_{2}$ condition at infinity which means that the underlying Orlicz-Sobolev spaces are not reflexive.
This paper is devoted to proving the almost global solvability of the Cauchy problem for the Kirchhoff equation in the Gevrey space $gamma^s_{eta,L^2}$. Furthermore, similar results are obtained for the initial-boundary value problems in bounded domains and in exterior domains with compact boundary.
This article is devoted to review the known results on global well-posedness for the Cauchy problem to the Kirchhoff equation and Kirchhoff systems with small data. Similar results will be obtained for the initial-boundary value problems in exterior domains with compact boundary. Also, the known results on large data problems will be reviewed together with open problems.
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 $lambda^{+}_{a}geq0$.
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