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Register a new userBounded Solutions to nonlinear problems involving the fractional laplacian depending on parameters
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The main goal of this paper is the study of two kinds of nonlinear problems depending on parameters in unbounded domains. Using a nonstandard variational approach, we first prove the existence of bounded solutions for nonlinear eigenvalue problems involving the fractional Laplace operator and nonlinearities that have subcritical growth. In the second part, based on a variational principle of Ricceri [16], we study a fractional nonlinear problem with two parameters and prove the existence of multiple solutions.
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We establish the existence and multiplicity of positive solutions to the problems involving the fractional Laplacian: begin{equation*} left{begin{array}{lll} &(-Delta)^{s}u=lambda u^{p}+f(u),,,u>0 quad &mbox{in},,Omega, &u=0quad &mbox{in},,mathbb{R}^{N}setminusOmega, end{array}right. end{equation*} where $Omegasubset mathbb{R}^{N}$ $(Ngeq 2)$ is a bounded smooth domain, $sin (0,1)$, $p>0$, $lambdain mathbb{R}$ and $(-Delta)^{s}$ stands for the fractional Laplacian. When $f$ oscillates near the origin or at infinity, via the variational argument we prove that the problem has arbitrarily many positive solutions and the number of solutions to problem is strongly influenced by $u^{p}$ and $lambda$. Moreover, various properties of the solutions are also described in $L^{infty}$- and $X^{s}_{0}(Omega)$-norms.
We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,mu_X)$ satisfying a $2$-Poincare inequality. Given a bounded domain $Omegasubset X$ with $mu_X(XsetminusOmega)>0$, and a function $f$ in the Besov class $B^theta_{2,2}(X)cap L^2(X)$, we study the problem of finding a function $uin B^theta_{2,2}(X)$ such that $u=f$ in $XsetminusOmega$ and $mathcal{E}_theta(u,u)le mathcal{E}_theta(h,h)$ whenever $hin B^theta_{2,2}(X)$ with $h=f$ in $XsetminusOmega$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally Holder continuous on $Omega$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups.
In this paper, we study existence of boundary blow-up solutions for elliptic equations involving regional fractional Laplacian. We also discuss the optimality of our results.
In this paper we deal with the multiplicity of positive solutions to the fractional Laplacian equation begin{equation*} (-Delta)^{frac{alpha}{2}} u=lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{alpha}-2}u, quadtext{in},,Omega, u=0,text{on},,partialOmega, end{equation*} where $Omegasubset mathbb{R}^{N}(Ngeq 2)$ is a bounded domain with smooth boundary, $0<alpha<2$, $(-Delta)^{frac{alpha}{2}}$ stands for the fractional Laplacian operator, $fin C(Omegatimesmathbb{R},mathbb{R})$ may be sign changing and $lambda$ is a positive parameter. We will prove that there exists $lambda_{*}>0$ such that the problem has at least two positive solutions for each $lambdain (0,,,lambda_{*})$. In addition, the concentration behavior of the solutions are investigated.
In this paper, we establish a new improved Sobolev inequality based on a weighted Morrey space. To be precise, there exists $C=C(n,m,s,alpha)>0$ such that for any $u,v in {dot{H}}^s(mathbb{R}^{n})$ and for any $theta in (bar{theta},1)$, it holds that begin{equation} label{eq0.3} Big( int_{ mathbb{R}^{n} } frac{ |(uv)(y)|^{frac{2^*_{s}(alpha)}{2} } } { |y|^{alpha} } dy Big)^{ frac{1}{ 2^*_{s} (alpha) }} leq C ||u||_{{dot{H}}^s(mathbb{R}^{n})}^{frac{theta}{2}} ||v||_{{dot{H}}^s(mathbb{R}^{n})}^{frac{theta}{2}} ||(uv)||^{frac{1-theta}{2}}_{ L^{1,n-2s+r}(mathbb{R}^{n},|y|^{-r}) }, end{equation} where $s !in! (0,1)$, $0!<!alpha!<!2s!<!n$, $2s!<!m!<!n$, $bar{theta}=max { frac{2}{2^*_{s}(alpha)}, 1-frac{alpha}{s}cdotfrac{1}{2^*_{s}(alpha)}, frac{2^*_{s}(alpha)-frac{alpha}{s}}{2^*_{s}(alpha)-frac{2alpha}{m}} }$, $r=frac{2alpha}{ 2^*_{s}(alpha) }$ and $y!=!(y,y) in mathbb{R}^{m} times mathbb{R}^{n-m}$. By using mountain pass lemma and (ref{eq0.3}), we obtain a nontrivial weak solution to a doubly critical system involving fractional Laplacian in $mathbb{R}^{n}$ with partial weight in a direct way. Furthermore, we extend inequality (ref{eq0.3}) to more general forms on purpose of studying some general systems with partial weight, involving p-Laplacian especially.
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