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Register a new userMultiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity
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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.
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We investigate the existence of infinitely many radially symmetric solutions to the following problem $$(-Delta_p)^s u=g(u) textrm{ in } mathbb{R}^N, uin W^{s,p}(mathbb{R}^N),$$ where $sin (0,1)$, $2 leq p < infty$, $sp leq N $, $2 leq N in mathbb{N}$ and $(-Delta_p)^s$ is the fractional $p$-Laplacian operator. We treat both of cases $sp=N$ and $sp<N.$ The nonlinearity $g$ is a function of Berestycki-Lions type with critical exponential growth if $sp=N$ and critical polynomial growth if $sp<N$. We also prove the existence of a ground state solution for the same problem.
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.
It is well known that a single nonlinear fractional Schrodinger equation with a potential $V(x)$ and a small parameter $varepsilon $ may have a positive solution that is concentrated at the nondegenerate minimum point of $V(x)$. In this paper, we can find two different positive solutions for two weakly coupled fractional Schrodinger systems with a small parameter $varepsilon $ and two potentials $V_{1}(x)$ and $V_{2}(x)$ having the same minimum point are concentrated at the same point minimum point of $V_{1}(x)$ and $V_{2}left(xright) $. In fact that by using the energy estimates, Nehari manifold technique and the Lusternik-Schnirelmann theory of critical points, we obtain the multiplicity results for a class of fractional Laplacian system. Furthermore, the existence and nonexistence of least energy positive solutions are also explored.
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.
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.
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