252 research outputs found
Equations involving fractional Laplacian operator: Compactness and application
In this paper, we consider the following problem involving fractional
Laplacian operator: \begin{equation}\label{eq:0.1} (-\Delta)^{\alpha} u=
|u|^{2^*_\alpha-2-\varepsilon}u + \lambda u\,\, {\rm in}\,\, \Omega,\quad u=0
\,\, {\rm on}\, \, \partial\Omega, \end{equation} where is a smooth
bounded domain in , ,
. We show that for any
sequence of solutions of \eqref{eq:0.1} corresponding to
, satisfying in the
Sobolev space defined in \eqref{eq:1.1a}, converges strongly in
provided that and . An application of this compactness
result is that problem \eqref{eq:0.1} possesses infinitely many solutions under
the same assumptions.Comment: 34 page
Local uniqueness of vortices for 2D steady Euler flow
We study the steady planar Euler flow in a bounded simply connected domain,
where the vortex function is with and the vorticity strength is
prescribed. By studying the location and local uniqueness of vortices, we prove
that the vorticity method and the stream function method actually give the same
solution. We also show that if the vorticity of flow is located near an
isolated minimum point and non-degenerate critical point of the Kirchhoff-Routh
function, it must be stable in the nonlinear sense.Comment: 47 pages. arXiv admin note: text overlap with arXiv:1703.0986
New solutions for nonlinear Schrödinger equations with critical nonlinearity
AbstractWe consider the following nonlinear Schrödinger equations in Rn{Δ2ÎuâV(r)u+up=0in Rn;u>0in Rn and uâH1(Rn), where V(r) is a radially symmetric positive function. In [A. Ambrosetti, A. Malchiodi, W.-M. Ni, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, Part I, Comm. Math. Phys. 235 (2003) 427â466], Ambrosetti, Malchiodi and Ni proved that if M(r)=rnâ1(V(r))p+1pâ1â12 has a nondegenerate critical point r0â 0, then a layered solution concentrating near r0 exists. In this paper, we show that if p=n+2nâ2 and the dimension n=3,4 or 5, another new type of solution exists: this solution has a layer near r0 and a bubble at the origin
EFFECT OF THE DOMAIN GEOMETRY ON THE EXISTENCE OF MULTIPEAK SOLUTIONS FOR AN ELLIPTIC PROBLEM
Abstract. In this paper, we construct multipeak solutions for a singularly perturbed Dirichlet problem. Under the conditions that the distance function d(x, ââŠ) has k isolated compact connected critical sets T1,..., Tk satisfying d(x, ââŠ) = cj = const., for all x â Tj, miniÌž=j d(Ti, Tj)> 2 max1â€jâ€k d(Tj, ââŠ), and the critical group of each critical set Ti is nontrivial, we construct a solution which has exactly one local maximum point in a small neighbourhood of Ti, i = 1,..., k. (1.1
Infinitely many solutions for p-Laplacian equation involving critical Sobolev growth
AbstractIn this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth:âÎpu=|u|pââ2u+ÎŒ|u|pâ2uin Ω,u=0on âΩ, provided N>p2+p, where Îp is the p-Laplacian operator, 1<p<N, pâ=pNNâp, ÎŒ>0 and Ω is an open bounded domain in RN
Infinitely many solutions for the Schrödinger equations in RN with critical growth
AbstractWe consider the following nonlinear problem in RN(0.1){âÎu+V(|y|)u=uN+2Nâ2,u>0, in RN;uâH1(RN), where V(r) is a bounded non-negative function, Nâ©Ÿ5. We show that if r2V(r) has a local maximum point, or local minimum point r0>0 with V(r0)>0, then (0.1) has infinitely many non-radial solutions, whose energy can be made arbitrarily large. As an application, we show that the solution set of the following problemâÎu=λu+uN+2Nâ2,u>0 on SN has unbounded energy, as long as λ<âN(Nâ2)4, Nâ©Ÿ5
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