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 $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$, $\varepsilon\in [0, 2^*_\alpha-2)$, $0<\alpha<1,\, 2^*_\alpha = \frac {2N}{N-2\alpha}$. We show that for any sequence of solutions $u_n$ of \eqref{eq:0.1} corresponding to $\varepsilon_n\in [0, 2^*_\alpha-2)$, satisfying $\|u_n\|_{H}\le C$ in the Sobolev space $H$ defined in \eqref{eq:1.1a}, $u_n$ converges strongly in $H$ provided that $N>6\alpha$ and $\lambda>0$. 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 $f=t_+^p$ with $p>0$ 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{ε2Δu−V(r)u+up=0in Rn;u>0in Rn and u∈H1(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) 427–466], Ambrosetti, Malchiodi and Ni proved that if M(r)=rn−1(V(r))p+1p−1−12 has a nondegenerate critical point r0≠0, then a layered solution concentrating near r0 exists. In this paper, we show that if p=n+2n−2 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, ∂Ω) has k isolated compact connected critical sets T1,..., Tk satisfying d(x, ∂Ω) = cj = const., for all x ∈ Tj, mini̸=j d(Ti, Tj)&gt; 2 max1≤j≤k d(Tj, ∂Ω), 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:−Δpu=|u|p⁎−2u+μ|u|p−2uin Ω,u=0on ∂Ω, provided N>p2+p, where Δp is the p-Laplacian operator, 1<p<N, p⁎=pNN−p, μ>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){−Δu+V(|y|)u=uN+2N−2,u>0, in RN;u∈H1(RN), where V(r) is a bounded non-negative function, N⩾5. 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 problem−Δu=λu+uN+2N−2,u>0 on SN has unbounded energy, as long as λ<−N(N−2)4, N⩾5