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On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials
Part of
- Elliptic equations and systems
- Partial differential equations
Jianfu Yang, Jinge Yang
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 150 / Issue 6 / December 2020

Published online by Cambridge University Press:

02 December 2019, pp. 3187-3215

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December 2020
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In this paper, we study the existence and concentration of normalized solutions to the supercritical nonlinear Schrödinger equation
\[ \left\{\begin{array}{@{}ll} -\Delta u + V(x) u = \mu_q u + a \vert u \vert ^q u & {\rm in}\ \mathbb{R}^2,\\ \int_{\mathbb{R}^2} \vert u \vert ^2\,{\rm d}x =1, & \end{array} \right.\]
where μq is the Lagrange multiplier. For ellipse-shaped potentials V(x), we show that for q > 2 close to 2, the equation admits an excited solution uq, and furthermore, we study the limiting behaviour of uq when q → 2+. Particularly, we describe precisely the blow-up formation of the excited state uq.

Variational analysis for a nonlinear elliptic problem on theSierpiński gasket
Gabriele Bonanno, Giovanni Molica Bisci, Vicenţiu Rădulescu
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 18 / Issue 4 / October 2012

Published online by Cambridge University Press:

16 January 2012, pp. 941-953

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October 2012
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Under an appropriate oscillating behaviour either at zero or at infinity of the nonlinearterm, the existence of a sequence of weak solutions for an eigenvalue Dirichlet problem onthe Sierpiński gasket is proved. Our approach is based on variational methods and on someanalytic and geometrical properties of the Sierpiński fractal. The abstract results areillustrated by explicit examples.

BOUNDARY BEHAVIOR OF SUPERHARMONIC FUNCTIONS SATISFYING NONLINEAR INEQUALITIES IN A PLANAR SMOOTH DOMAIN
Part of
- Two-dimensional theory
- Elliptic equations and systems
KENTARO HIRATA
Journal:

Journal of the Australian Mathematical Society / Volume 87 / Issue 2 / October 2009

Published online by Cambridge University Press:

09 October 2009, pp. 253-261

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October 2009
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This paper presents a sharp boundary growth estimate for all positive superharmonic functions u in a smooth domain Ω in ℝ2 satisfying the nonlinear inequality where c>0, α∈ℝ and p>0, and δΩ(x) stands for the distance from a point x to the boundary of Ω. A result is applied to show the existence of nontangential limits of such superharmonic functions.

EXISTENCE OF POSITIVE EVANESCENT SOLUTIONS TO SOME QUASILINEAR ELLIPTIC EQUATIONS
Part of
- General theory in ordinary differential equations
- Elliptic equations and systems
OCTAVIAN G. MUSTAFA
Journal:

Bulletin of the Australian Mathematical Society / Volume 78 / Issue 1 / August 2008

Published online by Cambridge University Press:

01 August 2008, pp. 157-162

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August 2008
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We establish that the elliptic equation defined in an exterior domain of ℝn, n≥3, has a positive solution which decays to 0 as under quite general assumptions upon f and g.

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