Quantization for an elliptic equation of order 2m with critical exponential non-linearity

On a smoothly bounded domain ${\Omega\subset\mathbb{R}^{2m}}$ we consider a sequence of positive solutions ${u_k\stackrel{w}{\rightharpoondown}0}$ in H ^m (Ω) to the equation ${(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}}$ subject to Dirichlet boundary conditions, where 0 < λ _k → 0. Assuming that $$0 < \Lambda:=\lim_{k\to\infty}\int\limits_\Omega u_k(-\Delta)^m u_k dx < \infty,$$ we prove that Λ is an integer multiple of Λ₁ := (2m ? 1)! vol(S ^2m ), the total Q-curvature of the standard 2m-dimensional sphere.     

Quantization for an evolution equation with critical exponential growth on a closed Riemann surface

In this paper, we analyze the concentration behavior of a positive solution to an evolution equation with critical exponential growth on a closed Riemann surface, and particularly derive an energy identity for such a solution. This extends a result of Lamm-Robert-Struwe and complements that of Yang.     

Morse theory for a fourth order elliptic equation with exponential nonlinearity

Given a Hilbert space (H,á·,·?){(\mathcal H,\langle\cdot,\cdot\rangle)}, and interval L ì (0,+￥){\Lambda\subset(0,+\infty)} and a map K ? C²(H,\mathbb R){K\in C^2(\mathcal H,\mathbb R)} whose gradient is a compact mapping, we consider the family of functionals of the type:

Multiplicity of solutions for a fourth order problem with exponential nonlinearity

Let B be the unit ball in RN, N?5 and n be the exterior unit normal vector on the boundary. We consider radial solutions to

     

Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity

We investigate entire radial solutions of the semilinear biharmonic equation Δ²u=λexp(u) in Rn, n?5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s=log|x|. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as |x|→∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x?−4log|x| plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n=5.     

Multiplicity of solutions for a fourth order equation with power-type nonlinearity

Let B be the unit ball in ${\mathbb{R}^N}Let B be the unit ball in \mathbbR^N{\mathbb{R}^N}, N ≥ 3 and n be the exterior unit normal vector on the boundary. We consider radial solutions to

D2 u = l(1+ sign(p)u)p     in  B,     u = 0,     \frac?u?n = 0     on  ?B\Delta^2 u = \lambda(1+ {\rm sign}(p)u)^{p} \quad {\rm in} \, B, \quad u = 0, \quad \frac{\partial{u}}{\partial{n}} = 0 \quad {\rm on} \, \partial B      10. Semiclassical ground state solutions for a Schrödinger equation in with critical exponential growth          下载免费PDF全文 Bo Tao  Zhibin Liu  Minbo Yang 《Mathematische Nachrichten》2016,289(5-6):727-747 Consider a Schrödinger equation where and are two continuous real functions on , ε is a positive parameter, the nonlinearity f is assumed to be of critical exponential growth in the sense of the Trudinger‐Moser inequality. By truncating the potentials and , we are able to establish some new existence and concentration results for critical Schrödinger equation in by variational methods. As a particular case, we observe that the concentration appears at the maximum set of the nonlinear potential which complements the results in 6 , 23 .      11. Monotone positive solutions for a fourth order equation with nonlinear boundary conditions      Edson Alves  Maurício Luciano Pelicer 《Nonlinear Analysis: Theory, Methods & Applications》2009,71(9):3834-3841 This work is concerned with the existence of monotone positive solutions for a class of beam equations with nonlinear boundary conditions. The results are obtained by using the monotone iteration method and they extend early works on beams with null boundary conditions. Numerical simulations are also presented.      12. Two homoclinic solutions for a nonperiodic fourth order differential equation with a perturbation      Juntao Sun  Tsung-fang Wu 《Journal of Mathematical Analysis and Applications》2014 In this paper, we study multiple homoclinic solutions for a class of fourth order differential equations with a perturbation. By establishing a compactness lemma and using variational methods, the existence result of two homoclinic solutions is obtained under some suitable assumptions, but not requiring the periodicity condition. Some recent results are improved and extended.      13. Fractional Kirchhoff equation with critical exponential nonlinearity      J. Giacomoni  Pawan Kumar Mishra  K. Sreenadh 《复变函数与椭圆型方程》2016,61(9):1241-1266       14. A boundedness theorem for a certain fourth order differential equation      Ezeilo  J. O. C.  Tejumola  H. O. 《Annali di Matematica Pura ed Applicata》1971,88(1):207-216 Summary The paper investigates conditions under which solutions of equations of the form(1.1) are bounded as t→+∞. Entrata in Redazione il 31 agosto 1970.      15. Perturbation at resonance for a fourth order ordinary differential equation      Jerry D Schuur 《Journal of Mathematical Analysis and Applications》1978,65(1):20-25       16. Qualitative analysis of a fourth order difference equation          下载免费PDF全文 Haza Saleh Alayachi  M. S. M. Noorani  E. M. Elsayed 《Journal of Applied Analysis & Computation》2020,10(4):1343-1354 In this paper, we will investigate some qualitative behavior of solutions of the following fourth order difference equation $x_{n+1}=ax_{n-1}+\frac{bx_{n-1}}{cx_{n-1}-dx_{n-3}},$ \ $n=0,1,...,$ where the initial conditions $x_{-3,}x_{-2},\ x_{-1}$\ and\ $x_{0}\ $are arbitrary real numbers and the values $a,\ b,\ c\ $and$\;d$ are\ defined as positive real numbers.      17. Blow-up phenomena for a class of fourth order parabolic equation      Xiangkun Shao  Guo-ji Tang 《Journal of Mathematical Analysis and Applications》2022,505(1):125445 A class of fourth order parabolic equation is studied in this paper. Some related blow-up results are obtained by applying the potential well theory, the concavity method and a series of differential-integral inequality techniques. More precisely, under some proper assumptions, the upper and lower bounds of the blow-up time and the growth rate for blow-up solutions are estimated. Moreover, a new blow-up condition independent of the depth of the potential well is found. These results complement the recent results obtained in Han (2018).      18. Positive solutions for a fourth order equation invariant under isometries      Fré  dé  ric Robert 《Proceedings of the American Mathematical Society》2003,131(5):1423-1431 Let be a smooth compact Riemannian manifold of dimension . We consider the problem where , , . We require to be positive and invariant under isometries. We prove existence results for on arbitrary compact manifolds. This includes the case of the geometric Paneitz-Branson operator on the sphere.       19. Nonlinear oscillation of a fourth order elliptic equation      Yuichi Kitamura  Takaŝi Kusano 《Journal of Differential Equations》1978,30(2):280-286       20. Multiple positive solutions for a quasilinear elliptic equation with critical exponential nonlinearity      Bhatia Sumit Kaur 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(8):2368-2382 Let Ω be a bounded domain in RN,N≥2, with C2 boundary. In this work, we study the existence of multiple positive solutions of the following problem:

Semiclassical ground state solutions for a Schrödinger equation in with critical exponential growth

Consider a Schrödinger equation where and are two continuous real functions on , ε is a positive parameter, the nonlinearity f is assumed to be of critical exponential growth in the sense of the Trudinger‐Moser inequality. By truncating the potentials and , we are able to establish some new existence and concentration results for critical Schrödinger equation in by variational methods. As a particular case, we observe that the concentration appears at the maximum set of the nonlinear potential which complements the results in 6 , 23 .     

Monotone positive solutions for a fourth order equation with nonlinear boundary conditions

This work is concerned with the existence of monotone positive solutions for a class of beam equations with nonlinear boundary conditions. The results are obtained by using the monotone iteration method and they extend early works on beams with null boundary conditions. Numerical simulations are also presented.     

Two homoclinic solutions for a nonperiodic fourth order differential equation with a perturbation

In this paper, we study multiple homoclinic solutions for a class of fourth order differential equations with a perturbation. By establishing a compactness lemma and using variational methods, the existence result of two homoclinic solutions is obtained under some suitable assumptions, but not requiring the periodicity condition. Some recent results are improved and extended.     

A boundedness theorem for a certain fourth order differential equation

Summary The paper investigates conditions under which solutions of equations of the form(1.1) are bounded as t→+∞. Entrata in Redazione il 31 agosto 1970.     

Qualitative analysis of a fourth order difference equation

In this paper, we will investigate some qualitative behavior of solutions of the following fourth order difference equation $x_{n+1}=ax_{n-1}+\frac{bx_{n-1}}{cx_{n-1}-dx_{n-3}},$ \ $n=0,1,...,$ where the initial conditions $x_{-3,}x_{-2},\ x_{-1}$\ and\ $x_{0}\ $are arbitrary real numbers and the values $a,\ b,\ c\ $and$\;d$ are\ defined as positive real numbers.     

Blow-up phenomena for a class of fourth order parabolic equation

A class of fourth order parabolic equation is studied in this paper. Some related blow-up results are obtained by applying the potential well theory, the concavity method and a series of differential-integral inequality techniques. More precisely, under some proper assumptions, the upper and lower bounds of the blow-up time and the growth rate for blow-up solutions are estimated. Moreover, a new blow-up condition independent of the depth of the potential well is found. These results complement the recent results obtained in Han (2018).     

Positive solutions for a fourth order equation invariant under isometries

Let be a smooth compact Riemannian manifold of dimension . We consider the problem

where , , . We require to be positive and invariant under isometries. We prove existence results for on arbitrary compact manifolds. This includes the case of the geometric Paneitz-Branson operator on the sphere.

     

Multiple positive solutions for a quasilinear elliptic equation with critical exponential nonlinearity

Let Ω be a bounded domain in R^N,N≥2, with C² boundary. In this work, we study the existence of multiple positive solutions of the following problem: