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Register a new userOn Uniqueness of Boundary Blow-up Solutions of a Class of Nonlinear Elliptic Equations
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We study boundary blow-up solutions of semilinear elliptic equations $Lu=u_+^p$ with $p>1$, or $Lu=e^{au}$ with $a>0$, where $L$ is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.
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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.
We are concerned with the focusing $L^2$-critical nonlinear Schrodinger equations in $mathbb{R}^d$ for $d=1,2$. The uniqueness is proved for a large energy class of multi-bubble blow-up solutions, which converge to a sum of $K$ pseudo-conformal blow-up solutions particularly with low rate $(T-t)^{0+}$, as $tto T$, $1leq K<infty$. Moreover, we also prove the uniqueness in the energy class of multi-solitons which converge to a sum of $K$ solitary waves with convergence rate $(1/t)^{2+}$, as $tto infty$. The uniqueness class is further enlarged to contain the multi-solitons with even lower convergence rate $(1/t)^{frac 12+}$ in the pseudo-conformal space. The proof is mainly based on the pseudo-conformal invariance and the monotonicity properties of several functionals adapted to the multi-bubble case, the latter is crucial towards the upgradation of the convergence to the fast exponential decay rate.
In this short note, we present a construction for the log-log blow up solutions to focusing mass-critical stochastic nonlinear Schroidnger equations with multiplicative noises. The solution is understood in the sense of controlled rough path as in cite{SZ20}.
We describe the asymptotic behavior of positive solutions $u_epsilon$ of the equation $-Delta u + au = 3,u^{5-epsilon}$ in $Omegasubsetmathbb{R}^3$ with a homogeneous Dirichlet boundary condition. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon and the functions $u_epsilon$ are assumed to be an optimizing sequence for the Sobolev inequality. Under a natural nondegeneracy assumption we derive the exact rate of the blow-up and the location of the concentration point, thereby proving a conjecture of Brezis and Peletier (1989). Similar results are also obtained for solutions of the equation $-Delta u + (a+epsilon V) u = 3,u^5$ in $Omega$.
We show that blow up of solutions with arbitrary positive initial energy of the Cauchy problem for the abstract wacve eqation of the form $Pu_{tt}+Au=F(u) (*)$ in a Hilbert space, where $P,A$ are positive linear operators and $F(cdot)$ is a continuously differentiable gradient operator can be obtained from the result of H.A. Levine on the growth of solutions of the Cauchy problem for (*). This result is applied to the study of inital boundary value problems for nonlinear Klein-Gordon equations, generalized Boussinesq equations and nonlinear plate equations. A result on blow up of solutions with positive initial energy of the initial boundary value problem for wave equation under nonlinear boundary condition is also obtained.
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