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Register a new userOn a bifurcation value related to quasi-linear Schrodinger equations
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By virtue of numerical arguments we study a bifurcation phenomenon occurring for a class of minimization problems associated with the quasi-linear Schrodinger equation.
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Let $Omega subset {mathbb R}^N$ ($N geq 3$) be a $C^2$ bounded domain and $F subset partial Omega$ be a $C^2$ submanifold of dimension $0 leq k leq N-2$. Put $delta_F(x)=dist(x,F)$, $V=delta_F^{-2}$ in $Omega$ and $L_{gamma V}=Delta + gamma V$. Denote by $C_H(V)$ the Hardy constant relative to $V$ in $Omega$. We study positive solutions of equations (LE) $-L_{gamma V} u = 0$ and (NE) $-L_{gamma V} u+ f(u) = 0$ in $Omega$ when $gamma < C_H(V)$ and $f in C({mathbb R})$ is an odd, monotone increasing function. We establish the existence of a normalized boundary trace for positive solutions of (LE) - first studied by Marcus and Nguyen for the case $F=partial Omega$ - and employ it to investigate the behavior of subsolutions and super solutions of (LE) at the boundary. Using these results we study boundary value problems for (NE) and derive a-priori estimates. Finally we discuss subcriticality of (NE) at boundary points of $Omega$ and establish existence and stability results when the data is concentrated on the set of subcritical points.
This paper is devoted to prove the existence and nonexistence of positive solutions for a class of fractional Schrodinger equation in RN of the We apply a new methods to obtain the existence of positive solutions when f(u) is asymptotically linear with respect to u at infinity.
This paper is concerned with the quasi-linear reflected backward stochastic partial differential equation (RBSPDE for short). Basing on the theory of backward stochastic partial differential equation and the parabolic capacity and potential, we first associate the RBSPDE to a variational problem, and via the penalization method, we prove the existence and uniqueness of the solution for linear RBSPDE with Lapalacian leading coefficients. With the continuity approach, we further obtain the well-posedness of general quasi-linear RBSPDEs. Related results, including It^o formulas for backward stochastic partial differential equations with random measures, the comparison principle for solutions of RBSPDEs and the connections with reflected backward stochastic differential equations and optimal stopping problems, are addressed as well.
For a class of divergence type quasi-linear degenerate parabolic equations with a Radon measure on the right hand side we derive pointwise estimates for solutions via nonlinear Wolff potentials.
We construct a local in time, exponentially decaying solution of the one-dimensional variable coefficient Schrodinger equation by solving a nonstandard boundary value problem. A main ingredient in the proof is a new commutator estimate involving the projections P+ and P- onto the positive and negative frequencies.
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