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Register a new userA classification result for the quasi-linear Liouville equation
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Authors
Pierpaolo Esposito
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Entire solutions of the $n-$Laplace Liouville equation in $mathbb{R}^n$ with finite mass are completely classified.
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In dimension n isolated singularities -- at a finite point or at infinity -- for solutions of finite total mass to the n-Liouville equation are of logarithmic type. As a consequence, we simplify the classification argument in arXiv:1609.03608 and establish a quantization result for entire solutions of the singular n-Liouville equation.
We present a new method of investigating the so-called quasi-linear strongly damped wave equations $$ partial_t^2u-gammapartial_tDelta_x u-Delta_x u+f(u)= abla_xcdot phi( abla_x u)+g $$ in bounded 3D domains. This method allows us to establish the existence and uniqueness of energy solutions in the case where the growth exponent of the non-linearity $phi$ is less than 6 and $f$ may have arbitrary polynomial growth rate. Moreover, the existence of a finite-dimensional global and exponential attractors for the solution semigroup associated with that equation and their additional regularity are also established. In a particular case $phiequiv0$ which corresponds to the so-called semi-linear strongly damped wave equation, our result allows to remove the long-standing growth restriction $|f(u)|leq C(1+ |u|^5)$.
The aim of this article is to prove new ill-posedness results concerning the nonlinear good Boussinesq equation, for both the periodic and non-periodic initial value problems. Specifically, we prove that the associated flow map is not continuous in Sobolev spaces $H^s$, for all $s<-1/2$.
We prove that the Dirichlet problem for the Lane-Emden equation in a half-space has no positive solution which is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution which is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions, or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.
We give blow-up analysis for the solutions of an elliptic equation under some conditions. Also, we derive a compactness result for this equation.
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