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Register a new userSome remarks on sign changing solutions of a quasilinear elliptic equation in two variables
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We consider planar solutions to certain quasilinear elliptic equations subject to the Dirichlet boundary conditions; the boundary data is assumed to have finite number of relative maximum and minimum values. We are interested in certain vanishing properties of sign changing solutions to such a Dirichlet problem. Our method is applicable in the plane.
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We study the existence of sign-changing solutions to the nonlinear heat equation $partial _t u = Delta u + |u|^alpha u$ on ${mathbb R}^N $, $Nge 3$, with $frac {2} {N-2} < alpha <alpha _0$, where $alpha _0=frac {4} {N-4+2sqrt{ N-1 } }in (frac {2} {N-2}, frac {4} {N-2})$, which are singular at $x=0$ on an interval of time. In particular, for certain $mu >0$ that can be arbitrarily large, we prove that for any $u_0 in mathrm{L} ^infty _{mathrm{loc}} ({mathbb R}^N setminus { 0 }) $ which is bounded at infinity and equals $mu |x|^{- frac {2} {alpha }}$ in a neighborhood of $0$, there exists a local (in time) solution $u$ of the nonlinear heat equation with initial value $u_0$, which is sign-changing, bounded at infinity and has the singularity $beta |x|^{- frac {2} {alpha }}$ at the origin in the sense that for $t>0$, $ |x|^{frac {2} {alpha }} u(t,x) to beta $ as $ |x| to 0$, where $beta = frac {2} {alpha } ( N -2 - frac {2} {alpha } ) $. These solutions in general are neither stationary nor self-similar.
We prove nonexistence of nontrivial, possibly sign changing, stable solutions to a class of quasilinear elliptic equations with a potential on Riemannian manifolds, under suitable weighted volume growth conditions on geodesic balls.
The authors of this paper deal with the existence and regularities of weak solutions to the homogenous $hbox{Dirichlet}$ boundary value problem for the equation $-hbox{div}(| abla u|^{p-2} abla u)+|u|^{p-2}u=frac{f(x)}{u^{alpha}}$. The authors apply the method of regularization and $hbox{Leray-Schauder}$ fixed point theorem as well as a necessary compactness argument to prove the existence of solutions and then obtain some maximum norm estimates by constructing three suitable iterative sequences. Furthermore, we find that the critical exponent of $m$ in $|f|_{L^{m}(Omega)}$. That is, when $m$ lies in different intervals, the solutions of the problem mentioned belongs to different $hbox{Sobolev}$ spaces. Besides, we prove that the solution of this problem is not in $W^{1,p}_{0}(Omega)$ when $alpha>2$, while the solution of this problem is in $W^{1,p}_{0}(Omega)$ when $1<alpha<2$.
We consider the nonlinear heat equation $u_t - Delta u = |u|^alpha u$ on ${mathbb R}^N$, where $alpha >0$ and $Nge 1$. We prove that in the range $0 < alpha <frac {4} {N-2}$, for every $mu >0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0 (x)= mu |x|^{-frac {2} {alpha }}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution.
In this article we discuss the maximum principle for the linear equation and the sign changing solutions of the semilinear equation with the Higgs potential. Numerical simulations indicate that the bubbles for the semilinear Klein-Gordon equation in the de Sitter spacetime are created and apparently exist for all times.
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