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Register a new userAsymptotic behavior of positive solutions of semilinear elliptic problems with increasing powers
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We prove existence results of two solutions of the problem [ begin{cases} L(u)+u^{m-1}=lambda u^{p-1} & text{ in $Omega$}, quad u>0 &text{ in $Omega$}, quad u=0 & text{ on $partial Omega$}, end{cases} ] where $L(v)=-{rm div}(M(x) abla v)$ is a linear operator, $pin (2,2^{*}]$ and $lambda$ and $ m$ sufficiently large. Then their asymptotical limit as $mto +infty$ is investigated showing different behaviors.
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Let $Omega subset mathbb{R}^N$ be a bounded domain and $delta(x)$ be the distance of a point $xin Omega$ to the boundary. We study the positive solutions of the problem $Delta u +frac{mu}{delta(x)^2}u=u^p$ in $Omega$, where $p>0, ,p e 1$ and $mu in mathbb{R},,mu e 0$ is smaller then the Hardy constant. The interplay between the singular potential and the nonlinearity leads to interesting structures of the solution sets. In this paper we first give the complete picture of the radial solutions in balls. In particular we establish for $p>1$ the existence of a unique large solution behaving like $delta^{- frac2{p-1}}$ at the boundary. In general domains we extend results of arXiv:arch-ive/1407.0288 and show that there exists a unique singular solutions $u$ such that $u/delta^{beta_-}to c$ on the boundary for an arbitrary positive function $c in C^{2+gamma}(partialOmega) , (gamma in (0,1)), c ge 0$. Here $beta_-$ is the smaller root of $beta(beta-1)+mu=0$.
This paper deals with existence and regularity of positive solutions of singular elliptic problems on a smooth bounded domain with Dirichlet boundary conditions involving the $Phi$-Laplacian operator. The proof of existence is based on a variant of the generalized Galerkin method that we developed inspired on ideas by Browder and a comparison principle. By using a kind of Moser iteration scheme we show $L^{infty}(Omega)$-regularity for positive solutions
We consider positive singular solutions to semilinear elliptic problems with possibly singular nonlinearity. We deduce symmetry and monotonicity properties of the solutions via the moving plane procedure.
We consider a sequence of blowup solutions of a two dimensional, second order elliptic equation with exponential nonlinearity and singular data. This equation has a rich background in physics and geometry. In a work of Bartolucci-Chen-Lin-Tarantello it is proved that the profile of the solutions differs from global solutions of a Liouville type equation only by a uniformly bounded term. The present paper improves their result and establishes an expansion of the solutions near the blowup points with a sharp error estimate.
This paper is concerned with the initial value problem for semilinear wave equation with structural damping $u_{tt}+(-Delta)^{sigma}u_t -Delta u =f(u)$, where $sigma in (0,frac{1}{2})$ and $f(u) sim |u|^p$ or $u |u|^{p-1}$ with $p> 1 + {2}/(n - 2 sigma)$. We first show the global existence for initial data small in some weighted Sobolev spaces on $mathcal R^n$ ($n ge 2$). Next, we show that the asymptotic profile of the solution above is given by a constant multiple of the fundamental solution of the corresponding parabolic equation, provided the initial data belong to weighted $L^1$ spaces.
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