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We study the existence and multiplicity of nonnegative solutions, as well as the behaviour of corresponding parameter-dependent branches, to the equation $-Delta u = (1-u) u^m - lambda u^n$ in a bounded domain $Omega subset mathbb{R}^N$ endowed with the zero Dirichlet boundary data, where $0<m leq 1$ and $n>0$. When $lambda > 0$, the obtained solutions can be seen as steady states of the corresponding reaction-diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination. In addition to the main new results, we formulate a few relevant conjectures.
We consider a class of semilinear nonlocal problems with vanishing exterior condition and establish a Ambrosetti-Prodi type phenomenon when the nonlinear term satisfies certain conditions. Our technique makes use of the probabilistic tools and heat kernel estimates.
In this paper, we prove some pointwise comparison results between the solutions of some second-order semilinear elliptic equations in a domain $Omega$ of $R^n$ and the solutions of some radially symmetric equations in the equimeasurable ball $Omega^*
For a general class of autonomous quasi-linear elliptic equations on R^n we prove the existence of a least energy solution and show that all least energy solutions do not change sign and are radially symmetric up to a translation in R^n.
A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. Thi
Given a smooth domain $OmegasubsetRR^N$ such that $0 in partialOmega$ and given a nonnegative smooth function $zeta$ on $partialOmega$, we study the behavior near 0 of positive solutions of $-Delta u=u^q$ in $Omega$ such that $u = zeta$ on $partialOm