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We prove existence results of two solutions of the problem
where 
\[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{in}\ \Omega,\\ u>0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial \Omega, \end{cases} \]
$L(v)=-\textrm {div}(M(x)\nabla v)$ is a linear operator, 
$p\in (2,2^{*}]$ and 
$\lambda$ and 
$m$ sufficiently large. Then their asymptotical limit as 
$m\to +\infty$ is investigated showing different behaviours.
This paper deals with solutions of semilinear elliptic equations of the type
where Ω is a radially symmetric domain of the plane that can be bounded or unbounded. We consider solutions u that are invariant by rotations of a certain angle θ and which have a bound on their Morse index in spaces of functions invariant by these rotations. We can prove that or u is radial, or, else, there exists a direction 
\[ \left\{\begin{array}{@{}ll} -\Delta u = f(|x|, u) \qquad & \text{ in } \Omega, \\ u= 0 & \text{ on } \partial \Omega, \end{array} \right. \]
$e\in \mathcal {S}$ such that u is symmetric with respect to e and it is strictly monotone in the angular variable in a sector of angle θ/2. The result applies to least-energy and nodal least-energy solutions in spaces of functions invariant by rotations and produces multiplicity results.
In this paper we study the problem
where Ω = ℝ
N
or Ω = B
1, N ⩾ 3, p > 1 and 
. Using a suitable map we transform problem (1) into another one without the singularity 1/|x|2. Then we obtain some bifurcation results from the radial solutions corresponding to some explicit values of λ.
In this paper we consider the following semilinear elliptic equation
where n ≥ 3, 
and β ≥ 0, γ ≥ 0, q > p ≥ 1, μ and ν are real constants. We note that if γ = 0, β > 0 and ν ≥ 2, then the equation above is called the Matukuma-type equation. If β = 0, γ > 0 and ν > 2, then the complete classification of all possible positive solutions had been conducted by Cheng and Ni. If β > 0, γ > 0 and μ ≥ ν ≥ 2, then some results about the maximal solution and positive solution structures can be found in Chern. The purpose of this paper is to discuss and investigate the blow-up and positive entire solutions of the equation above for the μ ≥ 2 ≥ ν case.
We consider a control constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during themodeling of diffuse-gray conductive-radiative heat transfer. After stating first-order necessary conditions, second-order sufficient conditions are derived that account for strongly active sets. These conditions ensure local optimality in an Ls -neighborhood of a reference function whereby the underlying analysis allows to use weaker norms than $L^\infty$ 
.
Let f be an odd function of a class C2 such that ƒ(1) = 0,ƒ'(0) < 0,ƒ'(1) > 0 and $x\mapsto f(x)/x$ 
increases on [0,1]. We approximate the positive solution of Δu + ƒ(u) = 0, on $\xR_{+}^{2}$ 
with homogeneous Dirichlet boundary conditions by thesolution of $-\Delta u_{L}+f(u_{L})=0,$ 
on ]0,L[2 with adequatenon-homogeneous Dirichlet conditions. We show that the error uL - utends to zero exponentially fast, in the uniform norm.
