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In this article, we consider the nonlinear elliptic equation
Here, $\mathcal {M^{+}}_{\lambda , \Lambda }$ denotes Pucci’s extremal operator with parameters $\Lambda \ge \lambda > 0$ and $-1<\beta <0$. We prove the existence of a critical exponent $p^{*}_+$ that determines the range of $p>1$ for which we have the existence or nonexistence of a positive radial solution to $(\star )$. In addition, we describe the solution set in terms of the parameter p and find two new critical exponents $1<p^{*}_{+}<{\tilde{p}}_{\beta }$ for the equation $(\star )$, where the solution set sharply changes its qualitative properties when the value of p exceeds these critical exponents.

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... are associated to (11), (13) for M − . We recover (see [14]) the expression for u from the dynamical system by setting ...

... Thus, we only need to prove periodic orbits lying entirely in R − λ ∪ + are not admissible for p = p p,a + . This is accomplished through the arguments in Dulac's criterium in [1,11,14]. Indeed, by defining ϕ(X, Z) = X α Z β , where β = 3−p p−1 and α as in (5) ...

... Both expressions are positive if 1 < p < min(p p,a + , p a ∆ ) = p a ∆ ; and both are negative if p > max(p p,a + , p a ∆ ) = p p,a + . Then one concludes by the same argument as in the classical Bendixson-Dulac criterion, see also [11,Theorem 3.1]. Indeed, the vector field F = (f, g) is Lipschitz continuous in (X, Z), so Green's area formula for the domain D enclosed by a periodic trajectory applies such as ...

In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. In particular we recover known results on regular solutions for the fully nonlinear non weighted problem by alternative proofs. We also slightly improve the classification of the solutions for the extremal operator $\mathcal{M}^-$.

... The case 0 < q < 1 seems more difficult to deal with, even with radial symmetry. We refer the reader to [27] for an Emden-Fowler analysis in the particular case f (t) = t p , p > 0. ...

In this paper we consider positive supersolutions of the elliptic equation, posed in exterior domains of RN where f is continuous in and positive in We classify supersolutions u into four types depending on the function m(R) = inf for large R, and give necessary and suffcient conditions in order to have supersolutions of each of these types. As a consequence, we also obtain Liouville theorems for supersolutions depending on the values of N, q and on some integrability properties on f at zero or infinity. We also describe these questions when the equation is posed in the whole RN.

In this paper we study existence, nonexistence and classification of radial positive solutions of some weighted fully nonlinear equations involving Pucci extremal operators. Our results are entirely based on the analysis of the dynamics induced by an autonomous quadratic system which is obtained after a suitable transformation. This method allows to treat both regular and singular solutions in a unified way, without using energy arguments. In particular we recover known results on regular solutions for the fully nonlinear non weighted problem by alternative proofs. We also slightly improve the classification of the solutions for the extremal operator M − .

We give a structure result for the positive radial solutions of the following equation: Δpu + K(r)u|u|q-1 = 0 with some monotonicity assumptions on the positive function K(r). Here r = |x|, x ∈ ℝn; we consider the case when n > p > 1, and g > p* = n(p-1)/n-p. We continue the discussion started by Kawano et al. in [11], refining the estimates on the asymptotic behavior of Ground States with slow decay and we state the existence of S.G.S., giving also for them estimates on the asymptotic behavior, both as r → 0 and as r →. We make use of a Emden-Fowler transform which allow us to give a geometrical interpretation to the functions used in [11] and related to the Pohozaev identity. Moreover we manage to use techniques taken from dynamical systems theory, in particular the ones developed in [10] for the problems obtained by substituting the ordinary Laplacian Δ for the p-Laplacian Δp in the preceding equations.

We consider a large class of degenerate or singular operators F defined on ℝ×(ℝ N ) * ×S, where S denotes the space of symmetric matrices on ℝ n , F is continuous. We give a new definition of viscosity sub and super solutions for F(x,∇u,∇ 2 u)=0. We prove a comparison theorem between sup and supersolutions for F(x,∇u,∇ 2 u)=b(u) where b is an increasing function, and a Liouville type result.

In this article we present the analysis of critical exponents for a large class of extremal operators, in the case of radially symmetric solutions. More precisely, for such an operator M, we consider the nonlinear equation (*) M(D2u) + up = 0, u > 0 in ℝN and we prove the existence of a critical exponent p* that determines the range of p > 1 for which we have existence or non-existence of a positive radial solution to (*). In the case of maximal operators, we define two dimension-like numbers N∞ and N0, depending on M and N, that satisfy 0 < N∞ ≤N0. We prove that our critical exponent satisfies max {N∞/N∞-2, p0} ≤ p * ≤ p∞ where p0 = (N0 + 2)/(N0 - 2) and p∞ = (N ∞ + 2)/(N∞ - 2). In the non-trivial case, N∞ < N0 and both inequalities above are strict. Indiana University Mathematics Journal

In this paper we shall establish some regularity results of solutions of a
class of fully nonlinear equations, with a first order term which is
sub-linear. We prove local H\"older regularity of the gradient both in the
interior and up to the boundary.

For a large class of nonlinear second order elliptic differential operators, we define a concept of dimension, upon which we construct a fundamental solution. This allows us to prove two properties associated to these operators, which are generalizations of properties for the Laplacian and Pucci’s operators. If ℳ denotes such an operator, the first property deals with the possibility of removing singularities of solutions to the equation ℳ(D 2 u)-u p =0inB∖{0}, where B is a ball in ℝ N . The second property has to do with existence or nonexistence of solutions in ℝ N to the inequality ℳ(D 2 u)+u p ≤0inℝ N · In both cases a common critical exponent defined upon the dimension number is obtained, which plays the role of N/(N-2) for the Laplacian.

Several problems for the differential equation LpÃŽÂ±u=g(r,u)Ã¢Â€Âƒ withÃ¢Â€Âƒ LpÃŽÂ±u=rÃ¢ÂˆÂ’ÃŽÂ±(rÃŽÂ±|uÃ¢Â€Â²|pÃ¢ÂˆÂ’2uÃ¢Â€Â²)Ã¢Â€Â² are considered. For ÃŽÂ±=NÃ¢ÂˆÂ’1, the operator LpÃŽÂ± is the radially symmetric p-Laplacian in Ã¢Â„ÂN. For the initial value problem with given data u(r0)=u0,uÃ¢Â€Â²(r0)=uÃ¢Â€Â²0 various uniqueness conditions and counterexamples to uniqueness are given. For the case where g is increasing in u, a sharp comparison theorem is established; it leads to maximal solutions, nonuniqueness and uniqueness results, among others. Using these results, a strong comparison principle for the boundary value problem and a number of properties of blow-up solutions are proved under weak assumptions on the nonlinearity g(r,u).

We consider the problem of uniqueness of positive solutions to boundary value problems containing the equation: -\Delta_p u =K(|x|)f(u), p>1. f is positive, is locally Lipschitz and satisfies some superlinear growth condition after u_0, a zero of f before which it is non positive and not identically 0. We show that the Sturmnian theory arguments used by Coffman and Kwong are valid for the equation containing the p-Laplacian operator, even though they were thought to be unextendable beyond the semilinear equation. We obtain a monotone separation result which finally yields the desired uniqueness results.

This text provides an introduction to the numerical solution of initial and boundary value problems in ordinary differential equations on a firm theoretical basis. This book strictly presents numerical analysis as a part of the more general field of scientific computing. Important algorithmic concepts are explained down to questions of software implementation. For initial value problems, a dynamical systems approach is used to develop Runge-Kutta, extrapolation, and multistep methods. For boundary value problems including optimal control problems, both multiple shooting and collocation methods are worked out in detail.
Graduate students and researchers in mathematics, computer science, and engineering will find this book useful. Chapter summaries, detailed illustrations, and exercises are contained throughout the book with many interesting applications taken from a rich variety of areas.
Peter Deuflhard is founder and president of the Zuse Institute Berlin (ZIB) and full professor of scientific computing at the Free University of Berlin, Department of Mathematics and Computer Science.
Folkmar Bornemann is full professor of scientific computing at the Center of Mathematical Sciences, Technical University of Munich.
This book was translated by Werner Rheinboldt, professor emeritus of numerical analysis and scientific computing at the Department of Mathematics, University of Pittsburgh.

This book provides a self-contained development of the regularity theory for solutions of fully nonlinear elliptic equations. Caffarelli and Cabré offer a detailed presentation of all techniques needed to extend the classical Schauder and Calderón-Zygmund regularity theories for linear elliptic equations to the fully nonlinear context.
The authors present the key ideas and prove all the results needed for the regularity theory of viscosity solutions of fully nonlinear equations. The book contains the study of convex fully nonlinear equations and fully nonlinear equations with variable coefficients. This book is suitable as a text for graduate courses in nonlinear elliptic partial differential equations.

In this article we study some results on the existence of radially symmetric, non-negative solutions for the nonlinear elliptic equation
\tag{$*$} \mathcal M_{\lambda ,\Lambda }^{ + }\left(D^{2}u\right) + u^{P} = 0 \quad \text{in }\mathbb{R}^{N}.
Here N⩾3 , p>1 and \mathcal M_{\lambda ,\Lambda }^{ + } denotes the Pucci’s extremal operators with parameters 0<λ⩽Λ . The goal is to describe the solution set in function of the parameter p . We find critical exponents 1 < p_{ + }^{*} < p_{ + }^{p} that satisfy: (i) If 1 < p < p_{ + }^{*} then there is no non-trivial radial solution of ( * ). (ii) If p = p_{ + }^{*} then there is a unique fast decaying radial solution of ( * ). (iii) If p_{ + }^{*} < p \leq p_{ + }^{p} then there is a unique pseudo-slow decaying radial solution to ( * ). (iv) If p^p_+
then there is a unique slow decaying radial solution to ( * ). Similar results are obtained for the operator \mathcal M_{\lambda ,\Lambda }^{ - } .
Résumé
Dans cet article nous avons étudié quelques résultats d’existence des solutions radiales non négatives pour l’équation elliptique non linéaire
\tag{$*$} \mathcal M_{\lambda ,\Lambda }^{ + }\left(D^{2}u\right) + u^{P} = 0 \quad u \geq 0 \text{ dans }\mathbb R^{N}.
Ici N⩾3 , p>1 et \mathcal M_{\lambda ,\Lambda }^{ + } est l’opérateur extrémal de Pucci avec les paramètres 0<λ⩽Λ . L’objectif est de décrire l’ensemble des solutions en fonction de p . On trouve des exposants critiques 1 < p_{ + }^{*} < p_{ + }^{p} tels que : (i) Si 1 < p < p_{ + }^{*} , alors il n’existe pas de solution radiale non triviale de ( * ). (ii) Si p = p^∗_+ , il existe une unique solution radiale de ( * ) à décroissance rapide. (iii) Si p^*
, il existe une unique solution radiale de ( * ) à décroissance pseudo-lente. (iv) Si p^p_+
, il existe une unique solution radiale de ( * ) à décroissance lente.
Un résultat similaire est obtenu pour l’opérateur \mathcal M_{\lambda ,\Lambda }^{ - } .

We study the positive radial solutions of a semilinear elliptic equationΔu+f(u)=0, wheref(u) has a supercritical growth order for smallu>0 and a subcritical growth order for largeu. By showing the uniqueness of positive solutions behaving likeO(|x|2−n) at infinity, we give an almost complete description for the structure of positive radial solutions. As a consequence, we also prove the uniqueness of positive solutions of the nonlinear Dirichlet problem for the equation in a finite ball.

In this article we study some results on the existence of radially symmetric, non-negative solutions for the nonlinear elliptic equation M λ,Λ+(D2u)+up = 0inRN. Here N≥3, p > 1 and Mλ,Λ+ denotes the Pucci's extremal operators with parameters 0<λ ≤ Λ. The goal is to describe the solution set in function of the parameter p. We find critical exponents 1<p *+<pp+ that satisfy: (i) If 1<p<p*+ then there is no non-trivial radial solution of (*). (ii) If p = p*+ then there is a unique fast decaying radial solution of (*). (iii) If p*+<p ≤ pp+ then there is a unique pseudo-slow decaying radial solution to (*). (iv) If pp+<p then there is a unique slow decaying radial solution to (*). Similar results are obtained for the operator Mλ,Λ-.

In this paper we prove some Liouville theorems for nonnegative viscosity supersolutions of a class of fully nonlinear uniformly elliptic problems in RN.

It is shown that, for a large class of nonlinearities, positive
solutions respect the symmetry of a second order nonlinear elliptic
differential equation with a certain symmetry (e.g., spherical
symmetry). In addition, positive, spherically symmetric solutions are
obtained for a specified nonlinear elliptic equation; and some recent
results on the partial classification of isolated singularities are
presented.

Rotations of a heavy string with one endpoint free are considered. According to the linear theory, a string of a given length can rotate only at certain eigenvelocities of rotation w1 which form a discrete spectrum. It is shown that according to the more accurate non-linear theory, a string can rotate at any velocity ω > ω1, and that for each w in the range ωn < ω < ωn+1 there are exactly n distinct modes of rotation.

Si provano alcune proprietà generali degli operatori ellittici estremanti. Mediante particolari soluzioni delle equazioni
estremanti si esaminano le singolarità isolate delle soluzioni di equazioni ellittiche delII ordine e si stabiliscono alcune limitazioni.
Some general properties of the maximizing elliptic operators are established. Particular solutions of the maximizing equation
are used to get information about bounds and isolated singularities of solutions of second order elliptic equations.

In this article we study basic properties for a class of nonlinear integral operators related to their fundamental solutions. Our goal is to establish Liouville type theorems: non-existence theorems for positive entire solutions for Iu⩽0 and for Iu+up⩽0, p>1.We prove the existence of fundamental solutions and use them, via comparison principle, to prove the theorems for entire solutions. The non-local nature of the operators poses various difficulties in the use of comparison techniques, since usual values of the functions at the boundary of the domain are replaced here by values in the complement of the domain. In particular, we are not able to prove the Hadamard Three Spheres Theorem, but we still obtain some of its consequences that are sufficient for the arguments.

In this paper we prove some Liouville theorems for nonnegative viscosity supersolutions of a class of fully nonlinear uniformly elliptic problems in ℝN.
Résumé
Dans ce travail nous démontrons des théorèmes de Liouville pour des sur-solutions de viscosité positives de problèmes uniformement elliptique complètement non linéaires dans ℝN.

We study fully nonlinear elliptic equations such as $F(D^2u) = u^p, \quad p>1,$ in $\R^n$ or in exterior domains, where $F$ is any uniformly elliptic, positively homogeneous operator. We show that there exists a critical exponent, depending on the homogeneity of the fundamental solution of $F$, that sharply characterizes the range of $p>1$ for which there exist positive supersolutions or solutions in any exterior domain. Our result generalizes theorems of Bidaut-V\'eron \cite{B} as well as Cutri and Leoni \cite{CL}, who found critical exponents for supersolutions in the whole space $\R^n$, in case $-F$ is Laplace's operator and Pucci's operator, respectively. The arguments we present are new and rely only on the scaling properties of the equation and the maximum principle. Comment: 16 pages, new existence results added

Partial differential equations

- L C Evans

L.C. Evans, Partial differential equations, Providence, RI : American Mathematical Society 2010,
2nd ed.

Symmetry and isolated singularities of positive solution of nonlinear elliptic equations, in: Nonlinear Partial Differention Equations in Engineering and Applied Science

- B Gidas

B. Gidas, Symmetry and isolated singularities of positive solution of nonlinear elliptic equations,
in: Nonlinear Partial Differention Equations in Engineering and Applied Science( Proc. Conf.,
Univ. Rhode Island, Kingston, RI, 1979), in: Lecture Notes in Pure Appl. Math., Vol. 54, Dekker,
New York 1980, pp. 255-273.

Anwendungen der mechanischen Warmentheorie auf Kosmologie und metheorologische Probleme, Leibzig

- R Emden

R. Emden, Gaskugeln, Anwendungen der mechanischen Warmentheorie auf Kosmologie und
metheorologische Probleme, Leibzig, 1907.