Entire radial solutions of elliptic systems and inequalities of the mean curvature type

In this paper we study first nonexistence of radial entire solutions of elliptic systems of the mean curvature type with a singular or degenerate diffusion depending on the solution u. In particular we extend a previous result given in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389]. Moreover, in the scalar case we obtain nonexistence of all entire solutions, radial or not, of differential inequalities involving again operators of the mean curvature type and a diffusion term. We prove that in the scalar case, nonexistence of entire solutions is due to the explosion of the derivative of every nonglobal radial solution in the right extremum of the maximal interval of existence, while in that point the solution is bounded. This behavior is qualitatively different with respect to what happens for the m-Laplacian operator, studied in [R. Filippucci, Nonexistence of radial entire solutions of elliptic systems, J. Differential Equations 188 (2003) 353-389], where nonexistence of entire solutions is due, even in the vectorial case, to the explosion in norm of the solution at a finite point. Our nonexistence theorems for inequalities extend previous results given by Naito and Usami in [Y. Naito, H. Usami, Entire solutions of the inequality div(A(|Du|)Du)?f(u), Math. Z. 225 (1997) 167-175] and Ghergu and Radulescu in [M. Ghergu, V. Radulescu, Existence and nonexistence of entire solutions to the logistic differential equation, Abstr. Appl. Anal. 17 (2003) 995-1003].     

Radial solutions for a prescribed mean curvature equation with exponential nonlinearity

We establish an existence result for radial solutions for a prescribed mean curvature equation with exponential nonlinearity. Our methods are based on degree theory combined with a time map analysis. We also obtain two nonexistence results for positive solutions for more general f; one of them is not limited to radial solutions.     

Oscillatory radial solutions for subcritical biharmonic equations

It is well known that the biharmonic equation Δ²u=u|u|^p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball.     

Radial symmetry and monotonicity for an integral equation

In this paper we study radial symmetry and monotonicity of positive solutions of an integral equation arising from some higher-order semilinear elliptic equations in the whole space Rn. Instead of the usual method of moving planes, we use a new Hardy-Littlewood-Sobolev (HLS) type inequality for the Bessel potentials to establish the radial symmetry and monotonicity results.     

An optimal Liouville-type theorem for radial entire solutions of the porous medium equation with source

We consider nonnegative (continuous) weak solutions of the porous medium equation with source ut−Δum=up, with p>m>1. We address the question of existence of nontrivial entire solutions, that is, solutions defined for all x∈Rn and t∈R. Such solutions do exist for critical and supercritical p (positive bounded stationary solutions). Our main result asserts that for subcritical p there are no bounded radial entire solutions u?0. This parabolic Liouville-type theorem is the first of its kind for reaction-diffusion equations involving porous medium operators. On the other hand, it will be the main tool in the study of universal bounds for global and nonglobal solutions in the forthcoming article [K. Ammar, Ph. Souplet, Liouville-type results and universal bounds for positive solutions of the porous medium equation with source, in preparation]. The proof is based on intersection-comparison arguments. A key step is to first show the positivity of possible bounded radial entire solutions. Among other auxiliary results, we establish pointwise gradient estimates of possible independent interest.     

On solving the potential equation

We study radial solutions to the generalized Swift-Hohenberg equation on the plane with an additional quadratic term. We find stationary localized radial solutions that decay at infinity and solutions that tend to constants as the radius increases unboundedly (“droplets”). We formulate existence theorems for droplets and sketch the proofs employing the properties of the limit system as r → ∞. This system is a Hamiltonian system corresponding to a spatially one-dimensional stationary Swift-Hohenberg equation. We analyze the properties of this system and also discuss concentric-wave-type solutions. All the results are obtained by combining the methods of the theory of dynamical systems, in particular, the theory of homo-and heteroclinic orbits, and numerical simulation.     

Regularity of radial minimizers and extremal solutions of semilinear elliptic equations

We consider a special class of radial solutions of semilinear equations −Δu=g(u) in the unit ball of Rn. It is the class of semi-stable solutions, which includes local minimizers, minimal solutions, and extremal solutions. We establish sharp pointwise, Lq, and Wk,q estimates for semi-stable radial solutions. Our regularity results do not depend on the specific nonlinearity g. Among other results, we prove that every semi-stable radial weak solution is bounded if n?9 (for every g), and belongs to H³=W3,2 in all dimensions n (for every g increasing and convex). The optimal regularity results are strongly related to an explicit exponent which is larger than the critical Sobolev exponent.     

Closed-form solutions for a mode III radial matrix crack penetrating a circular inhomogeneity

In this research we address in detail a mode III radial matrix crack penetrating a circular inhomogeneity. One tip of the radial crack lies in the matrix, while the other tip of the radial crack lies in the circular inhomogeneity. In addition the two tips of the crack are mutually image points (or inverse points) with respect to the circular inhomogeneity-matrix interface. First we conformally map the crack onto a unit circle C_a in the new ζ-plane. Meanwhile the inhomogeneity-matrix interface is mapped onto C_b, a part of another circle in the ζ-plane. In addition C_a and C_b intersect at a vertex angle π/2. By using the method of image in the ζ-plane, closed-form solutions in terms of elementary functions are derived for three loading cases: (1) remote uniform antiplane shearing; (2) a screw dislocation located in the unbounded matrix; and (3) a radial Zener–Stroh crack.     

Radially symmetric boundary value problems for real and complex elliptic Monge-Ampére equations

Radially symmetric Dirichlet and Neumann problems for real and complex Monge-Ampére equations are considered. Existence of radially symmetric solutions is proved by transforming the differential equations into integral ones, solvable by means of fixed point arguments. Then, taking advantage of integral formulae, regularity and convexity of the radial solutions are checked. Fairly weak assumptions are required in that process. In the real case, a priori radial symmetry is also discussed.     

Entire solutions for a semilinear fourth order elliptic problem with exponential nonlinearity

We investigate entire radial solutions of the semilinear biharmonic equation Δ²u=λexp(u) in Rn, n?5, λ>0 being a parameter. We show that singular radial solutions of the corresponding Dirichlet problem in the unit ball cannot be extended as solutions of the equation to the whole of Rn. In particular, they cannot be expanded as power series in the natural variable s=log|x|. Next, we prove the existence of infinitely many entire regular radial solutions. They all diverge to −∞ as |x|→∞ and we specify their asymptotic behaviour. As in the case with power-type nonlinearities [F. Gazzola, H.-Ch. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006) 905-936], the entire singular solution x?−4log|x| plays the role of a separatrix in the bifurcation picture. Finally, a technique for the computer assisted study of a broad class of equations is developed. It is applied to obtain a computer assisted proof of the underlying dynamical behaviour for the bifurcation diagram of a corresponding autonomous system of ODEs, in the case n=5.     

A necessary and a sufficient condition for the existence of the positive radial solutions to Hessian equations and systems with weights

In this article, we consider the existence of positive radial solutions for Hessian equations and systems with weights and we give a necessary condition as well as a sufficient condition for a positive radial solution to be large. The method of proving theorems is essentially based on a successive approximation technique. Our results complete and improve a work published recently by Zhang and Zhou(existence of entire positive k-convex radial solutions to Hessian equations and systems with weights. Applied Mathematics Letters,Volume 50, December 2015, Pages 48–55).     

Conservation laws and symmetries of semilinear radial wave equations

Classifications of symmetries and conservation laws are presented for a variety of physically and analytically interesting wave equations with power nonlinearities in n spatial dimensions: a radial hyperbolic equation, a radial Schrödinger equation and its derivative variant, and two proposed radial generalizations of modified Korteweg-de Vries equations, as well as Hamiltonian variants. The mains results classify all admitted local point symmetries and all admitted local conserved densities depending on up to first order spatial derivatives, including any that exist only for special powers or dimensions. All such cases for which these wave equations admit, in particular, dilational energies or conformal energies and inversion symmetries are determined. In addition, potential systems arising from the classified conservation laws are used to determine nonlocal symmetries and nonlocal conserved quantities admitted by these equations. As illustrative applications, a discussion is given of energy norms, conserved Hs norms, critical powers for blow-up solutions, and one-dimensional optimal symmetry groups for invariant solutions.     

Boundary blow-up solutions in the unit ball: Asymptotics, uniqueness and symmetry

We calculate the full asymptotic expansion of boundary blow-up solutions (see Eq. (1) below), for any nonlinearity f. Our approach enables us to state sharp qualitative results regarding uniqueness and radial symmetry of solutions, as well as a characterization of nonlinearities for which the blow-up rate is universal. Lastly, we study in more detail the standard nonlinearities f(u)=up, p>1.     

The Liouville-Bratu-Gelfand Problem for Radial Operators

We determine precise existence and multiplicity results for radial solutions of the Liouville-Bratu-Gelfand problem associated with a class of quasilinear radial operators, which includes perturbations of k-Hessian and p-Laplace operators.     

Non-simultaneous blowup in heat equations with nonstandard growth conditions

This paper cares about blowup solutions for a system of n-componential heat equations coupled via localized reactions and with variable exponents. The criteria for non-simultaneous and simultaneous blowup are established for radial solutions with or without assumptions on initial data, including the existence of non-simultaneous blowup for n components; any blowup must be simultaneous or non-simultaneous.     

Strongly elliptic pseudodifferential equations on the sphere with radial basis functions

Spherical radial basis functions are used to define approximate solutions to strongly elliptic pseudodifferential equations on the unit sphere. These equations arise from geodesy. The approximate solutions are found by the Galerkin and collocation methods. A salient feature of the paper is a unified theory for error analysis of both approximation methods.     

Uniqueness of large radial solutions and existence of nonradial solutions for a superlinear Dirichlet problem in annulii

Here we establish the existence of infinitely many nonradial solutions for a superlinear Dirichlet problem in annulii. Our proof relies on estimating the number of radial solutions having a prescribed number of nodal regions. We prove that, for k>0 large, there exist exactly two radial solutions with k nodal regions (connected components of ). The problem need not be homogeneous.     

Existence of radial solutions for an asymptotically linear p-Laplacian problem

We prove that an asymptotically linear Dirichlet problem which involves the p-Laplacian operator has multiple radial solutions when the nonlinearity has a positive zero and the range of the ‘p-derivative’ of the nonlinearity includes at least the first j radial eigenvalues of the p-Laplacian operator. The main tools that we use are a uniqueness result for the p-Laplacian operator and bifurcation theory.     

Radial solutions of Dirichlet problems with concave-convex nonlinearities

We prove the existence of a double infinite sequence of radial solutions for a Dirichlet concave-convex problem associated with an elliptic equation in a ball of Rⁿ. We are interested in relaxing the classical positivity condition on the weights, by allowing the weights to vanish. The idea is to develop a topological method and to use the concept of rotation number. The solutions are characterized by their nodal properties.     

Uniqueness and exact multiplicity of solutions for a class of Dirichlet problems

We apply the “monotone separation of graphs” technique of L.A. Peletier and J. Serrin [L.A. Peletier, J. Serrin, Uniqueness of positive solutions of semilinear equations in Rn, Arch. Ration. Mech. Anal. 81 (2) (1983) 181-197; L.A. Peletier, J. Serrin, Uniqueness of nonnegative solutions of semilinear equations in Rn, J. Differential Equations 61 (3) (1986) 380-397], as developed further by L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179-202], to the question of exact multiplicity of positive solutions for a class of semilinear equations on a unit ball in Rn. We also observe that using P. Pucci and J. Serrin [P. Pucci, J. Serrin, Uniqueness of ground states for quasilinear elliptic operators, Indiana Univ. Math. J. 47 (2) (1998) 501-528] improvement of a certain identity of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179-202] produces a short proof of L. Erbe and M. Tang [L. Erbe, M. Tang, Structure of positive radial solutions of semilinear elliptic equations, J. Differential Equations 133 (2) (1997) 179-202] result on the uniqueness of positive solution of (1<p, )