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We study a Klein-Gordon-Maxwell system, in a bounded spatial domain, under Neumann boundary conditions on the electric potential. We allow a nonconstant coupling coefficient. For sufficiently small data, we find infinitely many standing waves., 18 pages

We study a Klein-Gordon-Maxwell system, in a bounded spatial domain, under Neumann boundary conditions on the electric potential. We allow a nonconstant coupling coefficient. For sufficiently small data, we find infinitely many static solutions., 13 pages

In the present paper we analyze and discuss some mathematical aspects of the fluid-static configurations of a self-gravitating perfect gas enclosed in a spherical solid shell. The mathematical model we consider is based on the well-known Lane-Emden equation, albeit under boundary conditions that differ from those usually assumed in the astrophysical literature. The existence of multiple solutions requires particular attention in devising appropriate numerical schemes apt to deal with and catch the solution multiplicity as efficiently and accurately as possible. In sequence, we describe some analytical properties of the model, the two algorithms used to obtain numerical solutions, and the numerical results for two selected cases., Comment: Revision accepted for publication in "Communications in Computational Physics"; 23 pages, 9 figures

We open the paper with introductory considerations describing the motivations of our long-term research plan targeting gravitomagnetism, illustrating the fluid-dynamics numerical test case selected for that purpose, that is, a perfect-gas sphere contained in a solid shell located in empty space sufficiently away from other masses, and defining the main objective of this study: the determination of the gravitofluid-static field required as initial field ($t=0$) in forthcoming fluid-dynamics calculations. The determination of the gravitofluid-static field requires the solution of the isothermal-sphere Lane-Emden equation. We do not follow the habitual approach of the literature based on the prescription of the central density as boundary condition; we impose the gravitational field at the solid-shell internal wall. As the discourse develops, we point out differences and similarities between the literature's and our approach. We show that the nondimensional formulation of the problem hinges on a unique physical characteristic number that we call gravitational number because it gauges the self-gravity effects on the gas' fluid statics. We illustrate and discuss numerical results; some peculiarities, such as gravitational-number upper bound and multiple solutions, lead us to investigate the thermodynamics of the physical system, particularly entropy and energy, and preliminarily explore whether or not thermodynamic-stability reasons could provide justification for either selection or exclusion of multiple solutions. We close the paper with a summary of the present study in which we draw conclusions and describe future work., Comment: 32 pages, 26 figures

This paper concerns the blow-up behavior of large radial solutions of polyharmonic equations with power nonlinearities and positive radial weights. Specifically, we consider radially symmetric solutions of mu = c(|x|)|u| p on an annulus {x ∈ Rn | σ ≤ |x| < ρ}, with ρ ∈ (0,∞) and σ ∈ [0, ρ), that diverge to infinity as |x| → ρ. Here n,m ∈ N, p ∈ (1,∞), and c is a positive continuous function on the interval [σ, ρ]. Letting φρ(r) := QCρ/(ρ −r)q for r ∈ [σ, ρ), with q := 2m/(p−1), Q := (q(q +1)···(q +2m−1))1/(p−1), and Cρ := c(ρ)−1/(p−1), we show that, as |x| → ρ, the ratio u(x)/φρ(|x|) remains between positive constants that depend only on m and p. Extending well-known results for the second-order problem, we prove in the fourth-order case that u(x)/φρ(|x|) → 1 as |x| → ρ and obtain precise asymptotic expansions if c is sufficiently smooth at ρ. In certain higher-order cases, we find solutions for which the ratio u(x)/φρ(|x|)does not converge, but oscillates about 1 with non-vanishing amplitude.

The PDE Δmu = u|u|p–1, with \({m\in\mathbb{N}}\) and \({p\in(1,\infty)}\), serves as a paradigm for a large class of higher-order elliptic equations with power-like nonlinearities. In studying radially symmetric solutions of this PDE and their asymptotic behavior, it is of critical importance to understand the dynamics of the associated ODE, u(2m) = u|u|p–1, which is the subject of the present paper. Most solutions of the ODE blow up in finite time, diverging to either ∞ or −∞. In the phase space \({\mathbb{R}^{2m}}\), the orbits of these two types of solutions are separated by a (2m−1)-dimensional manifold \({\fancyscript{M}}\), which is unordered and homeomorphic to each of the coordinate hyperplanes under orthogonal projection. Solutions with orbits on \({\fancyscript{M}}\) do not blow up or, else, are unbounded from above and from below (oscillatory blow-up). In the second-order case, \({\fancyscript{M}}\) coincides with the stable manifold of the trivial equilibrium. In the fourth-order case, which is our main focus, \({\fancyscript{M}}\) contains a two-dimensional manifold \({\fancyscript{M}_0}\) of periodic orbits. There exists a unique periodic solution \({\bar u}\) with \({\bar u(0)=1}\) and \({\bar u'(0)=0}\), which shares most of the symmetry properties of the common cosine; all nontrivial periodic solutions are obtained from \({\bar u}\) via scaling and phase-shifting. Every solution with orbit on \({\fancyscript{M}\setminus\fancyscript{M}_0}\) converges to either the trivial equilibrium or one of the nontrivial periodic orbits; oscillatory blow-up does not occur. In higher-order cases, the structure of the manifold \({\fancyscript{M}}\) remains largely open.

We classify the regular radial solutions of the equation Δ m u = u | u | p − 1 with m a positive integer and p ∈ ( 1 , ∞ ) . Any such solution u is uniquely determined by the center values of u and its iterated Laplacians up to order m − 1 , that is, by the “center point” ( u ( 0 ) , Δ u ( 0 ) , … , Δ m − 1 u ( 0 ) ) . Most solutions have finite exit radius (that is, they are unbounded and defined on open balls of finite radius), are eventually monotonic with respect to the radial variable, and diverge to ∞ or − ∞ . The center points of these solutions form two open half-spaces in R m , separated by an ( m − 1 ) -dimensional manifold M . Solutions with center points on M are either global, or they have finite exit radius and oscillate about 0 with unbounded amplitude. Our results yield precise information regarding the existence, multiplicity, and nature of large radial solutions of Δ m u = u | u | p − 1 on a ball of given radius.

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.

In this work we prove some multiplicity results for solutions of a system of elliptic quasilinear equations, involving the p-Laplace operator (p > 2). The proof are based on variational and topological arguments and makes use of new perturbation results in Morse theory for the Banach space [Formula: see text].

In this paper we obtain multiple positive solutions of the nonlinear elliptic equation −h2Δu+V(x)u=K(x)|u|p−2u+Q(x)|u|q−2u, x∈RN, where V, K, Q are competing potential functions. We relate the number of solutions with the topology of the global minima set of a suitable ground energy function and improve a recent existence result of X. Wang and B. Zeng (1997, SIAM J. Math. Anal.28, 633–655).

We study the equation −dΔu+u=up−1in a bounded domain Ω, with Neumann boundary conditions. Precisely, we obtain Morse-like relations for the energy functional associated to the problem; as a consequence, we get some multiplicity results when the topology of the boundary ∂Ω is rich.

We study a parameter-dependent single-loop positive-feedback system in the nonnegative orthant of $\mathbb{R}^n$, with $n\in\mathbb{N}$, that arises in the analysis of the blow-up behavior of large radial solutions of polyharmonic PDEs with power nonlinearities. We describe the global dynamics of the system for arbitrary $n$ and prove that, in every dimension $n\ = 12$.

We study a parameter-dependent single-loop positive-feedback system in the nonnegative orthant of $\mathbb{R}^n$, with $n\in\mathbb{N}$, that arises in the analysis of the blow-up behavior of large radial solutions of polyharmonic PDEs with power nonlinearities. In Part 1 of the paper we showed that, in every dimension $n\ =12$. For $12\ =12$, followed by at least a second one if $n\>=62$, and that the number of successive bifurcations increases without bound as $n\to\infty$.

This paper is concerned with the elliptic system (0.1) Delta upsilon=phi, Delta phi=vertical bar del upsilon vertical bar(2) posed in a bounded domain Omega subset of R-N, N is an element of N. Specifically, we are interested in the existence and uniqueness or multiplicity of "large solutions," that is, classical solutions of (0.1) that approach infinity at the boundary of Omega. Assuming that Omega is a ball, we prove that the system (0.1) has a unique radially symmetric and nonnegative large solution with v(0) = 0 (obviously, v is determined only up to an additive constant). Moreover, if the space dimension N is sufficiently small, there exists exactly one additional radially symmetric large solution with v(0) = 0 (which, of course, fails to be nonnegative). We also study the asymptotic behavior of these solutions near the boundary of Omega and determine the exact blow-up rates; those are the same for all radial large solutions and independent of the space dimension. Our investigation is motivated by a problem in fluid dynamics. Under certain assumptions, the unidirectional flow of a viscous, heat-conducting fluid is governed by a pair of parabolic equations of the form (0.2) upsilon(t) -Delta upsilon=theta, theta t-Delta theta=vertical bar del upsilon vertical bar(2), where v and theta represent the fluid velocity and temperature, respectively. The system (0.1), with phi = -theta, is the stationary version of (0.2).

We consider an autonomous conservative second order system defined by a potential which admits a connected set $\Gamma$ of critical points at level zero. We prove the existence of multiple periodic solutions of large period which are located near $\Gamma$.

We study radial solutions u=(u1, u2) in an exterior domain ofR N (N⩾3)of the elliptic system−Δu+V⊂u)=0, where V is a positive and singular potential. We look for solutions which satisfy Dirichlet boundary conditions and vanish at infinity. We prove existence of infinitely many radial solutions, which can be topologically classified by their winding numbers around the singularity of V. Furthermore, we study some qualitative properties of such solutions.