Existence and asymptotic behavior for elliptic equations with singular anisotropic potentials

We study the equation Δu+u|u|^p−1+V(x)u+f(x)=0 in Rn, where n?3 and p>n/(n−2). The forcing term f and the potential V can be singular at zero, change sign and decay polynomially at infinity. We can consider anisotropic potentials of form h(x)|x|⁻² where h is not purely angular. We obtain solutions u which blow up at the origin and do not belong to any Lebesgue space Lr. Also, u is positive and radial, in case f and V are. Asymptotic stability properties of solutions, their behavior near the singularity, and decay are addressed.     

A note on positive evanescent solutions for a certain class of elliptic problems

This paper is devoted to the existence and properties of solutions of the following class of nonlinear elliptic differential equations Δu(x)+f(x,u(x))+g(‖x‖)x⋅∇u(x)=0, x∈Rn, ‖x‖>R. We prove existence of positive solutions vanishing at positive infinity. Our approach is based on the subsolution and supersolution method. The nonlinearity f covers both sublinear and superlinear cases and does not necessarily satisfy f(x,0)≡0. The asymptotic behavior of solutions is also described.     

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.     

Potential well method for Cauchy problem of generalized double dispersion equations

In this paper we study Cauchy problem of generalized double dispersion equations utt−uxx−uxxtt+uxxxx=f(u)_xx, where f(u)=p|u|, p>1 or u2k, . By introducing a family of potential wells we not only get a threshold result of global existence and nonexistence of solutions, but also obtain the invariance of some sets and vacuum isolating of solutions. In addition, the global existence and finite time blow up of solutions for problem with critical initial conditions E(0)=d, I(u₀)?0 or I(u₀)<0 are proved.     

Positive evanescent solutions of nonlinear elliptic equations

In this note we investigate the existence of positive solutions vanishing at +∞ to the elliptic equation Δu+f(x,u)+g(|x|)x⋅∇u=0, |x|>A>0, in Rn (n?3) under mild restrictions on the functions f, g.     

Uniqueness of positive radial solutions for quasilinear elliptic equations in an annulus

Extending a previous result of Tang [1] we prove the uniqueness of positive radial solutions of Δ_pu+f(u)=0, subject to Dirichlet boundary conditions on an annulus in Rⁿ with 2<p≤n, under suitable hypotheses on the nonlinearity f. This argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball (see [9]), in the complement of a ball or in the whole space Rⁿ (see [10], [3] and [11]).     

An application of the maximum principle to describe the layer behavior of large solutions and related problems

This work is devoted to the analysis of the asymptotic behavior of positive solutions to some problems of variable exponent reaction-diffusion equations, when the boundary condition goes to infinity (large solutions). Specifically, we deal with the equations ??u = u ^p(x), ??u = ?m(x)u?+?a(x)u ^p(x) where a(x)??? a ₀ >?0, p(x)??? 1 in ??, and ??u = e ^p(x) where p(x)??? 0 in ??. In the first two cases p is allowed to take the value 1 in a whole subdomain ${\Omega_c\subset \Omega}$ , while in the last case p can vanish in a whole subdomain ${\Omega_c\subset \Omega}$ . Special emphasis is put in the layer behavior of solutions on the interphase ?? _i :?= ??? _c ???. A similar study of the development of singularities in the solutions of several logistic equations is also performed. For example, we consider ???u = ?? m(x)u?a(x) u ^p(x) in ??, u = 0 on ???, being a(x) and p(x) as in the first problem. Positive solutions are shown to exist only when the parameter ?? lies in certain intervals: bifurcation from zero and from infinity arises when ?? approaches the boundary of those intervals. Such bifurcations together with the associated limit profiles are analyzed in detail. For the study of the layer behavior of solutions the introduction of a suitable variant of the well-known maximum principle is crucial.     

Isolated singularities of solutions to quasi-linear elliptic equations with absorption

We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a₀(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|^p−2+u|u|^q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.     

Symmetry breaking for a class of semilinear elliptic problems and the bifurcation diagram for a 1-dimensional problem

We study the behaviour of the positive solutions to the Dirichlet problem IR ⁿ in the unit ball in IR ^R wherep<(N+2)/(N−2) ifN≥3 and λ varies over IR. For a special class of functionsg viz.,g(x)=u ₀ ^p (x) whereu ₀ is the unique positive solution at λ=0, we prove that for certain λ’s nonradial solutions bifurcate from radially symmetric positive solutions. WhenN=1, we obtain the complete bifurcation diagram for the positive solution curve.     

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.     

Infinite Multiplicity of Positive Entire Solutions for a Semilinear Elliptic Equation

We establish that the elliptic equation Δu+K(x)u^p+μf(x)=0 in Rⁿ has infinitely many positive entire solutions for small μ?0 under suitable conditions on K, p, and f.     

Very weak solutions with boundary singularities for semilinear elliptic Dirichlet problems in domains with conical corners

Let Ω⊂Rn be a bounded Lipschitz domain with a cone-like corner at 0∈∂Ω. We prove existence of at least two positive unbounded very weak solutions of the problem −Δu=up in Ω, u=0 on ∂Ω, which have a singularity at 0, for any p slightly bigger that the generalized Brezis-Turner exponent p^*. On an example of a planar polygonal domain the actual size of the p-interval on which the existence result holds is computed. The solutions are found variationally as perturbations of explicitly constructed singular solutions in cones. This approach also makes it possible to find numerical approximations of the two very weak solutions on Ω following a gradient flow of a suitable functional and using the mountain pass algorithm. Two-dimensional examples are presented.     

Entire solutions of quasilinear elliptic equations

We study entire solutions of non-homogeneous quasilinear elliptic equations, with Eqs. (1) and (2) below being typical. A particular special case of interest is the following: Let u be an entire distribution solution of the equation Δpu=|u|^q−1u, where p>1. If q>p−1 then u≡0. On the other hand, if 0<q<p−1 and u(x)=o(|x|^{p/(p−q−1)}) as |x|→∞, then again u≡0. If q=p−1 then u≡0 for all solutions with at most algebraic growth at infinity.     

Existence and uniqueness of nonnegative solutions for a boundary blow-up problem

Assume that Ω is a bounded domain in RN (N?3) with smooth boundary ∂Ω. In this work, we study existence and uniqueness of blow-up solutions for the problem −Δp(u)+c(x)|∇u|^p−1+F(x,u)=0 in Ω, where 2?p. Under some conditions related to the function F, we give a sufficient condition for existence and nonexistence of nonnegative blow-up solutions. We study also the uniqueness of these solutions.     

Existence and Nonexistence of Entire Positive Solutions of Semilinear Elliptic Systems

We show that entire positive solutions exist for the semilinear elliptic system Δu = p(x)v^α, Δv = q(x)u^β on R^N, N ≥ 3, for positive α and β, provided that the nonnegative functions p and q are continuous and satisfy appropriate decay conditions at infinity. We also show that entire solutions fail to exist if the functions p and q are of slow decay.     

Asymptotic behavior of positive solutions for heat equation with nonlinear term

We study the nonlinear parabolic equation , in Rn×(0,∞) with boundary condition u(x,0)=u₀(x), not necessarily bounded function. The nonlinearity φ((x,t),u) is required to satisfy some conditions related to the parabolic Kato class P^∞(Rn) while allowing existence of positive solutions of the equation and continuity of such solutions. Our approach is based on potential theory tools.     

On positive entire solutions of indefinite semilinear elliptic equations

We establish that the elliptic equation Δu+K(x)up+μf(x)=0 in Rn has a continuum of positive entire solutions for small μ?0 under suitable conditions on K, p and f. In particular, K behaves like l|x| at ∞ for some l?−2, but may change sign in a compact region. For given l>−2, there is a critical exponent pc=pc(n,l)>1 in the sense that the result holds for p?pc and involves partial separation of entire solutions. The partial separation means that the set of entire solutions possesses a non-trivial subset in which any two solutions do not intersect. The observation is well known when K is non-negative. The point of the paper is to remove the sign condition on compact region. When l=−2, the result holds for any p>1 while pc is decreasing to 1 as l decreases to −2.     

Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|^q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|^p−2∇u), with initial data u₀∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T₀] in which the problem admits a solution. More precisely, T₀ depends only on Lr|u₀| and f.     

Study of weak solutions for parabolic equations with nonstandard growth conditions

The authors of this paper study the existence and uniqueness of weak solutions of the initial and boundary value problem for ut=div((uσ+d₀)|∇u|^p(x,t)−2∇u)+f(x,t). Localization property of weak solutions is also discussed.