Lane-Emden-Fowler equations with convection and singular potential

We are concerned with singular elliptic problems of the form −Δu±p(d(x))g(u)=λf(x,u)+μa|∇u| in Ω, where Ω is a smooth bounded domain in RN, d(x)=dist(x,∂Ω), λ>0, μ∈R, 0<a?2, and f is a nondecreasing function. We assume that p(d(x)) is a positive weight with possible singular behavior on the boundary of Ω and that the nonlinearity g is unbounded around the origin. Taking into account the competition between the anisotropic potential p(d(x)), the convection term a|∇u|, and the singular nonlinearity g, we establish various existence and nonexistence results.     

Boundary behaviour of the unique solution to a singular Dirichlet problem with a convection term

By Karamata regular variation theory and constructing comparison functions, we derive that the boundary behaviour of the unique solution to a singular Dirichlet problem −Δu=b(x)g(u)+λq|∇u|, u>0, x∈Ω, u_|∂Ω=0, which is independent of λq|∇uλ|, where Ω is a bounded domain with smooth boundary in RN, λ∈R, q∈(0,2], lims→⁰⁺g(s)=+∞, and b is non-negative on Ω, which may be vanishing on the boundary.     

Asymptotic behavior of large solution for boundary blowup problems with non-linear gradient terms

Let Ω⊂R^N(N?3) be a bounded domain with smooth boundary. We show the asymptotic behavior of boundary blowup solutions to non-linear elliptic equation Δu±|∇u|^q=b(x)f(u) in Ω, subject to the singular boundary condition u(x)=∞ as dist(x,∂Ω)→0,f is Γ-varying at ∞. Our analysis is based on the Karamata regular variation theory combined with the method of lower and supper solution.     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

Coincidence sets in quasilinear elliptic problems of monostable type

This paper concerns the formation of a coincidence set for the positive solution of the boundary value problem: −εΔpu=uq−1f(a(x)−u) in Ω with u=0 on ∂Ω, where ε is a positive parameter, Δpu=div(|∇u|^p−2∇u), 1<q?p<∞, f(s)∼|s|^θ−1s(s→0) for some θ>0 and a(x) is a positive smooth function satisfying Δpa=0 in Ω with infΩ|∇a|>0. It is proved in this paper that if 0<θ<1 the coincidence set Oε={x∈Ω:uε(x)=a(x)} has a positive measure for small ε and converges to Ω with order O(ε1/p) as ε→0. Moreover, it is also shown that if θ?1, then Oε is empty for any ε>0. The proofs rely on comparison theorems and the energy method for obtaining local comparison functions.     

Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term

We are concerned with singular elliptic equations of the form −Δu=p(x)(g(u)+f(u)+a|∇u|) in RN (N?3), where p is a positive weight and 0<a<1. Under the hypothesis that f is a nondecreasing function with sublinear growth and g is decreasing and unbounded around the origin, we establish the existence of a ground state solution vanishing at infinity. Our arguments rely essentially on the maximum principle.     

Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting

We study the boundary value problem −div(log(1+q|∇u|)|∇u|^p−2∇u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary. We distinguish the cases where either f(u)=−λ|u|^p−2u+|u|^r−2u or f(u)=λ|u|^p−2u−|u|^r−2u, with p, q>1, p+q<min{N,r}, and r<(Np−N+p)/(N−p). In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.     

Existence of a unique solution to a quasilinear elliptic equation

The purpose of this paper is to prove the existence of a unique classical solution u(x) to the quasilinear elliptic equation −∇⋅(a(u)∇u)+v⋅∇u=f, where u(x₀)=u₀ at x₀∈Ω and where n⋅∇u=g on the boundary ∂Ω. We prove that if the functions a, f, v, g satisfy certain conditions, then a unique classical solution u(x) exists. Applications include stationary heat/diffusion problems with convection and with a source/sink, where the value of the solution is known at a spatial location x₀∈Ω, and where n⋅∇u is known on the boundary.     

Bifurcation and asymptotics for the Lane–Emden–Fowler equation

We are concerned with the Lane–Emden–Fowler equation ?Δu=λf(u)+a(x)g(u) in $\text{Ω}$, subject to the Dirichlet boundary condition u=0 on $\text{?Ω,}$ where $\text{Ω?}\text{R}{}^{\mathrm{N}}$ is a smooth bounded domain, λ is a positive parameter, $\text{a}\text{:}\text{Ω}\text{→[0,∞)}$ is a Hölder function, and f is a positive nondecreasing continuous function such that f(s)/s is nonincreasing in (0,∞). The singular character of the problem is given by the nonlinearity g which is assumed to be unbounded around the origin. In this Note we discuss the existence and the uniqueness of a positive solution of this problem and we also describe the precise decay rate of this solution near the boundary. The proofs rely essentially on the maximum principle and on elliptic estimates. To cite this article: M. Ghergu, V.D. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

Multiple positive and 2-nodal symmetric solutions of elliptic problems with critical nonlinearity

We consider the problem −Δu+a(x)u=f(x)|u|^2*−2u in Ω, u=0 on ∂Ω, where Ω is a bounded smooth domain in RN, N?4, is the critical Sobolev exponent, and a,f are continuous functions. We assume that Ω, a and f are invariant under the action of a group of orthogonal transformations. We obtain multiplicity results which contain information about the symmetry and symmetry-breaking properties of the solutions, and about their nodal domains. Our results include new multiplicity results for the Brezis-Nirenberg problem −Δu+λu=|u|^2*−2u in Ω, u=0 on ∂Ω.     

Existence and nonexistence of blow-up boundary solutions for sublinear elliptic equations

In this paper we consider the semilinear elliptic problem Δu=a(x)f(u), u?0 in Ω, with the boundary blow-up condition u_|∂Ω=+∞, where Ω is a bounded domain in RN(N?2), a(x)∈C(Ω) may blow up on ∂Ω and f is assumed to satisfy (f₁) and (f₂) below which include the sublinear case f(u)=um, m∈(0,1). For the radial case that Ω=B (the unit ball) and a(x) is radial, we show that a solution exists if and only if . For Ω a general domain, we obtain an optimal nonexistence result. The existence for nonradial solutions is also studied by using sub-supersolution method.     

Boundary blow-up elliptic problems with nonlinear gradient terms

By Karamata regular variation theory and perturbation method, we show the exact asymptotical behaviour of solutions near the boundary to nonlinear elliptic problems Δu±q|∇u|=b(x)g(u), u>0 in Ω, u_|∂Ω=+∞, where Ω is a bounded domain with smooth boundary in RN, q?0, g∈C¹[0,∞),g(0)=0, g^′ is regularly varying at infinity with index ρ with ρ>0 and b is nonnegative nontrivial in Ω, which may be vanishing on the boundary.     

Nondegeneracy and uniqueness for boundary blow-up elliptic problems

In this paper, we use for the first time linearization techniques to deal with boundary blow-up elliptic problems. After introducing a convenient functional setting, we show that the problem Δu=λa(x)u^p+g(x,u) in Ω, with u=+∞ on ∂Ω, has a unique positive solution for large enough λ, and determine its asymptotic behavior as λ→+∞. Here p>1, a(x) is a continuous function which can be singular near ∂Ω and g(x,u) is a perturbation term with potential growth near zero and infinity. We also consider more general problems, obtained by replacing u^p by e^u or a “logistic type” function f(u).     

Sublinear singular elliptic problems with two parameters

We establish several existence and nonexistence results for the boundary value problem −Δu+K(x)g(u)=λf(x,u)+μh(x) in Ω, u=0 on ∂Ω, where Ω is a smooth bounded domain in , λ and μ are positive parameters, h is a positive function, while f has a sublinear growth. The main feature of this paper is that the nonlinearity g is assumed to be unbounded around the origin. Our analysis shows the importance of the role played by the decay rate of g combined with the signs of the extremal values of the potential K(x) on . The proofs are based on various techniques related to the maximum principle for elliptic equations.     

Large time behavior of solutions to degenerate parabolic equations with weights

We study the degenerate parabolic equation t_∂u=a(δ(x))upΔu−g(u) in Ω×(0,∞), where Ω⊂RN (N?1) is a smooth bounded domain, p?1, δ(x)=dist(x,∂Ω) and a is a continuous nondecreasing function such that a(0)=0. Under some suitable assumptions on a and g we prove the existence and the uniqueness of a classical solution and we study its asymptotic behavior as t→∞.     

Isolated singularities for weighted quasilinear elliptic equations

We classify all the possible asymptotic behavior at the origin for positive solutions of quasilinear elliptic equations of the form div(|∇u|^p−2∇u)=b(x)h(u) in Ω?{0}, where 1<p?N and Ω is an open subset of RN with 0∈Ω. Our main result provides a sharp extension of a well-known theorem of Friedman and Véron for h(u)=uq and b(x)≡1, and a recent result of the authors for p=2 and b(x)≡1. We assume that the function h is regularly varying at ∞ with index q (that is, limt→∞h(λt)/h(t)=λq for every λ>0) and the weight function b(x) behaves near the origin as a function b₀(|x|) varying regularly at zero with index θ greater than −p. This condition includes b(x)=θ|x| and some of its perturbations, for instance, b(x)=θ|x|m(−log|x|) for any m∈R. Our approach makes use of the theory of regular variation and a new perturbation method for constructing sub- and super-solutions.     

Bifurcation for a class of singular elliptic problems with quadratic convection term

We study the bifurcation problem ?Δu=g(u)+λ|?u|²+μ in $\text{Ω,}\text{u=0}$ on $\text{?Ω}$, where λ,μ?0 and $\text{Ω}$ is a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. The singular character of the problem is given by the nonlinearity g which is assumed to be decreasing and unbounded around the origin. In this Note we prove that the above problem has a positive classical solution (which is unique) if and only if λ(a+μ)<λ₁, where a=lim_t→+∞g(t) and λ₁ is the first eigenvalue of the Laplace operator in $\text{H}{}^1{}_0\text{(Ω)}$. We also describe the decay rate of this solution, as well as a blow-up result around the bifurcation parameter. To cite this article: M. Ghergu, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

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.     

Existence of boundary blow-up solutions for a quasilinear elliptic equation

In this paper, the existence of solution for a class of quasilinear elliptic problem div(|? u| ^p?2 ? u)=a(x)f(u), u≥0 in Ω=B (the unit ball), with the boundary blow-up condition u| _?Ω=+∞ is established, where a(x)∈C(Ω) blows up on ?Ω,p>1 and f is assumed to satisfy (f ₁) and (f ₂). The results are obtained by using sub-supersolution methods.