Concentration on minimal submanifolds for a singularly perturbed Neumann problem

We consider the equation −ε²Δu+u=up in Ω⊆RN, where Ω is open, smooth and bounded, and we prove concentration of solutions along k-dimensional minimal submanifolds of ∂Ω, for N?3 and for k∈{1,…,N−2}. We impose Neumann boundary conditions, assuming 1<p<(N−k+2)/(N−k−2) and ε→⁰⁺. This result settles in full generality a phenomenon previously considered only in the particular case N=3 and k=1.     

On the Cauchy problem for the evolution p-Laplacian equations with gradient term and source

In this paper, the authors study the equation ut=div(|Du|^p−2Du)+|u|^q−1u−λl|Du| in RN with p>2. We first prove that for 1?l?p−1, the solution exists at least for a short time; then for , the existence and nonexistence of global (in time) solutions are studied in various situations.     

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.     

Linking solutions for p-Laplace equations with nonlinearity at critical growth

Under a suitable condition on n and p, the quasilinear equation at critical growth −Δpu=λ|u|^p−2u+|u|^p∗−2u is shown to admit a nontrivial weak solution for any λ?λ₁. Nonstandard linking structures, for the associated functional, are recognized.     

Improved decay rates for solutions to one-dimensional linear and semilinear dissipative wave equations in all space

Better decay estimates to the 1-dimensional Cauchy problem on to the linear equation □u+u_t=0 can be discussed under rather restricted conditions on the initial data. Furthermore, as applications we derive the small data global existence result to the equation □u+u_t=|u|^p−1u, which has the “odd” functions as the initial data. Furthermore, the new method (see R. Ikehata, T. Matsuyama, Sci. Math. Japon. 55 (2002) 33-42) used in the first half will be applied to the problem coming from Ehrenpreis (Sugaku 26 (1974) 168).     

Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

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.     

Nonlinear diffusions, hypercontractivity and the optimal L-Euclidean logarithmic Sobolev inequality

The equation u_t=Δ_p(u^1/(p−1)) for p>1 is a nonlinear generalization of the heat equation which is also homogeneous, of degree 1. For large time asymptotics, its links with the optimal L^p-Euclidean logarithmic Sobolev inequality have recently been investigated. Here we focus on the existence and the uniqueness of the solutions to the Cauchy problem and on the regularization properties (hypercontractivity and ultracontractivity) of the equation using the L^p-Euclidean logarithmic Sobolev inequality. A large deviation result based on a Hamilton-Jacobi equation and also related to the L^p-Euclidean logarithmic Sobolev inequality is then stated.     

Asymptotic behavior of solutions to Schrödinger equations with a subcritical dissipative nonlinearity

We study the asymptotic behavior in time of solutions to the initial value problem of the nonlinear Schrödinger equation with a subcritical dissipative nonlinearity λ|u|^p−1u, where 1<p<1+2/n, n is the space dimension and λ is a complex constant satisfying Imλ<0. We show the time decay estimates and the large-time asymptotics of the solution, when the space dimension n?3, p is sufficiently close to 1+2/n and the initial data is sufficiently small.     

Global attractors for p-Laplacian equation

The existence of a -global attractor is proved for the p-Laplacian equation ut−div(|∇u|^p−2∇u)+f(u)=g on a bounded domain Ω⊂Rn(n?3) with Dirichlet boundary condition, where p?2. The nonlinear term f is supposed to satisfy the polynomial growth condition of arbitrary order c₁q|u|−k?f(u)u?c₂q|u|+k and f^′(u)?−l, where q?2 is arbitrary. There is no other restriction on p and q. The asymptotic compactness of the corresponding semigroup is proved by using a new a priori estimate method, called asymptotic a priori estimate.     

An application of semigroups of locally Lipschitz operators to Carrier equations with acoustic boundary conditions

A generation theorem of semigroups of locally Lipschitz operators on a subset of a real Banach space is given and applied to the problem of the well-posedness of the Carrier equation utt−κ(‖u^‖2)Δu+γ|ut|^p−1ut=0 in Ω×(0,∞) with acoustic boundary condition, where p>2 and Ω is a bounded domain in an arbitrary dimensional space.     

Local existence and uniqueness for a semilinear accretive wave equation

We study local existence and uniqueness in the phase space Hμ×Hμ−1(RN) of the solution of the semilinear wave equation utt−Δu=ut|ut|^p−1 for p>1.     

Multiple solutions for a Hénon-like equation on the annulus

For the equation −Δu=||xα|−2|up−1, 1<|x|<3, we prove the existence of two solutions for α large, and of two additional solutions when p is close to the critical Sobolev exponent ^2∗=2N/(N−2). A symmetry-breaking phenomenon appears, showing that the least-energy solutions cannot be radial functions.     

Boundary singularities of solutions of N-harmonic equations with absorption

We study the boundary behaviour of solutions u of −ΔNu+|u|^q−1u=0 in a bounded smooth domain Ω⊂RN subject to the boundary condition u=0 except at one point, in the range q>N−1. We prove that if q?2N−1 such an u is identically zero, while, if N−1<q<2N−1, u inherits a boundary behaviour which either corresponds to a weak singularity, or to a strong singularity. Such singularities are effectively constructed.     

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.     

Nontrivial solutions for perturbations of a Hardy-Sobolev operator on unbounded domains

In this paper we study the existence of nontrivial solution of the problem −Δ_pu−(μ/[d(x)]^p)|u|^p−2u=f(u) in Ω and u=0 on ∂Ω, where is a bounded domain with smooth boundary in Existence is established using mountain-pass lemma and concentration of compactness principle.     

Solutions concentrating on higher dimensional subsets for singularly perturbed elliptic equations II

We construct spike layered solutions for the semilinear elliptic equation −ε²Δu+V(x)u=K(x)up−1 on a domain Ω⊂RN which may be bounded or unbounded. The solutions concentrate simultaneously on a finite number of m-dimensional spheres in Ω. These spheres accumulate as ε→0 at a prescribed sphere in Ω whose location is determined by the potential functions V,K.     

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.     

Propagation property for anisotropic nonlinear diffusion equation with convection

We consider propagation property for anisotropic diffusion equation with convection in 2 dimension,

t_∂(um)−x_1∂(|x_1∂u|^p₁−1x_1∂u)−x_2∂(|x_2∂u|^p₂−1x_2∂u)+uα−1x_1∂u=0,