Existence of positive solutions for the Hénon equation involving critical Sobolev terms

Let N?3, ^2*=2N/(N−2) and Ω⊂RN be a bounded domain with a smooth boundary ∂Ω and 0∈Ω. Our purpose in this paper is to consider the existence of solutions of Hénon equation:

     

Gradient estimates for Dirichlet parabolic problems in unbounded domains

We consider autonomous parabolic Dirichlet problems in a regular unbounded open set Ω⊂R^N involving second-order operator A with (possibly) unbounded coefficients. We determine new conditions on the coefficients of A yielding global gradient estimates for the bounded classical solution.     

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.     

Concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent

We consider the boundary value problem Δu+up=0 in a bounded, smooth domain Ω in R² with homogeneous Dirichlet boundary condition and p a large exponent. We find topological conditions on Ω which ensure the existence of a positive solution up concentrating at exactly m points as p→∞. In particular, for a nonsimply connected domain such a solution exists for any given m?1.     

Asymptotic stability for a free boundary problem arising in a tumor model

We consider a tumor model in which all cells are proliferating at a rate μ and their density is proportional to the nutrient concentration. The model consists of a coupled system of an elliptic equation and a parabolic equation, with the tumor boundary as a free boundary. It is known that for an appropriate choice of parameters, there exists a unique spherically symmetric stationary solution with radius RS which is independent of μ. It was recently proved that there is a function μ_∗(RS) such that the spherical stationary solution is linearly stable if μ<μ_∗(RS) and linearly unstable if μ>μ_∗(RS). In this paper we prove that the spherical stationary solution is nonlinearly stable (or, asymptotically stable) if μ<μ_∗(RS).     

The heat equation with nonlinear generalized Robin boundary conditions

Let Ω⊂RN be a bounded domain and let μ be an admissible measure on ∂Ω. We show in the first part that if Ω has the H¹-extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by on ∂Ω, generates a strongly continuous nonlinear submarkovian semigroup SB=(SB(t))_t?0 on L²(Ω). We also obtain that this semigroup is ultracontractive in the sense that for every u,v∈Lp(Ω), p?2 and every t>0, one has

     

Blowing up boundary conditions for a nonlocal nonlinear diffusion equation in several space dimensions

We analyze boundary value problems prescribing Dirichlet or Neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation in a bounded smooth domain Ω∈R^N with N≥1. First, we prove existence and uniqueness of solutions and the validity of a comparison principle for these problems. Next, we impose boundary data that blow up in finite time and study the behavior of the solutions.     

Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems

We consider the 2m-th order elliptic boundary value problem Lu=f(x,u) on a bounded smooth domain Ω⊂RN with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic linear operator of order 2m whose principle part is of the form . We assume that f is superlinear at the origin and satisfies , , where are positive functions and q>1 is subcritical. By combining degree theory with new and recently established a priori estimates, we prove the existence of a nontrivial solution.     

Attractors for reaction-diffusion equations on thin domains whose linear part is non-self-adjoint

For a bounded smooth domain Ω⊂R^N_x+N_y let Ω_?, 0<?, be a family of domains squeezed in y∈R^N_y direction. On Ω_? we consider a reaction-diffusion equation with nonsymmetrical linear part. We show that under natural conditions on the nonlinearity the generated semi-flows have global attractors which in a certain sense have limits, as ?↓0.     

Concentration phenomena for a fourth-order equation with exponential growth: The radial case

We let Ω be a smooth bounded domain of R⁴ and a sequence of functions (Vk)_k∈N∈C⁰(Ω) such that limk→+∞Vk=1 in . We consider a sequence of functions (uk)_k∈N∈C⁴(Ω) such that Δ²uk=Vke4uk in Ω for all k∈N. We address in this paper the question of the asymptotic behavior of the (uk)'s when k→+∞. The corresponding problem in dimension 2 was considered by Brézis and Merle, and Li and Shafrir (among others), where a blow-up phenomenon was described and where a quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi, Struwe and the author in [Adimurthi, F. Robert and M. Struwe, Concentration phenomena for Liouville equations in dimension four, J. Eur. Math. Soc., in press, available on http://www-math.unice.fr/~frobert], a similar quantization phenomenon does not hold for this fourth-order problem. Assuming that the uk's are radially symmetrical, we push further the analysis of the mentioned work. We prove that there are exactly three types of blow-up and we describe each type in a very detailed way.     

Boundary behavior of large solutions to semilinear elliptic equations with nonlinear gradient terms

In this paper, we study the boundary behavior of solutions to boundary blow-up elliptic problems , where Ω is a bounded domain with smooth boundary in R^N, q>0, , which is positive in Ω and may be vanishing on the boundary and rapidly varying near the boundary, and f is rapidly varying or normalized regularly varying at infinity.     

A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing in the class of functions W1,G(Ω) with , for a given φ₀?0 and bounded. W1,G(Ω) is the class of weakly differentiable functions with . The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω∩∂{u>0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1,α regularity of their free boundaries near “flat” free boundary points.     

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.     

Existence of radial solutions with prescribed number of zeros for elliptic equations and their Morse index

In this paper we consider radially symmetric solutions of the nonlinear Dirichlet problem Δu+f(|x|,u)=0 in Ω, where Ω is a ball in RN, N?3 and f satisfies some appropriate assumptions. We prove existence of radially symmetric solutions with k prescribed number of zeros. Moreover, when f(|x|,u)=K(|x|)|u|^p−1u, using the uniqueness result due to Tanaka (2008) [21], we verify that these solutions are non-degenerate and we prove that their radial Morse index is exactly k.     

Nonlinear diffusion with a bounded stationary level surface

We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C²$C^2$. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C²$C^2$-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole RN${\mathbb{R}}^N$ of some substance whose density is initially a characteristic function of the complement of a domain with bounded C²$C^2$ boundary, and obtain similar results. These results are also extended to the heat flow in the sphere SN${\mathbb{S}}^N$ and the hyperbolic space HN${\mathbb{H}}^N$.     

Extremal equilibria for monotone semigroups in ordered spaces with application to evolutionary equations

We consider monotone semigroups in ordered spaces and give general results concerning the existence of extremal equilibria and global attractors. We then show some applications of the abstract scheme to various evolutionary problems, from ODEs and retarded functional differential equations to parabolic and hyperbolic PDEs. In particular, we exhibit the dynamical properties of semigroups defined by semilinear parabolic equations in RN with nonlinearities depending on the gradient of the solution. We consider as well systems of reaction-diffusion equations in RN and provide some results concerning extremal equilibria of the semigroups corresponding to damped wave problems in bounded domains or in RN. We further discuss some nonlocal and quasilinear problems, as well as the fourth order Cahn-Hilliard equation.     

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.     

Isoperimetric inequalities for the first Neumann eigenvalue in Gauss space

We provide isoperimetric Szegö–Weinberger type inequalities for the first nontrivial Neumann eigenvalue _μ1(Ω)${\mu }_1(\Omega )$ in Gauss space, where Ω   is a possibly unbounded domain of R^N${\mathbb{R}}^N$. Our main result consists in showing that among all sets Ω   of R^N${\mathbb{R}}^N$ symmetric about the origin, having prescribed Gaussian measure, _μ1(Ω)${\mu }_1(\Omega )$ is maximum if and only if Ω is the Euclidean ball centered at the origin.     

An optimization problem with volume constraint for a degenerate quasilinear operator

We consider the optimization problem of minimizing with a constraint on the volume of {u>0}. We consider a penalization problem, and we prove that for small values of the penalization parameter, the constrained volume is attained. In this way we prove that every solution u is locally Lipschitz continuous and that the free boundary, ∂{u>0}∩Ω, is smooth.     

Bubble towers for supercritical semilinear elliptic equations

We construct positive solutions of the semilinear elliptic problem with Dirichet boundary conditions, in a bounded smooth domain Ω⊂R^N(N?4), when the exponent p is supercritical and close enough to and the parameter λ∈R is small enough. As , the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. Our result extends the result of Del Pino et al. (J. Differential Equations 193(2) (2003) 280) when Ω is a ball and the solutions are radially symmetric.