Volume 20, Issue 2 - Special Issue on Nonlinear Partial Differential Equations. A Tribute to L. Véron for his 70th Anniversary, Guest Editors: J. López-Gómez and P. Pucci

We consider the elliptic equation -Δ⁢u=uq⁢|∇⁡u|p{-\Delta u=u^{q}|\nabla u|^{p}} in ℝn{\mathbb{R}^{n}} for any p>2{p>2} and q>0{q>0}. We prove a Liouville-type theorem, which asserts that any positive bounded solution is constant. The proof technique is based on monotonicity properties for the spherical averages of sub- and super-harmonic functions, combined with a gradient bound obtained by a local Bernstein argument. This solves, in the case of bounded solutions, a problem left open in [2], where the case 0<p<2{0<p<2} is considered. Some extensions to elliptic systems are also given.

This paper characterizes whether or not Σ ∞ ≡ lim λ ↑ ∞ ⁡ σ ⁢ [ 𝒫 + λ ⁢ m ⁢ ( x , t ) , 𝔅 , Q T ] \Sigma_{\infty}\equiv\lim_{\lambda\uparrow\infty}\sigma[\mathcal{P}+\lambda m(% x,t),\mathfrak{B},Q_{T}] is finite, where m⪈0{m\gneq 0} is T -periodic and σ⁢[𝒫+λ⁢m⁢(x,t),𝔅,QT]{\sigma[\mathcal{P}+\lambda m(x,t),\mathfrak{B},Q_{T}]} stands for the principal eigenvalue of the parabolic operator 𝒫+λ⁢m⁢(x,t){\mathcal{P}+\lambda m(x,t)} in QT≡Ω×[0,T]{Q_{T}\equiv\Omega\times[0,T]} subject to a general boundary operator of mixed type, 𝔅{\mathfrak{B}}, on ∂⁡Ω×[0,T]{\partial\Omega\times[0,T]}. Then this result is applied to discuss the nature of the territorial refuges in periodic competitive environments.

For exponents in the subcritical range, we revisit some optimal interpolation inequalities on the sphere with carré du champ methods and use the remainder terms to produce improved inequalities. The method provides us with lower estimates of the optimal constants in the symmetry breaking range and stability estimates for the optimal functions. Some of these results can be reformulated in the Euclidean space using the stereographic projection.

In this paper we deal with positive radially symmetric solutions for a boundary value problem containing a strongly nonlinear operator. The proof of existence of positive solutions that we give uses the blow-up method as a main ingredient for the search of a-priori bounds of solutions. The blow-up argument is one by contradiction and uses a sort of scaling, reminiscent to the one used in the theory of minimal surfaces, see [B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations 6 1981, 883–901], and therefore the homogeneity of the operators, Laplacian or p -Laplacian, and second members powers or power like functions play a fundamental role in the method. Thus, when the differential operators are no longer homogeneous, and similarly for the second members, applying the blow-up method to obtain a-priori bounds of solutions seems an almost impossible task. In spite of this fact, in [M. García-Huidobro, I. Guerra and R. Manásevich, Existence of positive radial solutions for a weakly coupled system via blow up, Abstr. Appl. Anal. 3 1998, 1–2, 105–131], we were able to overcome this difficulty and obtain a-priori bounds for a certain (simpler) type of problems. We show in this paper that the asymptotically homogeneous functions provide, in the same sense, a nonlinear rescaling, that allows us to generalize the blow-up method to our present situation. After the a-priori bounds are obtained, the existence of a solution follows from Leray–Schauder topological degree theory.

In this paper, we study global well-posedness and long-time asymptotic behavior of solutions to the nonlinear heat equation with absorption, ut-Δ⁢u+|u|α⁢u=0{u_{t}-\Delta u+\lvert u\rvert^{\alpha}u=0}, where u=u⁢(t,x)∈ℝ{u=u(t,x)\in\mathbb{R}}, (t,x)∈(0,∞)×ℝN{(t,x)\in(0,\infty)\times\mathbb{R}^{N}} and α>0{\alpha>0}. We focus particularly on highly singular initial values which are antisymmetric with respect to the variables x1,x2,…,xm{x_{1},x_{2},\ldots,x_{m}} for some m∈{1,2,…,N}{m\in\{1,2,\ldots,N\}}, such as u0=(-1)m∂1∂2⋯∂m|⋅|-γ∈𝒮′(ℝN){u_{0}=(-1)^{m}\partial_{1}\partial_{2}\cdots\partial_{m}\lvert\,{\cdot}\,% \rvert^{-\gamma}\in\mathcal{S}^{\prime}(\mathbb{R}^{N})}, 0<γ<N{0<\gamma<N}. In fact, we show global well-posedness for initial data bounded in an appropriate sense by u0{u_{0}} for any α>0{\alpha>0}. Our approach is to study well-posedness and large time behavior on sectorial domains of the form Ωm={x∈ℝN:x1,…,xm>0}{\Omega_{m}=\{x\in\mathbb{R}^{N}:x_{1},\ldots,x_{m}>0\}}, and then to extend the results by reflection to solutions on ℝN{\mathbb{R}^{N}} which are antisymmetric. We show that the large time behavior depends on the relationship between α and 2γ+m{\frac{2}{\gamma+m}}, and we consider all three cases, α equal to, greater than, and less than 2γ+m{\frac{2}{\gamma+m}}. Our results include, among others, new examples of self-similar and asymptotically self-similar solutions.

Our purpose in this paper is to study positive solutions of the Lane–Emden equation - Δ ⁢ u = V ⁢ u p   in  ⁢ ℝ N ∖ { 0 } , -\Delta u=Vu^{p}\quad\text{in }\mathbb{R}^{N}\setminus\{0\}, perturbed by a nonhomogeneous potential V , with p∈(NN-2,pc)p\in(\frac{N}{N-2},p_{c}), where pc{p_{c}} is the Joseph–Ludgren exponent. We construct a sequence of fast and slow decaying solutions with appropriated restrictions for V .

It is known that the Sobolev space W1,p⁢(ℝN){W^{1,p}(\mathbb{R}^{N})} is embedded into LN⁢p/(N-p)⁢(ℝN){L^{Np/(N-p)}(\mathbb{R}^{N})} if p<N{p<N} and into L∞⁢(ℝN){L^{\infty}(\mathbb{R}^{N})} if p>N{p>N}. There is usually a discontinuity in the proof of those two different embeddings since, for p>N{p>N}, the estimate ∥u∥∞≤C⁢∥D⁢u∥pN/p⁢∥u∥p1-N/p{\lVert u\rVert_{\infty}\leq C\lVert Du\rVert_{p}^{N/p}\lVert u\rVert_{p}^{1-N% /p}} is commonly obtained together with an estimate of the Hölder norm. In this note, we give a proof of the L∞{L^{\infty}}-embedding which only follows by an iteration of the Sobolev–Gagliardo–Nirenberg estimate ∥u∥N/(N-1)≤C⁢∥D⁢u∥1{\lVert u\rVert_{N/(N-1)}\leq C\lVert Du\rVert_{1}}. This kind of proof has the advantage to be easily extended to anisotropic cases and immediately exported to the case of discrete Lebesgue and Sobolev spaces; we give sample results in case of finite differences and finite volumes schemes.

We characterize the existence of solutions to the quasilinear Riccati-type equation { - div ⁡ 𝒜 ⁢ ( x , ∇ ⁡ u ) = | ∇ ⁡ u | q + σ in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}\mathcal{A}(x,\nabla u)% &\displaystyle=|\nabla u|^{q}+\sigma&&\displaystyle\phantom{}\text{in }\Omega,% \\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. with a distributional or measure datum σ. Here div⁡𝒜⁢(x,∇⁡u){\operatorname{div}\mathcal{A}(x,\nabla u)} is a quasilinear elliptic operator modeled after the p -Laplacian (p>1{p>1}), and Ω is a bounded domain whose boundary is sufficiently flat (in the sense of Reifenberg). For distributional data, we assume that p>1{p>1} and q>p{q>p}. For measure data, we assume that they are compactly supported in Ω, p>3⁢n-22⁢n-1{p>\frac{3n-2}{2n-1}}, and q is in the sub-linear range p-1<q<1{p-1<q<1}. We also assume more regularity conditions on 𝒜{\mathcal{A}} and on ∂⁡Ω⁢Ω{\partial\Omega\Omega} in this case.

We obtain sufficient conditions for solutions of the m th-order differential inequality ∑ | α | = m ∂ α ⁡ a α ⁢ ( x , u ) ≥ f ⁢ ( x ) ⁢ g ⁢ ( | u | )   in  ⁢ B 1 ∖ { 0 } \sum_{|\alpha|=m}\partial^{\alpha}a_{\alpha}(x,u)\geq f(x)g(|u|)\quad\text{in % }B_{1}\setminus\{0\} to have a removable singularity at zero, where aα{a_{\alpha}}, f , and g are some functions, and B1={x:|x|<1}{B_{1}=\{x:|x|<1\}} is a unit ball in ℝn{{\mathbb{R}}^{n}}. We show in some examples the sharpness of these conditions.

Let Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} (N≥3{N\geq 3}) be a C2{C^{2}} bounded domain, and let δ be the distance to ∂⁡Ω{\partial\Omega}. We study equations (E±){(E_{\pm})}, -Lμ⁢u±g⁢(u,|∇⁡u|)=0{-L_{\mu}u\pm g(u,\lvert\nabla u\rvert)=0} in Ω, where Lμ=Δ+μδ2{L_{\mu}=\Delta+\frac{\mu}{\delta^{2}}}, μ∈(0,14]{\mu\in(0,\frac{1}{4}]} and g:ℝ×ℝ+→ℝ+{g\colon\mathbb{R}\times\mathbb{R}_{+}\to\mathbb{R}_{+}} is nondecreasing and locally Lipschitz in its two variables with g⁢(0,0)=0{g(0,0)=0}. We prove that, under some subcritical growth assumption on g , equation (E+){(E_{+})} with boundary condition u=ν{u=\nu} admits a solution for any nonnegative bounded measure on ∂⁡Ω{\partial\Omega}, while equation (E-){(E_{-})} with boundary condition u=ν{u=\nu} admits a solution provided that the total mass of ν is small. Then we analyze the model case g⁢(s,t)=|s|p⁢tq{g(s,t)=\lvert s\rvert^{p}t^{q}} and obtain a uniqueness result, which is even new with μ=0{\mu=0}. We also describe isolated singularities of positive solutions to (E+){(E_{+})} and establish a removability result in terms of Bessel capacities. Various existence results are obtained for (E-){(E_{-})}. Finally, we discuss existence, uniqueness and removability results for (E±){(E_{\pm})} in the case g⁢(s,t)=|s|p+tq{g(s,t)=\lvert s\rvert^{p}+t^{q}}.

The classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator -Δ+V{-\Delta+V} with a nonnegative potential V which merely belongs to Lloc1⁢(Ω){L_{\mathrm{loc}}^{1}(\Omega)}. More precisely, if u∈W01,2⁢(Ω)∩L2⁢(Ω;V⁢d⁢x){u\in W_{0}^{1,2}(\Omega)\cap L^{2}(\Omega;V\mathop{}\!\mathrm{d}{x})} satisfies -Δ⁢u+V⁢u=f{-\Delta u+Vu=f} on Ω for some nonnegative datum f∈L∞⁢(Ω){f\in L^{\infty}(\Omega)}, f≢0{f\not\equiv 0}, then we show that at every point a∈∂⁡Ω{a\in\partial\Omega} where the classical normal derivative ∂⁡u⁢(a)∂⁡n{\frac{\partial u(a)}{\partial n}} exists and satisfies the Poisson representation formula, one has ∂⁡u⁢(a)∂⁡n>0{\frac{\partial u(a)}{\partial n}>0} if and only if the boundary value problem { - Δ ⁢ v + V ⁢ v = 0 in  Ω , v = ν on  ∂ ⁡ Ω , \begin{dcases}\begin{aligned} \displaystyle-\Delta v+Vv&\displaystyle=0&&% \displaystyle\phantom{}\text{in ${\Omega}$,}\\ \displaystyle v&\displaystyle=\nu&&\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\end{dcases} involving the Dirac measure ν=δa{\nu=\delta_{a}} has a solution. More generally, we characterize the nonnegative finite Borel measures ν on ∂⁡Ω{\partial\Omega} for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.

We consider the high-dimensional equation ∂t⁡u-Δ⁢um+u-β⁢χ{u>0}=0{\partial_{t}u-\Delta u^{m}+u^{-\beta}{\chi_{\{u>0\}}}=0}, extending the mathematical treatment made in 1992 by B. Kawohl and R. Kersner for the one-dimensional case. Besides the existence of a very weak solution u∈𝒞⁢([0,T];Lδ1⁢(Ω)){u\in\mathcal{C}([0,T];L_{\delta}^{1}(\Omega))}, with u-β⁢χ{u>0}∈L1⁢((0,T)×Ω){u^{-\beta}\chi_{\{u>0\}}\in L^{1}((0,T)\times\Omega)}, δ⁢(x)=d⁢(x,∂⁡Ω){\delta(x)=d(x,\partial\Omega)}, we prove some pointwise gradient estimates for a certain range of the dimension N , m≥1{m\geq 1} and β∈(0,m){\beta\in(0,m)}, mainly when the absorption dominates over the diffusion (1≤m<2+β{1\leq m<2+\beta}). In particular, a new kind of universal gradient estimate is proved when m+β≤2{m+\beta\leq 2}. Several qualitative properties (such as the finite time quenching phenomena and the finite speed of propagation) and the study of the Cauchy problem are also considered.

In this paper, dedicated to Laurent Veron, we prove that the Strong Maximum Principle holds for solutions of some quasilinear elliptic equations having lower order terms with quadratic growth with respect to the gradient of the solution.