Volume 21, Issue 2

This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏:[a,b]×[c,d]→Φ0⁢(U,V){\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)}, (λ,μ)↦𝔏⁢(λ,μ){(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)}, depends continuously on the perturbation parameter , μ, and holomorphically, as well as nonlinearly, on the spectral parameter , λ, where Φ0⁢(U,V){\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V . The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

We study of the regularizing effect of the interaction between the coefficient of the zero-order term and the lower-order term in quasilinear Dirichlet problems whose model is ∫ Ω M ⁢ ( x , u ) ⁢ ∇ ⁡ u ⋅ ∇ ⁡ φ + ∫ Ω a ⁢ ( x ) ⁢ u ⁢ φ = ∫ Ω b ⁢ ( x ) ⁢ | ∇ ⁡ u | q ⁢ φ + ∫ Ω f ⁢ ( x ) ⁢ φ   for all  ⁢ φ ∈ W 0 1 , 2 ⁢ ( Ω ) ∩ L ∞ ⁢ ( Ω ) , \int_{\Omega}M(x,u)\nabla u\cdot\nabla\varphi+\int_{\Omega}a(x)u\varphi=\int_{% \Omega}b(x)|\nabla u|^{q}\varphi+\int_{\Omega}f(x)\varphi\quad\text{for all }% \varphi\in W_{0}^{1,2}(\Omega)\cap L^{\infty}(\Omega), where Ω is a bounded open set of ℝN{\mathbb{R}^{N}}, M⁢(x,s){M(x,s)} is a Carathéodory matrix on Ω×ℝ{\Omega\times\mathbb{R}} which is elliptic (that is, M⁢(x,s)⁢ξ⋅ξ≥α⁢|ξ|2>0{M(x,s)\xi\cdot\xi\geq\alpha|\xi|^{2}>0} for every (x,s,ξ)∈Ω×ℝ×(ℝN∖{0}){(x,s,\xi)\in\Omega\times\mathbb{R}\times(\mathbb{R}^{N}\setminus\{0\})}) and bounded (that is, |M⁢(x,s)|≤β{|M(x,s)|\leq\beta} for every (x,s)∈Ω×ℝ{(x,s)\in\Omega\times\mathbb{R}}), b⁢(x)∈L22-q⁢(Ω){b(x)\in L^{\frac{2}{2-q}}(\Omega)}, 1<q<2{1<q<2} and 0≤a⁢(x)∈L1⁢(Ω){0\leq a(x)\in L^{1}(\Omega)}. We prove the existence of a weak solution u belonging to W01,2⁢(Ω){W_{0}^{1,2}(\Omega)} and to L∞⁢(Ω){L^{\infty}(\Omega)} when either b ∈ L 2 ⁢ m 2 - q ⁢ ( Ω ) ⁢  for some  ⁢ m > N 2 ⁢  and \displaystyle b\in L^{\frac{2m}{2-q}}(\Omega)\text{ for some }m>\frac{N}{2}% \text{ and} (0.1) ∃ Q > 0 ⁢  such that  ⁢ | f ⁢ ( x ) | ≤ Q ⁢ a ⁢ ( x ) \displaystyle\exists\,Q>0\text{ such that }|f(x)|\leq Qa(x) or f ∈ L m ⁢ ( Ω ) ⁢  for some  ⁢ m > N 2 ⁢  and \displaystyle f\in L^{m}(\Omega)\text{ for some }m>\frac{N}{2}\text{ and} (0.2) ∃ R > 0 ⁢  such that  ⁢ | b ⁢ ( x ) | 2 2 - q ≤ R ⁢ a ⁢ ( x ) . \displaystyle\exists\,R>0\text{ such that }|b(x)|^{\frac{2}{2-q}}\leq Ra(x). In addition, we also prove the existence for every f∈L1⁢(Ω){f\kern-1.0pt\in\kern-1.0ptL^{1}(\Omega)} and b⁢(x)∈L22-q⁢(Ω){b(x)\kern-1.0pt\in\kern-1.0ptL^{\frac{2}{2-q}}(\Omega)} satisfying both conditions (0.1) and (0.2) jointly.

In the present paper, we study the existence of nonnegative solutions to the Dirichlet problem ℒp,qM⁢u:=-Δ⁢u+up-M⁢|∇⁡u|q=μ{{\mathcal{L}}^{{M}}_{p,q}u:=-\Delta u+u^{p}-M|\nabla u|^{q}=\mu} in a domain Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} where μ is a nonnegative Radon measure, when p>1{p>1}, q>1{q>1} and M≥0{M\geq 0}. We also give conditions under which nonnegative solutions of ℒp,qM⁢u=0{{\mathcal{L}}^{{M}}_{p,q}u=0} in Ω∖K{\Omega\setminus K}, where K is a compact subset of Ω, can be extended as a solution of the same equation in Ω.

In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.

We consider a degenerate scalar parabolic equation, in one spatial dimension, of flux-saturated type. The equation also contains a convective term. We study the existence and regularity of traveling-wave solutions; in particular we show that they can be discontinuous. Uniqueness is recovered by requiring an entropy condition, and entropic solutions turn out to be the vanishing-diffusion limits of traveling-wave solutions to the equation with an additional non-degenerate diffusion. Applications to crowds dynamics, which motivated the present research, are also provided.

Let ℍ2{\mathbb{H}^{2}} be the hyperbolic space of dimension 2. Denote by Mn=ℍ2×ℝn-2{M^{n}=\mathbb{H}^{2}\times\mathbb{R}^{n-2}} the product manifold of ℍ2{\mathbb{H}^{2}} and ℝn-2(n≥3){\mathbb{R}^{n-2}(n\geq 3)}. In this paper we establish some sharp Hardy–Adams inequalities on Mn{M^{n}}, though Mn{M^{n}} is not with strictly negative sectional curvature. We also show that the sharp constant in the Poincaré–Sobolev inequality on Mn{M^{n}} coincides with the best Sobolev constant, which is of independent interest.

We consider the following Hénon-type problem with critical growth: (H) { - Δ ⁢ u = K ⁢ ( | y ′ | , y ′′ ) ⁢ u 2 * - 1 , u > 0 in  ⁢ B 1 , u = 0 on  ⁢ ∂ ⁡ B 1 , \displaystyle{}\left\{\begin{aligned} \displaystyle{}-\Delta u&\displaystyle=K% (|y^{\prime}|,y^{\prime\prime})u^{2^{*}-1},\quad u>0&&\displaystyle\text{in }B% _{1},\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial B_{1},\end{% aligned}\right. where 2*=2⁢NN-2{2^{*}\kern-1.0pt=\kern-1.0pt\frac{2N}{N-2}}, N≥5{N\kern-1.0pt\geq\kern-1.0pt5}, B1{B_{1}} is the unit sphere in ℝN{\mathbb{R}^{N}}, y=(y′,y′′)∈ℝ2×ℝN-2y\kern-1.0pt=\kern-1.0pt(y^{\prime},y^{\prime\prime})\kern-1.0pt\in\kern-1.0pt% \mathbb{R}^{2}\times\mathbb{R}^{N-2}, r=|y′|{r\kern-1.0pt=\kern-1.0pt|y^{\prime}|} and K⁢(y)=K⁢(r,y′′)∈C2⁢(B1){K(y)\kern-1.0pt=\kern-1.0ptK(r,y^{\prime\prime})\kern-1.0pt\in\kern-1.0ptC^{2% }(B_{1})} is a bounded non-negative function. By using a finite reduction argument and local Pohozaev-type identities, we prove that if N≥5{N\geq 5} and K⁢(r,y′′){K(r,y^{\prime\prime})} has a stable critical point y0=(r0,y0′′)∈∂⁡B1{y_{0}=(r_{0},y_{0}^{\prime\prime})\in\partial B_{1}}, then the above problem has infinitely many solutions, whose energy can be arbitrarily large.

In this article, we are interested in multi-bump solutions of the singularly perturbed problem - ε 2 ⁢ Δ ⁢ v + V ⁢ ( x ) ⁢ v = f ⁢ ( v )   in  ⁢ ℝ N . -\varepsilon^{2}\Delta v+V(x)v=f(v)\quad\text{in }\mathbb{R}^{N}. Extending previous results, we prove the existence of multi-bump solutions for an optimal class of nonlinearities f satisfying the Berestycki–Lions conditions and, notably, also for more general classes of potential wells than those previously studied. We devise two novel topological arguments to deal with general classes of potential wells. Our results prove the existence of multi-bump solutions in which the centers of bumps converge toward potential wells as ε→0{\varepsilon\rightarrow 0}. Examples of potential wells include the following: the union of two compact smooth submanifolds of ℝN{\mathbb{R}^{N}} where these two submanifolds meet at the origin and an embedded topological submanifold of ℝN{\mathbb{R}^{N}}.

The aim of this paper is to extend the theory of lower and upper solutions to the periodic problem associated with planar systems of differential equations. We generalize previously given definitions and we are able to treat both the well-ordered case and the non-well-ordered case. The proofs involve topological degree arguments, together with a detailed analysis of the solutions in the phase plane.

We consider the following nonlinear Schrödinger equation involving supercritical growth: { - Δ ⁢ u + V ⁢ ( y ) ⁢ u = Q ⁢ ( y ) ⁢ u 2 * - 1 + ε in  ⁢ ℝ N , u > 0 , u ∈ H 1 ⁢ ( ℝ N ) , \left\{\begin{aligned} &\displaystyle{-}\Delta u+V(y)u=Q(y)u^{2^{*}-1+% \varepsilon}&&\displaystyle\phantom{}\text{in }\mathbb{R}^{N},\\ &\displaystyle u>0,\quad u\in H^{1}(\mathbb{R}^{N}),\end{aligned}\right.{} where 2*=2⁢NN-2{2^{*}=\frac{2N}{N-2}} is the critical Sobolev exponent, N≥5{N\geq 5}, and V⁢(y){V(y)} and Q⁢(y){Q(y)} are bounded nonnegative functions in ℝN{\mathbb{R}^{N}}. By using the finite reduction argument and local Pohozaev-type identities, under some suitable assumptions on the functions V and Q , we prove that for ε>0{\varepsilon>0} is small enough, problem (*){(*)} has large number of bubble solutions whose functional energy is in the order ε-N-4(N-2)2.{\varepsilon^{-\frac{N-4}{(N-2)^{2}}}.}

We are concerned with the following Schrödinger–Newton problem: - ε 2 ⁢ Δ ⁢ u + V ⁢ ( x ) ⁢ u = 1 8 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 u 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ u , x ∈ ℝ 3 . -\varepsilon^{2}\Delta u+V(x)u=\frac{1}{8\pi\varepsilon^{2}}\Bigg{(}\int_{% \mathbb{R}^{3}}\frac{u^{2}(\xi)}{|x-\xi|}\,d\xi\Bigg{)}u,\quad x\in\mathbb{R}^% {3}. For ε small enough, we prove the non-degeneracy of the positive solution to the above problem, that is, the corresponding linear operator ℒ ε ⁢ ( η ) = - ε 2 ⁢ Δ ⁢ η ⁢ ( x ) + V ⁢ ( x ) ⁢ η ⁢ ( x ) - 1 8 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 u ε 2 ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ η ⁢ ( x ) - 1 4 ⁢ π ⁢ ε 2 ⁢ ( ∫ ℝ 3 u ε ⁢ ( ξ ) ⁢ η ⁢ ( ξ ) | x - ξ | ⁢ 𝑑 ξ ) ⁢ u ε ⁢ ( x ) \mathcal{L}_{\varepsilon}(\eta)=-\varepsilon^{2}\Delta\eta(x)+V(x)\eta(x)-% \frac{1}{8\pi\varepsilon^{2}}\Bigg{(}\int_{\mathbb{R}^{3}}\frac{u_{\varepsilon% }^{2}(\xi)}{|x-\xi|}\,d\xi\Bigg{)}\eta(x)-\frac{1}{4\pi\varepsilon^{2}}\Bigg{(% }\int_{\mathbb{R}^{3}}\frac{u_{\varepsilon}(\xi)\eta(\xi)}{|x-\xi|}\,d\xi\Bigg% {)}u_{\varepsilon}(x) is non-degenerate, i.e., ℒε⁢(ηε)=0⇒ηε=0{\mathcal{L}_{\varepsilon}(\eta_{\varepsilon})=0\Rightarrow\eta_{\varepsilon}=0} for small ε>0{\varepsilon>0}. The main tools are the local Pohozaev identities and the blow-up analysis. This may be the first non-degeneracy result on the peak solutions to the Schrödinger–Newton system.

The aim of this paper is investigating the existence of one or more weak solutions of the coupled quasilinear elliptic system of gradient type \textup{(P)} { - div ⁡ ( A ⁢ ( x , u ) ⁢ | ∇ ⁡ u | p 1 - 2 ⁢ ∇ ⁡ u ) + 1 p 1 ⁢ A u ⁢ ( x , u ) ⁢ | ∇ ⁡ u | p 1 = G u ⁢ ( x , u , v ) in  Ω , - div ⁡ ( B ⁢ ( x , v ) ⁢ | ∇ ⁡ v | p 2 - 2 ⁢ ∇ ⁡ v ) + 1 p 2 ⁢ B v ⁢ ( x , v ) ⁢ | ∇ ⁡ v | p 2 = G v ⁢ ( x , u , v ) in  Ω , u = v = 0 on  ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}(A(x,u)|\nabla u|^{p_{1% }-2}\nabla u)+\frac{1}{p_{1}}A_{u}(x,u)|\nabla u|^{p_{1}}&\displaystyle=G_{u}(% x,u,v)&&\displaystyle\phantom{}\text{in~{}${\Omega}$,}\\ \displaystyle-\operatorname{div}(B(x,v)|\nabla v|^{p_{2}-2}\nabla v)+\frac{1}{% p_{2}}B_{v}(x,v)|\nabla v|^{p_{2}}&\displaystyle=G_{v}(x,u,v)&&\displaystyle% \phantom{}\text{in~{}${\Omega}$,}\\ \displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\right. where Ω⊂ℝN{\Omega\subset\mathbb{R}^{N}} is an open bounded domain, p1{p_{1}}, p2>1{p_{2}>1} and A⁢(x,u){A(x,u)}, B⁢(x,v){B(x,v)} are 𝒞1{\mathcal{C}^{1}}-Carathéodory functions on Ω×ℝ{\Omega\times\mathbb{R}} with partial derivatives Au⁢(x,u){A_{u}(x,u)}, respectively Bv⁢(x,v){B_{v}(x,v)}, while Gu⁢(x,u,v){G_{u}(x,u,v)}, Gv⁢(x,u,v){G_{v}(x,u,v)} are given Carathéodory maps defined on Ω×ℝ×ℝ{\Omega\times\mathbb{R}\times\mathbb{R}} which are partial derivatives of a function G⁢(x,u,v){G(x,u,v)}. We prove that, even if the coefficients make the variational approach more difficult, under suitable hypotheses functional 𝒥{{\mathcal{J}}}, related to problem (P), admits at least one critical point in the “right” Banach space X . Moreover, if 𝒥{{\mathcal{J}}} is even, then (P) has infinitely many weak bounded solutions. The proof, which exploits the interaction between two different norms, is based on a weak version of the Cerami–Palais–Smale condition, a “good” decomposition of the Banach space X and suitable generalizations of the Ambrosetti–Rabinowitz Mountain Pass Theorems.