Showing papers in "Nodea-nonlinear Differential Equations and Applications in 2018"

Stable solutions in potential mean field game systems

Abstract: We introduce the notion of stable solution in mean field game theory: they are locally isolated solutions of the mean field game system. We prove that such solutions exist in potential mean field games and are local attractors for learning procedures.

Nehari method for locally Lipschitz functionals with examples in problems in the space of bounded variation functions

Abstract: In this work we prove some abstract results about the existence of a minimizer for locally Lipschitz functionals, over a set which has its definition inspired in the Nehari manifold. As applications we present a result of existence of ground state bounded variation solutions of problems involving the 1-laplacian and the 1-biharmonic operator, where the nonlinearity satisfies mild assumptions.

Existence and uniqueness results for possibly singular nonlinear elliptic equations with measure data

Abstract: We study existence and uniqueness of solutions to a nonlinear elliptic boundary value problem with a general, and possibly singular, lower order term, whose model is $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\Delta _p u = H(u)\mu &{}\quad \text {in}\ \Omega ,\\ u>0 &{}\quad \text {in}\ \Omega ,\\ u=0 &{}\quad \text {on}\ \partial \Omega . \end{array}\right. } \end{aligned}$$ Here $$\Omega $$ is an open bounded subset of $${\mathbb {R}}^N$$ ( $$N\ge 2$$ ), $$\Delta _p u:= {\text {div}}(| abla u|^{p-2} abla u)$$ ( $$1

The concentration-compactness principle for fractional order Sobolev spaces in unbounded domains and applications to the generalized fractional Brezis–Nirenberg problem

Abstract: In this paper we extend the well-known concentration-compactness principle for the Fractional Laplacian operator in unbounded domains. As an application we show sufficient conditions for the existence of solutions to some critical equations involving the fractional p-Laplacian in the whole $${\mathbb {R}}^n$$ .

The prescribed mean curvature equation in weakly regular domains

Abstract: We show that the characterization of existence and uniqueness up to vertical translations of solutions to the prescribed mean curvature equation, originally proved by Giusti in the smooth case, holds true for domains satisfying very mild regularity assumptions. Our results apply in particular to the non-parametric solutions of the capillary problem for perfectly wetting fluids in zero gravity. Among the essential tools used in the proofs, we mention a generalized Gauss–Green theorem based on the construction of the weak normal trace of a vector field with bounded divergence, in the spirit of classical results due to Anzellotti, and a weak Young’s law for $$(\Lambda ,r_{0})$$ -minimizers of the perimeter.

On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives

Abstract: Initial-boundary value problems for second order fully nonlinear PDEs with Caputo time fractional derivatives of order less than one are considered in the framework of viscosity solution theory. Associated boundary conditions are Dirichlet and Neumann, and they are considered in the strong sense and the viscosity sense, respectively. By a comparison principle and Perron’s method, unique existence for the Cauchy–Dirichlet and Cauchy–Neumann problems are proved.

About reaction-diffusion systems involving the Holling-type II and the Beddington-DeAngelis functional responses for predator-prey models

Abstract: We consider in this paper a microscopic model (that is, a system of three reaction–diffusion equations) incorporating the dynamics of handling and searching predators, and show that its solutions converge when a small parameter tends to 0 towards the solutions of a reaction–cross diffusion system of predator–prey type involving a Holling-type II or Beddington–DeAngelis functional response. We also provide a study of the Turing instability domain of the obtained equations and (in the case of the Beddington–DeAngelis functional response) compare it to the same instability domain when the cross diffusion is replaced by a standard diffusion.

Liouville-type theorems with finite Morse index for semilinear $${\varvec{\Delta }}_{{\varvec{\lambda }}}$$ Δ λ -Laplace operators

Abstract: In this paper we study solutions, possibly unbounded and sign-changing, of the following equation $$\begin{aligned} -\Delta _{\lambda } u=|x|_{\lambda }^a |u|^{p-1}u \quad \text{ in }\; {\mathbb {R}}^n, \end{aligned}$$ where $$n\ge 1$$ , $$p>1$$ , $$a \ge 0$$ and $$\Delta _{\lambda }$$ is a strongly degenerate elliptic operator, the functions $$\lambda =(\lambda _1, \ldots , \lambda _k) : \; {\mathbb {R}}^n \rightarrow {\mathbb {R}}^k$$ satisfies some certain conditions, and $$|.|_{\lambda }$$ the homogeneous norm associated to the $$\Delta _{\lambda }$$ -Laplacian. We prove various Liouville-type theorems for smooth solutions under the assumption that they are stable or stable outside a compact set of $${\mathbb {R}}^n$$ . First, we establish the standard integral estimates via stability property to derive the nonexistence results for stable solutions. Next, by mean of the Pohozaev identity, we deduce the Liouville-type theorem for solutions stable outside a compact set.

Existence and asymptotic behavior of sign-changing solutions for fractional Kirchhoff-type problems in low dimensions

Abstract: This paper is dedicated to studying the following fractional Kirchhoff-type equation $$\begin{aligned} \left( a+b\int _{\mathbb {R}^N}|(-\triangle )^{\alpha /2}{u}|^2\mathrm {d}x\right) (-\triangle )^{\alpha }u+V(|x|)u =f(|x|, u), \ \ \ \ x\in {\mathbb {R}}^{N}, \end{aligned}$$ where $$a, b>0$$ , either $$N=2$$ and $$\alpha \in (1/2,1)$$ or $$N=3$$ and $$\alpha \in (3/4,1)$$ holds, $$V\in \mathcal {C}(\mathbb {R}^{N}, [0,\infty ))$$ and $$f\in \mathcal {C}(\mathbb {R}^N\times \mathbb {R}, \mathbb {R})$$ . By combining the constraint variational method with some new inequalities, we prove that the above problem possesses a radial sign-changing solution $$u_b$$ for $$b\ge 0$$ without the usual Nehari-type monotonicity condition on f, and its energy is strictly larger than twice that of the ground state radial solutions of Nehari-type. Moreover, we establish the convergence property of $$u_b$$ as $$b\searrow 0$$ . In particular, our results unify both asymptotically cubic and super-cubic cases, which improve and complement the existing ones in the literature.

Well-posedness and fast-diffusion limit for a bulk–surface reaction–diffusion system

Abstract: We analyze a certain class of coupled bulk–surface reaction–drift–diffusion systems arising in the modeling of signalling networks in biological cells. The coupling is by a nonlinear Robin-type boundary condition for the bulk variable and a corresponding source term on the cell boundary. For reaction terms with at most linear growth and under different regularity assumptions on the data we prove the existence of weak and classical solutions. In particular, we show that solutions grow at most exponentially with time. Furthermore, we rigorously derive an asymptotic reduction to a non-local reaction–drift–diffusion system on the membrane in the fast-diffusion limit.

Regularity estimates for BMO-weak solutions of quasilinear elliptic equations with inhomogeneous boundary conditions

Abstract: This paper studies regularity estimates in Lebesgue spaces for gradients of weak solutions of a class of general quasilinear equations of p-Laplacian type in bounded domains with inhomogeneous conormal boundary conditions. In the considered class of equations the leading terms are vector-valued functions that are measurable in the x-variable and that depend in a nonlinear way on the solution and on its gradient. This class of equations consists of the well-known class of degenerate p-Laplace equations for $$p >1$$ . Under some sufficient conditions, we establish local interior, local boundary, and global $$W^{1,q}$$ -regularity estimates for weak solutions with $$q>p$$ , assuming that the weak solutions are in the John–Nirenberg BMO space. The paper therefore improves available results because it removes the boundedness or continuity assumptions on solutions. Our results also unify and cover known results for equations in which the principals are only allowed to depend on x-variable and gradient of solution variable. More than that, this paper gives a method to treat non-homogeneous boundary value problems directly without using any form of translations that is sometimes complicated due to the nonlinearities.

$$\varvec{p}$$ p -Laplacian problems involving critical Hardy–Sobolev exponents

Abstract: We prove existence, multiplicity, and bifurcation results for p-Laplacian problems involving critical Hardy–Sobolev exponents. Our results are mainly for the case $$\lambda \ge \lambda _1$$ and extend results in the literature for $$0< \lambda < \lambda _1$$ . In the absence of a direct sum decomposition, we use critical point theorems based on a cohomological index and a related pseudo-index.

On a dynamic boundary condition for singular degenerate parabolic equations in a half space

Abstract: We consider the initial value problem for a fully-nonlinear degenerate parabolic equation with a dynamic boundary condition in a half space. Our setting includes geometric equations with singularity such as the level-set mean curvature flow equation. We establish a comparison principle for a viscosity sub- and supersolution. We also prove existence of solutions and Lipschitz regularity of the unique solution. Moreover, relation to other types of boundary conditions is investigated by studying the asymptotic behavior of the solution with respect to a coefficient of the dynamic boundary condition.

A boundary estimate for singular parabolic diffusion equations

Abstract: We prove an estimate on the modulus of continuity at a boundary point of a cylindrical domain for local weak solutions to singular parabolic equations of p-Laplacian type. The estimate is given in terms of a Wiener-type integral, defined by a proper elliptic p-capacity.

Formation of singularities of spherically symmetric solutions to the 3D compressible Euler equations and Euler–Poisson equations

Abstract: By introducing a new averaged quantity with a fast decay weight to perform Sideris’s argument (Commun Math Phys 101:475–485, 1985) developed for the Euler equations, we extend the formation of singularities of classical solution to the 3D Euler equations established in Makino et al. (Jpn J Appl Math 3:249–257, 1986) and Sideris (1985) for the initial data with compactly supported disturbances to the spherically symmetric solution with general initial data in Sobolev space. Moreover, we also prove the formation of singularities of the spherically symmetric solutions to the 3D Euler–Poisson equations, but remove the compact support assumptions on the initial data in Makino and Perthame (Jpn J Appl Math 7:165–170, 1990) and Perthame (Jpn J Appl Math 7:363–367, 1990). Our proof also simplifies that of Lei et al. (Math Res Lett 20:41–50, 2013) for the Euler equations and is undifferentiated in dimensions.

Dual variational methods for a nonlinear Helmholtz system

Abstract: This paper considers a pair of coupled nonlinear Helmholtz equations $$\begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2} \right) |u|^{\frac{p}{2} - 2}u,\\ -\,\Delta v - u v = a(x) \left( |v|^\frac{p}{2} + b(x) |u|^\frac{p}{2} \right) |v|^{\frac{p}{2} - 2}v \end{array}\right. } \end{aligned}$$ on \(\mathbb {R}^N\) where \(\frac{2(N+1)}{N-1}< p < 2^*\). The existence of nontrivial strong solutions in \(W^{2, p}(\mathbb {R}^N)\) is established using dual variational methods. The focus lies on necessary and sufficient conditions on the parameters deciding whether or not both components of such solutions are nontrivial.

Annular rearrangements, incompressible axi-symmetric whirls and $$L^1$$ L 1 -local minimisers of the distortion energy

Abstract: In this paper we consider a variational problem consisting of an energy functional defined by the integral, $$\begin{aligned} \mathbb {F}[u,\mathbf{X}] = \frac{1}{2}\int _{\mathbf{X}} \frac{| abla u|^2}{|u|^2} \,dx, \end{aligned}$$ and an associated mapping space, here, the space of incompressible Sobolev mappings of the symmetric annular domain in the Euclidean n-space $$\mathbf{X}= \lbrace x \in {\mathbb {R}}^n{:}\,a<|x|

Positive solutions of an elliptic Neumann problem with a sublinear indefinite nonlinearity

Abstract: Let $$\Omega \subset \mathbb {R}^{N}$$ ( $$N\ge 1$$ ) be a bounded and smooth domain and $$a:\Omega \rightarrow \mathbb {R}$$ be a sign-changing weight satisfying $$\int _{\Omega }a<0$$ . We prove the existence of a positive solution $$u_{q}$$ for the problem if $$q_{0}0$$ . In doing so, we improve the existence result previously established in Kaufmann et al. (J Differ Equ 263:4481–4502, 2017). In addition, we provide the asymptotic behavior of $$u_{q}$$ as $$q\rightarrow 1^{-}$$ . When $$\Omega $$ is a ball and a is radial, we give some explicit conditions on q and a ensuring the existence of a positive solution of $$(P_{a,q})$$ . We also obtain some properties of the set of q’s such that $$(P_{a,q})$$ admits a solution which is positive on $$\overline{\Omega }$$ . Finally, we present some results on nonnegative solutions having dead cores. Our approach combines bifurcation techniques, a priori bounds and the sub-supersolution method.

Carleman estimate for a linearized bidomain model in electrocardiology and its applications

Abstract: This paper concerns Carleman estimate and its applications for a linearized bidomain model in electrocardiology, which describes the electrical activity in the cardiac tissue. We first establish a new Carleman estimate for this reaction–diffusion system. By means of this Carleman estimate, we study two problems for the linearized bidomian model, a Cauchy problem and an inverse conductivities problem. We prove a conditional stability result for the Cauchy problem and a Holder stability result for the inverse conductivities problem.

A Reshetnyak-type lower semicontinuity result for linearised elasto-plasticity coupled with damage in $W^{1,n}$

Abstract: In this paper we prove a lower semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. To do so we characterise the limit of measures $$\alpha _k\,{\mathrm {E}}u_k$$ with respect to the weak convergence $$\alpha _k\rightharpoonup \alpha $$ in $$W^{1,n}(\Omega )$$ and the weak $$^*$$ convergence $$u_k{\mathop {\rightharpoonup }\limits ^{*}}u$$ in $$BD(\Omega )$$ , $${\mathrm {E}}$$ denoting the symmetrised gradient. A concentration compactness argument shows that the limit has the form $$\alpha \,{\mathrm {E}}u+\eta $$ , with $$\eta $$ supported on an at most countable set.

Finite time blow-up for damped wave equations with space–time dependent potential and nonlinear memory

Abstract: In this paper, we consider the Cauchy problem in $$\mathbb {R}^n,$$ $$n\ge 1,$$ for semilinear damped wave equations with space–time dependent potential and nonlinear memory. A blow-up result under some positive data in any dimensional space is obtained. Moreover, the local existence in the energy space is also studied.

Liouville type theorems for stable solutions of the weighted elliptic system with the advection term: $$ \varvec{p \ge \vartheta >1}$$

Abstract: In this paper, we are concerned with the weighted elliptic system with the advection term $$\begin{aligned} {\left\{ \begin{array}{ll} -\omega (x)\Delta u(x)- abla \omega (x)\cdot abla u(x)=\omega _1 v^{\vartheta },\\ -\omega (x)\Delta v(x)- abla \omega (x)\cdot abla v(x)=\omega _2 u^p, \end{array}\right. } \quad \text{ in }\;\ \mathbb {R}^N, \end{aligned}$$ where $$N \ge 3$$ , $$p \ge \vartheta >1$$ and $$\omega , \omega _1, \omega _2 e 1$$ satisfy some suitable conditions. We establish Liouville type theorems for stable solutions with two cases $$\omega _1 e \omega _2$$ and $$\omega _1 \equiv \omega _2$$ , respectively. Based on a delicate application of some new techniques, these difficulties caused by the advection term and the weighted term are overcome, and the sharp results are obtained.

Robust Stackelberg controllability for the Navier–Stokes equations

Abstract: In this paper we deal with a robust Stackelberg strategy for the Navier–Stokes system. The scheme is based in considering a robust control problem for the “follower control” and its associated disturbance function. Afterwards, we consider the notion of Stackelberg optimization (which is associated to the “leader control”) in order to deduce a local null controllability result for the Navier–Stokes system.

The Cauchy problem for the Ostrovsky equation with positive dispersion

Abstract: This paper is devoted to studying the Cauchy problem for the Ostrovsky equation $$\begin{aligned} \partial _{x}\left( u_{t}-\beta \partial _{x}^{3}u +\frac{1}{2}\partial _{x}(u^{2})\right) -\gamma u=0, \end{aligned}$$ with positive $$\beta $$ and $$\gamma $$ . This equation describes the propagation of surface waves in a rotating oceanic flow. We first prove that the problem is locally well-posed in $$H^{-\frac{3}{4}}(\text{ R })$$ . Then we reestablish the bilinear estimate, by means of the Strichartz estimates instead of calculus inequalities and Cauchy–Schwartz inequalities. As a byproduct, this bilinear estimate leads to the proof of the local well-posedness of the problem in $$H^{s}(\text{ R })$$ for $$ s>-\frac{3}{4}$$ , with help of a fixed point argument.

Uniqueness of closed self-similar solutions to $$\varvec{\sigma _k^{\alpha }}$$ σ k α -curvature flow

Abstract: By adapting the test functions introduced by Choi–Daskaspoulos (Uniqueness of closed self-similar solutions to the Gauss curvature flow. arXiv:1609.05487 , 2016) and Brendle–Choi–Daskaspoulos (Acta Math 219(1):1–16, 2017) and exploring the properties of the k-th elementary symmetric function $$\sigma _{k}$$ intensively, we show that for any fixed k with $$1\le k\le n-1$$ , any strictly convex closed hypersurface in $${\mathbb {R}}^{n+1}$$ satisfying $$\sigma _{k}^{\alpha }=\langle X, u \rangle $$ , with $$\alpha \ge \frac{1}{k}$$ , must be a round sphere. In fact, we prove a uniqueness result for any strictly convex closed hypersurface in $${\mathbb {R}}^{n+1}$$ satisfying $$F+C=\langle X, u \rangle $$ , where F is a positive homogeneous smooth symmetric function of the principal curvatures and C is a constant.

Capacity solution to a coupled system of parabolic–elliptic equations in Orlicz–Sobolev spaces

Abstract: The existence of a capacity solution to a coupled nonlinear parabolic–elliptic system is analyzed, the elliptic part in the parabolic equation being of the form $$-\,\mathrm{div}\, a(x,t,u, abla u)$$ . The growth and the coercivity conditions on the monotone vector field a are prescribed by an N-function, M, which does not have to satisfy a $$\Delta _2$$ condition. Therefore we work with Orlicz–Sobolev spaces which are not necessarily reflexive. We use Schauder’s fixed point theorem to prove the existence of a weak solution to certain approximate problems. Then we show that some subsequence of approximate solutions converges in a certain sense to a capacity solution.

Weighted variable exponent Sobolev estimates for elliptic equations with non-standard growth and measure data

Abstract: Consider the following nonlinear elliptic equation of p(x)-Laplacian type with nonstandard growth $$\begin{aligned} {\left\{ \begin{array}{ll} \mathrm{div}a(Du, x)=\mu &{}\quad \text {in} \quad \Omega ,\\ u=0 &{} \quad \text {on} \quad \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega $$ is a Reifenberg domain in $$\mathbb {R}^n$$ , $$\mu $$ is a Radon measure defined on $$\Omega $$ with finite total mass and the nonlinearity $$a: \mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}^n$$ is modeled upon the $$p(\cdot )$$ -Laplacian. We prove the estimates on weighted variable exponent Lebesgue spaces for gradients of solutions to this equation in terms of Muckenhoupt–Wheeden type estimates. As a consequence, we obtain some new results such as the weighted $$L^q-L^r$$ regularity (with constants $$q < r$$ ) and estimates on Morrey spaces for gradients of the solutions to this non-linear equation.

On semilinear elliptic equations with diffuse measures

Abstract: We consider semilinear equation of the form $$-Lu=f(x,u)+\mu $$ , where L is the operator corresponding to a transient symmetric regular Dirichlet form $${\mathcal {E}}$$ , $$\mu $$ is a diffuse measure with respect to the capacity associated with $${\mathcal {E}}$$ , and the lower-order perturbing term f(x, u) satisfies the sign condition in u and some weak integrability condition (no growth condition on f(x, u) as a function of u is imposed). We prove the existence of a solution under mild additional assumptions on $${\mathcal {E}}$$ . We also show that the solution is unique if f is nonincreasing in u.

On a fractional quasilinear parabolic problem: the influence of the Hardy potential

Abstract: The aim goal of this paper is to treat the following problem where $$\Omega $$ is a bounded domain containing the origin, with $$1

Symmetry-breaking bifurcation for the Moore–Nehari differential equation

Abstract: We study the bifurcation problem of positive solutions for the Moore-Nehari differential equation, $$u''+h(x,\lambda )u^p=0$$ , $$u>0$$ in $$(-1,1)$$ with $$u(-1)=u(1)=0$$ , where $$p>1$$ , $$h(x,\lambda )=0$$ for $$|x|<\lambda $$ and $$h(x,\lambda )=1$$ for $$\lambda \le |x| \le 1$$ and $$\lambda \in (0,1)$$ is a bifurcation parameter. We shall show that the problem has a unique even positive solution $$U(x,\lambda )$$ for each $$\lambda \in (0,1)$$ . We shall prove that there exists a unique $$\lambda _*\in (0,1)$$ such that a non-even positive solution bifurcates at $$\lambda _*$$ from the curve $$(\lambda , U(x,\lambda ))$$ , where $$\lambda _*$$ is explicitly represented as a function of p.