Volume 11, Issue 1

We consider a degenerate chemotaxis model with two-species and two-stimuli in dimension d ≥ 3 and find two critical curves intersecting at one point which separate the global existence and blow up of weak solutions to the problem. More precisely, above these curves (i.e. subcritical case), the problem admits a global weak solution obtained by the limits of strong solutions to an approximated system. Based on the second moment of solutions, initial data are constructed to make sure blow up occurs in finite time on and below these curves (i.e. critical and supercritical cases). In addition, the existence or non-existence of minimizers of free energy functional is discussed on the critical curves and the solutions exist globally in time if the size of initial data is small. We also investigate the crossing point between the critical lines in which a refined criteria in terms of the masses is given again to distinguish the dichotomy between global existence and blow up. We also show that the blow ups is simultaneous for both species.

We consider a nonlinear Dirichlet problem driven by the ( p , q )-Laplacian and with a reaction which is dependent on the gradient. We look for positive solutions and we do not assume that the reaction is nonnegative. Using a mixture of variational and topological methods (the "frozen variable" technique), we prove the existence of a positive smooth solution.

This paper studies the mass-critical variable coefficient nonlinear Schrödinger equation. We first get the existence of the ground state by solving a minimization problem. Then we prove a compactness result by the variational characterization of the ground state solutions. In addition, we construct the blow-up solutions at the minimal mass threshold and further prove the uniqueness result on the minimal mass blow-up solutions which are pseudo-conformal transformation of the ground states.

This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ L q (S n −1 ) ( q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.

We consider the following critical nonlocal Schrödinger problem with general nonlinearities − ε 2a+ε b∫ R3|∇ u|2Δ u+V(x)u=f(u)+u5,x∈ R3,u∈ H1(R3), $$\begin{array}{} \displaystyle \left\{\begin{array}{} &-\left(\varepsilon^{2}a+\varepsilon b\displaystyle\int\limits_{\mathbb{R}^{3}}|\nabla u|^{2}\right){\it\Delta} u+V(x)u=f(u)+u^{5}, &x \in \mathbb{R}^{3},\\ &u\in H^{1}(\mathbb{R}^{3}), \end{array}\right. \end{array}$$( SK ε ) and study the existence of semiclassical ground state solutions of Nehari-Pohožaev type to ( SK ε ), where f ( u ) may behave like | u | q –2 u for q ∈ (2, 4] which is seldom studied. With some decay assumption on V , we establish an existence result which improves some exiting works which only handle q ∈ (4, 6). With some monotonicity condition on V , we also get a ground state solution v̄ ε and analysis its concentrating behaviour around global minimum x ε of V as ε → 0. Our results extend some related works.

Our purpose of this paper is to study positive solutions of Lane-Emden equation − Δ u=VupinRN∖ {0} $$\begin{array}{} -{\it\Delta} u = V u^p\quad {\rm in}\quad \mathbb{R}^N\setminus\{0\} \end{array}$$(0.1) perturbed by a non-homogeneous potential V when p∈ [pc,N+2N− 2), $\begin{array}{} p\in [p_c, \frac{N+2}{N-2}), \end{array}$ where p c is the Joseph-Ludgren exponent. When p∈ (NN− 2,pc), $\begin{array}{} p\in (\frac{N}{N-2}, p_c), \end{array}$ the fast decaying solution could be approached by super and sub solutions, which are constructed by the stability of the k -fast decaying solution w k of − Δ u = u p in ℝ N ∖ {0} by authors in [9]. While the fast decaying solution w k is unstable for p∈ (pc,N+2N− 2), $\begin{array}{} p\in (p_c, \frac{N+2}{N-2}), \end{array}$ so these fast decaying solutions seem not able to disturbed like (0.1) by non-homogeneous potential V . A surprising observation that there exists a bounded sub solution of (0.1) from the extremal solution of − Δ u=uN+2N− 2 $\begin{array}{} -{\it\Delta} u = u^{\frac{N+2}{N-2}} \end{array}$ in ℝ N and then a sequence of fast decaying solutions and slow decaying solutions could be derived under appropriated restrictions for V .

We are concerned with a class of Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent − Δ u+V(x)u=Iα ∗ [Q(x)|u|N+α N]Q(x)|u|α N− 1u,x∈ RN. $$\begin{array}{} \displaystyle -{\it\Delta} u+V(x)u=\left(I_{\alpha}\ast [Q(x)|u|^{\frac{N+\alpha}{N}}]\right)Q(x)|u|^{\frac{\alpha}{N}-1}u, \quad x\in \mathbb R^N. \end{array}$$ By using variational approaches, we investigate the existence of groundstates relying on the asymptotic behaviour of weighted potentials at infinity. Moreover, non-existence of non-trivial solutions is also considered. In particular, we give a partial answer to some open questions raised in [D.~Cassani, J. Van Schaftingen and J. J. Zhang, Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent, Proceedings of the Royal Society of Edinburgh, Section A Mathematics , 150 (2020), 1377–1400].

A cost function involving the eigenvalues of an elastic structure is optimized using a phase-field approach, which allows for topology changes and multiple materials.We show continuity and differentiability of simple eigenvalues in the phase-field context. Existence of global minimizers can be shown, for which first order necessary optimality conditions can be obtained in generic situations. Furthermore, a combined eigenvalue and compliance optimization is discussed.

Apartial discrete Dirichlet boundary value problem involving mean curvature operator is concerned in this paper. Under proper assumptions on the nonlinear term, we obtain some feasible conditions on the existence of multiple solutions by the method of critical point theory. We further separately determine open intervals of the parameter to attain at least two positive solutions and an unbounded sequence of positive solutions with the help of the maximum principle.

In this paper, we study a discrete model on Wolbachia infection frequency. Assume that a periodic and impulsive release strategy is implemented, where infected males are released during the first N generations with the release ratio α , and the release is terminated from ( N + 1)-th generation to T -th generation. We find a release ratio threshold denoted by α * ( N , T ), and prove the existence of a T -periodic solution for the model when α ∈ (0, α * ( N , T )). For the special case when N = 1 and T = 2, we prove that the model has a unique T -periodic solution which is unstable when α ∈ (0, α * ( N , T )). While α ≥ α * ( N , T ), no periodic phenomenon occurs and the Wolbachia fixation equilibrium is globally asymptotically stable. Numerical simulations are also provided to illustrate our theoretical results. One main contribution of this work is to offer a new method to determine the exact number of periodic orbits to discrete models.

Asymptotic properties of the sequences ( a ) {Pj}j=1∞ $\{P^{j}\}_{j=1}^{\infty}$and ( b ) {j−1∑i=0j−1Pi}j=1∞ $\{ j^{-1} \sum _{i=0}^{j-1} P^{i}\}_{j=1}^{\infty}$ are studied for g ∈ G = { f ∈ L 1 ( I ) : f ≥ 0 and ‖ f ‖ = 1}, where P : L 1 ( I ) → L 1 ( I ) is a Markov operator defined by Pf:=∫Pyfdp(y) $Pf:= \int P_{y}f\, dp(y) $for f ∈ L 1 ; { P y } y∈Y is the family of the Frobenius-Perron operators associated with a family { φ y } y∈Y of nonsingular Markov maps defined on a subset I ⊆ ℝ d ; and the index y runs over a probability space ( Y , Σ ( Y ), p ). Asymptotic properties of the sequences ( a ) and ( b ), of the Markov operator P , are closely connected with the asymptotic properties of the sequence of random vectors xj=φξj(xj−1) $x_{j}=\varphi_{\xi_{j}}(x_{j-1})$for j = 1,2, . . .,where {ξj}j=1∞ $\{\xi_{j}\}_{j=1}^{\infty}$is a sequence of Y -valued independent random elements with common probability distribution p . An operator-theoretic analogue of Rényi’s Condition is introduced for the family { P y } y∈Y of the Frobenius-Perron operators. It is proved that under some additional assumptions this condition implies the L 1 - convergence of the sequences ( a ) and ( b ) to a unique g 0 ∈ G . The general result is applied to some families { φ y } y∈Y of smooth Markov maps in ℝ d .

We show existence and concentration results for a class of p & q critical problems given by −divaϵp|∇u|pϵp|∇u|p−2∇u+V(z)b|u|p|u|p−2u=f(u)+|u|q⋆−2uinRN, $$-div\left(a\left(\epsilon^{p}|\nabla u|^{p}\right) \epsilon^{p}|\nabla u|^{p-2} \nabla u\right)+V(z) b\left(|u|^{p}\right)|u|^{p-2} u=f(u)+|u|^{q^{\star}-2} u\, \text{in} \,\mathbb{R}^{N},$$ where u ∈ W 1, p (ℝ N ) ∩ W 1, q (ℝ N ), ϵ > 0 is a small parameter, 1 < p ≤ q < N , N ≥ 2 and q * = Nq /( N − q ). The potential V is positive and f is a superlinear function of C 1 class. We use Mountain Pass Theorem and the penalization arguments introduced by Del Pino & Felmer’s associated to Lions’ Concentration and Compactness Principle in order to overcome the lack of compactness.

In this paper we provide a numerical approximation of bifurcation branches from nodal radial solutions of the Lane Emden Dirichlet problem in the unit ball in ℝ 2 , as the exponent of the nonlinearity varies. We consider solutions with two or three nodal regions. In the first case our numerical results complement the analytical ones recently obtained in [11]. In the case of solutions with three nodal regions, for which no analytical results are available, our analysis gives numerical evidence of the existence of bifurcation branches. We also compute additional approximations indicating presence of an unexpected branch of solutions with six nodal regions. In all cases the numerical results allow to formulate interesting conjectures.

We give a new non-smooth Clark’s theorem without the global symmetric condition. The theorem can be applied to generalized quasi-linear elliptic equations with small continous perturbations. Our results improve the abstract results about a semi-linear elliptic equation in Kajikiya [10] and Li-Liu [11].

In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The obtained solutions depend on the first eigenvalues of the Robin and Steklov eigenvalue problems for the p -Laplacian.

In this paper, we establish the second boundary behavior of the unique strictly convex solution to a singular Dirichlet problem for the Monge-Ampère equation  det(D2u)=b(x)g(−u),u<0 in Ω and u=0 on ∂Ω, $$\mbox{ det}(D^{2} u)=b(x)g(-u),\,u<0 \mbox{ in }\Omega \mbox{ and } u=0 \mbox{ on }\partial\Omega, $$ where Ω is a bounded, smooth and strictly convex domain in ℝ N ( N ≥ 2), b ∈ C ∞ (Ω) is positive and may be singular (including critical singular) or vanish on the boundary, g  ∈  C 1 ((0, ∞), (0, ∞)) is decreasing on (0, ∞) with lim t → 0 + g ( t ) = ∞ $ \lim\limits_{t\rightarrow0^{+}}g(t)=\infty $ and g is normalized regularly varying at zero with index − γ ( γ >1). Our results reveal the refined influence of the highest and the lowest values of the ( N  − 1)-th curvature on the second boundary behavior of the unique strictly convex solution to the problem.

We consider a nonlinear boundary value problem with degenerate nonlocal term depending on the L q -norm of the solution and the p -Laplace operator. We prove the multiplicity of positive solutions for the problem, where the number of solutions doubles the number of “positive bumps” of the degenerate term. The solutions are also ordered according to their L q -norms.

For nonautonomous linear difference equations, we introduce the notion of the so-called nonuniform dichotomy spectrum and prove a spectral theorem. As an application of the spectral theorem, we prove a reducibility result.

In the present paperwe study the existence of nontrivial solutions of a class of static coupled nonlinear fractional Hartree type system. First, we use the direct moving plane methods to establish the maximum principle(Decay at infinity and Narrow region principle) and prove the symmetry and nonexistence of positive solution of this nonlocal system. Second, we make complete classification of positive solutions of the system in the critical case when some parameters are equal. Finally, we prove the existence of multiple nontrivial solutions in the critical case according to the different parameters ranges by using variational methods. To accomplish our results we establish the maximum principle for the fractional nonlocal system.

We are concerned with the following Schrödinger system with coupled quadratic nonlinearity −ε2Δv+P(x)v=μvw,x∈RN,−ε2Δw+Q(x)w=μ2v2+γw2,x∈RN,v>0,w>0,v,w∈H1RN, $$\begin{equation}\left\{\begin{array}{ll}-\varepsilon^{2} \Delta v+P(x) v=\mu v w, & x \in \mathbb{R}^{N}, \\ -\varepsilon^{2} \Delta w+Q(x) w=\frac{\mu}{2} v^{2}+\gamma w^{2}, & x \in \mathbb{R}^{N}, \\ v>0, \quad w>0, & v, w \in H^{1}\left(\mathbb{R}^{N}\right),\end{array}\right. \end{equation}$$ which arises from second-harmonic generation in quadratic media. Here ε > 0 is a small parameter, 2 ≤ N < 6, μ > 0 and μ > γ, P ( x ), Q ( x ) are positive function potentials. By applying reduction method, we prove that if x 0 is a non-degenerate critical point of Δ ( P + Q ) on some closed N − 1 dimensional hypersurface, then the system above has a single peak solution ( v ε , w ε ) concentrating at x 0 for ε small enough.

In this work, we study the existence of a positive solution to an elliptic equation involving the fractional Laplacian (− Δ ) s in ℝ n , for n ≥ 2, such as (0.1) (−Δ)su+E(x)u+V(x)uq−1=K(x)f(u)+u2s⋆−1. $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=K(x) f(u)+u^{2_{s}^{\star}-1}.$$ Here, s ∈ (0, 1), q∈2,2s⋆ $q \in\left[2,2_{s}^{\star}\right)$with 2s⋆:=2nn−2s $2_{s}^{\star}:=\frac{2 n}{n-2 s}$being the fractional critical Sobolev exponent, E ( x ), K ( x ), V ( x ) > 0 : ℝ n → ℝ are measurable functions which satisfy joint “vanishing at infinity” conditions in a measure-theoretic sense, and f ( u ) is a continuous function on ℝ of quasi-critical, super- q -linear growth with f ( u ) ≥ 0 if u ≥ 0. Besides, we study the existence of multiple positive solutions to an elliptic equation in ℝ n such as (0.2) (−Δ)su+E(x)u+V(x)uq−1=λK(x)ur−1, $$(-\Delta)^{s} u+E(x) u+V(x) u^{q-1}=\lambda K(x) u^{r-1},$$ where 2 < r < q < ∞(both possibly (super-)critical), E ( x ), K ( x ), V ( x ) > 0 : ℝ n → ℝ are measurable functions satisfying joint integrability conditions, and λ > 0 is a parameter. To study (0.1)-(0.2), we first describe a family of general fractional Sobolev-Slobodeckij spaces M s ; q , p (ℝ n ) as well as their associated compact embedding results.

We consider a double-phase non-Newtonian fluid, described by a stress tensor which is the sum of a p -Stokes and a q -Stokes stress tensor, with 1 <  p <2 <  q <∞. For a wide range of parameters ( p , q ), we prove the uniqueness of small solutions. We use the p  < 2 features to obtain quadratic-type estimates for the stress-tensor, while we use the improved regularity coming from the term with q  > 2 to justify calculations for weak solutions. Results are obtained through a careful use of the symmetries of the convective term and are also valid for rather general (even anisotropic) stress-tensors.

This paper is concerned with the following Hamiltonian elliptic system −Δu+V(x)u=Wv(x,u,v),    x∈RN,−Δv+V(x)v=Wu(x,u,v),    x∈RN, $$ \left\{ \begin{array}{ll} -\Delta u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ -\Delta v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\ \end{array} \right. $$ where z = ( u , v ) : ℝ N → ℝ 2 , V ( x ) and W ( x , z ) are 1-periodic in x . By making use of variational approach for strongly indefinite problems, we obtain a new existence result of nontrivial solution under new conditions that the nonlinearity W(x,z):=12V∞(x)|Az|2+F(x,z) $ W(x,z):=\frac{1}{2}V_{\infty}(x)|Az|^2+F(x, z) $ is general asymptotically quadratic, where V ∞ ( x ) ∈ (ℝ N , ℝ) is 1-periodic in x and inf ℝ N V ∞ ( x ) > min ℝ N V ( x ), and A is a symmetric non-negative definite matrix.

In this paper, we are concerned with the following a new critical nonlocal Schrödinger-Poisson system on the Heisenberg group: −a−b∫Ω|∇Hu|2dξΔHu+μϕu=λ|u|q−2u+|u|2u,inΩ,−ΔHϕ=u2,inΩ,u=ϕ=0,on∂Ω, $$\begin{equation*}\begin{cases} -\left(a-b\int_{\Omega}|\nabla_{H}u|^{2}d\xi\right)\Delta_{H}u+\mu\phi u=\lambda|u|^{q-2}u+|u|^{2}u,\quad &\mbox{in} \, \Omega,\\ -\Delta_{H}\phi=u^2,\quad &\mbox{in}\, \Omega,\\ u=\phi=0,\quad &\mbox{on}\, \partial\Omega, \end{cases} \end{equation*}$$ where Δ H is the Kohn-Laplacian on the first Heisenberg group H1 $ \mathbb{H}^1 $ , and Ω⊂H1 $ \Omega\subset \mathbb{H}^1 $ is a smooth bounded domain, a , b  > 0, 1 <  q < 2 or 2 <  q < 4, λ > 0 and μ∈R $ \mu\in \mathbb{R} $ are some real parameters. Existence and multiplicity of solutions are obtained by an application of the mountain pass theorem, the Ekeland variational principle, the Krasnoselskii genus theorem and the Clark critical point theorem, respectively. However, there are several difficulties arising in the framework of Heisenberg groups, also due to the presence of the non-local coefficient ( a  −  b ∫ Ω ∣∇ H u ∣ 2 dx ) as well as critical nonlinearities. Moreover, our results are new even on the Euclidean case.

In this paper, we study an anomalous pseudo-parabolic Kirchhoff-type dynamical model aiming to reveal the control problem of the initial data on the dynamical behavior of the solution in dynamic control system. Firstly, the local existence of solution is obtained by employing the Contraction Mapping Principle. Then, we get the global existence of solution, long time behavior of global solution and blowup solution for J ( u 0 ) ⩽ d , respectively. In particular, the lower and upper bound estimates of the blowup time are given for J ( u 0 )< d . Finally, we discuss the blowup of solution in finite time and also estimate an upper bound of the blowup time for high initial energy.

Let w be a Muckenhoupt A 2 (ℝ n ) weight and Ω a bounded Reifenberg flat domain in ℝ n . Assume that p (·):Ω → (1, ∞) is a variable exponent satisfying the log-Hölder continuous condition. In this article, the authors investigate the weighted W 1, p (·) (Ω,  w )-regularity of the weak solutions of second order degenerate elliptic equations in divergence form with Dirichlet boundary condition, under the assumption that the degenerate coefficients belong to weighted BMO spaces with small norms.

We are devoted to the study of a semilinear time fractional Rayleigh-Stokes problem on ℝ N , which is derived from a non-Newtonain fluid for a generalized second grade fluid with Riemann-Liouville fractional derivative. We show that a solution operator involving the Laplacian operator is very effective to discuss the proposed problem. In this paper, we are concerned with the global/local well-posedness of the problem, the approaches rely on the Gagliardo-Nirenberg inequalities, operator theory, standard fixed point technique and harmonic analysis methods. We also present several results on the continuation, a blow-up alternative with a blow-up rate and the integrability in Lebesgue spaces.

We study the following Kirchhoff type equation: −a+b∫RN|∇u|2dxΔu+u=k(x)|u|p−2u+m(x)|u|q−2u  in  RN, $$\begin{equation*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\right)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R}^{N}, \end{array} \end{equation*}$$ where N =3, a,b>0 $ a,b \gt 0 $ , 1<q<2<p<min{4,2∗} $ 1 \lt q \lt 2 \lt p \lt \min\{4, 2^{*}\} $ , 2≤=2 N /( N  − 2), k  ∈  C (ℝ N ) is bounded and m  ∈  L p /( p − q ) (ℝ N ). By imposing some suitable conditions on functions k ( x ) and m ( x ), we firstly introduce some novel techniques to recover the compactness of the Sobolev embedding H1(RN)↪Lr(RN)(2≤r<2∗) $ H^{1}(\mathbb{R}^{N})\hookrightarrow L^{r}(\mathbb{R}^{N}) (2\leq r \lt 2^{*}) $ ; then the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three positive solutions for the above problem.

This paper considers an initial-boundary value problem for a class of singular quasilinear second-order ordinary differential equations with the constraint condition stemming from fluid mechanics. We prove that the existence of positive Sobolev regular solutions for this kind of singular quasilinear ODEs by means of a suitable Nash-Moser iteration scheme Meanwhile, asymptotic expansion of those positive solutions is shown.

In this paper we are concerned with the existence, nonexistence and bifurcation of nontrivial solution of the nonlinear Schrödinger-Korteweg-de Vries type system(NLS-NLS-KdV). First, we find some conditions to guarantee the existence and nonexistence of positive solution of the system. Second, we study the asymptotic behavior of the positive ground state solution. Finally, we use the classical Crandall-Rabinowitz local bifurcation theory to get the nontrivial positive solution. To get these results we encounter some new challenges. By combining the Nehari manifolds constraint method and the delicate energy estimates, we overcome the difficulties and find the two bifurcation branches from one semitrivial solution. This is an new interesting phenomenon but which have not previously been found.

We first prove De Giorgi type level estimates for functions in W 1, t (Ω), Ω⊂RN $ \Omega\subset{\mathbb R}^N $ , with t>N≥2 $ t \gt N\geq 2 $ . This augmented integrability enables us to establish a new Harnack type inequality for functions which do not necessarily belong to De Giorgi’s classes as obtained in Di Benedetto–Trudinger [10] for functions in W 1,2 (Ω). As a consequence, we prove the validity of the strong maximum principle for uniformly elliptic operators of any even order, in fairly general domains in dimension two and three, provided second order derivatives are taken into account.

We consider quasilinear elliptic systems in divergence form. In general, we cannot expect that weak solutions are locally bounded because of De Giorgi’s counterexample. Here we assume that off-diagonal coefficients have a “butterfly support”: this allows us to prove local boundedness of weak solutions.

In this paper, we consider the following modified quasilinear problem: − Δ u− κ uΔ u2=λ a(x)u− α +b(x)uβ inΩ ,u> 0inΩ ,u=0on∂ Ω , $$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array} $$ where Ω ⊂ ℝ N is a smooth bounded domain, N ≥ 3, a , b are two bounded continuous functions, α > 0, 1 < β ≤ 22 * − 1 and λ > 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.

In this paper, the Cauchy problem for the one-dimensional compressible isentropic magnetohydrodynamic (MHD) equations with no vacuum at infinity is considered, but the initial vacuum can be permitted inside the region. By deriving a priori ν (resistivity coefficient)-independent estimates, we establish the non-resistive limit of the global strong solutions with large initial data. Moreover, as a by-product, the global well-posedness of strong solutions for the compressible resistive MHD equations is also established.

The paper studies absolute stability of neutral differential nonlinear systems x˙(t)=Axt+Bxt−τ+Dx˙t−τ+bf(σ(t)),σ(t)=cTx(t),t⩾0 $$ \begin{align}\dot x(t)=Ax\left ( t \right )+Bx\left ( {t-\tau} \right ) +D\dot x\left ( {t-\tau} \right ) +bf({\sigma (t)}),\,\, \sigma (t)=c^Tx(t), \,\, t\geqslant 0 \end{align} $$ where x is an unknown vector, A , B and D are constant matrices, b and c are column constant vectors, 𝜏 > 0 is a constant delay and f is a Lurie-type nonlinear function satisfying Lipschitz condition. Absolute stability is analyzed by a general Lyapunov-Krasovskii functional with the results compared with those previously known.

The existence of a positive entire weak solution to a singular quasi-linear elliptic system with convection terms is established, chiefly through perturbation techniques, fixed point arguments, and a priori estimates. Some regularity results are then employed to show that the obtained solution is actually strong.

In the paper, nonlinear systems of mixed-type functional differential equations are analyzed and the existence of semi-global and global solutions is proved. In proofs, the monotone iterative technique and Schauder-Tychonov fixed-point theorem are used. In addition to proving the existence of global solutions, estimates of their co-ordinates are derived as well. Linear variants of results are considered and the results are illustrated by selected examples.

We study distributional solutions of semilinear biharmonic equations of the type Δ2u+f(u)=0 onℝN, {\Delta ^2}u + f(u) = 0\quad on\;{{\mathbb R} ^N}, where f is a continuous function satisfying f ( t ) t ≥ c | t | q +1 for all t ∈ ℝ with c > 0 and q > 1. By using a new approach mainly based on careful choice of suitable weighted test functions and a new version of Hardy- Rellich inequalities, we prove several Liouville theorems independently of the dimension N and on the sign of the solutions.

We investigate optimal decay rates for higher–order spatial derivatives of strong solutions to the 3D Cauchy problem of the compressible viscous quantum magnetohydrodynamic model in the H 5 × H 4 × H 4 framework, and the main novelty of this work is three–fold: First, we show that fourth order spatial derivative of the solution converges to zero at the L2-rate (1+t)-114 {L^2} - {\rm{rate}}\,{(1 + t)^{- {{11} \over 4}}} , which is same as one of the heat equation, and particularly faster than the L2-rate (1+t)-54 {L^2} - {\rm{rate}}\,{(1 + t)^{- {5 \over 4}}} in Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and the L2-rate (1+t)-94 {L^2} - {\rm{rate}}\,{(1 + t)^{- {9 \over 4}}} , in Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Second, we prove that fifth–order spatial derivative of density ρ converges to zero at the L2-rate (1+t)-134 {L^2} - {\rm{rate}}\,{(1 + t)^{- {{13} \over 4}}} , which is same as that of the heat equation, and particularly faster than ones of Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Third, we show that the high-frequency part of the fourth order spatial derivatives of the velocity u and magnetic B converge to zero at the L2-rate (1+t)-134 {L^2} - {\rm{rate}}\,{(1 + t)^{- {{13} \over 4}}} , which are faster than ones of themselves, and totally new as compared to Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019].

This paper provides the Carleson characterization of the extension of fractional Sobolev spaces and Lebesgue spaces to Lq(ℝ+n+1,μ) L^q (\mathbb{R}_ + ^{n + 1} ,\mu ) via space-time fractional equations. For the extension of fractional Sobolev spaces, preliminary results including estimates, involving the fractional capacity, measures, the non-tangential maximal function, and an estimate of the Riesz integral of the space-time fractional heat kernel, are provided. For the extension of Lebesgue spaces, a new L p –capacity associated to the spatial-time fractional equations is introduced. Then, some basic properties of the L p –capacity, including its dual form, the L p –capacity of fractional parabolic balls, strong and weak type inequalities, are established.

Let X be the direct product of X i where X i is smooth manifold for 1 ≤ i ≤ k . As is known, if every X i satisfies the doubling volume condition, then the centered Hardy-Littlewood maximal function M on X is weak (1,1) bounded. In this paper, we consider the product manifold X where at least one X i does not satisfy the doubling volume condition. To be precise, we first investigate the mapping properties of M when X 1 has exponential volume growth and X 2 satisfies the doubling condition. Next, we consider the product space of two weighted hyperbolic spaces X 1 = (ℍ n +1 , d , y α − n −1 dydx ) and X 2 = (ℍ n +1 , d , y β − n −1 dydx ) which both have exponential volume growth. The mapping properties of M are discussed for every α,β≠n2 \alpha,\beta \ne {n \over 2} . Furthermore, let X = X 1 × X 2 × … X k where X i = (ℍ n i +1 , y α i − n i −1 dydx ) for 1 ≤ i ≤ k . Under the condition αi>ni2 {\alpha_i} > {{{n_i}} \over 2} , we also obtained the mapping properties of M .

In the present paper, a class of Schrödinger equations is investigated, which can be stated as −Δu+V(x)u=f(u),    x∈ℝN. - \Delta u + V(x)u = f(u),\;\;\;\;x \in {{\rm{\mathbb R}}^N}. If the external potential V is radial and coercive, then we give the local Ambrosetti-Rabinowitz super-linear condition on the nonlinearity term f ∈ C (ℝ, ℝ) which assures the problem has not only infinitely many radial sign-changing solutions, but also infinitely many non-radial sign-changing solutions.

We consider on M (ℝ d ) (the set of all finite measures on ℝ d ) the evolution equation associated with the nonlinear operator F↦ΔF′+∑k⩾1bkFk F \mapsto \Delta F' + \sum\nolimits_{k \geqslant 1} b_k F^k , where F′ is the variational derivative of F and we show that it has a solution represented by means of the distribution of the d -dimensional Brownian motion and the non-local branching process on the finite configurations of M (ℝ d ), induced by the sequence ( b k ) k ⩾1 of positive numbers such that ∑k⩾1bk⩽1 \sum\nolimits_{k \geqslant 1} b_k \leqslant 1 . It turns out that the representation also holds with the same branching process for the solution to the equation obtained replacing the Laplace operator by the generator of a Markov process on ℝ d instead of the d -dimensional Brownian motion; more general, we can take the generator of a right Markov process on a Lusin topological space. We first investigate continuous flows driving branching processes. We show that if the branching mechanism of a superprocess is independent of the spatial variable, then the superprocess is obtained by introducing the branching in the time evolution of the right continuous flow on measures, canonically induced by a right continuous flow as spatial motion. A corresponding result holds for non-local branching processes on the set of all finite configurations of the state space of the spatial motion.

We consider density solutions for gradient flow equations of the form u t = ∇ · ( γ ( u )∇ N( u )), where N is the Newtonian repulsive potential in the whole space ℝ d with the nonlinear convex mobility γ ( u ) = u α , and α > 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ ( u ) = u . For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u = c 1 t −1 supported in a ball that spreads in time like c 2 t 1/ d , thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form γ ( u ) = u α , with 0 < α < 1 studied in [10]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detailed analysis allowing us to show a waiting time phenomena. This is a typical behaviour for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations illustrating the proven results and showcasing some open problems.

In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P) −Δu+u=F(u)+κμ  in  RN, u>0  in  RN, u(x)→0  as  |x|→∞, - \Delta u + u = F(u) + \kappa \mu \quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u > 0\quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u(x) \to 0\quad {\kern 1pt} {\rm as}{\kern 1pt} \quad |x| \to \infty , where F = F ( t ) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ∈Lc1(RN)\{0} \mu \in L_{\rm{c}}^1({{\bf R}^N})\backslash \{ 0\} is nonnegative. Then, under a suitable integrability condition on μ , there exists a threshold parameter κ * > 0 such that problem (P) possesses a solution if 0 < κ < κ * and it does not possess no solutions if κ > κ * . Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ * .

This article deals with the degenerate fractional Kirchhoff wave equation with structural damping or strong damping. The well-posedness and the existence of global attractor in the natural energy space by virtue of the Faedo-Galerkin method and energy estimates are proved. It is worth mentioning that the results of this article cover the case of possible degeneration (or even negativity) of the stiffness coefficient. Moreover, under further suitable assumptions, the fractal dimension of the global attractor is shown to be infinite by using Z 2 {{\mathbb{Z}}}_{2} index theory.

In this article, we are interested in the following Kirchhoff-type problem (0.1) − a + b ∫ R N ∣ ∇ u ∣ 2 d x Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u in R N , u ∈ H 1 ( R N ) , \left\{\begin{array}{l}-\left(a+b\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| \nabla u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right)\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\end{array}\right. where a , b > 0 , N = 2 a,b\gt 0,N=2 or 3, the potential function V V is radial and bounded from below by a positive number. Because the nonlocal b ∣ ∇ u ∣ L 2 ( R N ) 2 Δ u b| \nabla u\hspace{-0.25em}{| }_{{L}^{2}\left({{\mathbb{R}}}^{N})}^{2}\Delta u is 3-homogeneous which is in complicated competition with the nonlinear term ∣ u ∣ 2 u | u\hspace{-0.25em}{| }^{2}u . This causes that not all function in H 1 ( R N ) {H}^{1}\left({{\mathbb{R}}}^{N}) can be projected on the Nehari manifold and thereby the classical Nehari manifold method does not work. By introducing the Gersgorin Disk theorem and the Miranda theorem, via a limit approach and subtle analysis, we prove that for each positive integer k k , equation (0.1) admits a radial nodal solution U k , 4 b {U}_{k,4}^{b} having exactly k k nodes. Moreover, we show that the energy of U k , 4 b {U}_{k,4}^{b} is strictly increasing in k k and for any sequence { b n } \left\{{b}_{n}\right\} with b n → 0 + , {b}_{n}\to {0}_{+}, up to a subsequence, U k , 4 b n {U}_{k,4}^{{b}_{n}} converges to U k , 4 0 {U}_{k,4}^{0} in H 1 ( R N ) {H}^{1}\left({{\mathbb{R}}}^{N}) , which is a radial nodal solution with exactly k k nodes of the classical Schrödinger equation − a Δ u + V ( ∣ x ∣ ) u = ∣ u ∣ 2 u in R N , u ∈ H 1 ( R N ) . \left\{\begin{array}{l}-a\Delta u+V\left(| x| )u=| u\hspace{-0.25em}{| }^{2}u\hspace{1.0em}{\rm{in}}\hspace{0.33em}{{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}).\end{array}\right. Our results extend the existence result from the super-cubic case to the cubic case.

In this article, we consider the parabolic system ( u i ) t = ∇ ⋅ ( m U m − 1 A ( ∇ u i , u i , x , t ) + ℬ ( u i , x , t ) ) , ( 1 ≤ i ≤ k ) {({u}^{i})}_{t}=\nabla \cdot (m{U}^{m-1}{\mathcal{A}}(\nabla {u}^{i},{u}^{i},x,t)+{\mathcal{ {\mathcal B} }}({u}^{i},x,t)),\hspace{1.0em}(1\le i\le k) in the range of exponents m > n − 2 n m\gt \frac{n-2}{n} where the diffusion coefficient U U depends on the components of the solution u = ( u 1 , … , u k ) {\bf{u}}=({u}^{1},\ldots ,{u}^{k}) . Under suitable structure conditions on the vector fields A {\mathcal{A}} and ℬ {\mathcal{ {\mathcal B} }} , we first showed the uniform L ∞ {L}^{\infty } boundedness of the function U U for t ≥ τ > 0 t\ge \tau \gt 0 . We also proved the law of L 1 {L}^{1} mass conservation and the local continuity of solution u {\bf{u}} . In the last result, all components of the solution u {\bf{u}} have the same modulus of continuity if the ratio between U U and u i {u}^{i} , ( 1 ≤ i ≤ k 1\le i\le k ), is uniformly bounded above and below.

In this article, we consider the non-linear Choquard equation − Δ u + V ( ∣ x ∣ ) u = ∫ R 3 ∣ u ( y ) ∣ 2 ∣ x − y ∣ d y u in R 3 , -\Delta u+V\left(| x| )u=\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{3}}\frac{| u(y){| }^{2}}{| x-y| }{\rm{d}}y\right)u\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{3}, where V ( r ) V\left(r) is a positive bounded function. Under some proper assumptions on V ( r ) V\left(r) , we are able to establish the existence of infinitely many non-radial solutions.

This article deals with the following fractional Kirchhoff problem with critical exponent a + b ∫ R N ∣ ( − Δ ) s 2 u ∣ 2 d x ( − Δ ) s u = ( 1 + ε K ( x ) ) u 2 s ∗ − 1 , in R N , \left(a+b\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}| {\left(-\Delta )}^{\tfrac{s}{2}}u\hspace{-0.25em}{| }^{2}{\rm{d}}x\right){\left(-\Delta )}^{s}u=\left(1+\varepsilon K\left(x)){u}^{{2}_{s}^{\ast }-1},\hspace{1.0em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where a , b > 0 a,b\gt 0 are given constants, ε \varepsilon is a small parameter, 2 s ∗ = 2 N N − 2 s {2}_{s}^{\ast }=\frac{2N}{N-2s} with 0 < s < 1 0\lt s\lt 1 and N ≥ 4 s N\ge 4s . We first prove the nondegeneracy of positive solutions when ε = 0 \varepsilon =0 . In particular, we prove that uniqueness breaks down for dimensions N > 4 s N\gt 4s , i.e., we show that there exist two nondegenerate positive solutions which seem to be completely different from the result of the fractional Schrödinger equation or the low-dimensional fractional Kirchhoff equation. Using the finite-dimensional reduction method and perturbed arguments, we also obtain the existence of positive solutions to the singular perturbation problems for ε \varepsilon small.

Using elements of the theory of linear operators in Hilbert spaces and monotonicity tools we obtain the existence and uniqueness results for a wide class of nonlinear problems driven by the equation T x = N ( x ) Tx=N\left(x) , where T T is a self-adjoint operator in a real Hilbert space ℋ {\mathcal{ {\mathcal H} }} and N N is a nonlinear perturbation. Both potential and nonpotential perturbations are considered. This approach is an extension of the results known for elliptic operators.

In this article, we consider the upper critical Choquard equation with a local perturbation − Δ u = λ u + ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u + μ ∣ u ∣ q − 2 u , x ∈ R N , u ∈ H 1 ( R N ) , ∫ R N ∣ u ∣ 2 = a , \left\{\begin{array}{l}-\Delta u=\lambda u+\left({I}_{\alpha }\ast | u\hspace{-0.25em}{| }^{p})| u\hspace{-0.25em}{| }^{p-2}u+\mu | u\hspace{-0.25em}{| }^{q-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\\ u\in {H}^{1}\left({{\mathbb{R}}}^{N}),\hspace{1em}{\displaystyle \int }_{{{\mathbb{R}}}^{N}}| u\hspace{-0.25em}{| }^{2}=a,\end{array}\right. where N ≥ 3 N\ge 3 , μ > 0 \mu \gt 0 , a > 0 a\gt 0 , λ ∈ R \lambda \in {\mathbb{R}} , α ∈ ( 0 , N ) \alpha \in \left(0,N) , p = p ¯ ≔ N + α N − 2 p=\bar{p}:= \frac{N+\alpha }{N-2} , q ∈ 2 , 2 + 4 N q\in \left(2,2+\frac{4}{N}\right) , and I α = C ∣ x ∣ N − α {I}_{\alpha }=\frac{C}{| x{| }^{N-\alpha }} with C > 0 C\gt 0 . When μ a q ( 1 − γ q ) 2 ≤ ( 2 K ) q γ q − 2 p ¯ 2 ( p ¯ − 1 ) \mu {a}^{\tfrac{q\left(1-{\gamma }_{q})}{2}}\le {\left(2K)}^{\tfrac{q{\gamma }_{q}-2\bar{p}}{2\left(\bar{p}-1)}} with γ q = N 2 − N q {\gamma }_{q}=\frac{N}{2}-\frac{N}{q} and K K being some positive constant, we prove (1) Existence and orbital stability of the ground states. (2) Existence, positivity, radial symmetry, exponential decay, and orbital instability of the “second class” solutions. This article generalized and improved parts of the results obtained for the Schrödinger equation.

In this article, a new type of comparison theorem for some second-order nonlinear parabolic systems with nonlinear boundary conditions is given, which can cover classical linear boundary conditions, such as the homogeneous Dirichlet or Neumann boundary condition. The advantage of our comparison theorem over the classical ones lies in the fact that it enables us to compare two solutions satisfying different types of boundary conditions. As an application of our comparison theorem, we can give some new results on the existence of blow-up solutions of some parabolic equations and systems with nonlinear boundary conditions.

We consider the Dirichlet problem for partial trace operators which include the smallest and the largest eigenvalue of the Hessian matrix. It is related to two-player zero-sum differential games. No Lipschitz regularity result is known for the solutions, to our knowledge. If some eigenvalue is missing, such operators are nonlinear, degenerate, non-uniformly elliptic, neither convex nor concave. Here we prove an interior Lipschitz estimate under a non-standard assumption: that the solution exists in a larger, unbounded domain, and vanishes at infinity. In other words, we need a condition coming from far away. We also provide existence results showing that this condition is satisfied for a large class of solutions. On the occasion, we also extend a few qualitative properties of solutions, known for uniformly elliptic operators, to partial trace operators.

In this article, we consider the following Schrödinger-Poisson system: − Δ u + u + k ( x ) ϕ ( x ) u = f ( x ) ∣ u ∣ p − 1 u + g ( x ) , x ∈ R 3 , − Δ ϕ = k ( x ) u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+u+k\left(x)\phi \left(x)u=f\left(x)| u{| }^{p-1}u+g\left(x),& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =k\left(x){u}^{2},& x\in {{\mathbb{R}}}^{3},\end{array}\right. with p ∈ ( 3 , 5 ) p\in \left(3,5) . Under suitable assumptions on potentials f ( x ) f\left(x) , g ( x ) g\left(x) and k ( x ) k\left(x) , then at least four positive solutions for the above system can be obtained for sufficiently small ‖ g ‖ H − 1 ( R 3 ) \Vert g{\Vert }_{{H}^{-1}\left({{\mathbb{R}}}^{3})} by taking advantage of variational methods and Lusternik-Schnirelman category.

We consider Hamiltonian functions of the classical type, namely, even and convex with respect to the generalized momenta. A brake orbit is a periodic solution of Hamilton’s equations such that the generalized momenta are zero on two different points. Under mild assumptions, this paper reduces the multiplicity problem of the brake orbits for a Hamiltonian function of the classical type to the multiplicity problem of orthogonal geodesic chords in a concave Finslerian manifold with boundary. This paper will be used for a generalization of a Seifert’s conjecture about the multiplicity of brake orbits to Hamiltonian functions of the classical type.

In this article, we consider a class of optimal control problems governed by state equations of Kobayashi-Warren-Carter-type. The control is given by physical temperature. The focus is on problems in dimensions less than or equal to 4. The results are divided into four Main Theorems, concerned with: solvability and parameter dependence of state equations and optimal control problems; the first-order necessary optimality conditions for these regularized optimal control problems. Subsequently, we derive the limiting systems and optimality conditions and study their well-posedness.

In this article, we study a large class of evolutionary variational–hemivariational inequalities of hyperbolic type without damping terms, in which the functional framework is considered in an evolution triple of spaces. The inequalities contain both a convex potential and a locally Lipschitz superpotential. The results on existence, uniqueness, and regularity of solution to the inequality problem are provided through the Rothe method.

We study robust regularity estimates for a class of nonlinear integro-differential operators with anisotropic and singular kernels. In this paper, we prove a Sobolev-type inequality, a weak Harnack inequality, and a local Hölder estimate.

In this article, we aimed to study a class of nonhomogeneous fractional ( p , q )-Laplacian systems with critical nonlinearities as well as critical Hardy nonlinearities in R N {{\mathbb{R}}}^{N} . By appealing to a fixed point result and fractional Hardy-Sobolev inequality, the existence of nontrivial nonnegative solutions is obtained. In particular, we also consider Choquard-type nonlinearities in the second part of this article. More precisely, with the help of Hardy-Littlewood-Sobolev inequality, we obtain the existence of nontrivial solutions for the related systems based on the same approach. Finally, we obtain the corresponding existence results for the fractional ( p , q )-Laplacian systems in the case of N = s p = l q N=sp=lq . It is worth pointing out that using fixed point argument to seek solutions for a class of nonhomogeneous fractional ( p , q )-Laplacian systems is the main novelty of this article.

In this article, we study discrete Kirchhoff-type problems when the nonlinearity is resonant at both zero and infinity. We establish a series of results on the existence of nontrivial solutions by combining variational method with Morse theory. Several examples are provided to illustrate applications of our results.

We study positive solutions to the one-dimensional generalized double phase problems of the form: − ( a ( t ) φ p ( u ′ ) + b ( t ) φ q ( u ′ ) ) ′ = λ h ( t ) f ( u ) , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) , \left\{\begin{array}{l}-(a\left(t){\varphi }_{p}\left(u^{\prime} )+b\left(t){\varphi }_{q}\left(u^{\prime} ))^{\prime} =\lambda h\left(t)f\left(u),\hspace{1em}t\in \left(0,1),\\ u\left(0)=0=u\left(1),\end{array}\right. where 1 < p < q < ∞ 1\lt p\lt q\lt \infty , φ m ( s ) ≔ ∣ s ∣ m − 2 s {\varphi }_{m}\left(s):= | s{| }^{m-2}s , a , b ∈ C ( [ 0 , 1 ] , [ 0 , ∞ ) ) a,b\in C\left(\left[0,1],{[}0,\infty )) , h ∈ L 1 ( ( 0 , 1 ) , ( 0 , ∞ ) ) ∩ C ( ( 0 , 1 ) , ( 0 , ∞ ) ) , h\in {L}^{1}\left(\left(0,1),\left(0,\infty ))\cap C\left(\left(0,1),\left(0,\infty )), and f ∈ C ( [ 0 , ∞ ) , R ) f\in C\left({[}0,\infty ),{\mathbb{R}}) is nondecreasing. More precisely, we show various existence results including the existence of at least two or three positive solutions according to the behaviors of f ( s ) f\left(s) near zero and infinity. Both positone (i.e., f ( 0 ) ≥ 0 f\left(0)\ge 0 ) and semipositone (i.e., f ( 0 ) < 0 f\left(0)\lt 0 ) problems are considered, and the results are obtained through the Krasnoselskii-type fixed point theorem. We also apply these results to show the existence of positive radial solutions for high-dimensional generalized double phase problems on the exterior of a ball.

We prove long-time existence for the negative L 2 {L}^{2} -gradient flow of the p -elastic energy, p ≥ 2 p\ge 2 , with an additive positive multiple of the length of the curve. To achieve this result, we regularize the energy by cutting off the degeneracy at points with vanishing curvature and add a small multiple of a higher order energy, namely, the square of the L 2 {L}^{2} -norm of the normal gradient of the curvature κ \kappa . Long-time existence is proved for the gradient flow of these new energies together with the smooth subconvergence of the evolution equation’s solutions to critical points of the regularized energy in W 2 , p {W}^{2,p} . We then show that the solutions to the regularized evolution equations converge to a weak solution of the negative gradient flow of the p -elastic energies. These latter weak solutions also subconverge to critical points of the p -elastic energy.

This article is concerned with the following nonlinear supercritical elliptic problem: − M ( ‖ ∇ u ‖ 2 2 ) Δ u = f ( x , u ) , in B 1 ( 0 ) , u = 0 , on ∂ B 1 ( 0 ) , \left\{\begin{array}{ll}-M(\Vert \nabla u{\Vert }_{2}^{2})\Delta u=f\left(x,u),& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{B}_{1}\left(0),\\ u=0,& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial {B}_{1}\left(0),\end{array}\right. where B 1 ( 0 ) {B}_{1}\left(0) is the unit ball in R 2 {{\mathbb{R}}}^{2} , M : R + → R + M:{{\mathbb{R}}}^{+}\to {{\mathbb{R}}}^{+} is a Kirchhoff function, and f ( x , t ) f\left(x,t) has supercritical exponential growth on t t , which behaves as exp [ ( β 0 + ∣ x ∣ α ) t 2 ] \exp {[}({\beta }_{0}+| x\hspace{-0.25em}{| }^{\alpha }){t}^{2}] and exp ( β 0 t 2 + ∣ x ∣ α ) \exp ({\beta }_{0}{t}^{2+| x{| }^{\alpha }}) with β 0 {\beta }_{0} , α > 0 \alpha \gt 0 . Based on a deep analysis and some detailed estimate, we obtain Nehari-type ground state solutions for the above problem by variational method. Moreover, we can determine a fine upper bound for the minimax level under weaker assumption on liminf t → ∞ t f ( x , t ) exp [ ( β 0 + ∣ x ∣ α ) t 2 ] {\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp {[}({\beta }_{0}+| \hspace{-0.25em}x\hspace{-0.25em}{| }^{\alpha }){t}^{2}]} and liminf t → ∞ t f ( x , t ) exp ( β 0 t 2 + ∣ x ∣ α ) {\mathrm{liminf}}_{t\to \infty }\frac{tf\left(x,t)}{\exp ({\beta }_{0}{t}^{2+| x{| }^{\alpha }})} , respectively. Our results generalize and improve the ones in G. M. Figueiredo and U. B. Severo ( Ground state solution for a Kirchhoff problem with exponential critical growth , Milan J. Math. 84 (2016), no. 1, 23–39.) and Q. A. Ngó and V. H. Nguyen ( Supercritical Moser-Trudinger inequalities and related elliptic problems , Calc. Var. Partial Differ. Equ. 59 (2020), no. 2, Paper No. 69, 30.) for M ( t ) = 1 M(t)=1 . In particular, if the weighted term ∣ x ∣ α | x\hspace{-0.25em}{| }^{\alpha } is vanishing, we can obtain the ones in S. T. Chen, X. H. Tang, and J. Y. Wei (2021) ( Improved results on planar Kirchhoff-type elliptic problems with critical exponential growth , Z. Angew. Math. Phys. 72 (2021), no. 1, Paper No. 38, Theorem 1.3 and Theorem 1.4) immediately.

In this work, we handle a time-dependent Navier-Stokes problem in dimension three with a mixed boundary conditions. The variational formulation is written considering three independent unknowns: vorticity, velocity, and pressure. We use the backward Euler scheme for time discretization and the spectral method for space discretization. We present a complete numerical analysis linked to this variational formulation, which leads us to a priori error estimate.

In this article, we consider nondiffusive variational problems with mixed boundary conditions and (distributional and weak) gradient constraints. The upper bound in the constraint is either a function or a Borel measure, leading to the state space being a Sobolev one or the space of functions of bounded variation. We address existence and uniqueness of the model under low regularity assumptions, and rigorously identify its Fenchel pre-dual problem. The latter in some cases is posed on a nonstandard space of Borel measures with square integrable divergences. We also establish existence and uniqueness of solution to this pre-dual problem under some assumptions. We conclude the article by introducing a mixed finite-element method to solve the primal-dual system. The numerical examples illustrate the theoretical findings.

Let n ≥ 2 n\ge 2 and Ω ⊂ R n \Omega \subset {{\mathbb{R}}}^{n} be a bounded nontangentially accessible domain. In this article, the authors investigate (weighted) global gradient estimates for Dirichlet boundary value problems of second-order elliptic equations of divergence form with an elliptic symmetric part and a BMO antisymmetric part in Ω \Omega . More precisely, for any given p ∈ ( 2 , ∞ ) p\in \left(2,\infty ) , the authors prove that a weak reverse Hölder inequality with exponent p p implies the global W 1 , p {W}^{1,p} estimate and the global weighted W 1 , q {W}^{1,q} estimate, with q ∈ [ 2 , p ] q\in \left[2,p] and some Muckenhoupt weights, of solutions to Dirichlet boundary value problems. As applications, the authors establish some global gradient estimates for solutions to Dirichlet boundary value problems of second-order elliptic equations of divergence form with small BMO {\rm{BMO}} symmetric part and small BMO {\rm{BMO}} antisymmetric part, respectively, on bounded Lipschitz domains, quasi-convex domains, Reifenberg flat domains, C 1 {C}^{1} domains, or (semi-)convex domains, in weighted Lebesgue spaces. Furthermore, as further applications, the authors obtain the global gradient estimate, respectively, in (weighted) Lorentz spaces, (Lorentz–)Morrey spaces, (Musielak–)Orlicz spaces, and variable Lebesgue spaces. Even on global gradient estimates in Lebesgue spaces, the results obtained in this article improve the known results via weakening the assumption on the coefficient matrix.

This paper is devoted to investigate the existence and multiplicity of the normalized solutions for the following fractional Schrödinger equation: (P) ( − Δ ) s u + λ u = μ ∣ u ∣ p − 2 u + ∣ u ∣ 2 s ∗ − 2 u , x ∈ R N , u > 0 , ∫ R N ∣ u ∣ 2 d x = a 2 , \left\{\begin{array}{l}{\left(-\Delta )}^{s}u+\lambda u=\mu | u{| }^{p-2}u+| u{| }^{{2}_{s}^{\ast }-2}u,\hspace{1em}x\in {{\mathbb{R}}}^{N},\hspace{1.0em}\\ u\gt 0,\hspace{1em}\mathop{\displaystyle \int }\limits_{{{\mathbb{R}}}^{N}}| u{| }^{2}{\rm{d}}x={a}^{2},\hspace{1.0em}\end{array}\right. where 0 < s < 1 0\lt s\lt 1 , a a , μ > 0 \mu \gt 0 , N ≥ 2 N\ge 2 , and 2 < p < 2 s ∗ 2\lt p\lt {2}_{s}^{\ast } . We consider the L 2 {L}^{2} -subcritical and L 2 {L}^{2} -supercritical cases. More precisely, in L 2 {L}^{2} -subcritical case, we obtain the multiplicity of the normalized solutions for problem ( P ) \left(P) by using the truncation technique, concentration-compactness principle, and genus theory. In L 2 {L}^{2} -supercritical case, we obtain a couple of normalized solution for ( P ) \left(P) by using a fiber map and concentration-compactness principle. To some extent, these results can be viewed as an extension of the existing results from Sobolev subcritical growth to Sobolev critical growth.

This paper is concerned with existence and concentration properties of ground-state solutions to the following fractional Choquard equation with indefinite potential: ( − Δ ) s u + V ( x ) u = ∫ R N A ( ε y ) ∣ u ( y ) ∣ p ∣ x − y ∣ μ d y A ( ε x ) ∣ u ( x ) ∣ p − 2 u ( x ) , x ∈ R N , {\left(-\Delta )}^{s}u+V\left(x)u=\left(\mathop{\int }\limits_{{{\mathbb{R}}}^{N}}\frac{A\left(\varepsilon y)| u(y){| }^{p}}{| x-y{| }^{\mu }}{\rm{d}}y\right)A\left(\varepsilon x)| u\left(x){| }^{p-2}u\left(x),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where s ∈ ( 0 , 1 ) s\in \left(0,1) , N > 2 s N\gt 2s , 0 < μ < 2 s 0\lt \mu \lt 2s , 2 < p < 2 N − 2 μ N − 2 s 2\lt p\lt \frac{2N-2\mu }{N-2s} , and ε \varepsilon is a positive parameter. Under some natural hypotheses on the potentials V V and A A , using the generalized Nehari manifold method, we obtain the existence of ground-state solutions. Moreover, we investigate the concentration behavior of ground-state solutions that concentrate at global maximum points of A A as ε → 0 \varepsilon \to 0 .

In this article, we consider the fractional Rayleigh-Stokes problem with the nonlinearity term satisfies certain critical conditions. The local existence, uniqueness and continuous dependence upon the initial data of ε \varepsilon -regular mild solutions are obtained. Furthermore, a unique continuation result and a blow-up alternative result of ε \varepsilon -regular mild solutions are given in the end.

The existence of unbounded solutions and their asymptotic behavior is studied for higher order differential equations considered as perturbations of certain linear differential equations. In particular, the existence of solutions with polynomial-like or noninteger power-law asymptotic behavior is proved. These results give a relation between solutions to nonlinear and corresponding linear equations, which can be interpreted, roughly speaking, as an asymptotic proximity between the linear case and the nonlinear one. Our approach is based on the induction method, an iterative process and suitable estimates for solutions to the linear equation.

The article is concerned with systems of fractional discrete equations Δ α x ( n + 1 ) = F n ( n , x ( n ) , x ( n − 1 ) , … , x ( n 0 ) ) , n = n 0 , n 0 + 1 , … , {\Delta }^{\alpha }x\left(n+1)={F}_{n}\left(n,x\left(n),x\left(n-1),\ldots ,x\left({n}_{0})),\hspace{1em}n={n}_{0},{n}_{0}+1,\ldots , where n 0 ∈ Z {n}_{0}\in {\mathbb{Z}} , n n is an independent variable, Δ α {\Delta }^{\alpha } is an α \alpha -order fractional difference, α ∈ R \alpha \in {\mathbb{R}} , F n : { n } × R n − n 0 + 1 → R s {F}_{n}:\left\{n\right\}\times {{\mathbb{R}}}^{n-{n}_{0}+1}\to {{\mathbb{R}}}^{s} , s ⩾ 1 s\geqslant 1 is a fixed integer, and x : { n 0 , n 0 + 1 , … } → R s x:\left\{{n}_{0},{n}_{0}+1,\ldots \right\}\to {{\mathbb{R}}}^{s} is a dependent (unknown) variable. A retract principle is used to prove the existence of solutions with graphs remaining in a given domain for every n ⩾ n 0 n\geqslant {n}_{0} , which then serves as a basis for further proving the existence of bounded solutions to a linear nonhomogeneous system of discrete equations Δ α x ( n + 1 ) = A ( n ) x ( n ) + δ ( n ) , n = n 0 , n 0 + 1 , … , {\Delta }^{\alpha }x\left(n+1)=A\left(n)x\left(n)+\delta \left(n),\hspace{1em}n={n}_{0},{n}_{0}+1,\ldots , where A ( n ) A\left(n) is a square matrix and δ ( n ) \delta \left(n) is a vector function. Illustrative examples accompany the statements derived, possible generalizations are discussed, and open problems for future research are formulated as well.

We study the behaviour, when p → + ∞ p\to +\infty , of the first p -Laplacian eigenvalues with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that the limit of the eigenfunctions is a viscosity solution to an eigenvalue problem for the so-called ∞ \infty -Laplacian. Moreover, in the second part of the article, we focus our attention on the p -Poisson equation when the datum f f belongs to L ∞ ( Ω ) {L}^{\infty }\left(\Omega ) and we study the behaviour of solutions when p → ∞ p\to \infty .