We study the multiplicity of solutions for a two-point boundary value problem of Neumann type associated with a Hamiltonian system which couples a system with periodic Hamiltonian in the space variable with a second one with positively-(p, q)-homogeneous Hamiltonian. The periodic problem is also treated.
We prove a local higher integrability result for the gradient of a weak solution to parabolic double-phase systems of p-Laplace type when (tfrac{2n}{n+2}< ple 2) . The result is based on a reverse Hölder inequality in intrinsic cylinders combining p-intrinsic and (p, q)-intrinsic geometries. A singular scaling deficits affects the range of q.
In this paper, we shall discuss singular solutions of semilinear elliptic equations with general supercritical growth on spherically symmetric Riemannian manifolds. More precisely, we shall prove the existence, uniqueness and asymptotic behavior of the singular radial solution, and also show that regular radial solutions converges to the singular solution. In particular, we shall provide these properties on spherically symmetric Riemannian manifolds including the hyperbolic space as well as the sphere.
In this paper we study the problem of prescribing fractional Q-curvature of order (2sigma ) for a conformal metric on the standard sphere (mathbb {S}^n) with (sigma in (0,n/2)) and (nge 3). Compactness and existence results are obtained in terms of the flatness order (beta ) of the prescribed curvature function K. Making use of integral representations and perturbation result, we develop a unified approach to obtain these results when (beta in [n-2sigma ,n)) for all (sigma in (0,n/2)). This work generalizes the corresponding results of Jin-Li-Xiong (Math Ann 369:109–151, 2017) for (beta in (n-2sigma ,n)).
We study approximate solutions to a hyperbolic system of conservation laws, constructed by a backward Euler scheme, where time is discretized while space is still described by a continuous variable (xin {mathbb R}). We prove the global existence and uniqueness of these approximate solutions, and the invariance of suitable subdomains. Furthermore, given a left and a right state (u_l, u_r) connected by an entropy-admissible shock, we construct a traveling wave profile for the backward Euler scheme connecting these two asymptotic states in two main cases. Namely: (1) a scalar conservation law, where the jump (u_l-u_r) can be arbitrarily large, and (2) a strictly hyperbolic system, assuming that the jump (u_l-u_r) occurs in a genuinely nonlinear family and is sufficiently small.
In this paper, we obtain the rate (O(varepsilon ^{1/2})) of convergence in periodic homogenization of forced graphical mean curvature flows in the laminated setting. We also discuss with an example that a faster rate cannot be obtained by utilizing Lipscthiz estimates.
In this paper, we study an optimal control problem associated to the conformal CR sub-Laplacian obstacle problem on a compact pseudohermitian manifold. When the CR Yamabe constant is positive, we show that the optimal controls are equal to their associated optimal states and show the existence of a smooth optimal control which induces a conformal contact form with constant Webster scalar curvature.
In this paper, we deal with the boundary value problem (-Delta u= |u|^{4/(n-2)}u/[ln (e+|u|)]^varepsilon ) in a bounded smooth domain ( Omega ) in ({mathbb {R}}^n), (nge 3) with homogenous Dirichlet boundary condition. Here (varepsilon >0). Clapp et al. (J Differ Equ 275:418–446, 2021) built a family of solution blowing up if (nge 4) and (varepsilon ) small enough. They conjectured in their paper the existence of sign changing solutions which blow up and blow down at the same point. Here we give a confirmative answer by proving that our slightly subcritical problem has a solution with the shape of sign changing bubbles concentrating on a stable critical point of the Robin function for (varepsilon ) sufficiently small.
We consider the 3D Euler equations for incompressible homogeneous fluids and we study the problem of energy conservation for weak solutions in the space-periodic case. First, we prove the energy conservation for a full scale of Besov spaces, by extending some classical results to a wider range of exponents. Next, we consider the energy conservation in the case of conditions on the gradient, recovering some results which were known, up to now, only for the Navier–Stokes equations and for weak solutions of the Leray-Hopf type. Finally, we make some remarks on the Onsager singularity problem, identifying conditions which allow to pass to the limit from solutions of the Navier–Stokes equations to solution of the Euler ones, producing weak solutions which are energy conserving.
In this paper, we consider the following Choquard equation
$$begin{aligned} -varepsilon ^{N}Delta _{N}u+V(x)|u|^{N-2}u=varepsilon ^{mu -N}left( I_mu *F(u)right) f(u) quad {text{ in }quad mathbb {R}^N}, end{aligned}$$where (Nge 3), (I_mu =|x|^{-mu }) with (0<mu <N), (Delta _{N}u=textrm{div}(|nabla u|^{N-2}nabla u)) denotes the N-Laplacian operator, V(x) is a continuous real function on (mathbb {R}^N), F(s) is the primitive of f(s) and (varepsilon ) is a positive parameter. Assuming that the nonlinearity f(s) has critical exponential growth in the sense of Trudinger–Moser inequality, we establish the existence, multiplicity and concentration of solutions by variational methods and Ljusternik–Schnirelmann theory, which extends the works of Alves and Figueiredo (J Differ Equ 246:1288–1311, 2009) to the problem with Choquard nonlinearity, Alves et al. (J Differ Equ 261:1933–1972, 2016) to higher dimension.
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