By applying the method of monotone operators and techniques related to the method of sub-super solutions, we conduct a thorough examination of the existence and uniqueness of solutions to mixed local-nonlo cal Dirichlet boundary value problems. These problems are characterized by the presence of both local and nonlo cal operators, which introduce significant complexity into the analysis. In addition, we also investigate the continuous dependence of these solutions on various parameters. To achieve this, we derive a parameter-dependent surjectivity theorem in presence of non-uniqueness of solutions. Furthermore, we extend our analysis to include applications where the problem depends on a bounded weight.
We are concerned with the following gauged nonlinear Schrodinger equations {-triangle u + (1 + |x|(2))u + lambda (integral(infinity )(|x|)h(u)(s)/su(2 )(s)ds + h(u)(2)(|x|)/|x|(2))u = g(x, u) in R-2, u(x) = u(|x|), where lambda > 0 is a parameter, h(u)(s) = integral(s )(0)r/2u(2)(r)dr and the nonlinearity g has the supercritical exponential growth at infinity in the Trudinger-Moser sense. When the continuous function g(x, & sdot;) is odd for every x is an element of R-2, thanks to a new type of critical point theorem, we conclude the existence of multiple nontrivial solutions for a sufficiently small lambda > 0 with the help of a powerful truncation procedure and the elliptic regularity theory. In particular, the analyses developed in this article allow us to contemplate the case that g(x, u) = |u|(p-2)u with 2 < p <= 6. Moreover, we also demonstrate that the "almost optimal" growth condition lim(|t|->+infinity) G(x, t)e(-4xt2 )> 0 uniformly in x is an element of R-2 makes the existence of a mountain-pass solution available, where and in the whole paper G(x, & sdot;) is denoted by the primitive of g(x, & sdot;).
In this paper, we focus on prescribed mass standing waves for the nonlinear Schrodinger equation with rotation and it is natural to consider the normalized solutions for the stationary equation with rotation -Delta u +V(x)u-Omega Lu = & micro;qu +a|u|qu in R-2, having prescribed mass integral(2)(R) |u|(2)dx = 1, where a > 0, q > 2, & micro;(q) is an element of R, V(x) = |x|(2) is a trapping potential, Omega is an element of (0,2) characterizes the rotational velocity of Bose gas rotates around the origin, and L := -i(x1 partial derivative x(2)-x(2)partial derivative x(1)) is the angular momentum. It is shown that there exist two solutions for q > 2 approaching to the mass critical exponent 2, one of which is a local minimizer and the other is mountain pass type. In particular, we prove that the local minimizer u(q) converges to the ground state of above problem with q = 2 as q -> 2(+). Moreover, we verify that the mountain pass type solution occurs blow-up as q -> 2(+.)
We study the following type of magnetic system-(del + iA(x))2u + V(x)u + lambda phi|u|(x)u = |u|p-1u, x is an element of I3, (0.1) where p is an element of (1, 2), lambda is a parameter, and phi|u|(x) is a nonlo cal convolution potential. Under suitable assumptions on the potentials A(x) and V (x), we show that problem (0.1) has a ground state by using variational methods. Moreover, the asymptotical behavior of ground states as lambda -> 0 has also been discussed.
In this paper, we study the existence of bound states with prescribed mass for the NLS energy with localized nonlinearities which combine L2-subcritical and supercritical terms on a noncompact metric graph G. The study employs a min-max principle for constrained functionals, integrating the monotonicity technique with second-order details of Palais-Smale sequences, along with an analysis of blow-up in bound states characterized by a given mass and a uniformly bounded Morse index. Our results extend the previous results of J. Borthwick, X. Chang, L. Jeanjean, and N. Soave to nonlinear terms with mixed L2-subcritical and supercritical growth.
In this paper, we investigate the following planar quasilinear Schr & ouml;dinger equations characterized by distinct potentials -Delta u+lambda V(x)u+kappa/2 Delta(u(2))u=f(u), x is an element of R-2, where lambda>0 and kappa>0 are two parameters and the nonlinearity f:R -> R has critical exponential growth. By employing the dual approach, we first establish the existence of nontrivial solutions for the equation featuring both periodic and radial potentials. Subsequently, we demonstrate the existence and concentration of nontrivial solutions for the equation with a steep potential well. A pivotal aspect of our methodology involves a meticulous analysis of the L infinity-estimation. A significant aspect of the results derived in this research lies in the fact that the nonlinear term satisfies more generalized conditions than those typically required.
In this paper, we look for normalized solutions to the following nonlinear Schrodinger-Poisson system with doubly critical growth and non-constant potential {-Delta u+V(x)u+lambda u-phi|u|(3)u=mu|u|(q-2)u+|u|(4)u, x is an element of R-3, -Delta phi=|u|(5), x is an element of R-3, having prescribed mass integral(3)(R)|u|(2)dx=a(2), where u is an element of H-1(R-3), mu,a>0,2
In this paper, we investigate the normalized solutions searching the conditions (with rho, N, p) for the existence of solutions to the following elliptic equation { -triangle U +lambda U = |x|(alpha)|U|Up-1, in ohm U = 0, on partial derivative ohm integral (ohm) U-2 dx = rho, where N >3, ohm C R-N is is a ball or an annulus, rho> 0, 2 < alpha < + infinity and 1 < p < p(alpha) := N+2+2 alpha /N-2 . More precisely, we classify the problem into three cases, i.e., p is L-2-sub critical (p < 1 + 4+2 alpha/N), L-2-supercritical (p = 1 + 4+2 alpha /N ) based on a general Gagliardo-Nirenberg inequality and an adapted pointwise blow-up analysis. When pis L-2-subcritical, the problem admits solutions for every rho >0. In the L-2-critical and supercritical case, we show that for any k is an element of N the problem admits solutions with Morse index bounded above bykonlyif rho is sufficiently small. Furthermore, we present existence results for certain ranges of rho, which can be estimated in terms of the Dirichlet eigenvalues of-triangle inH(0)(1),(rad)(ohm).
We study the initial-boundary value problem of the threedimensional full compressible Euler equations with general damping and heat conduction under the Dirichlet or Neumann boundary condition for the temperature. When the initial data are near some constant equilibrium state, we prove the global existence and uniqueness of the classical solution. Moreover, we show that the classical solution converges to the steady state exponentially fast in time. To the best of our knowledge, this is the first global well-posedness result on the three-dimensional full compressible Euler equations with general damping and heat conduction in a bounded domain.
This paper investigates the Cahn-Hilliard equation equipped with Robin-type dynamic boundary conditions, with bulk-surface coupling governed by parameters (K, L) is an element of (0, infinity) x [0, infinity], while allowing for possible degeneracy in both the bulk diffusion mobility and the surface diffusion mobility. By incorporating the physical Gibbs-Thomson effect, we establish the global existence of weak solutions for the degenerate Cahn-Hilliard system with a smooth potential. This result is achieved by analyzing the limiting behavior of non-degenerate Cahn- Hilliard system, through a Faedo-Galerkin approximation scheme.
We establish the turnpike property for linear quadratic control problems for which the control operator is admissible and may be unbounded, under quite general and natural assumptions. The turnpike property has been well studied for bounded control operators, based on the theory of differential and algebraic Riccati equations. For unbounded control operators, there are only few results, limited to some special cases of hyperbolic systems in dimension one or to analytic semigroups. Our analysis is inspired by the pioneering work of Porretta and Zuazua [28]. We start by approximating the admissible control operator with a sequence of bounded ones. We then prove the convergence of the approximate problems to the initial one in a suitable sense. Establishing this convergence is the core of the paper. It requires to revisit in some sense the linear quadratic optimal control theory with admissible control operators, in which the roles of energy and adjoint states, and the connection between infinite-horizon and finite-horizon optimal control problems with an appropriate final cost are investigated.
This paper is concerned with the nonlinear SchrodingerPoisson system with a doping profile. We are interested in the strong instability of standing waves associated with ground state solutions in the L2-supercritical case. The presence of a doping profile causes several difficulties, especially in examining geometric shapes of fibering maps along an L2-invariant scaling curve. Furthermore, the classical approach by Berestycki-Cazenave for the strong instability cannot be applied to our problem due to a remainder term caused by the doping profile. To overcome these difficulties, we establish a new energy inequality associated with the L2-invariant scaling and adopt the strong instability result developed in [19]. When the doping profile is a characteristic function supported on a bounded smooth domain, some geometric quantities related to the domain, such as the mean curvature, are responsible for the strong instability of standing waves.
The definition of generalized random processes in Gel'fand sense allows to extend well-known stochastic models, such as the fractional Brownian motion, and study the related fractional pde's, as well as stochastic differential equations in distributional sense. By analogy with the construction (in the infinite-dimensional white-noise space) of the latter, we introduce two processes defined by means of Hadamard-type fractional operators. When used to replace the time derivative in the governing p.d.e.'s, the Hadamard-type derivatives are usually associated with ultra-slow diffusions. On the other hand, in our construction, they directly determine the memory properties of the so-called Hadamard fractional Brownian motion (H-fBm) and its long-time behaviour. Still, for any finite time horizon, the H-fBm displays a standard diffusing feature. We then extend the definition of the H-fBm from the white noise space to an infinite dimensional grey-noise space built on the Le Roy measure, so that our model represents an alternative to the generalized grey Brownian motion. In this case, we prove that the one-dimensional distribution of the process satisfies a heat equation with non-constant coefficients and fractional Hadamard time-derivative. Finally, once proved the existence of the distributional derivative of the above defined processes and derived an integral formula for it, we construct an Ornstein-Uhlenbeck type process and evaluate its distribution.
In this paper, we study the Hartree-Fock type system as follows: (-triangle u + u + lambda phi(u,v) u = |u|(p-2) u + beta |v|(p/2) |u|(p/2-2) u in R-3, -triangle v + v +lambda phi(u,v) v = |v|(p-2) v +beta|u|(p/2) |v|(p/2-2) v in R-3, where phi(u,v)(x) = integral(3)(R) u(2)(y)+v(2)(y)/|x-y| dy, the parameters lambda, beta > 0 and 2 < p < 4. Such a system is viewed as an approximation of the Coulomb system with two particles appeared in quantum mechanics, taking into account the Pauli principle. Its characteristic feature lies on the presence of the double coupled terms. When 2 < p < 3, we establish the existence and multiplicity of nontrivial radial solutions, including vectorial ones, in the radial space H-r := H-rad(1)(R-3) & times; H-rad(1)(R-3) by describing the internal relationship between the coupling constants lambda and beta. When 2 < p < 4, we study the existence of vectorial solutions in the non-radial space H := H-1(R-3)& times;H-1(R-3) by developing a novel constraint method, together with some new analysis techniques. In particular, when 3 <= p < 4, a vectorial ground state solution is found in H, which is innovative as it was not discussed at all in any previous results. Our study can be regarded as an entire supplement in d'Avenia et al. [J. Differential Equations 335 (2022) 580-614].
This paper is dedicated to the local existence theory of the Cauchy problem for a general class of symmetrizable hyperbolic partially diffusive systems (also called hyperbolic-parabolic systems) in the whole space Rd with d >= 1. We address the question of well-posedness for large data having critical Besov regularity in the spirit of previous works by the second author on the compressible Navier-Stokes equations. Compared to the pioneering work of Kawashima in [14] and to the more recent paper by Serre in [18], we take advantage of the partial parabolicity of the system to consider data in functional spaces that need not be embedded in the set of Lipschitz functions. This is in sharp contrast with the classical well-posedness theory of (multi-dimensional) hyperbolic systems where it is mandatory. A leitmotiv of our analysis is to require less regularity for the components experiencing a direct diffusion, than for the hyperbolic components. We then use an energy method that is performed on the system after spectral localization and a suitable Ga & ring;rding inequality. As examples, we consider the Navier-Stokes-Fourier and Euler-Fourier systems.
We establish the local Lipschitz regularity for solutions to an orthotropic q-Laplacian-type equation within the Heisenberg group. Our approach is largely inspired by the works of X. Zhong, who investigated the q-Laplacian in the same setting and proved the H & ouml;lder regularity for the gradient of solutions. Due to the degeneracy of the current equation, such regularity for the gradient of solutions is not even known in the Euclidean setting for dimensions greater than 2, where only boundedness is expected.
We introduce a fairly general dispersive-dissipative nonlinear equation, which is characterized by fractional Laplacian operators in both the dispersive and dissipative terms. This equation includes some physically relevant models of fluid dynamics as particular cases. Among them are the \emph{dispersive Kuramoto-Velarde}, the \emph{Kuramoto-Sivashinsky} equation, and some nonlocal perturbations of the \emph{KdV} and the \emph{Benjamin-Ono} equations. We thoroughly study the effects of the fractional Laplacian operators in the qualitative study of solutions: on the one hand, we prove a sharp well-posedness result in the framework of Sobolev spaces of negative order, and on the other hand, we investigate the pointwise decaying properties of solutions in the spatial variable, which are optimal in some cases. These last results are of particular interest for the corresponding physical models. Precisely, they align with previous numerical works on the spatial decay of a particular kind of solutions, commonly referred to as solitary waves.
We consider the initial value problem for the incompressible magnetohydro dynamics system with the Coriolis force in a threedimensional infinite layer. We prove the unique existence of global solutions for initial data in the scaling invariant space when the speed of rotation is sufficiently high. Moreover, we show that the global solution converges to that of the coupled system of the 2D incompressible magnetohydro dynamics equations and the 3D incompressible Maxwell equations as the rotation speed tends to infinity.
This paper deals with the following Choquard equation with doubly critical exponents and a local nonlinear perturbation: {-triangle u +u = gamma(I-alpha * |u|(alpha/N+1))|u| (alpha/N-1)u+& micro;|u|(q-2)u+| u|(2 & lowast;-2)u, x is an element of R-N; u is an element of H-1(R-N), where N >= 3, alpha is an element of (0, N), gamma > 0, & micro; >= 0 and I-alpha : R-N -> R is the Riesz potential. The exponent 1 + alpha/N is the lower critical with respect to the Hardy-Littlewood-Sobolev inequality. We demonstrate that the aforementioned equation admits no nontrivial solutions when & micro; =0; conversely, it possesses a ground state solution under mild conditions on N, q, alpha and gamma when & micro; > 0.
