In this paper, we consider the following Schr & ouml;dinger equation with Hartree nonlinearities -Delta u+lambda u-alpha integral(& Ropf;)3(|u(y)|p)/(|x-y|)dy|u|p(-2)u-beta|u|(q-2)u= 0, x is an element of & Ropf;(3). When both alpha and beta are negative, a non-existence result is established. Provided that alpha and beta are not both negative, for each alpha > 0, we establish the existence of normalized solutions for the elliptic equation above with prescribed mass & Vert;u & Vert;(2)(2) =alpha and show that these are nonnegative and bell-shaped. Moreover, we obtain the asymptotic behaviors of the ground state energy and the Lagrange multiplier as the mass alpha approaches zero, and show that lambda= 0 is a bifurcation point. Our analysis provides a comprehensive characterization of the ground state energy function e, including local Lipschitz continuity, strict monotonicity and strict subadditivity. If alpha, beta > 0, we further prove that these normalized solutions are strictly positive and investigate their regularity by applying the Moser iteration scheme. We also consider the corresponding linearized problem, analyze the spectral structure of the linearized operators, and prove the non-degeneracy of the solutions. Finally, we rigorously prove that these solutions are orbitally stable.
In this work, we obtain a short time solution for a geometric flow on noncompact affine Riemannian manifolds. Using this result, we can construct a Hessian metric with nonnegative bounded Hessian sectional curvature on some Hessian manifolds with nonnegative Hessian sectional curvature. Our results can be regarded as a real version of Lee-Tam [1]. As an application, we prove that a complete noncompact Hessian manifold with nonnegative Hessian sectional curvature is diffeomorphic to Rn if its tangent bundle has maximal volume growth. This is an improvement of Theorem 1.3 in Jiao-Yin [2].
In this paper we prove that any distribution u is an element of D '(R-N) satisfying u >= 0, -Delta u >= 0, and (-Delta)(m) u >= 0, must be constant whenever N <= 2m. The essential feature is that no requirement is made on the intermediate iterates (-Delta)(j)u for 2 <= j <= m - 1. We further prove that the reverse inequality (-Delta)(m) u <= 0, together with u >= 0 and -Delta u >= 0, forces u to be constant, with no restriction on m or N. Both results extend to operators of the form P(-Delta), where P is a polynomial whose roots all lie in (-infinity, 0]. We also establish a more general version replacing the sign conditions on u and -Delta u by a single averaged vanishing condition at infinity: any u is an element of L-loc(1)(R-N) satisfying such a condition together with (-Delta)(m) u >= 0 or <= 0 (and more generally P(-Delta)u >= 0 or <= 0) must be constant, provided N <= 2m.
We study the interior Lipschitz continuity and higher weak differentiability of local minimizers of functionals with (p, q)-growth in the Heisenberg group for the full range 1 < p <= q< infinity under a suitable bound on the gap p/q. Our results cover both degenerate and nondegenerate cases.
The global existence of smooth solutions to the three-dimensional compressible viscous non-resistive magnetohydrodynamic (MHD) equations in R3 remains a significant open problem. In this paper, we investigate the global existence and stability of classical solutions to the 3D compressible MHD equations with horizontal magnetic dissipation near a background magnetic field. The absence of vertical dissipation of the magnetic field poses substantial challenges for stability analysis. By exploiting the wave structure and performing anisotropic estimates, we establish the global existence of solutions and their algebraic decay-in-time behavior. Moreover, using the time-frequency method, we obtain optimal time-decay rates when the initial perturbation is bounded in L1.
In this paper we study weak solutions to general non-homogeneous parabolic obstacle problems, whose obstacle function is time-dependent irregular. We prove the well-posedness of a global solution to the obstacle problem and we describe the asymptotic behavior of such a solution. In particular, we measure the distance in time of the solutions to two parabolic problems with different initial data and forcing terms. Moreover, in the autonomous case, we prove that, as the time approaches infinity, the global solution of our obstacle problem converges to the solution of the corresponding elliptic obstacle problem. Finally, we compare the solution to a quasi-harmonic evolution problem from the solution to the stationary case.
We study nonlinear parabolic systems with inhomogeneous right-hand side and nonstandard (p, q)-growth. Under the natural structural conditions 2 <= p <= q < p + 4/N + 2, we establish local Lipschitz regularity for weak solutions. The proof is first developed for smooth approximations, where all estimates can be justified directly, and then extended to general weak solutions by a stability and limit argument. The resulting quantitative bounds capture the precise interplay between the data and the nonlinear structure of the system.
This paper investigates a quasilinear Schrodinger-Poisson system with a general subcritical or critical growth: {-Delta u + V(x)u + phi u = f(u) + mu|u|(4)u, x is an element of R-3, -Delta phi - epsilon(4) Delta(4)phi = u(2), x is an element of R-3, where mu = 0 (subcritical case) or mu = 1 (critical case), epsilon > 0 V(x) is a radially symmetric potential, and f is a continuous nonlinearity without the Ambrosetti-Rabinowitz condition. In the subcritical case (mu = 0) , we prove the existence of a radially symmetric ground state solution by using a perturbation method. We also study the asymptotic behavior of the ground state solution as epsilon -> 0, showing convergence to a solution of the classical Schrodinger-Poisson system. Furthermore, by applying the symmetric mountain pass theorem, we establish the existence of infinitely many radially symmetric solutions. In the critical case (mu = 1), we similarly prove the existence of a radially symmetric ground state solution and its asymptotic behavior as epsilon -> 0. Meanwhile, a multiplicity result is obtained by applying an abstract critical point theorem due to Perera [25].
For functions in Sobolev-Slobodeckij trace spaces, we prove a necessary and sufficient condition for membership to the class of functions of bounded variation.
Optimal local Lipschitz regularity for scalar almost minimizers of two-phase free-boundary func-tionals F(02) := (x, [VD]) + 2X1000) + 2X100) + min(41.421210 dx. (Vo)+ with growth function is established, ass assuming a a generalized Orlicz function and 2, 4, nonnegative bounded functions, a "small" density for either the positivity or the negativity set.
The existence and uniqueness of weak solutions to a size-structured growth-coagulation-fragmentation (GCF) equation with a renewal boundary condition are shown for a class of unbounded coagulation and fragmentation kernels. The existence proof is based on a weak compactness framework in the weighted L1-space. This result extends the existence results of Banasiak and Lamb [1] and Ackleh et al. [2, 3]. Furthermore, we establish a stability result and derive uniqueness as a direct consequence of it. Moreover, this study establishes the decay of the zeroth and first moments of weak solutions for a specific class of kernels.
We establish an interior C-1,C-sigma estimate for viscosity solutions to the following degenerate Bellman type equation: sup(alpha is an element of A) {-vertical bar Du vertical bar(theta)a Delta(N)(Pa) u - f(alpha)(x)} = 0 in B-1. We prove an equicontinuity estimate of rescaled solutions via doubling argument and weak Harnack inequality, and then apply improvement of flatness iteration to obtain the interior C-1,C-sigma estimate for some sigma is an element of(0, 1).
We will study a quasilinear problem with critical exponent and two different kinds of discontinuous nonlinearity. By applying the critical point theory for nondifferentiable functionals, we will prove that our problem has at least one nontrivial weak solution for any value of positive parameters associated with the problem. We proved new results when one of the nonlinearities satisfied the Ambrosetti-Rabinowitz (AR) condition.
In this paper, we study the following p-Laplace type equation -div(V(x)|del u|(p-2)del u) =V(x)f(x,u), where 2< p < N,f is an element of C(& Ropf;(N)& times;& Ropf;,& Ropf;),V(x) =e(p theta(x))and theta(x) is a non-negative function. On the one hand,we prove the existence of ground state solution for p-Laplace type equation by variational methods. On the other hand, if f(x,u) =g(u) +eta|u|p & lowast;-2u,by working in the weighted Sobolev spaces, and using variational method again,we prove the multiplicity of self-similar solutions to the equation under appropriate conditions forg and eta, where p & lowast;=NpN-
The radial map u(x)= (x)/parallel to x parallel to is a well-known example of a harmonic map from R-m - {0} into the spheres Sm-1 with a point singularity at x =0. Several studies are given for this special example of harmonic maps ([1,8,13,17], etc.). In [25] the author constructed, for any positive integer m, n satisfying n <= m, a family of harmonic maps u((n)) from R-m - {0} into the sphere Smn-1 with one point singularity at the origin, such that u((1)) is the above radial map. It is known that the radial map u((1)) is stable as a harmonic map, and furthermore for any m >= 3, it is a minimizer of the energy of harmonic maps [1,13]. On the other hand in [26], the author proved that for n >= 2, the map root u(n) is unstable if m >= 3 and n> (root 3-1) /2 (m-1). Furthermore in [27], we proved that for n >= 2, the map u(n) is unstable as a p-harmonic map if m > p and n >= 1/2 m -m p/m - 2(m - p + 1), while the radial u((1)) is stable as a p-harmonic map and furthermore p-minimizing for all real number p satisfying 1 < p < m ([2,6,7]). It is remarkable that u((n)) may be unstable in the case of n >= 2. The concept of symphonic maps was introduced in a study of variational problem for the con-formality of maps, and are in a position to be a counter part of harmonic maps from the viewpoint of the pullbacks of maps. In this paper we study the symphonicity and stability as a symphonic map for the map u((n)). We prove that u((n)) is a symphonic map for all positive integer n (Theorem 1), and that it is unstable as a symphonic map for any n >= 2 (Theorem 2). Our main tool is a calculation of the differential quantity D(i)u((n)) D(j)u((n)). The calculation is slightly long, but the result is simple (Proposition 1). It is useful for calculations of other quantities. Our results give many examples of unstable symphonic maps into the spheres with a point singularities at the origin.
We consider a quasilinear system with a drift term of N >= 2 equations in a bounded domain Omega subset of Rn with n >= 3. Assuming that the support of the off-diagonal coefficients is contained in a crossed staircase of squares with geometrically increasing side lengths, it is proved that weak solutions enjoy Lp-regularity for any p is an element of [1, +infinity). The existence of at least a weak solution is established provided that the Lp-norm of the drift term is sufficiently small. to Gioconda Moscariello on occasion of her birthday
As the temperature decreases, liquid crystals transition from the nematic phase, characterized by orientational order, to the smectic phase, which also exhibits positional order. This transition can be described by the de Gennes model using a coupled system between a heat flow of harmonic maps and a time-dependent Ginzburg-Landau equation. In this paper, we establish the global existence of smectic liquid crystal flows in R2 within the simplified de Gennes model.
The object of this work is to study the bi-harmonic Hartree problem {Delta(2)u = (integral(Omega epsilon) u(2 alpha*)/|x-y|alpha dy)u(2 alpha*- 1 )in Omega(epsilon, ) u = Delta u = 0, on partial derivative Omega(epsilon,) where Omega(epsilon) is a smooth bounded domain in R-N with a small hole. N >= 9, 0 < alpha < 9 with alpha sufficiently close to 0, 2(alpha)* = 2N - alpha/N - 4 is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We construct the new solutions whose profile is a superposition of bubbles with alternate sign which concentrate at the center of the hole. This seems to be the first existence result of tower of bubbles for the critical Hartree equation.
This paper deals with the following doubly degenerate cross-diffusion system with logistic source { u(t)= del & sdot;(uv del u) - del & sdot;(u(2)v del v) +u-u(kappa), x is an element of Omega, t >0, v(t)= Delta v-uv, x is an element of Omega, t >0 in a bounded smooth domain Omega subset of R-n (n >= 2) with no-flux boundary conditions. It is shown that under the assumption that for all suitably smooth initial data satisfying nu(0) > 0 in Omega and integral(Omega) ln u(0) > -infinity, if kappa> (n+ 2)/2, for some null set N-star subset of (0, infinity), a corresponding global weak solution satisfies & varphi;(u(& sdot;,t))(star)-> 0 in L-infinity(Omega)and v(& sdot;,t)-> 0in L-p(Omega) for p is an element of [1,infinity) as(0,infinity)N-star is not an element of t ->infinity,where & varphi;(xi) & ratio;=(xi-1)(2)/(xi+1)(2) for xi >= 0.
In this paper, we study the minimizer of Bernoulli-type problems with unbounded or degenerate weights. In the first part, we give the H & ouml;lder or BMO estimates for the minimizer of Alt-Caffarelli functional J(phi)(u) & ratio;=integral(Omega)(|del u|(p )+ phi(x)chi({u>0)) ) dx where phi is an element of Lp(Omega) with 1< p <+infinity Omega) with 1< p + infinity and we provide a rough description of its free boundary. Furthermore, we solve a shape optimization problem involving rho-Laplacian. In the second part, we consider the degenerate case that rho = 2 and phi is H & ouml;lder continuous but vanishes on a C(1,alpha)submanifold Gamma of dimension 0 <= k <= n- 1. We show that if x(0 ) is an element of Gamma is a free boundary point, then the blow-up at x(0) is homogeneous when phi satisfies some growth condition.
