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We prove the existence of compact spacelike hypersurfaces with prescribed k-curvature in de Sitter space, where the prescription function depends on both space and the tilt function.
This paper deals with the following non-linear equation with a fractional Laplacian operator and almost critical exponents:
\[ (-\Delta)^{s} u=K(|y'|,y'')u^{({N+2s})/(N-2s)\pm\epsilon},\quad u > 0,\quad u\in D^{1,s}(\mathbb{R}^{N}), \]
where N ⩾ 4, 0 < s < 1, (y′, y″) ∈ ℝ2 × ℝN−2, ε > 0 is a small parameter and K(y) is non-negative and bounded. Under some suitable assumptions of the potential function K(r, y″), we will use the finite-dimensional reduction method and some local Pohozaev identities to prove that the above problem has a large number of bubble solutions. The concentration points of the bubble solutions include a saddle point of K(y). Moreover, the functional energies of these solutions are in the order $\epsilon ^{-(({N-2s-2})/({(N-2s)^2})}$.
In this paper, we look at a linear system of ordinary differential equations as derived from the two-dimensional Ginzburg–Landau equation. In two cases, it is known that this system admits bounded solutions coming from the invariance of the Ginzburg–Landau equation by translations and rotations. The specific contribution of our work is to prove that in the other cases, the system does not admit any bounded solutions. We show that this bounded solution problem is related to an eigenvalue problem.
We study the existence of entropy solutions by assuming the right-hand side function f to be an integrable function for some elliptic nonlocal p-Laplacian type problems. Moreover, the existence of weak solutions for the corresponding parabolic cases is also established. The main aim of this paper is to provide some positive answers for the two questions proposed by Chipot and de Oliveira (Math. Ann., 2019, 375, 283-306).
where b > 0, p ∈ (4, 6), the potential $V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., $\inf _{\mathbb R^N} V=0$. As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.
We consider the fractional critical problem
$A_{s}u=K(x)u^{(n+2s)/(n-2s)},u>0$
in
$\unicode[STIX]{x1D6FA},u=0$
on
$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$
, where
$A_{s},s\in (0,1)$
, is the fractional Laplace operator and
$K$
is a given function on a bounded domain
$\unicode[STIX]{x1D6FA}$
of
$\mathbb{R}^{n},n\geq 2$
. This is based on A. Bahri’s theory of critical points at infinity in Bahri [Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182 (Longman Scientific & Technical, Harlow, 1989)]. We prove Bahri’s estimates in the fractional setting and we provide existence theorems for the problem when
$K$
is close to 1.
We prove that if two
$C^{1,1}(\unicode[STIX]{x1D6FA})$
solutions of the second boundary value problem for the generated Jacobian equation intersect in
$\unicode[STIX]{x1D6FA}$
then they are the same solution. In addition, we extend this result to
$C^{2}(\overline{\unicode[STIX]{x1D6FA}})$
solutions intersecting on the boundary, via an additional convexity condition on the target domain.
We consider the problem of minimizing the weighted Dirichlet energy between homeomorphisms of planar annuli. A known challenge lies in the case when the weight λ depends on the independent variable z. We prove that for an increasing radial weight λ(z) the infimal energy within the class of all Sobolev homeomorphisms is the same as in the class of radially symmetric maps. For a general radial weight λ(z) we establish the same result in the case when the target is conformally thin compared to the domain. Fixing the admissible homeomorphisms on the outer boundary we establish the radial symmetry for every such weight.
where σD is a modulus of continuity defined in the interval [0, |D|] and depends only on the domain D. The inequality is optimal for any domain D and for any values of $\|f\|_1$ and $\|f\|_\infty .$ We also show that
\sigma_D(t)\leq\sigma_B(t),\text{ for }t\in[0,|D|],
where B is a ball and |B| = |D|. Using this optimality property of σD, we derive Brezis–Galloute–Wainger type inequalities on the L∞ norm of u in terms of the L1 and L∞ norms of f. As an application we derive L∞ − L1 estimates on the k-th Laplace eigenfunction of the domain D.
where $\unicode[STIX]{x1D6FA}$ is a smooth bounded domain in $\mathbb{R}^{N}$, $N\geq 2$. This work generalizes the well-known works on the Lyapunov inequality for extremal Pucci’s equations with gradient nonlinearity.
where $\beta \in {\mathbb R}$ is a coupling constant, $\mu ,\nu $ are positive constants, P,Q are weight functions decaying exponentially to zero at infinity, α can be regarded as a parameter. This type of system arises, in particular, in models in Bose–Einstein condensates theory and Kerr-like photo refractive media.
We prove that, for any positive integer k > 1, there exists a suitable range of α such that the above problem has a non-radial positive solution with exactly k maximum points which tend to infinity as $\alpha \to +\infty $ (or $0^+$). Moreover, we also construct prescribed number of sign-changing solutions.
In this paper we study a general eigenvalue problem for the so called (p, 2)-Laplace operator on a smooth bounded domain Ω ⊂ ℝN under a nonlinear Steklov type boundary condition, namely
For positive weight functions a and b satisfying appropriate integrability and boundedness assumptions, we show that, for all p>1, the eigenvalue set consists of an isolated null eigenvalue plus a continuous family of eigenvalues located away from zero.
where
$\unicode[STIX]{x1D6FA}\subset \mathbb{R}^{N}(N\geqslant 1)$
is a bounded domain with smooth boundary
$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$
,
$h\in L^{1}(\unicode[STIX]{x1D6FA})$
,
$h>0$
almost everywhere in
$\unicode[STIX]{x1D6FA}$
,
$k\in L^{\infty }(\unicode[STIX]{x1D6FA})$
is a non-negative function,
$B:[0,+\infty )\rightarrow [m,+\infty )$
is continuous for some positive constant
$m$
,
$p>1$
,
$0\leqslant q\leqslant p-1$
, and
$\unicode[STIX]{x1D6FE}>1$
. A “compatibility condition” on the couple
$(h(x),\unicode[STIX]{x1D6FE})$
will be given for the problem to admit at least one solution. To be a little more precise, it is shown that the problem admits at least one solution if and only if there exists a
$u_{0}\in W_{0}^{1,p}(\unicode[STIX]{x1D6FA})$
such that
$\int _{\unicode[STIX]{x1D6FA}}h(x)u_{0}^{1-\unicode[STIX]{x1D6FE}}\text{d}x<\infty$
. When
$k(x)\equiv 0$
, the weak solution is unique.
We introduce the notion of a distributional
$k$
-Hessian (
$k=2,\ldots ,n$
) associated with fractional Sobolev functions on
$\unicode[STIX]{x1D6FA}$
, a smooth bounded open subset in
$\mathbb{R}^{n}$
. We show that the distributional
$k$
-Hessian is weakly continuous on the fractional Sobolev space
$W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$
and that the weak continuity result is optimal, that is, the distributional
$k$
-Hessian is well defined in
$W^{s,p}(\unicode[STIX]{x1D6FA})$
if and only if
$W^{s,p}(\unicode[STIX]{x1D6FA})\subseteq W^{2-2/k,k}(\unicode[STIX]{x1D6FA})$
.
Similar to how Hopf–Lax–Oleinik-type formula yield variational solutions for Hamilton–Jacobi equations on Euclidean space, optimal mass transportations can sometimes provide variational formulations for solutions of certain mean-field games. We investigate here the particular case of transports that maximize and minimize the following ‘ballistic’ cost functional on phase space TM, which propagates Brenier’s transport along a Lagrangian L,
where
$\mathcal{A}$
is the set of stochastic processes satisfying dX = βX (t, X) dt + dWt, for some drift βX (t, X), and where Wt is σ(Xs: 0 ≤ s ≤ t)-Brownian motion. Both cases lead to Lax–Oleinik-type formulas on Wasserstein space that relate optimal ballistic transports to those associated with dynamic fixed-end transports studied by Bernard–Buffoni and Fathi–Figalli in the deterministic case, and by Mikami–Thieullen in the stochastic setting. While inf-convolution easily covers cost minimizing transports, this is not the case for total cost maximizing transports, which actually are sup-inf problems. However, in the case where the Lagrangian L is jointly convex on phase space, Bolza-type dualities – well known in the deterministic case but novel in the stochastic case – transform sup-inf problems to sup–sup settings. We also write Eulerian formulations and point to links with the theory of mean-field games.
We consider semistable, radially symmetric and increasing solutions of Sk(D2u) = g(u) in the unit ball of ℝn, where Sk(D2u) is the k-Hessian operator of u and g ∈ C1 is a general positive nonlinearity. We establish sharp pointwise estimates for such solutions in a proper weighted Sobolev space, which are optimal and do not depend on the specific nonlinearity g. As an application of these results, we obtain pointwise estimates for the extremal solution and its derivatives (up to order three) of the equation Sk(D2u) = λg(u), posed in B1, with Dirichlet data
$u\arrowvert _{B_1}=0$
, where g is a continuous, positive, nonincreasing function such that lim t→−∞g(t)/|t|k = +∞.
where Ω ⊂ ℝ2 is a bounded domain. In the second place, we present existence results for the following stationary Schrödinger systems defined in the whole plane
0.2
$$\left\{ {\matrix{ {-\Delta u + u = g(v)\;\;\;{\rm in}\;{\open R}^2,} \cr {-\Delta v + v = f(u)\;\;\;{\rm in}\;{\open R}^2.} \cr } } \right.$$
We assume that the nonlinearities f, g have critical growth in the sense of Trudinger–Moser. By using a suitable variational framework based on the generalized Nehari manifold method, we obtain the existence of ground state solutions of both systems (0.1) and (0.2).
In this paper, we study the existence of positive solutions to a semilinear nonlocal elliptic problem with the fractional α-Laplacian on Rn, 0 < α < n. We show that the problem has infinitely many positive solutions in
$ {C^\tau}({R^n})\bigcap H_{loc}^{\alpha /2}({R^n}) $
. Moreover, each of these solutions tends to some positive constant limit at infinity. We can extend our previous result about sub-elliptic problem to the nonlocal problem on Rn. We also show for α ∊ (0, 2) that in some cases, by the use of Hardy’s inequality, there is a nontrivial non-negative
$ H_{loc}^{\alpha /2}({R^n}) $
weak solution to the problem