Published in Advanced Nonlinear Studies
We prove some results about existence of connecting and closed geodesics in a manifold endowed with a Kropina metric. These have applications to both null geodesics of spacetimes endowed with a null Killing vector field and Zermelo’s navigation problem with critical wind.
Published in Advanced Nonlinear Studies
In this article, we prove the upper weighted critical exponents for some embeddings from weighted Sobolev spaces of radial functions into weighted Lebesgue spaces. We also consider the lower critical exponent for certain embedding.
Published in Advanced Nonlinear Studies
We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. The families of equations that we consider include well-known operato...
Published in Advanced Nonlinear Studies
The chemotaxis–Stokes system n t + u ⋅ ∇ n = ∇ ⋅ ( D ( n ) ∇ n ) − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , c t + u ⋅ ∇ c = Δ c − n c , u t = Δ u + ∇ P + n ∇ Φ , ∇ ⋅ u = 0 , \left\{\begin{array}{l}{n}_{t}+u\cdot \nabla n=\nabla \cdot (D\left(n)\nabla n)-\nabla \cdot (nS\left(x,n,c)\cdot \nabla c),\\ {c}_{t}+u\cdot \nabla c=\Delta c-nc,\\ {u}_{t}=\Delta u+...
Published in Advanced Nonlinear Studies
In this article, we investigate the boundary blow-up problem Δ ∞ h u = f ( x , u ) , in Ω , u = ∞ , on ∂ Ω , \left\{\begin{array}{ll}{\Delta }_{\infty }^{h}u=f\left(x,u),& {\rm{in}}\hspace{0.33em}\Omega ,\\ u=\infty ,& {\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Δ ∞ h u = ∣ D u ∣ h − 3 ⟨ D 2 u D u , D u ⟩ {\Delta }_{\infty }^{...
Published in Advanced Nonlinear Studies
We study the following fractional logarithmic Schrödinger equation: ( − Δ ) s u + V ( x ) u = u log u 2 , x ∈ R N , {\left(-\Delta )}^{s}u+V\left(x)u=u\log {u}^{2},\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ≥ 1 N\ge 1 , ( − Δ ) s {\left(-\Delta )}^{s} denotes the fractional Laplace operator, 0
Published in Advanced Nonlinear Studies
In this article, we are concerned with the following prescribed curvature problem involving polyharmonic operator on S N {{\mathbb{S}}}^{N} : D m u = K ( ∣ y ∣ ) u m ∗ − 1 , u > 0 in S N , u ∈ H m ( S N ) , {D}^{m}u=K\left(| y| ){u}^{{m}^{\ast }-1},\hspace{1.0em}u\gt 0\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{S}}}...
Published in Advanced Nonlinear Studies
We consider Monge-Ampère type equations involving the gradient that are elliptic in the framework of convex functions. Through suitable symmetrization we find sharp estimates to solutions of such equations. An overdetermined problem related to our Monge-Ampère type operators is also considered and we show that such a problem may admit a solution on...
Published in Advanced Nonlinear Studies
In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first-order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving plane method of Alexandrov–Serrin and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions.
Published in Advanced Nonlinear Studies
In this paper we study the existence as well as uniform decay rates of the energy associated with the nonlinear damped Schrödinger equation, i u t + Δ u + | u | α u - g ( u t ) = 0 in Ω × ( 0 , ∞ ) , iu_{t}+\Delta u+|u|^{\alpha}u-g(u_{t})=0\quad\text{in }\Omega\times(0,\infty), subject to Dirichlet boundary conditions, where Ω ⊂ ℝ n {\Ome...
