The Cauchy problem for the stochastic nonlinear Schrödinger equation with multiplicative noise is considered where the nonlinear term is of power type and the noise coefficients are purely imaginary numbers. The main purpose of this paper is to construct classical solutions in $H^2(\mathbb{R}^d)$ for the problem. The techniques of Kato [21, 22] work well in overcoming smoothness problems even for the stochastic equations.

The nonlocal differential equation \begin{equation} -A \big ( \big (b*u^{p(\cdot)} \big )(1) \big )u''(t) =\lambda f\big(t,u(t)\big)\text{, }t\in(0,1)\notag \end{equation} is considered, where $$ \displaystyle(a*b)(t):=\int_0^ta(t-s)b(s)\ ds , \ \ \ t\ge0 , $$ is a finite convolution. The existence of at least one positive solution is considered when the equation is subject to boundary conditions. A model case of the above problem occurs when $b(t)\equiv1$, as in this case the problem reduces to \begin{equation} -A \Big (\int_0^1\big(u(s)\big)^{p(s)}\ ds \Big )u''(t) =\lambda f\big(t,u(t)\big)\text{, }t\in(0,1).\notag \end{equation} As a consequence nonlocal coefficients possessing $p(x)$-growth are considered. By using a specialized order cone together with topological fixed point theory we are able to provide existence theorems with a minimum of assumptions on the coefficient function $A$. In particular, $A$ can vanish or change sign on intervals of infinite measure.

In this paper, we consider the Choquard-Kirchhoff involving fractional $p$-Laplace equation of the form: \begin{align*} M \Big (\varepsilon^{sp-N}[u]^p_{s,p}+\varepsilon^{-N} \int_{\mathbb{R}^{N}}V(x)|u|^pdx \Big ) & \big(\varepsilon^{sp}(-\Delta)_p^su +V(x)|u|^{p-2}u)\big)\\ & = \varepsilon^{\mu -N} (\frac {1}{|x|^\mu}*F(u))f(u) \end{align*} in $\mathbb{R}^N,$ where $\varepsilon > 0$ is a parameter, $s\in (0 , 1)$, $0 < \mu < ps$, $(-\Delta)_p^s$ is the fractional $p$-Laplacian operator, $M$ represents Kirchhoff function, $V$ is a continuous potential function and $f$ is a continuous function involving subcritical growth. With the help of well-known penalization methods and Ljusternik-Schnirelmann theory, we obtain the existence, multiplicity and concentration of solutions for the above equations.

In this paper, we focus on the study of Holder regularity in time and space variables, to solutions of a parabolic Hamilton-Jacobi problem, where the standard time derivative is replaced by a variable-order fractional derivative of Caputo type.

We investigate multi-bump positive and nodal solutions to the logarithmic scalar field equation $$ -\Delta u + V(y) u = u\log |u|,\quad u\in H^1(\mathbb R^N), $$ where $N\geq2$ and the potential $V$ is bounded radially symmetric, which is a class of important Schrödinger equations in mathematical physics. The main difficulties to apply Lyapunov-Schmidt reduction to logarithmic scalar equations are caused by the non-smooth property and sub-linear growth of the logarithmic nonlinearity. To overcome these difficulties, we develop a new approach to carry out the Lyapunov-Schmidt reduction, which can be used to construct not only positive but also nodal solutions to the logarithmic equations. Finally, both infinitely many positive and infinitely many nodal solutions with an arbitrarily large number of bumps are constructed for scalar field equations when the potential satisfies different proper decay conditions.

The chemotaxis-Navier-Stokes system $$ \begin{cases} n_t+u\cdot\nabla n = \Delta \big(n\phi(c)\big),\\ c_t+ u\cdot\nabla c = \Delta c -c+n,\\ u_t + (u\cdot\nabla) u =\Delta u+\nabla P + n\nabla\Phi, \qquad \nabla\cdot u=0, \end{cases} $$ is considered along with homogeneous boundary conditions of no-flux type for $n$ and $c$, and of Dirichlet type for $u$, in a smoothly bounded planar domain $\Omega$, where $\Phi \in W^{2,\infty}(\Omega)$ is given. A concept of generalized solvability is introduced, and in this framework a statement on global existence is derived for all reasonably regular initial data $(n_0,c_0,u_0)$, provided that $\phi\in C^3((0,\infty))$ is positive and bounded on $(\xi_0,\infty)$ for each $\xi_0>0$, and such that $$ \xi\phi(\xi) \to +\infty \qquad \mbox{and} \qquad \frac{\xi\phi'^2(\xi)}{\phi(\xi)} \to 0 \qquad \mbox{as } \xi\to\infty. $$ These solutions particularly satisfy $\int_0^T \Omega n\ln (n+1) < \infty$ for all $T > 0$ as well as $\int_\Omega n(\cdot,t)=\int_\Omega n_0$ for a.e. $t > 0$.

We study the asymptotics of solutions of logistic type equations with fractional Laplacian as time goes to infinity and as the exponent in nonlinear part goes to infinity. We prove strong convergence of solutions in the energy space and uniform convergence to the solution of an obstacle problem. As a by-product, we also prove the cut-off property for eigenvalues of the Dirichlet fractional Laplace operator perturbed by exploding potentials.

Our aim of this paper is to study qualitative properties of isolated singular solutions to Choquard equation \begin{equation}\label{eq 0.1} -\Delta u+ u =I_\alpha[u^p] u^q+k\delta_0\quad {\rm in}\ \, \mathcal{D}'(\mathbb{R}^N), \tag{0.1} \end{equation} where $p, \, q\ge 1$, $N\ge2$, $\alpha\in(0,N)$, $k > 0$, $\delta_0$ is the Dirac mass concentrated at the origin and $I_\alpha[u^p](x)=\int_{\mathbb{R}^N} \frac{u(y)^p}{|x-y|^{N-\alpha}}\, dy.$ Multiple properties of very weak solutions of (0.1) are considered: (i) to obtain the existence of minimal solutions and extremal solutions for $N=2$, which are derived in [8] when $N\geq3$; (ii) to analyze the stability of minimal solutions and the semi stability of extremal solutions; (iii) to derive a second solution by the Mountain Pass theorem when $q=p-1$ and $N=2,3$; (iv) to obtain the radial symmetry of the positive singular solutions by the method of moving planes.

In this paper, we study the following class of fractional Hénon problems involving exponents critical or supercritical \begin{equation*} \begin{cases} \displaystyle (-\Delta)^s u = \lambda |x|^{\mu} u +|x|^{\alpha}|u|^{ (p_{\alpha,s}^*+ \varepsilon) -1} u & \mbox{in } \Omega, \\ u = 0 & \mbox{in } \mathbb{R}^N\setminus\Omega, \end{cases} \end{equation*} where $p_{\alpha,s}^*= \frac{N + 2 \alpha + 2s}{N-2s}$ is the critical exponent for a nonlinearity with Hénon weight in nonlocal context, $\varepsilon \geq 0$, $\Omega$ is either a ball or an annulus in $\mathbb R^N$, $s \in (0,1)$ and $\mu, \alpha > -2s$. We used the Emden--Fowler transformation to make the one-dimensional reduction of problems and under appropriate hypotheses on the constant $\lambda$, we prove the existence of at least one non-trivial radial solution for these problems using the concentration compactness principle or Linking Theorem.

We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and this condition is expressed via a differential equation along the curve. In the classical differential geometry setting, the analogous condition was considered by Bryant and Hsu in [7, 21] who proved that it is equivalent to the surjectivity of a holonomy map. The purpose of this paper is to extend this deformation theory to curves of fixed degree providing several examples and applications. In particular, we give a useful sufficient condition to guarantee the possibility of deforming a curve.