In this paper, we study quasiconcavity properties of solutions of Dirichlet problems related to modified nonlinear Schrödinger equations of the type$$-{\rm div}\big(a(u) \nabla u\big) + \frac{a'(u)}{2} |\nabla u|^2 = f(u) \quad \hbox{in $\Omega$},$$where $\Omega$ is a convex bounded domain of $\mathbb{R}^N$.In particular, we search for a function $\varphi:\mathbb{R} \to \mathbb{R}$, modeled on $f\in C^1$ and $a\in C^1$, which makes $\varphi(u)$ concave. Moreover, we discuss the optimality of the conditions assumed on the source.

The time-domain fluorescence diffuse optical tomography (FDOT) problem is to recover the distribution of fluorophores in biological tissue from the time-domain measurement on the boundary. The measurement is conducted by several pairs (S-D pairs) of a point source and a point detector. In this paper, we identify the location of the distribution of fluorophores over a point, refer as a point target. We first express a solution for the forward problem in a dimensionless form and consider its asymptotic expansion. Then, we theoretically investigate the minimal number of S-D pairs to determine the point target location, analyzing the sensitivity matrix. Finally, we numerically verify the invertibility of the matrix and demonstrate the local solvability for locating well-separated multiple point targets.

In this paper, we study a singularKirchhoff-type problem with a nonlinearity$h(x)|u|^{q-2} u$$(2 < q < 4)$on an unbounded domain.Since the (PS) sequence may not be boundedon the associated Nehari manifold and theassociated energy functional is notdifferentiable because of the singular term,we cannot apply the variationalmethod according to a standard way.By analyzing the structure of theNehari manifold and developingsome approximation techniques,the above obstacles are overcome insubtle ways. As a result,two positive solutions of that problemare obtained with negative and positive energy,respectively.

We are interested in the stability of a class of totally geodesic wave maps, as recently studied in [1, 7, 8]The relevant equations of motion are a system of coupled semilinear wave and Klein-Gordon equations in$\mathbb R ^{1+n}$ whose nonlinearities have critical time decay when$n=2$.In this paper, we use a pure energy method to show global existence when$n = 2$. By carefully examining the structure of the nonlinear terms, we are able to obtain uniform energy bounds at lower orders. This allows us to prove pointwise decay estimates and also to reduce the required regularity.

We study existence and asymptotic behavior of entire positive bounded solutions for the following class of semilinear elliptic problem \begin{equation*} \left\{ \begin{aligned} L(u) & = \varrho(x) f(x,u) & \text{ in } & \mathbb{R}^N,\ N \geq 3,\\ u & > 0, & \text{ in } & \mathbb{R}^N, \end{aligned} \right. \end{equation*}where$0 \leq \varrho \in L^p_{loc}(\mathbb{R}^N),$ for some$ N < p \leq \infty.$ Here,$L$ is a local uniform elliptic operator and$f(x,s)$ is a nonlinearity with sublinear behavior at zero and at$+\infty$. This type of result has already been studied in the celebrated work by H. Brezis and S. Kamin for the case when$L= -\Delta$ and$\varrho \in L^{\infty}_{loc}(\mathbb{R}^N)$. Our approach allows us to include for instance$ - \text{dive} \left ( (1+|x|^\mu )^\nu\nabla u \right ) = u^q(|x|^\alpha + |x| ^\beta)^{-1}$ with suitable$\alpha,\ \beta > 0,$$\mu,\ \nu \in \mathbb{R}$ and$0 < q < 1.$ Here, we include two local uniform elliptic situations:$\mu > 0$ with$\nu =1$ or$\nu =-1.$

In this paper, we study the existence of nontrivial solutions tothe critical fractional elliptic system on the Heisenberg group:$$\begin{gathered}\begin{cases}(-\Delta_{\mathbb{H}^{n}})^{s} u+\gamma u=\lambda f(\xi)|u|^{r-2} u+\frac{2 \alpha}{\alpha+\beta}|u|^{\alpha-2} u|v|^{\beta} & \text { in } \Omega, \\(-\Delta_{\mathbb{H}^{n}})^{s} v+\nu v=\mu g(\xi)|v|^{r-2} v+\frac{2 \beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2} v & \text { in } \Omega, \\u = v =0 & \text { in } \mathbb{H}^{n}\setminus \Omega,\end{cases}\end{gathered}$$where ${\Delta_{\mathbb{H}^{n}}}$ is the Kohn-Laplacian, $s\in(0,1)$, $ Q > 2 s$ and $\Omega\subset\mathbb{H}^{n}$ is a boundeddomain with smooth boundary. $\lambda, \mu,\gamma,\nu$ are positive real parameters and $ Q=2n+2$ is the homogeneousdimension of ${\mathbb{H}^{n}}$. The exponent $r$ satisfies$1 < r < 2^{*}_{s}$, $\alpha > 1, \beta > 1$ satisfy$2 < \alpha+\beta=2_{s}^{*}$, $2_{s}^{*}= 2Q/(Q-2s)$ being thefractional critical Sobolev exponent. The results presented hereextend or complete recent papers and are new to critical fractionalelliptic system on Heisenberg group.