Anisotropic singularities to semilienar elliptic equations in a measure framework

The purpose of this article is to study very weak solutions of elliptic equation $$-\Delta u+g(u)=2k\frac{\partial \delta_0}{\partial x_N }+j\delta_0\quad {\rm in}\quad\ \ B_1(0),\qquad u=0\quad {\rm on}\quad\ \ \partial B_1(0),$$ where $k>0$, $j\ge0$, $B_1(0)$ denotes the unit ball centered at the origin in $\mathbb{R}^N$ with $N\geq2$, $g:\mathbb{R}\to\mathbb{R}$ is an odd, nondecreasing and $C^1$ function, $\delta_0$ is the Dirac mass concentrated at the origin and $\frac{\partial\delta_0}{\partial x_N}$ is defined in the distribution sense that $$\langle\frac{\partial \delta_0}{\partial x_N},\zeta\rangle=\frac{\partial\zeta(0)}{\partial x_N} , \qquad \forall \zeta\in C^1_0(B_1(0)).$$ We obtain that the above problem admits a unique very weak solution $u_{k,j}$ under the integral subcritical assumption $$\int_1^{\infty}g(s)s^{-1-\frac{N+1}{N-1}}ds<+\infty.$$ Furthermore, we prove that $u_{k,j}$ has anisotropic singularity at the origin and we consider the odd property $u_{k,0}$ and limit of $\{u_{k,0}\}_k$ as $k\to\infty$. We pose the constraint on nonlienarity $g(u)$ that we only require integrability in the principle value sense, due to the singularities only at the origin. This makes us able to search the very weak solutions in a larger scope of the nonlinearity.