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.
Categories: math.AP