Existence of singular solutions for a Dirichlet problem containing a Dirac mass. (English) Zbl 1214.35022
Summary: We give general existence results of solutions \((u,v)\) to the Dirichlet problem
\[ \begin{cases} \begin{aligned} -\Delta u&= f(x,u,v)+ c\delta_0\\ -\Delta v&= g(x,u,v)+ d\delta_0 \end{aligned} &\text{in }{\mathcal D}'(B),\\ u=v=0 &\text{on }\partial B, \end{cases} \tag{P} \]
where \(B\) is the unit ball centered at zero in \(\mathbb R^N\), \(N\geq 3\), \(\delta_0\) is the Dirac mass at 0 and \(c,d\) are nonnegative constants. No assumptions on the sign of the functions \(f\) and \(g\) are required. We also characterize the set of \((c,d)\) such that problem (P) admits a solution in some particular cases of the nonlinearities \(f\) and \(g\).
\[ \begin{cases} \begin{aligned} -\Delta u&= f(x,u,v)+ c\delta_0\\ -\Delta v&= g(x,u,v)+ d\delta_0 \end{aligned} &\text{in }{\mathcal D}'(B),\\ u=v=0 &\text{on }\partial B, \end{cases} \tag{P} \]
where \(B\) is the unit ball centered at zero in \(\mathbb R^N\), \(N\geq 3\), \(\delta_0\) is the Dirac mass at 0 and \(c,d\) are nonnegative constants. No assumptions on the sign of the functions \(f\) and \(g\) are required. We also characterize the set of \((c,d)\) such that problem (P) admits a solution in some particular cases of the nonlinearities \(f\) and \(g\).
MSC:
| 35J57 | Boundary value problems for second-order elliptic systems |
| 35J47 | Second-order elliptic systems |
| 35J61 | Semilinear elliptic equations |
| 35D30 | Weak solutions to PDEs |
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