On the definition of the solution to a semilinear elliptic problem with a strong singularity at \(u = 0\)
Let \(\Omega\) be a bounded open set in \(\mathbb{R}^N\), and let \(A:\Omega\rightarrow \mathbb{R}^{N}\times \mathbb{R}^{N}\) be a matrix with entries in \(L^\infty(\Omega)\). The authors consider the semilinear elliptic problem \[ \begin{cases} -\text{div}A(x)D u=F(x,u) & \text{in }\Omega,\\ u\geq 0 & \text{in } \Omega,\\ u=0 & \text{on } \partial \Omega, \end{cases}\tag{\(P\)} \] where the nonlinearity \(F:\Omega \times [0,+\infty[\rightarrow [0,+\infty]\) is a real extended Carathéodory function satisfying the growth condition \[ F(x,s)\leq \frac{h(x)}{\Gamma(s)}, \;\;\;\text{for almost all} \;\;x\in \Omega, \;\;\text{and for all} \;\;s\in ]0,+\infty[,\tag{1} \] for some nonnegative function \(\Gamma \in C^1([0,+\infty[)\), with \(\Gamma(0)=0\) and \(\Gamma'(s)>0\) for all \(s>0\), and some nonnegative function \(h\in L^r(\Omega)\), with \(r=\frac{2N}{N+2}\) if \(N\geq 3\), \(r>1\) if \(N=2\), and \(r=1\) if \(N=1\). The growth condition \((1)\) allows the nonlinearity \(F\) to be strongly singular at \(u=0\), and when this occurs, an appropriate definition of solution for problem \((P)\) is to be considered. In a recent paper, after introducing a new test function space \(\mathcal{V}(\Omega)\), the authors gave a definition of solution for problem \((P)\) under condition \((1)\), which is suggested by the notion of ``solution defined by transposition'' introduced by J. L. Lions and E. Magenes and by G. Stampacchia. This definition involves the truncation operators \(T_k(s)=\max\{-k,\min\{k,s\}\}\) and \(G_k(s)=s-T_k(s)\) and states that a nonnegative \(u\in L^2(\Omega)\cap H^1_{loc}(\Omega)\) is a solution to problem \((P)\) if, for each \(k>0\), the functions \(T_k(u)\) and \(G_k(u)\) satisfy, in a weak sense, a corresponding equation in divergence form. In another paper, the authors also introduced the definition of solution to problem \((P)\) for the case of mild singularities, that is when \(\Gamma(s)=s^\gamma\), with \(\gamma \in ]0,1]\). In this paper, the authors prove that the above two definitions are equivalent and, moreover, they prove that the condition that the equation in divergence form is satisfied by \(T_k(u)\) and \(G_k(u)\) for each \(k>0\) (which is required in the definition of solution for the strongly singular case) holds if and only if the same equation is satisfied only by some \(T_{k_0}(u)\) and \(G_{k_0}(u)\), with \(k_0>0\) arbitrarily chosen. The proofs of these equivalences are based on a regularity result for the solutions (in the sense of the above definitions) of problem \((P)\). The authors also prove that the definition of solution for the strongly singular case can be simplified in the case of bounded solutions. Among other results, it is proved that the test function space \(\mathcal{V}(\Omega)\) can be replaced by the subspace \(\mathcal{W}(\Omega)\) of all function of the form \(w=\varphi^2\), where \(\varphi\in L^\infty(\Omega)\cap H_0^1(\Omega)\), in all the results established by the authors in a previous paper. Moreover, given a solution \(u\) to problem \((P)\), the authors study the set \(u^{-1}(0)\) and prove that, for almost every \(x\in u^{-1}(0)\), one has \(F(x,0)=0\), which means that if a solution \(u\) exists, then \(F(x,u)\) cannot be singular in a set of positive measure. Finally, the authors extend some existence, stability and uniqueness results established for problem \((P)\) to the case in which a term \(\mu u\) is added in the left-hand side of the equation, where \(\mu\) is a nonnegative finite Radon measure belonging to \(H^{-1}(\Omega)\).
scientific article; zbMATH DE number 6983311
scientific article