×

Existence and multiplicity results for semilinear equations with measure data. (English) Zbl 1136.35044

The authors study the elliptic problem
\[ -\Delta u =g(x,u)+\mu \text{ in } \Omega\subset \mathbb{R}^n,\quad n>2,\qquad u=0\text{ on } \partial\Omega, \]
where \(\mu\) is from the space \(\mathcal{M}(\Omega)\) of Radon measures and \(g:\Omega\times \mathbb{R}\to \mathbb{R}\) is a Carathéodory function. A solution is defined in a very weak form as a function \(u\in L^1(\Omega)\) with \(g(x,u)\in L^1(\Omega)\) satisfying \[ \int_\Omega-u\Delta \varphi \,dx= \int_\Omega g(x,u) \varphi \,dx+\int_\Omega \varphi \,d\mu\;\forall \varphi\in C_0^2(\overline \Omega), \]
where \(C_0^2(\overline \Omega)\) denotes the space of functions from \(C^2(\overline \Omega)\) which vanish on \(\partial\Omega\). Assuming that \(g(x,s)\) has a linear asymptotic behaviour as \(| s| \to \infty\) the existence of a solution \(u\in W_0^{1,q}(\Omega),\;q<n/(n-1),\) is proved. The second problem has the form
\[ -\Delta u =g(u)+\varepsilon\mu \text{ in } \Omega,\quad u\geq 0 \text{ in } \Omega,\quad u=0\text{ on } \partial\Omega,\quad \varepsilon>0. \]
Standard arguments from the critical point theory are used in the proofs. In order to find solutions, suitable functionals are introduced by means of approximation arguments and iterative schemes.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35J15 Second-order elliptic equations
PDFBibTeX XMLCite