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Existence results for nonlinear elliptic equations with degenerate coercivity. (English) Zbl 1105.35040

It is proved the existence of (at least) one solution in \(W^{1,p}_0(\Omega) \cap L^\infty(\Omega)\) for some nonlinear elliptic equations, whose model is \[ -\text{div}\left(|\nabla u|^{p-2}\nabla u \over{(1+|u|)^{\theta(p-1)}}\right)=f\quad\text{in } \Omega, \] with homogeneous Dirichlet boundary conditions, where \(\Omega\) is a bounded open subset of \(\mathbb R^N\), \(N \geq 2\), \(p>1\), \(\theta \in [0,1]\) and \(f \in L^m(\Omega)\) with \(m > N/p\). Let us point out that the differential operator may not be coercive on \(W^{1,p}_0(\Omega).\) Due to this lack of coercivity, standard existence theorems for solutions of nonlinear elliptic equations cannot be applied.
If the summability of \(f\) is decreased (i.e. \(\tilde{m}< m < N/p\), for some \(\tilde{m}\)) and \(\theta < 1\), the existence of solution is obtained in \(W^{1,p}_0(\Omega) \cap L^s(\Omega)\) for \(s = Nm (p-1)(1-\theta)/(N-mp)\).

MSC:

35J70 Degenerate elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
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