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Comparison and regularity results for a nonlinear elliptic equation. (English) Zbl 0786.35008

The following nonlinear elliptic problem is considered: \[ -(a_ i(x,u,\nabla u))_{x_ i}-(b_ i(x) | u |^{p-2}u)_{x_ i}+h(x,u)=(f_ i)_{x_ i} \quad \text{in } \Omega,\;u \in W_ 0^{1,p} (\Omega). \tag{*} \] The authors first compare a solution \(u\) of \((*)\) with a solution \(v(x)\) of a spherically symmetric elliptic problem whose structure depends on the hypotheses on the coefficients. Since \(v(x)\) can be calculated explicitly, it is possible to obtain a priori bounds for \(u\). Precisely the authors prove that, if the coefficients \(b_ i(x)\) are in the intermediate Lorentz spaces \(L(n/(p-1),r/(p-1))\) with \(n \leq r<\infty\) and \(f_ i(x)\) belonging to \(L(q,k/(p-1))\) with \(p' \leq q<n/(p-1)\) and \(p \leq k \leq q(p-1)\), then the solution \(u\) is in \(L(s,k)\). Furthermore the authors consider the limit case \(b_ i(x)\in L(n/(p-1),\infty)\): they show that \(u\) still belongs to \(L(s,k)\), provided \(q\) is less than a critical value depending on the norm of the coefficients \(b_ i(x)\). Similar results hold for problem \((*)\) with the left side equal to \((f_ i)_{x_ i}+g(x)\) with \(g\) in a suitable Lorentz space.

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35B45 A priori estimates in context of PDEs
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
35B65 Smoothness and regularity of solutions to PDEs
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