Regularizing effect of the interplay between coefficients in some noncoercive integral functionals

We are interested in regularizing effect of the interplay between the coefficient of zero order term and the datum in some noncoercive integral functionals of the type

\(\cal{J} (v)= \int_\Omega j(x,v,\nabla v)\, {\rm d}x +\int_\Omega a(x) \vert v\vert^{2} \, {\rm d} x -\int_\Omega fv \, {\rm d}x, \quad v\in W^{1,2}_{0}(\Omega),\) $$\mathcal{J}(v)={\int }_{\mathrm{\Omega }}j(x,v,\mathrm{\nabla }v)\mathrm{d}x+{\int }_{\mathrm{\Omega }}a(x)\mid v{\mid }^2\mathrm{d}x-{\int }_{\mathrm{\Omega }}fv\mathrm{d}x,v\in W_0^{1,2}(\mathrm{\Omega }),$$

where Ω ⊂ ℝ^N, j is a Carathéodory function such that ξ ↦ j(x, s, ξ) is convex, and there exist constants 0 ⩽ τ < 1 and M > 0 such that

\({\vert\xi\vert^{2}}{\over{{(1+\vert s\vert)^{\tau}}}}\leqslant j(x,s,\xi)\leqslant M\vert\xi\vert^2\) $$\mid \xi {\mid }^2\frac{}{(1+\mid s\mid {)}^{\tau }}⩽j(x,s,\xi )⩽M\mid \xi {\mid }^2$$

for almost all x ∈ Ω, all s ∈ ℝ and all ξ ∈ ℝ^N. We show that, even if 0 < a(x) and f(x) only belong to L¹(Ω), the interplay

\(\vert f(x)\vert\leqslant2 Qa(x)\) $$\mid f(x)\mid ⩽2Qa(x)$$

implies the existence of a minimizer u ∈ W ₀ ^1,2 (Ω) which belongs to L^∞(Ω).