We give necessary and sufficient conditions for the existence of weak solutions to the model equation
in the case
science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303147&_mathId=si2.gif&_user=111111111&_pii=S0022123616303147&_rdoc=1&_issn=00221236&md5=e0458b381fecf1c55353d8aa65959917" title="Click to view the MathML source">0<q<p−1, where
science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303147&_mathId=si3.gif&_user=111111111&_pii=S0022123616303147&_rdoc=1&_issn=00221236&md5=a5334407f8dcce92e6e7c68e450d8e92" title="Click to view the MathML source">σ≥0 is an arbitrary locally integrable function, or measure, and
science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303147&_mathId=si4.gif&_user=111111111&_pii=S0022123616303147&_rdoc=1&_issn=00221236&md5=9db76aa8285bbaf072e1f604b760d686" title="Click to view the MathML source">Δpu=div(∇u|∇u|p−2) is the
p-Laplacian. Sharp global pointwise estimates and regularity properties of solutions are obtained as well. As a consequence, we characterize the solvability of the equation
where
science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303147&_mathId=si111.gif&_user=111111111&_pii=S0022123616303147&_rdoc=1&_issn=00221236&md5=936a4155290d2e61ef833a386d73699e" title="Click to view the MathML source">b>0. These results are new even in the classical case
science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303147&_mathId=si7.gif&_user=111111111&_pii=S0022123616303147&_rdoc=1&_issn=00221236&md5=a97937f582932b0db9f705160ec78581" title="Click to view the MathML source">p=2.
Our approach is based on the use of special nonlinear potentials of Wolff type adapted for “sublinear” problems, and related integral inequalities. It allows us to treat simultaneously several problems of this type, such as equations with general quasilinear operators science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303147&_mathId=si771.gif&_user=111111111&_pii=S0022123616303147&_rdoc=1&_issn=00221236&md5=4eb424e4db98e43fbc457300128761b7">
script>
script>, fractional Laplacians science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616303147&_mathId=si9.gif&_user=111111111&_pii=S0022123616303147&_rdoc=1&_issn=00221236&md5=8b22f0d74ab99d4c02fd2749739d359b" title="Click to view the MathML source">(−Δ)α, or fully nonlinear k-Hessian operators.