Existence of renormalized solutions to nonlinear elliptic equations with two lower order terms and measure data

Abstract

In this paper we prove the existence of a renormalized solution to a class of nonlinear elliptic problems whose prototype is(P){−△pu−div⁡(c(x)|u|γ)+b(x)|∇u|λ=μamp;in Ω,u=0amp;in ∂Ω,\begin{equation}\tag {P} \begin {cases} - \bigtriangleup _p u -\operatorname {div}(c(x)|u|^{\gamma })+b(x)|\nabla u|^{\lambda } =\mu & \text {in $\Omega $},\\ u=0 & \text {in $\partial \Omega $}, \end{cases} \end{equation}whereΩ\Omegais a bounded open subset ofRN\mathbb {R}^N,N≥2N\geq 2,△p\bigtriangleup _pis the so-calledp−p-Laplace operator,1>p>N1> p> N,μ\muis a Radon measure with bounded variation onΩ\Omega,0≤γ≤p−10\le \gamma \le p-1,0≤λ≤p−10\le \lambda \le p-1, and|c||c|andbbbelong to the Lorentz spacesLNp−1,r(Ω)L^{\frac {N}{p-1},r}(\Omega ),Np−1≤r≤+∞\frac {N}{p-1}\leq r \leq +\infty, andLN,1(Ω)L^{N,1}(\Omega ), respectively. In particular we prove the existence under the assumptions thatγ=λ=p−1\gamma =\lambda =p-1,|c||c|belongs to the Lorentz spaceLNp−1,r(Ω)L^{\frac {N}{p-1},r}(\Omega ),Np−1≤r>+∞\frac {N}{p-1}\leq r>+\infty, and‖c‖LNp−1,r(Ω)\|c\|_{ L^{\frac {N}{p-1},r}(\Omega )}is small enough.