Adisak Seesanea

The paper is devoted to the existence problem for positive solutions <!-- MATH ${u \in L^{r}(\mathbb{R}^{n})}$ --> , <!-- MATH $0<r<\infty$ --> <img width="98" height="37" align="MIDDLE" border="0" src="images/img2.gif" alt="$ 0<r<\infty $">, to the quasilinear elliptic equation

We give necessary and sufficient conditions for the existence of a positive solution with zero boundary values to the elliptic equation \[ \mathcal{L}u = \sigma u^{q} + \mu \quad \text{in} \;\; \Omega, \] in the sublinear case $0<q<1$, with finite generalized energy: $\mathbb{E}_{\gamma}[u]:=\int_{\Omega} |\nabla u|^{2} u^{\gamma-1}dx<\infty$, for $\gamma >0$. In this case $u \in L^{\gamma+q}(\Omega, \sigma)\cap L^{\gamma}(\Omega, \mu)$, where $\gamma=1$ corresponds to finite energy solutions. Here $\mathcal{L} u:= -\,\text{div}(\mathcal{A}\nabla u)$ is a linear uniformly elliptic operator with bounded measurable coefficients, and $\sigma$, $\mu$ are nonnegative functions (or Radon measures), on an arbitrary domain $\Omega\subseteq \mathbb{R}^n$ which possesses a positive Green function associated with $\mathcal{L}$. When $0<\gamma\leq 1$, this result yields sufficient conditions for the existence of a positive solution to the above problem which belongs to the Dirichlet space $\dot{W}_{0}^{1,p}(\Omega)$ for $1<p\leq 2$.

We study the existence problem for positive solutions $u \in L^{r}(\mathbb{R}^{n})$, $0<r<\infty$, to the quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n \] in the sub-natural growth case $0<q< p-1$, where $\Delta_{p}u = \text{div}( |\nabla u|^{p-2} \nabla u )$ is the $p$-Laplacian with $1<p<\infty$, and $\sigma$ is a nonnegative measurable function (or measure) on $\mathbb{R}^n$. Our techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators such as the $\mathcal{A}$-Laplacian $\text{div} \mathcal{A}(x,\nabla u)$, and the fractional Laplacian $(-\Delta)^{\alpha}$ on $\mathbb{R}^n$, as well as linear uniformly elliptic operators with bounded measurable coefficients $\text{div}(\mathcal{A} \nabla u)$ on an arbitrary domain $\Omega \subseteq \mathbb{R}^n$ with a positive Green function.