Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials

Abstract

Given $N\ge 3$, $1<p<N$, two measurable functions $V\left(r\right)\ge 0$, $K\left(r\right)>0$ and a continuous function $A(r)>0$ ($r>0$), we study the quasilinear elliptic equation \[ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) u+V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}. \] We find existence of nonegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space $X$ into the sum of Lebesgue spaces $L_K^{q_1}+L_K^{q_2}$, and thus into $L_K^q$ ($=L_K^q+L_K^q$) as a particular case. Our results do not require any compatibility between how the potentials $A$, $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of $V$ and $K$, not of the potentials separately. The nonlinearity $f$ has a double-power behavior, whose standard example is $f(t)=\min \{t^{q_1-1},t^{q_2-1}\}$, recovering the usual case of a single-power behavior when $q_1=q_2$.

arXiv admin note: text overlap with arXiv:1609.05556, arXiv:1510.03879, arXiv:1403.3803