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- W3210181370 abstract "We study the existence, multiplicity, and certain qualitative properties of solutions to the zero Dirichlet problem for the equation $-Delta_p u = lambda |u|^{p-2}u + a(x)|u|^{q-2}u$ in a bounded domain $Omega subset mathbb{R}^N$, where $1<q<p$, $lambdainmathbb{R}$, and $a$ is a continuous sign-changing weight function. Our primary interest concerns ground states and nonnegative solutions which are positive in ${xin Omega: a(x)>0}$, when the parameter $lambda$ lies in a neighborhood of the critical value $lambda^* = infleft{int_Omega |nabla u|^p , dx/int_Omega |u|^p , dx: uin W_0^{1,p}(Omega) setminus {0}, int_Omega a|u|^q,dx geq 0,right}$. Among main results, we show that if $p>2q$ and either $int_Omega avarphi_p^q,dx=0$ or $int_Omega avarphi_p^q,dx>0$ is sufficiently small, then such solutions do exist in a right neighborhood of $lambda^*$. Here $varphi_p$ is the first eigenfunction of the Dirichlet $p$-Laplacian in $Omega$. This existence phenomenon is of a purely subhomogeneous and nonlinear nature, since either in the superhomogeneous case $q>p$ or in the sublinear case $q<p=2$ the nonexistence takes place for any $lambda geq lambda^*$. Moreover, we prove that if $p>2q$ and $int_Omega avarphi_p^q,dx>0$ is sufficiently small, then there exist three nonzero nonnegative solutions in a left neighborhood of $lambda^*$, two of which are strictly positive in ${xin Omega: a(x)>0}$." @default.
- W3210181370 created "2021-11-08" @default.
- W3210181370 creator A5005785237 @default.
- W3210181370 creator A5040580526 @default.
- W3210181370 date "2021-10-22" @default.
- W3210181370 modified "2023-10-16" @default.
- W3210181370 title "On subhomogeneous indefinite $p$-Laplace equations in supercritical spectral interval" @default.
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- W3210181370 doi "https://doi.org/10.48550/arxiv.2110.11849" @default.
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