Construction of pseudo-gradient vector field and sign-changing multiple solutions involving \(p\)-Laplacian
In the present study the authors consider the existence of multiple and sign-changing solutions of the problem \[ -\Delta_p u=f(u)\quad \text{in }\Omega, \qquad u(x)=0,\quad \text{on }\partial\Omega, \tag{1} \] where \(-\Delta_p u:= \text{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian, \(1< p<+\infty\), \(\Omega\) is a smooth bounded domain in \(\mathbb R^N\) \((N\geq 1)\), \(f(u)\) is the local Lipschitz continuous having ``jumping'' nonlinearities at zero or infinity. To this end, the authors carefully construct a pseudo-gradient vector field in \(C_0^1 (\overline{\Omega})\), by which they obtain the positive and negative cones in \(C_0^1 (\overline{\Omega})\), which are all the invariant sets of the descent flow of the corresponding functional and use (PS) condition.
scientific article; zbMATH DE number 2085854