Some bifurcation results and multiple solutions for the \(p\)-Laplacian equation
In this paper, the authors first give some bifurcation results near the origin for the following quasilinear elliptic equation \[ \begin{cases} -\Delta_pu = \lambda|u|^{p-2}u + f(x, u) & \mbox{in}\;\; \Omega\,, \cr u = 0 & \mbox{on}\;\; \partial \Omega\,, \cr \end{cases} \eqno{(P)} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(\Delta_p u=\) \(\mbox{div}(|\bigtriangledown u|^{p-2}\bigtriangledown u)\) with \(1 < p < \infty\), and \(\lambda \in\mathbb{R}\) is a parameter. Then they got multiple solutions to this problem. To obtain these results, the authors assume on \(f\): (\(f_1\)) \(f \in C(\Omega\times\mathbb{R},\mathbb{R}), f(x, 0) = 0\) and \[ \lim_{|t|\to 0} \frac{f(x,t)}{|t|^{p-2}t}= 0\,,\;\;\mbox{for}\;\; x\in \Omega\,; \] (\(f_2\)) there are \(c > 0\) and \(\gamma\in [1, p^\ast)\) such that \[ |f(x,t)| \le c\,(1 + |t|^{\gamma-1})\,,\;\;\mbox{for}\;\; x\in\Omega \;\;\mbox{and}\;\; t \in\mathbb{R}\,, \] where \(p^\ast = Np/(N-p)\) if \(p < N\) and \(p^\ast = \infty\) if \(N \le p\); (\(f_3\)) there exist constants \(\theta> p\) and \(M > 0\) such that \[ 0 < \theta F(x,t) := \theta \int^t_0 f(x, s)\,ds \le f(x,t)t\,\;\;\mbox{for}\;\; |t| \ge M\,,\; x\in\Omega\,; \] (\(f^-\)) there exists \(\alpha > 0\) such that \[ tf (x,t) < 0\,,\;\;\mbox{for}\;\; 0 < |t| \le \alpha\,, \;\; x\in \Omega\,. \] Let \(\lambda_1\) be the first eigenvalue of \(-\Delta_p\) in the space \(W^{1,p}_0 (\Omega)\). The authors are interested in the existence and multiplicity of nontrivial solutions of problem (P) when \(\lambda\) near \(\lambda_1\). The first result of this article is the following theorem. Theorem 1.2. Suppose that (\(f_1\)) and (\(f^-\)) hold, then there exists \(\delta > 0\) such that for \(\lambda\in (\lambda_1, \lambda_1 + \delta)\), problem (P) has at least two nontrivial solutions. Next, the authors aim to give some multiplicity results on existence of nontrivial solutions of problem (P) mainly under additional assumptions (\(f_3\)). Theorem 1.4. Suppose that (\(f_1\))--(\(f_3\)) and (\(f^-\)) hold, then there exists \(\delta > 0\) such that for \(\lambda\in (\lambda_1, \lambda_1 + \delta)\) and \[ \sup_{(x,u)\in\Omega\times\mathbb{R}}\max \{-F(x, u), 0\} < \delta\,, \] problem (P) has at least three nontrivial solutions. The authors apply Morse theory to prove the main theorems by computing and comparing critical groups of associated energy functional at isolated critical points. Specifically, their proofs are based on the fact that Gromoll-Meyer pairs for an isolated invariant set are stable under small perturbation. The solutions are constructed by a combination of bifurcation arguments and topological linking methods.
scientific article; zbMATH DE number 7474118
scientific article