Multiple solutions for quasilinear elliptic equations in unbounded cylinder domains (Q424593)

Let \(x=(y,z) \in {\mathbb R}^m \times {\mathbb R}^n = {\mathbb R}^N\), \(N=m+n\geq 3\), \(m\geq 0\), \(n\geq 1\), \(2\leq p<N\), \(2\leq p<q<p^*:=NP/(N-p)\). Then the authors concern with the existence of solutions of the quasilinear elliptic equation NEWLINE\[NEWLINE \begin{cases} -\Delta _p u + | u | ^{p-2} u = Q(x) | u | ^{q-2} & \mathrm{in}\,\, \Omega ,\\ u \in W^{1,p}_0 (\Omega ), u \not\equiv 0, \end{cases} \tag{1.1} NEWLINE\]NEWLINE where \(\omega \subset {\mathbb R}^m\) is a bounded smooth domain, \(0\in \Omega := \omega \times {\mathbb R}^n\) is an unbounded cylinder domain, \(\Delta _p\) is the \(p\)-Laplacian: NEWLINE\[NEWLINE \Delta _p = \sum _{i=1}^N \frac{\partial }{\partial x_i} \biggl( | \nabla u | ^{p-2} \frac{\partial u}{\partial x_i } \biggr). NEWLINE\]NEWLINE Here \(Q=Q(x)\) is a positive bounded, continuous function on \(\overline{\Omega }\) satisfying NEWLINE\[NEWLINE Q(x)\geq Q_{\infty }>0 \mathrm{ in } \overline{\Omega}, Q(x) \not\equiv Q_{\infty }\mathrm{ and } \lim _ {| z | \to \infty }Q(x) = Q_{\infty } \mathrm{ uniformly for } y \in \overline{\omega }. \tag{A.1} NEWLINE\]NEWLINE They get the following theorems.NEWLINENEWLINE\textbf{Theorem 1.1.} Assume that \(N\geq 3, 2\leq p <N\) and \(Q(x)\) satisfies (A.1), then equation (1.1) possesses a positive ground state solution in \(\Omega \).NEWLINENEWLINE\textbf{Theorem 1.2.} Assume that \(N\geq 3, 2\leq p <N\) and \(Q(x)\) satisfies (A.1) and there exists positive constants \(\delta < \bigl(\frac{1+\lambda _1}{p-1}\bigr)^{1/p}, C_0 \) and \(R_0\) such that NEWLINE\[NEWLINE Q(x) \geq Q_{\infty } + C_0 e^{-\delta | z | } NEWLINE\]NEWLINE for \(| z | \geq R_0\) uniformly for \(y \in \overline{\omega }\) where \(\lambda _1\) is the first eigenvalue of the Derichlet problem \(-\Delta _p\) in \(\omega \). Then equation (1.1) possesses a nodal solution in \(\Omega \) in addition to a positive solution.NEWLINENEWLINE\textbf{Theorem 1.3.} Assume that \(\Omega = {\mathbb R}^N, N\geq 3, 2\leq p <N\) and \(Q(x)\) satisfies (A.1), then the equation (1.1) possesses a positive ground state solution in \({\mathbb R}^N\).NEWLINENEWLINE\textbf{Theorem 1.4.} Assume that \(\Omega = {\mathbb R}^N, N\geq 3, 2\leq p <N\) and \(Q(x)\) satisfies (A.1) and there exist positive constants \(\delta < \bigl( \frac{1}{p-1} \bigr) ^{1/p}\), \(C_0\) and \(R_0\) such that NEWLINE\[NEWLINE Q(x) \geq Q_{\infty } + C_0e^{-\delta | x | }\quad \mathrm{for }\,\, | x | \geq R_0. NEWLINE\]NEWLINE Then equation (1.1) possesses a nodal solution in \({\mathbb R}^N\) in addition to a positive solution.