Pairs of sign-changing solutions for sublinear elliptic equations with Neumann boundary conditions (Q2877590)

The authors consider the Neumann problem for a semilinear elliptic equation NEWLINE\[NEWLINE \begin{cases} -\Delta u (x) = f(u(x)) & x \in \Omega ,\\ \frac{\partial u}{\partial n}\bigl| _{\partial \Omega }=0, & \end{cases}\tag{1} NEWLINE\]NEWLINE where \(\Omega \subset {\mathbb R}^N\) \((N\geq 1)\) is a convex and bounded domain with the smooth boundary \(\partial \Omega \) and \(n\) denotes the outward normal to the boundary, \(f(u) : {\mathbb R} \to {\mathbb R}\). The authors assume the followingNEWLINENEWLINE(F1) \(f\in C^{1}({\mathbb R} , {\mathbb R})\) is strictly increasing and \(f(0)=0\).NEWLINENEWLINE(F2) the limits \(f'(\pm \infty )= \lim _{u \to \pm \infty } f'(u)\) exist and \(0<f'(\pm \infty )< (\pi /d_{\Omega })^2\) where \(d_{\Omega } \) denotes the diameter of \(\Omega \).NEWLINENEWLINE(F3) \(F(u):= \int _0^u f(s)ds =F(-u)\) for all \(u\in {\mathbb R}\).NEWLINENEWLINE(F4) there exist \(p\in {\mathbb N}, M>0\) and \(\rho >0\) such that \(d_{\Omega }>2p \pi /\sqrt{M}, M> 4p^2 f'(\pm \infty )\) and NEWLINE\[NEWLINE F(u) \geq \frac12 M| u | ^2 ,\quad \forall | u | \leq \rho . NEWLINE\]NEWLINENEWLINENEWLINE(F5) for \(\Omega \) there are continuous functions \(e_1(x),\ldots , e_p(x) \in X_0 \setminus \{0\}\) where \(X_0= \{ u \in H^1(\Omega ); \int _{\Omega }u(x)dx=0\}\) which are orthogonal in \(H^1(\Omega )\) and \(L^2(\Omega )\), such that NEWLINE\[NEWLINE \int _{\Omega } | \nabla e_j(x) | ^2 dx \leq \left[ \frac{2(j+1)\pi }{d_{\Omega } } \right]^2 \int _{\Omega } | e_j(x) | ^2 dx, \forall 1\leq j \leq p. NEWLINE\]NEWLINENEWLINENEWLINEThen they get the following theorem.NEWLINENEWLINE{ Theorem 1.} Under the hypotheses (F1)-(F5), the equation (1) has \(p\) distinct pairs \((u(x), -u(x))\) of sign-changing classical solutions, and has no positive and negative solution, provided that \(d_{\Omega }\in (2p\pi /\sqrt{M}, \pi /\sqrt{f'(\pm \infty )})\).NEWLINENEWLINEAs an example, they apply the Theorem to \(B_r(0) \subset {\mathbb R}^2\).NEWLINENEWLINE{ Theorem 2.} Let \(f\) satisfy (F1)-(F4) with \(\Omega = \{(x_1,x_2)\in {\mathbb R}^2; x_1^2+ x_2^2< r^2\}\), then for \(r \in (2p\pi /\sqrt{M}, \pi /\sqrt{f'(\pm \infty )})\), the problem NEWLINE\[NEWLINE \begin{cases} -\Delta u (x) = f(u(x)), & x_1^2+x_2^2< r^2, \\ \frac{\partial u}{\partial n} \biggl| _{x_1^2+x_2^2 = r^2 }=0, & \end{cases}NEWLINE\]NEWLINE has \(p\)-distinct pairs \((u(x),-u(x))\) of sign-changing classical solutions.