A flower-shape geometry and nonlinear problems on strip-like domains
Let \(d,m\in \mathbb{N}\), with \(d\geq m+2\), let \(\omega\subset \mathbb{R}^d\) be an open smooth bounded set, and let \(\alpha \in L^1(\omega \times \mathbb{R}^{d-m})\cap L^\infty(\omega \times \mathbb{R}^{d-m})\) be a nonnegative function, radially symmetric in the second variable and such that \(\operatorname{ess}\inf_{\omega \times B(0,r)}\alpha>0\), for some \(r>0\). Finally, let \(f:\mathbb{R}\rightarrow \mathbb{R}\) be a continuous function satisfying \(\lim_{t\rightarrow 0}\frac{f(t)}{t}=0\) and the subcritical growth condition \(\sup_{t\in \mathbb{R}\setminus \{0\}}|f(t)|(|t|+|t|^q)^{-1}<+\infty\), where \(2<q< 2d/(d-2)\). The authors consider the elliptic problem \[ \left\{\begin{array}{ll} -\Delta u=\lambda \alpha(x,u)f(u), \ \ &\text{in} \ \ \omega \times \mathbb{R}^d,\\ u=0, &\text{on} \ \ \partial\omega \times \mathbb{R}^d, \end{array}\right. \] where \(\lambda\) is a real parameter, and prove that for each \(\lambda>0\), this problem admits a solution \(u_\lambda\) with cylindric symmetry, provided that \(f\) satisfies the superlinear Ambrosetti-Rabinowitz growth condition. Moreover, if, in addiction, \(d=m+4\) or \(d\geq m+6\) and \(f\) is odd, the existence of \(s_{d,m}\)-sequences of solutions with different symmetry is also proved, where \(s_{d,m}\) is an explicitly computed number. The authors also consider the case in which \(f\) is sublinear at infinity and, for some \(t_0>0\), satisfies \(F(t):=\int_0^t f(\tau)d\tau\geq 0\), for each \(t\in[0,t_0]\), and \(F(t_0)>0\). In this case, the authors establish a nonexistence result for \(\lambda>0\) small, and a multiplicity result for \(\lambda>0\) large. In particular, if, in addiction, \(f\) is odd, then, for \(\lambda>0\) large, the authors prove that there exist \(s_{d,m}\)-pairs of solutions with different symmetry. Finally, under the same conditions on \(f\), the authors establish a result similar to the previous ones for the following perturbation of the above problem \[ \left\{\begin{array}{ll} -\Delta u=\lambda \alpha(x,u)f(u)+\mu \beta(x,u)g(u), \ \ &\text{in} \ \ \omega \times \mathbb{R}^d,\\ u=0, &\text{on} \ \ \partial\omega \times \mathbb{R}^d, \end{array}\right. \] where \(\beta\in L^1(\omega \times \mathbb{R}^{d-m})\cap L^\infty(\omega \times \mathbb{R}^{d-m})\) is radially symmetric in the second variable, \(g:\mathbb{R}\rightarrow \mathbb{R}\) is a continuous function with subcritical growth, and \(\mu\) is a real parameter. The authors define suitable subgroups of the orthogonal group \(O(d-m)\) of \(\mathbb{R}^{d-m}\) to get mutually disjoint subspaces of partially symmetric functions of \(H^1_0(\omega\times\mathbb{R}^{d-m})\), and then use variational methods to prove their main results.
semilinear equation with Laplace operator
scientific article; zbMATH DE number 7388783
scientific article