An application of Ricceri theorem in solving boundary value problems (Q6060276)

The author considers the coupled system \(-\operatorname{div}(h_{1}(x)\nabla u)=\sigma _{2}(x)F_{2u}(x,u,v)+\varsigma _{2}(x)G_{2u}(x,u,v)+K_{2u}(x,u,v)\), \( -\operatorname{div}(h_{2}(x)\nabla v)=\sigma _{2}(x)F_{2v}(x,u,v)+\varsigma _{2}(x)G_{2v}(x,u,v)+K_{2v}(x,u,v)\), posed in a bounded domain \(\Omega \sqsubseteq \mathbb{R}^{n}\) (\(n\geq 2\)) with smooth boundary \(\partial \Omega \), completed with the homogeneous boundary conditions \(u=v=0\) on \( \partial \Omega \). The functions \(F_{2},G_{2},K_{2}\) are supposed to belong to the set \(\mathcal{A}_{2}\) of \(C^{2}\) functions \(A:\Omega \times \mathbb{R} ^{2}\rightarrow \mathbb{R}\) which satisfy \(\left\vert A_{t}(x,s,t)\right\vert \leq c_{1}\left\vert s\right\vert ^{\gamma }\left\vert t\right\vert ^{\delta +1}\), \(\left\vert A_{s}(x,s,t)\right\vert \leq c_{2}\left\vert s\right\vert ^{\gamma +1}\left\vert t\right\vert ^{\delta }\), for all \(x\in \Omega \), \(s,t\in \mathbb{R}\), for positive constants \(c_{1}\) and \(c_{2}\) and parameters \(\gamma ,\delta \) which satisfy appropriate inequalities involving further parameters \(p,q\).\ These functions also satisfy the conditions: \(K_{2}(x,0,0)=0\) for all \(x\in \Omega \); there exists \(a\in (0,\lambda _{1}\theta /2)\) such that \(K_{2}(x,s,t)\leq c_{2}\left\vert s\right\vert ^{\gamma +1}\left\vert t\right\vert ^{\delta +1} \), for all \(x\in \Omega \), \(s,t\in \mathbb{R}\), where \[ \lambda _{1}=\inf_{(u,v)\in X\setminus \{0,0\}}\frac{\int_{\Omega }\frac{\gamma +1}{p} h_{1}(x)\left\vert \nabla u\right\vert ^{2}+\frac{\delta +1}{q} h_{2}(x)\left\vert \nabla v\right\vert ^{2}dx}{\int_{\Omega }\left\vert u\right\vert ^{\gamma +1}\left\vert v\right\vert ^{\delta +1}dx}, \] with \( X=H_{0}^{1}(\Omega ,h_{1})\times H_{0}^{1}(\Omega ,h_{2})\); \( \lim_{s^{2}+t^{2}\rightarrow +\infty }\frac{\sup_{x\in \Omega }(\left\vert F_{1}(x,s,t)\right\vert +\left\vert G_{1}(x,s,t)\right\vert )}{s^{2}+t^{2}}=0 \); One has: \[ \lim_{s^{2}+t^{2}\rightarrow +\infty }\frac{\sup_{x\in \Omega }(\left\vert F_{2}(x,s,t)\right\vert +\left\vert G_{2}(x,s,t)\right\vert )}{ s^{2}+t^{2}}=0; \] One has \[ meas(\{x\in \Omega :\left\vert F_{1}(x,0,0)\right\vert ^{2}+\left\vert G_{1}(x,0,0)\right\vert ^{2}>0\})>0 \] and \[ \left\vert F_{1}(x,0,0)\right\vert ^{2}+\left\vert G_{1}(x,0,0)\right\vert ^{2}\leq \left\vert F_{1}(x,s,t)\right\vert ^{2}+\left\vert G_{1}(x,s,t)\right\vert ^{2},\text{ for all }x\in \Omega, \ s,t\in \mathbb{R}; \] One has \[ meas(\{x\in \Omega :\inf_{(s,t)\in \mathbb{R} ^{2}}(\left\vert F_{1}(x,0,0)\right\vert F_{1}(x,s,t)+\left\vert G_{1}(x,0,0)\right\vert G_{1}(x,s,t))<\left\vert F_{1}(x,0,0)\right\vert ^{2}+\left\vert G_{1}(x,0,0)\right\vert ^{2}\})>0. \] The function \( h_{1}:\Omega \rightarrow \lbrack 0,+\infty )\) belongs to \(L_{\mathrm{loc}}^{1}(\Omega )\) and there exists a positive constant \(\alpha \) such that \(\liminf_{x\rightarrow z}\left\vert x-z\right\vert ^{-\alpha }h_{1}(x)>0\) for all \(z\in \Omega \). The function \(h_{1}:\Omega \rightarrow \lbrack 0,+\infty )\) belongs to \(L_{\mathrm{loc}}^{1}(\Omega )\) and there exists a positive constant \( \beta \) such that \(\liminf_{x\rightarrow z}\left\vert x-z\right\vert ^{-\beta }h_{2}(x)>0\) for all \(z\in \Omega \). The main result proves that, for every convex set \(S_{2}\sqsubseteq L^{\infty }(\Omega )\times L^{\infty }(\Omega )\) which is dense in \(L^{2}(\Omega )\times L^{2}(\Omega )\), there exists \( (\sigma _{2},\varsigma _{2})\in S_{2}\) such that the above problem has at least three weak solutions, two of which are global minima in \( H_{0}^{1}(\Omega )\times H_{0}^{1}(\Omega )\) of the functional \( (u,v)\rightarrow \frac{1}{2}\int_{\Omega }h_{1}(x)\left\vert \nabla u\right\vert ^{2}dx+\frac{1}{2}\int_{\Omega }h_{2}(x)\left\vert \nabla v\right\vert ^{2}dx-\int_{\Omega }(\sigma _{2}(x)F_{2}(x,u,v)+\varsigma _{2}(x)G_{2}(x,u,v)+K_{2}(x,u(x),v(x)))dx\). For the proof, the author applies an existence result obtained by \textit{B. Ricceri} in [Stud. Univ. Babes-Bolyai, Math. 66, No. 1, 75-84 (2021; Zbl 1513.35221)] for some general functionals.