Existence and regularity of multiple solutions for infinitely degenerate nonlinear elliptic equations with singular potential
This paper is devoted to the study of the existence and regularity of solution to the following semilinear infinitely degenerate elliptic equation \[ \begin{cases} -\Delta_X u-\varepsilon V_n u= au \log |u|+b u+ g(x,u) &\text{in } \Omega, \\ u=0&\text{on } \partial\Omega, \end{cases}\eqno{(1.1)} \] where vector fields \(X=(\partial_{x_1}, \dots , \partial_{x_{n-1}}, \varphi(x')\partial_{x_n})=(X_1,\dots, X_n)\) defined on an open domain \(\widetilde{\Omega}\subset\mathbb R^n\) for \(n \geq 3\), \(\Omega\) is a bounded open subset in \(\widetilde{\Omega}\) which contains the origin, \(a>0, b>0; \varphi(x')\) is a non-negative \(C^\infty\)-smooth function in \(x'=(x_1, x_2, \dots , x_{n-1})\) and for \(\Gamma \subset \widetilde{\Omega}, \partial^\alpha_{x'} \varphi(x')|_{(x',x_n)\in \Gamma} =0\) for any \(|\alpha| \geq 0; g(x, u)\) is a Carathéodory function with primitive \(G(x, u) =\int_0^u g(x,v)\, dv\), which satisfies the following conditions: {\parindent=8mm \begin{itemize} \item[(\(\mathit{f}_1\))] \(g(x, u) =-g(x,-u)\), \item[(\(\mathit{f}_2\))] there exist \(q\in (1, 2)\), and \(c>0\), such that \(|g(x, u)| \leq c(1 +|u|^{q-1})\), \item[(\(\mathit{f}_3\))] if \(u \in L^2(\Omega )\) and \(u\not=0\), then \(\lim_{t \to 0^+}\frac{\int_\Omega G(x,tu)\, dx}{t^2 \log t}=-\infty \), \item[(\(\mathit{f}_4\))] there exists \(\varepsilon_0>0\), such that for \(0 \leq u \leq\varepsilon_0\), we have \(g(x,u)+au \log |u| +bu \geq 0\), \item[(\(\mathit{f}_5\))] \(g(x, u)\) is \(C^\infty\) in \(x\), and \(C^\infty\) in \(u\) except \(u =0\). \end{itemize}} The authors suppose that the vector fields \(X\) satisfies the following logarithmic regularity estimates \[ \|(\log\Lambda)^su \|^2_{L^2(\Omega)} \leq C_0 \Bigl[ \int_\Omega | Xu |^2 \,dx + \|u\|^2_{L^2(\Omega)}\Bigr] \eqno(1.2) \] for all \(u \in C^\infty_0(\widetilde{\Omega})\), where \(\Lambda =(e^2+|D|^2)^{\frac{1}{2}}=\langle D\rangle\). The potential term \(V_n(x) \geq 0\) satisfies the following Hardy's inequality \[ \int_\Omega V_n u^2 \,dx \leq \int_\Omega |Xu|^2 \, dx, \quad \text{for all} \; u \in H^1_{X,0}(\Omega ), \eqno(1.3) \] where \(H^1_{X,0}(\Omega )\) is a Hilbert space defined as a closure of \(C^\infty_0(\Omega)\) in the space \[ H^1_X(\widetilde{\Omega}) = \{u \in L^2(\widetilde{\Omega}) \bigm| X_j u \in L^2( \widetilde \Omega ), \; j = 1, \dots ,n\} \] with norm \(\|u\|^2_{H^1_X}=\|u\|^2_{L^2}+\|X u\|^2_{L^2}\), and \(\|X u\|^2_{L^2}=\sum^n_{j=1} \|X_j u\|^2_{L^2}\). Let \[ M(q,\Omega ) = \frac{(1-\varepsilon)\eta_1}{4} \Bigl(\frac{1}{q} C^{2-q} + C\Bigr)^{-1}, \eqno(1.4) \] where \(C=|\Omega | e^{C_1/a}, C_1=\frac{C_0a^2}{2(1-\varepsilon)}+\frac{(1-\varepsilon)}{2}+b-\frac{a}{2}, |\Omega |\) is the Lebesgue measure of \(\Omega , C_0\) is the positive constant appeared in (1.2) and \(\eta_1>0\) is the first eigenvalue of the operator \(-\Delta_X\). In this paper authors need the following hypothesis: {\parindent=8mm \begin{itemize}\item[(H-1)] \(\partial \Omega\) is \(C^\infty\) and non-characteristic for the system of vector fields \(X\); \item[(H-2)] \(X\) satisfies the finite type of Hörmander's condition with Hörmander index \(Q\) on \(\widetilde{\Omega}\) except a union of \(\qquad\) smooth surfaces \(\Gamma\) which are non-characteristic for \(X\); \item[(H-3)] \(X\) satisfies logarithmic regularity estimate (1.2) with \(s\geq \frac{3}{2}\). \item[(H-4)] The non-negative singular potential function \(V_n(x)\) is \(C^\infty(\Omega \setminus\Gamma_1)\), here \(\Gamma_1\subseteq\Gamma\) is the set on which \(\qquad V_n(x)\) is unbounded, and satisfies the inequality (1.3). \end{itemize}} Then authors introduce the energy functional \(J: H^1_{X,0}(\Omega )\to\mathbb R\) \[ \begin{multlined} J(u)\!=\!\frac{1}{2} \Bigl( \int_\Omega \! \sum_{j=1}^n (X_j u)^2 dx - \varepsilon \!\int_\Omega \! V_n u^2 dx - \!\int_\Omega\! au^2 \log|u| dx \\ + \!\int_\Omega\! \frac{au^2}{2} dx - \!\int_\Omega\! bu^2 dx\Bigr) -\!\int_\Omega\! G(x,u)dx \end{multlined} \] and prove the following main results. Theorem 1.1. Under the conditions (H-1), (H-2), (H-3) and (H-4), if \(0 <\varepsilon<1\), and \(0 \leq\varphi(x') \leq 1\), then we have {\parindent=6mm \begin{itemize}\item[(1)] If \(g(x, u)\) satisfies (\(f_2\)) with \(c\in (0, M(q, \Omega ))\), then the semi-linear Dirichlet problem (1.1) possesses at least one nonzero weak solution \(u_\varepsilon\) in \(H^1_{X,0}(\Omega )\) with \(J(u_\varepsilon) >0\). \item[(2)] If \(g(x, u)\) satisfies (\(f_2\)) with \(c\in (0, M(q, \Omega ))\) and (\(f_3\)), then the semi-linear Dirichlet problem (1.1) possesses at least two nonzero weak solutions \(u^1_\varepsilon\) and \(u^2_\varepsilon\) in \(H^1_{X,0}(\Omega )\) with \(J(u^1_\varepsilon) >0\) and \(J(u^2_\varepsilon) <0\). \item[(3)] If \(g(x, u)\) satisfies (\(f_1\)), (\(f_2\)) with \(c\in (0, M(q, \Omega ))\), then the semi-linear Dirichlet problem (1.1) possesses infinitely weak solutions \(u^k_\varepsilon\) in \(H^1_{X,0}(\Omega )\) for \(k=1, 2, \dots\). \end{itemize}} Theorem 1.2. Under the conditions (H-1), (H-2), (H-3) and (H-4), if \(0 <\varepsilon<1\), and \(0 \leq\varphi(x') \leq 1\), then we have {\parindent=6mm \begin{itemize}\item[(1)] If \(g(x, u)\) satisfies (\(f_2\)), \(u_\varepsilon\in H^1_{X,0}(\Omega )\), \(u_\varepsilon\geq 0\) and \(\|u_\varepsilon\|_{L^2(\Omega )}\not=0\), is a weak solution of (1.1), then for \(1 <p<\frac{1+\sqrt{1-\varepsilon}}{\varepsilon}\), one has \(u_\varepsilon\in L^{2p}(\Omega )\). \item[(2)] If \(\varepsilon\in (0, \frac{4}{(Q+n-1)}(1-\frac{1}{(Q+n-1)} )), g(x, u)\) satisfies \text{(\(f_2\)), (\(f_4\))} and \text{(\(f_5\))}, \(u_\varepsilon\in H^1_{X,0}(\Omega ), u_\varepsilon\geq 0\), and \(\|u_\varepsilon\|_{L^2(\Omega )}\not=0\) is a weak solution of (1.1), then \(u_\varepsilon\in C^\infty(\Omega \setminus\Gamma ) \cap C^0(\overline{\Omega} \setminus\Gamma )\) and \(u_\varepsilon(x) >0\) for all \(x\in \Omega \setminus\Gamma \). \end{itemize}} The authors use the symmetric Mountain Pass theorem and minmax-method to prove the existence of multiple solutions to the problem (1.1).
infinitely degenerate elliptic equations
scientific article; zbMATH DE number 6343733
scientific article