Existence and global behavior of positive solutions for some eigenvalue problems

Let \(f(t)=t^{p}+1\) for \(p\geq0\) and \(t\geq0\). Then we discuss three cases:

1. Case 1:

   \(p\in[0,1)\). In this case \(\beta=+\infty\) and the existence result for (1.1) holds for \(\lambda\in(0,\infty)\).

    
2. Case 2:

   \(p=1\). Then \(\beta=1\) and the existence result for (1.1) holds for \(\lambda\in(0,\frac{1}{\|Va\|_{\infty}})\).

    
3. Case 3:

   \(p>1\). In this case \(g(t)= \frac{t}{1+t^{p}}\) and \(\beta=g (\frac{1}{(p-1)^{\frac{1}{p}}} )=\frac{(p-1)^{1-\frac {1}{p}}}{p}\). So the existence result for (1.1) holds for \(\lambda\in(0,\frac {(p-1)^{1-\frac{1}{p}}}{p\|Va\|_{\infty}}]\).

    

Let Ω be a smooth domain (bounded or unbounded) with compact boundary or \(\Omega={\mathbb{R}}^{n}_{+}=\{x=(x_{1},x_{2},\ldots, x_{n}) \in{\mathbb{R}}^{n}: x_{n}>0\}\) (\(n\geq3\)) be the upper half space and let \(G(x,y)\) be the Green function of the Laplacian \((-\Delta)\) on Ω with Dirichlet boundary conditions. Assume that a is a nontrivial nonnegative Borel measurable function such that its Green potential \(Va(x)=\int_{\Omega}G(x,y)a(y)\,dy\) is continuous and satisfies \(\lim_{x\rightarrow\partial^{\infty} \Omega}Va(x)=0\), where \(\partial^{\infty} \Omega=\partial\Omega\) if Ω is bounded and \(\partial^{\infty}\Omega=\partial\Omega\cup\{\infty\}\) whenever Ω is unbounded. Then for each \(0<\lambda<\frac{1}{\|Va\|_{\infty }} \sup_{t>0} \frac{t}{f(t)}\), the problem has a positive continuous solution u satisfying for each \(x \in\Omega\)

Let m be a positive integer, \(\Omega=B(0,1)\) be the unit ball in \({\mathbb{R}}^{n}\) (\(n\geq2\)) and \(G_{m,n}(x,y)\) be the Green function of the polyharmonic Laplacian \((-\Delta)^{m}\) on \(B(0,1)\) with Dirichlet boundary conditions (see [11]). Let a be a nontrivial nonnegative function on \(B(0,1)\) such that its Green potential \(Va(x)= \int_{B(0,1)}G_{m,n}(x,y)a(y)\,dy\) is continuous on \(B(0,1)\) and \(\lim_{|x| \rightarrow1 }\frac {Va(x)}{(1-|x|)^{m-1}}=0\). Then, for each \(0<\lambda<\frac{1}{\|Va\|_{\infty}} \sup_{t>0}\frac {t}{f(t)}\), the problem has a positive continuous solution u satisfying for each \(x \in B(0,1)\)

For \(t>s\) and \(x,y\in{\mathbb{R}}^{n}_{+}\), we denote by the Green function of the heat operator \(\frac{\partial}{\partial t}-\Delta\) on \({\mathbb {R}}^{n}_{+}\times(0,\infty)\) with Dirichlet boundary conditions. Put \(P_{t}1(x)=P1(x,t)= \int_{{\mathbb{R}}^{n}_{+}} \Gamma(x,t,y,0)\,dy= \frac{1}{\sqrt{\pi t}} \int_{0}^{x_{n}}\exp(-\frac{\xi^{2}}{4t})\,d\xi\). Consider a nontrivial nonnegative measurable function a on \({\mathbb {R}}^{n}_{+}\times(0,\infty)\) such that the function \((x,t)\rightarrow \frac {a(x,t)}{P_{t}1(x)}\) belongs to the parabolic Kato class \(P^{\infty}({\mathbb{R}}^{n}_{+})\) introduced and studied in [12]. Then from [12] the function \((x,t)\rightarrow Va(x,t)= \int_{{\mathbb{R}}^{n}_{+}\times (0,\infty)} \Gamma(x,t,y,s)a(y,s) \,dy\,ds\) is continuous and bounded in \({\mathbb{R}}^{n}_{+}\times(0,\infty)\) and for each \((x,t)\in{\mathbb{R}}^{n}_{+}\times(0,\infty)\) we have Consequently, for \(0<\lambda< \frac{1}{\|Va\|_{\infty}} \sup_{t>0} \frac{t}{f(t)}\), the problem has a positive continuous solution u satisfying for each \((x,t)\in {\mathbb{R}}^{n}_{+}\times(0,\infty)\)

Let Ω be a smooth bounded of \({\mathbb{R}}^{n}\) (\(n\geq2\)), \(f:[0,\infty)\rightarrow(0,\infty)\) be a continuous function such that \(g(t)= \frac{t}{f(t)}\) is bounded and let \(a(x)= \frac{1}{ (\delta(x) )^{\lambda}}\) with \(\lambda<2\). Let \(\lambda_{1}>0\) be the first positive eigenvalue of the problem \(-\Delta u=\lambda a(x)u\) in Ω with Dirichlet boundary conditions. From [13], \(\lambda_{1}\) is a simple eigenvalue. Let \(\varphi_{1}\) be the positive normalized (\(\|\varphi_{1}\|_{\infty}=1\)) eigenfunction associated with \(\lambda_{1}\). Then \(\varphi_{1}\) satisfies the equation \(\lambda V(a \varphi_{1})=\varphi_{1}\). Moreover, it is well known that \(\varphi_{1}(x) \approx\delta(x)\). Namely, there exists \(C>0\) such that \(\frac{1}{C} \delta(x)\leq \varphi_{1}(x)\leq C \delta(x)\), for each \(x\in\Omega\). So \(\int_{\Omega}a(x) \varphi_{1}(x)\leq C \int _{\omega} \frac{1}{(\delta(x))^{\lambda-1}}\,dx<\infty\). Hence from Theorem 1.3, the problem has no positive bounded solution in Ω for each \(\lambda>\lambda _{1} \|g\|_{\infty}\).