Examples
Next, we give some examples where (H1) and (H2) are satisfied.
Example 3.1
Let \(f(t)=t^{p}+1\) for \(p\geq0\) and \(t\geq0\). Then we discuss three cases:
-
Case 1:
\(p\in[0,1)\). In this case \(\beta=+\infty\) and the existence result for (1.1) holds for \(\lambda\in(0,\infty)\).
-
Case 2:
\(p=1\). Then \(\beta=1\) and the existence result for (1.1) holds for \(\lambda\in(0,\frac{1}{\|Va\|_{\infty}})\).
-
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 \(\lambda\in(0,\frac {(p-1)^{1-\frac{1}{p}}}{p\|Va\|_{\infty}}]\).
Example 3.2
Let \(f(t)=e^{t}\). Then \(\beta= \sup_{t>0}g(t)=g(1)= \frac{1}{e}\). In this case, the existence result for (1.1) holds for \(\lambda \in(0,\frac{1}{e\|Va\|_{\infty}}]\).
Applications
Throughout the following first three applications, we assume that the function \(f:[0,\infty) \rightarrow(0,\infty)\) is continuous and nondecreasing.
Application 3.1
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
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} -\Delta u=\lambda a(x)f(u), \quad \textit{in } \Omega, \\ u_{/ \partial^{\infty} \Omega}=0 \end{array}\displaystyle \right . \end{aligned}$$
has a positive continuous solution
u
satisfying for each
\(x \in\Omega\)
$$\begin{aligned} \frac{1}{C} Va(x)\leq u(x)\leq C Va(x) \quad\textit{for some } C>0. \end{aligned}$$
Application 3.2
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
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} (-\Delta)^{m} u=\lambda a(x)f(u), \quad \textit{in } \Omega, \\ \lim_{|x|\rightarrow1}\frac{u(x)}{(1-|x|)^{m-1}}=0 \end{array}\displaystyle \right . \end{aligned}$$
has a positive continuous solution
u
satisfying for each
\(x \in B(0,1)\)
$$\begin{aligned} \frac{1}{C} Va(x)\leq u(x)\leq C Va(x) ,\quad \textit{for some } C>0. \end{aligned}$$
Application 3.3
For
\(t>s\)
and
\(x,y\in{\mathbb{R}}^{n}_{+}\), we denote by
$$\Gamma(x,t,y,s)= \bigl(4\pi(t-s)\bigr)^{-\frac{n}{2}} \biggl[1-\exp \biggl(- \frac{x_{n}y_{n}}{(t-s)} \biggr) \biggr] \exp \biggl(-\frac{|x-y|^{2}}{4(t-s)} \biggr) $$
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
$$\lim_{\xi\rightarrow\partial{\mathbb{R}}^{n}_{+}} Va(\xi,t)=\lim_{r \rightarrow0^{+} } Va(x,r)=0. $$
Consequently, for
\(0<\lambda< \frac{1}{\|Va\|_{\infty}} \sup_{t>0} \frac{t}{f(t)}\), the problem
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} \frac{\partial u}{\partial t}- \Delta u=\lambda a(x,t)f(u), \quad \textit{in } {\mathbb{R}}^{n}_{+}\times(0,\infty),\\ u(x,0)=0 \quad\textit{for } x \in{\mathbb{R}}^{n}_{+},\\ u_{/{\mathbb{R}}^{n}_{+}\times(0,\infty)}=0 \end{array}\displaystyle \right . \end{aligned}$$
has a positive continuous solution
u
satisfying for each
\((x,t)\in {\mathbb{R}}^{n}_{+}\times(0,\infty)\)
$$\begin{aligned} \frac{1}{C} Va(x,t)\leq u(x,t)\leq C Va(x,t) \quad\textit{for some } C>0. \end{aligned}$$
Application 3.4
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
$$\begin{aligned} \left \{ \textstyle\begin{array}{@{}l} -\Delta u=\lambda a(x)f(u), \quad \textit{in } \Omega, \\ u_{/ \partial\Omega}=0 \end{array}\displaystyle \right . \end{aligned}$$
has no positive bounded solution in Ω for each
\(\lambda>\lambda _{1} \|g\|_{\infty}\).