Abstract
In this paper, we study the existence and nonexistence of positive bounded solutions of the integral equation \(u=\lambda V(af(u))\), where λ is a positive parameter, a is a nontrivial nonnegative measurable function with bounded potential and V belongs to a class of positive kernels that contains in particular the potential kernel of the classical Laplacian \(V=(\Delta)^{1}\) or \(V=(\frac{\partial}{\partial t }\Delta)^{1}\) or the inverse of the polyharmonic Laplacian \((\Delta)^{m}\), \(m\geq2\).
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1 Introduction
Let Ω be a smooth domain of \({\mathbb{R}}^{n}\) (\(n\geq1\)) and \(f:[0,\infty) \rightarrow(0,\infty)\) be a nondecreasing continuous function. In this paper, we are interested in the existence of a positive solution of the following integral equation:
where λ is a positive parameter, a is a nontrivial nonnegative measurable function satisfying an appropriate condition and V belongs to a class of positive kernels that contains in particular the potential kernel of the classical Laplacian \(V=(\Delta)^{1}\) or \(V=(\frac{\partial}{\partial t }\Delta)^{1}\) or the inverse of the polyharmonic Laplacian \((\Delta)^{m}\), \(m\geq 1\).
We note that if V is the inverse of a differential operator −L (i.e. \(V=(L)^{1}\)) and u is a solution of (1.1), then u satisfies
under some appropriate Dirichlet conditions on the boundary of Ω.
For \(\lambda=1\), (1.2) has been extensively studied either for nonincreasing positive nonlinearities (see [1–5]) or for sublinear positive nonlinearities (see [3, 6–9]). In fact existence, uniqueness and global behavior of the solution are discussed in the above works.
Here we are interested in the general equation (1.1). In fact, we will show that there exists \(\lambda ^{\star}>0\) such that (1.1) has a positive solution for \(0<\lambda<\lambda^{\star}\) under the following hypotheses on the functions a and f:
 (H_{1}):

\(f:[0,\infty) \rightarrow(0,\infty)\) is a nondecreasing continuous function.
 (H_{2}):

a is a nontrivial nonnegative measurable function on Ω such that the potential function Va is bounded in Ω.
More precisely, let \(\Va\_{\infty}= \sup_{x\in\Omega }Va(x)\); \(g(t)=\frac{t}{f(t)}\) for \(t>0\) and \(\beta= \sup_{t>0}g(t) \in(0,\infty]\). So, we prove the following result.
Theorem 1.1
Assume (H_{1})(H_{2}). Then we have:

(i)
If \(g(t)<\beta\) for each \(t>0\), then (1.1) has a positive solution u for each \(0<\lambda<\frac{\beta}{\Va\_{\infty}}\).

(ii)
If there exists \(t_{0}>0\) such that \(g(t_{0})=\beta\), then equation (1.1) has a positive solution u for each \(0<\lambda\leq\frac {\beta}{\Va\_{\infty}}\).
Moreover, in the two cases there exists \(C>0\) such that for each \(x\in \Omega\),
Remark 1.2
Let Ω be a smooth bounded domain and let \(\delta(x)\) denote the Euclidean distance from \(x\in\Omega\) to the boundary ∂Ω. Then for each \(\lambda<2\), the function \(a(x)= \frac{1}{ (\delta(x) )^{\lambda}}\) satisfies (H_{2}).
Next, we give a nonexistence result under some restrictions on the function a. So we assume that the kernel V is selfadjoint, namely for each nonnegative measurable functions h, k in Ω we have \(\int_{\Omega}Vh(x) k(x)\,dx=\int_{\Omega}h(x) Vk(x)\,dx\). The functions f and a are required to satisfy the following hypotheses:
 (H_{3}):

The function \(g(t)= \frac{t}{f(t)}\) is nonnegative and bounded on \((0,\infty)\).
 (H_{4}):

There exist \(\lambda_{1}>0\) and a positive measurable function \(\varphi_{1}\) such that \(\lambda_{1} V(a\varphi_{1})=\varphi_{1}\) and \(\int_{\Omega }a(x)\varphi_{1}(x)\,dx<\infty\).
Then we have the following nonexistence result.
Theorem 1.3
Assume (H_{3})(H_{4}). Then for each \(\lambda>\lambda_{1} \g\_{\infty}\), (1.1) has no positive bounded solution in Ω.
Remark 1.4
The problem
has the positive bounded solution \(u(x)= \frac{2}{\cos (\frac{\sqrt{\lambda}}{2} )} \sin (\frac{\sqrt{\lambda}}{2}x )\sin (\frac{\sqrt{\lambda }}{2}(1x) )\) for \(0<\lambda<\pi^{2}=\lambda_{1}\). Hence the nonexistence result in Theorem 1.3 is optimal.
Our paper is organized as follows. Section 2 is devoted to the proof of Theorems 1.1 and 1.3. The last section is devoted to the study of some examples and applications.
2 Proof of main results
Proof of Theorem 1.1
Assume that a satisfies (H_{2}) and let \(0<\lambda< \frac{\beta}{\Va\_{\infty}}= \frac{1}{\Va\_{\infty}} \sup_{t>0}\frac{t}{f(t)}\). Consider \(0< c_{1}\leq\lambda f(0)\) and \(0< c_{2}\) such that \(0<\lambda\leq\frac{c_{2}}{f(c_{2}\Va\_{\infty})}\). Then, from hypothesis (H_{1}), we deduce that \(0< c_{1}\leq c_{2}\). Consider the nonempty closed convex set
where \(B^{+}(\Omega)\) denotes the set of nonnegative Borel measurable functions in Ω. Let \(T_{\lambda}\) be the operator defined on Λ by
Since f is nondecreasing on \([0,\infty)\), \(T_{\lambda}\) is nondecreasing on Λ. Moreover, for each \(u\in\Lambda\), we have
So \(T_{\lambda} \Lambda\subset\Lambda\). Consider the sequence \((u_{k})_{k\geq0}\) defined by \(u_{0}=c_{1} Va\) and \(u_{k+1}=T_{\lambda} u_{k}\) for \(k\geq0\). Since \(T_{\lambda}\Lambda \subset\Lambda\) and \(T_{\lambda}\) is nondecreasing, we find by induction that the sequence \((u_{k})_{k\geq0}\) is nondecreasing and satisfies for each \(k\geq0\)
Hence, it follows from the monotone convergence theorem that the sequence \((u_{k})_{k\geq0}\) converges to a function \(u\in\Lambda\) satisfying the integral equation
This proves assertion (i) of the theorem.
Now, assume that there exists \(t_{0}>0\) such that \(g(t_{0})=\beta\). Then for \(\lambda= \frac{\beta}{\Va\_{\infty}}\), we take \(c_{2}= \frac{t_{0}}{\Va\_{\infty}}\) and \(0< c_{1}<\lambda f(0)= \frac{t_{0} f(0)}{\Va\_{\infty}f(t_{0})}\), to adapt the previous proof. □
Proof of Theorem 1.3
Assume that (1.1) has a positive bounded solution u in Ω. Then we have
This shows that \(\lambda_{1} {\g\_{\infty}}\geq\lambda\). □
Example 2.1
As a consequence of Theorem 1.3, we deduce that the problem
has no positive bounded solution for \(\lambda> \frac{{\pi }^{2}}{e}\). Indeed, in this case we have \(\lambda_{1}={\pi}^{2}\) and \(\varphi_{1}(x)=c_{0} \frac{\sin(\pix)}{x}\), \(c_{0}>0\).
3 Examples and applications
3.1 Examples
Next, we give some examples where (H_{1}) and (H_{2}) 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 \(\beta=g (\frac{1}{(p1)^{\frac{1}{p}}} )=\frac{(p1)^{1\frac {1}{p}}}{p}\). So the existence result for (1.1) holds for \(\lambda\in(0,\frac {(p1)^{1\frac{1}{p}}}{p\Va\_{\infty}}]\).
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 \(\lambda \in(0,\frac{1}{e\Va\_{\infty}}]\).
3.2 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
has a positive continuous solution u satisfying for each \(x \in\Omega\)
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 \(\lim_{x \rightarrow1 }\frac {Va(x)}{(1x)^{m1}}=0\). 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 \(x \in B(0,1)\)
Application 3.3
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 \((x,t)\in {\mathbb{R}}^{n}_{+}\times(0,\infty)\)
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 \(\int_{\Omega}a(x) \varphi_{1}(x)\leq C \int _{\omega} \frac{1}{(\delta(x))^{\lambda1}}\,dx<\infty\). Hence from Theorem 1.3, the problem
has no positive bounded solution in Ω for each \(\lambda>\lambda _{1} \g\_{\infty}\).
References
Alsaedi, R, Mâagli, H, Zeddini, N: Exact behavior of the unique positive solution to some singular elliptic problem in exterior domains. Nonlinear Anal. 119, 186198 (2015)
Cheng, J, Zhang, Z: On the existence of positive solutions for a class of singular boundary value problems. Nonlinear Anal. TMA 44, 645655 (2001)
Lair, AV, Shaker, AW: Entire solution of a singular semilinear elliptic problem. J. Math. Anal. Appl. 200, 498505 (1996)
Mâagli, H, Zribi, M: On a new Kato class and singular solutions of a nonlinear elliptic equation in bounded domains of \({\mathbb{R}}^{n}\). Positivity 9, 667686 (2005)
Zeddini, N, Alsaedi, R, Mâagli, H: Exact boundary behavior of the unique positive solution to some singular elliptic problems. Nonlinear Anal. 89, 146156 (2013)
Alsaedi, R, Mâagli, H, Zeddini, N: Estimates on potential functions and boundary behavior of positive solutions for sublinear Dirichlet problems. Electron. J. Differ. Equ. 2014, 08 (2014)
Brezis, H, Kamin, S: Sublinear elliptic equations in \({\mathbb{R}}^{n}\). Manuscr. Math. 74, 87106 (1992)
Bachar, I, Zeddini, N: On the existence of positive solutions for a class of semilinear elliptic equations. Nonlinear Anal. 52, 12391247 (2003)
Ghergu, M, Rǎdulescu, VD: Sublinear singular elliptic problems with two parameters. J. Differ. Equ. 195, 520536 (2003)
Zeddini, N: Positive solutions for a singular nonlinear problem on a bounded domain in \({\mathbb{R}}^{2}\). Potential Anal. 18, 97118 (2003)
Bachar, I, Mâagli, H, Masmoudi, S, Zribi, M: Estimates for the Green function and singular solutions for polyharmonic nonlinear equations. Abstr. Appl. Anal. 12, 715741 (2003)
Mâagli, H, Masmoudi, S, Zribi, M: On a parabolic problem with nonlinear term in a half space and global behavior of solutions. J. Differ. Equ. 246, 34173447 (2009)
Hansen, W: Valeurs propres pour l’opérateur de Schrödinger. In: Séminaire de Théorie du Potentiel, No. 9, Paris. Lecture Notes in Mathematics, vol. 1393, pp. 117134 (1989)
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant No. 257/662/1434. The authors, therefore, acknowledge with thanks DSR technical and financial support.
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Zeddini, N., Alsaedi, R.S. & Mâagli, H. Existence and global behavior of positive solutions for some eigenvalue problems. Bound Value Probl 2015, 225 (2015). https://doi.org/10.1186/s1366101504693
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DOI: https://doi.org/10.1186/s1366101504693