Abstract
We consider a nonlinear parametric Dirichlet problem driven by the p-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carathéodory perturbation which is (\(p-1\))-linear near \(+\infty \). The problem is uniformly nonresonant with respect to the principal eigenvalue of \((-\Delta _p,W^{1,p}_0(\Omega ))\). We look for positive solutions and prove a bifurcation-type theorem describing in an exact way the dependence of the set of positive solutions on the parameter \(\lambda >0\).
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1 Introduction
Let \(\Omega \subseteq \mathbb R^N\) be a bounded domain with \(C^2\)-boundary \(\partial \Omega \). In this paper we study the following nonlinear parametric singular Dirichlet problem:
In this problem, \(\Delta _p\) denotes the p-Laplacian differential operator defined by
On the right-hand side of (\(P_{\lambda }\)) (the reaction of the problem), we have a parametric singular term \(u\mapsto \lambda u^{-\gamma }\) with \(\lambda >0\) being the parameter and \(0<\gamma <1\). Also, there is a Carathéodory perturbation f(z, x) (that is, for all \(x\in \mathbb R\) the mapping \(z\mapsto f(z,x)\) is measurable and for almost all \(z\in \Omega \) the mapping \(x\mapsto f(z,x)\) is continuous). We assume that \(f(z,\cdot )\) exhibits \((p-1)\)-linear growth near \(+\infty \).
We are looking for positive solutions of problem (\(P_{\lambda }\)). Our aim is to describe in a precise way the dependence on the parameter \(\lambda >0\) of the set of positive solutions.
We prove a bifurcation-type property, which is the main result of our paper. Concerning the hypotheses H(f) on the perturbation f(z, x) and the other notation used in the statement of the theorem, we refer to Sect. 2. The main result of the present paper is stated in the following theorem.
Theorem A
If hypotheses H(f) hold, then there exists \(\lambda ^*\in (0,+\infty )\) such that
-
(a)
for every \(\lambda \in (0,\lambda ^*)\), problem (\(P_{\lambda }\)) has at least two positive solutions
$$\begin{aligned} u_\lambda ,\hat{u}_\lambda \in \mathrm{int}\,C_+,\quad u_\lambda \ne \hat{u}_\lambda ,\quad u_\lambda \leqslant \hat{u}_\lambda ; \end{aligned}$$ -
(b)
for \(\lambda =\lambda ^*\), problem (\(P_{\lambda }\)) has at least one positive solution
$$\begin{aligned} u^*_\lambda \in \mathrm{int}\,C_+; \end{aligned}$$ -
(c)
for \(\lambda >\lambda ^*\), problem (\(P_{\lambda }\)) has no positive solutions.
In the past, singular problems were studied in the context of semilinear equations (that is, \(p=2\)). We mention the works of Coclite and Palmieri [2], Ghergu and Rădulescu [5], Hirano et al. [10], Lair and Shaker [11] and Sun et al. [21]. A detailed bibliography and additional topics on the subject, can be found in the book of Ghergu and Rădulescu [6]. For nonlinear equations driven by the p-Laplacian, we mention the works of Giacomoni et al. [7], Papageorgiou et al. [16, 17], Papageorgiou and Smyrlis [18] and Perera and Zhang [19]. Of the aforementioned papers, closest to our work here is that of Papageorgiou and Smyrlis [18], where the authors also deal with a parametric singular problem and prove a bifurcation-type result. In their problem, the perturbation f(z, x) is (\(p-1\))-superlinear in \(x\in \mathbb R\) near \(+\infty \). So, our present work complements the results of [18], by considering equations in which the reaction has the competing effects of a singular term and of a \((p-1)\)-linear term.
Our approach uses variational tools together with suitable truncation and comparison techniques.
2 Preliminaries and hypotheses
Let X be a Banach space and \(X^*\) its topological dual. By \(\left\langle \cdot ,\cdot \right\rangle \) we denote the duality brackets of the pair \((X^*,X)\). Given \(\varphi \in C^1(X,\mathbb R)\), we say that \(\varphi \) satisfies the “Cerami condition” (the “C-condition” for short), if the following property holds:
“Every sequence \(\{u_n\}_{n\geqslant 1}\subseteq X\) such that
admits a strongly convergent subsequence.”
Using this notion, we can state the “mountain pass theorem”.
Theorem 1
(Mountain pass theorem) Assume that \(\varphi \in C^1(X,\mathbb R)\) satisfies the C-condition, \(u_0,u_1\in X, ||u_1-u_0||>\rho >0\),
and \(c=\inf \nolimits _{\gamma \in \Gamma }\max \nolimits _{0\leqslant t\leqslant 1}\ \varphi (\gamma (t))\) with \(\Gamma =\{\gamma \in C([0,1],X):\gamma (0)=u_0,\gamma (1)=u_1\}\). Then \(c\geqslant m_{\rho }\) and c is a critical value of \(\varphi \) (that is, we can find \(\hat{u}\in X\) such that \(\varphi '(\hat{u})=0\) and \(\varphi (\hat{u})=c\)).
The analysis of problem (\(P_{\lambda }\)) will involve the Sobolev space \(W^{1,p}_0(\Omega )\) and the Banach space
We denote by \(||\cdot ||\) the norm of \(W^{1,p}_0(\Omega )\). On account of the Poincaré inequality, we have
The space \(C^1_0(\overline{\Omega })\) is an ordered Banach space with positive (order) cone
This cone has a nonempty interior given by
Here, \(n(\cdot )\) denotes the outward unit normal on \(\partial \Omega \).
Let \(h_1,h_2\in L^{\infty }(\Omega )\). We write \(h_1\prec h_2\), if for every compact \(K\subseteq \Omega \), we can find \(c_K>0\) such that \(c_K\leqslant h_2(z)-h_1(z)\) for almost all \(z\in K\). Note that, if \(h_1,h_2\in C(\Omega )\) and \(h_1(z)<h_2(z)\) for all \(z\in \Omega \), then \(h_1\prec h_2\).
The next strong comparison principle can be found in Papageorgiou and Smyrlis [18, Proposition 4] (see also Giacomoni et al. [7, Theorem 2.3]).
Proposition 2
If \(\hat{\xi }\geqslant 0,h_1,h_2\in L^{\infty }(\Omega ), h_1\prec h_2,u_1\in C_+\) with \(u_1(z)>0\) for all \(z\in \Omega , u_2\in \mathrm{int}\, C_+\) and
then \(u_2-u_1\in \mathrm{int}\, C_+.\)
We denote by \(A\,{:}\,W^{1,p}_0(\Omega )\rightarrow W^{-1,p'}(\Omega )=W^{1,p}_0(\Omega )^*\left( \frac{1}{p}+\frac{1}{p'}=1\right) \) the nonlinear map defined by
This map has the following properties (see Motreanu et al. [15, p. 40]).
Proposition 3
The map \(A\,{:}\,W^{1,p}_0(\Omega )\rightarrow W^{-1,p'}(\Omega )\) is bounded (that is, A maps bounded sets to bounded sets), continuous, strictly monotone and of type \((S)_+\), that is, if \(u_n{\mathop {\rightarrow }\limits ^{w}}u\) in \(W^{1,p}_0(\Omega )\) and \(\limsup \nolimits _{n\rightarrow \infty }\left\langle A(u_n),u_n-u\right\rangle \leqslant 0\), then \(u_n\rightarrow u\) in \(W^{1,p}_0(\Omega )\).
Consider the following nonlinear eigenvalue problem
We say that \(\hat{\lambda }\in \mathbb R\) is an “eigenvalue” of (\(-\Delta _p,W^{1,p}_0(\Omega )\)) if problem (1) admits a nontrivial solution \(\hat{u}\in W^{1,p}_0(\Omega )\), known as an “eigenfunction” corresponding to \(\hat{\lambda }\). The nonlinear regularity theory (see Gasinski and Papageorgiou [3, pp. 737–738]) implies that \(\hat{u}\in C^1_0(\overline{\Omega })\). There is a smallest eigenvalue \(\hat{\lambda }_1>0\) with the following properties:
-
\(\hat{\lambda }_1>0\) is isolated (that is, if \(\hat{\sigma }(p)\) denotes the spectrum of (\(-\Delta _p,W^{1,p}_0(\Omega )\)) then we can find \(\epsilon >0\) such that \((\hat{\lambda }_1,\hat{\lambda }_1+\epsilon )\cap \hat{\sigma }(p)=0\));
-
\(\hat{\lambda }_1\) is simple (that is, if \(\hat{u},\hat{v}\in C^1_0(\overline{\Omega })\) are eigenfunctions corresponding to \(\hat{\lambda }_1\), then \(\hat{u}=\xi \hat{v}\) for some \(\xi \in \mathbb R\backslash \{0\}\));
-
$$\begin{aligned} \hat{\lambda }_1=\inf \left\{ \frac{||Du||^p_p}{||u||^p_p}:u\in W^{1,p}_0(\Omega ),u\ne 0\right\} . \end{aligned}$$(2)
It follows from the above properties that the eigenfunctions corresponding to \(\hat{\lambda }_1\) do not change sign. We denote by \(\hat{u}_1\) the positive, \(L^p\)-normalized (that is, \(||\hat{u}_1||_p=1\)) eigenfunction corresponding to \(\hat{\lambda }_1>0\). From the nonlinear maximum principle (see, for example, Gasinski and Papageorgiou [3, p. 738]), we have \(\hat{u}_1\in \mathrm{int}\, C_+\). Any eigenfunction corresponding to an eigenvalue \(\hat{\lambda }\ne \hat{\lambda }_1\), is nodal (that is, sign-changing). More details about the spectrum of \((-\Delta _p,W^{1,p}_0(\Omega ))\) can be found in [3, 15].
We can also consider a weighted version of the eigenvalue problem (1). So, let \(m\in L^{\infty }(\Omega ), m(z)\geqslant 0\) for almost all \(z\in \Omega ,\ m\ne 0\). We consider the following nonlinear eigenvalue problem:
This problem has the same properties as (1). So, there is a smallest eigenvalue \(\tilde{\lambda }_1(m)>0\) which is isolated, simple and admits the following variational characterization
Also the eigenfunctions corresponding to \(\tilde{\lambda }_1(m)\) have a fixed sign and we denote by \(\tilde{u}_1(m)\) the positive, \(L^p\)-normalized eigenfunction. We have \(\tilde{u}_1(m)\in \mathrm{int}\, C_+\). These properties lead to the following monotonicity property of the map \(m\mapsto \tilde{\lambda }_1(m)\).
Proposition 4
If \(m_1,m_2\in L^{\infty }(\Omega ),0\leqslant m_1(z)\leqslant m_2(z)\) for almost all \(z\in \Omega \) and both inequalities are strict on sets of positive measure, then \(\tilde{\lambda }_1(m_2)<\tilde{\lambda }_1(m_1)\).
Given \(x\in \mathbb R\), we set \(x^{\pm }=\max \{\pm x, 0\}\). Then for \(u\in W^{1,p}_0(\Omega )\), we set \(u^{\pm }(\cdot )=u(\cdot )^{\pm }\). We know that
If \(g\,{:}\,\Omega \times \mathbb R\) is a measurable function (for example, a Carathéodory function), then by \(N_g(\cdot )\) we denote the Nemytski map corresponding to \(g(\cdot ,\cdot )\) defined by
Given \(v,u\in W^{1,p}_0(\Omega )\) with \(v\leqslant u\), we define the order interval [v, u] by
The hypotheses on the perturbation f(z, x) are the following:
\(H(f)\,{:}\,f\,{:}\,\Omega \times \mathbb R\leftarrow \mathbb R\) is a Carathéodory function such that \(f(z,0)=0\) for almost all \(z\in \Omega \) and
-
(i)
for every \(\rho >0\), there exists \(a_{\rho }\in L^{\infty }(\Omega )\) such that
$$\begin{aligned} |f(z,x)|\leqslant a_{\rho }(z)\ \hbox {for almost all}\ z\in \Omega ,\ \hbox {and all}\ 0\leqslant x\leqslant \rho ; \end{aligned}$$ -
(ii)
\(\hat{\lambda }_1<\eta \leqslant \liminf \nolimits _{x\rightarrow +\infty }\frac{f(z,x)}{x^{p-1}}\leqslant \limsup \nolimits _{x\rightarrow +\infty }\frac{f(z,x)}{x^{p-1}}\leqslant \hat{\eta }\) uniformly for almost all \(z\in \Omega ;\)
-
(iii)
there exists a function \(w\in C^1(\overline{\Omega })\) such that
$$\begin{aligned} w(z)\geqslant c_0>0\ \hbox {for all}\ z\in \overline{\Omega },\ \Delta _pw\in L^{\infty }(\Omega )\ \hbox {with}\ \Delta _pw(z)\leqslant 0\ \hbox {for almost all}\ z\in \Omega , \end{aligned}$$and for every compact \(K\subseteq \Omega \) we can find \(c_K>0\) such that
$$\begin{aligned} w(z)^{-\gamma }+f(z,w(z))\leqslant -c_K<0\quad \hbox {for almost all}\,\, z\in K; \end{aligned}$$ -
(iv)
there exists \(\delta _0\in (0,c_0)\) such that for every compact \(K\subseteq \Omega \)
$$\begin{aligned} f(z,x)\geqslant \hat{c}_K>0\quad \hbox {for almost all}\,\, z\in K,\quad \hbox {and all}\,\, x\in \left( 0,\delta _0\right] ; \end{aligned}$$ -
(v)
for every \(\rho >0\), there exists \(\hat{\xi }_{\rho }>0\) such that for almost all \(z\in \Omega \) the function
$$\begin{aligned} x\mapsto f(z,x)+\hat{\xi }_{\rho }x^{p-1} \end{aligned}$$is nondecreasing on \([0,\rho ]\).
Remark 1
Since we are looking for positive solutions and all the above hypotheses concern the positive semiaxis \(\mathbb R_+=\left[ 0,+\infty \right) \), we may assume without any loss of generality that
Hypothesis H(f)(iii) implies that asymptotically at \(+\infty \) we have uniform nonresonance with respect to the principal eigenvalue \(\hat{\lambda }_1>0\) of \((-\Delta _p,W^{1,p}_0(\Omega ))\). The resonant case was recently examined for nonparametric singular Dirichlet problems by Papageorgiou et al. [16].
Example 1
The following functions satisfy hypotheses H(f). For the sake of simplicity we drop the z-dependence:
with \(1<\tau<\vartheta , 1<q<p\) and \(\eta >\hat{\lambda }_1\); and
with \(\eta >\hat{\lambda }_1, 1<q<p\).
3 A purely singular problem
In this section we deal with the following purely singular parametric problem:
The next proposition establishes the existence and \(\lambda \)-dependence of the positive solutions for problem (\(Au_{\lambda }\)).
Proposition 5
For every \(\lambda >0\) problem (\(Au_{\lambda }\)) admits a unique solution \(\tilde{u}_{\lambda }\in \mathrm{int}\, C_+\), the map \(\lambda \mapsto \tilde{u}_{\lambda }\) is nondecreasing from \((0,\infty )\) into \(C^1_0(\overline{\Omega })\) (that is, if \(0<\vartheta <\lambda \), then \(\tilde{u}_{\vartheta }\leqslant \tilde{u}_{\lambda }\)) and \(||\tilde{u}_{\lambda }||_{C^1_0(\overline{\Omega })}\rightarrow 0\) as \(\lambda \rightarrow 0^+\).
Proof
The existence of a unique solution \(\tilde{u}_{\lambda }\in \mathrm{int}\, C_+\) follows from Proposition 5 of Papageorgiou and Smyrlis [18].
Let \(0<\vartheta <\lambda \) and let \(\tilde{u}_{\vartheta },\tilde{u}_{\lambda }\in \mathrm{int}\, C_+\) be the corresponding unique solutions of problem (\(Au_{\lambda }\)). Evidently, \(\tilde{u}^{p'}_{\vartheta }\in \mathrm{int}\, C_+\left( \frac{1}{p}+\frac{1}{p'}=1\right) \) and so by Proposition 2.1 of Marano and Papageorgiou [14], we can find \(c_1>0\) such that
The lemma of Lazer and McKenna [12, p. 726], implies that \(\hat{u}_1^{-\gamma /p'}\in L^{p'}(\Omega )\). Therefore \(\tilde{u}_{\vartheta }^{-\gamma }\in L^{p'}(\Omega )\). We introduce the Carathéodory function \(g_{\lambda }(z,x)\) defined by
We set \(G_{\lambda }(z,x)=\int ^x_0g_{\lambda }(z,s)ds\) and consider the functional \(\hat{\psi }_{\lambda }:W^{1,p}_0(\Omega )\rightarrow \mathbb R\) defined by
Proposition 3 of Papageorgiou and Smyrlis [18] implies that \(\hat{\psi }_{\lambda }\in C^1(W^{1,p}_0(\Omega ))\). From (5) and since \(\tilde{u}_{\vartheta }^{-\gamma }\in L^{p'}(\Omega )\) it follows that \(\hat{\psi }_{\lambda }(\cdot )\) is coercive. Also, via the Sobolev embedding theorem, we see that \(\hat{\psi }_{\lambda }(\cdot )\) is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find \(\bar{u}_{\lambda }\in W^{1,p}_0(\Omega )\) such that
In (3) we choose \(h=(\tilde{u}_{\vartheta }-\bar{u}_{\lambda })^+\in W^{1,p}_0(\Omega )\). We have
Therefore the map \(\lambda \mapsto \tilde{u}_{\lambda }\) is nondecreasing from \((0,+\infty )\) into \(C^1_0(\overline{\Omega })\).
We have
Choosing \(h=\tilde{u}_{\lambda }\in W^{1,p}_0(\Omega )\), we obtain
As in the first part of the proof, using Proposition 2.1 of Marano and Papageorgiou [14], we show that \(\tilde{u}_{\lambda }^{-\gamma }\in L^r(\Omega )\) for \(r>N\). Then Proposition 1.3 of Guedda and Véron [8] implies that
Let \(k_{\lambda }=\lambda \tilde{u}^{-\gamma }_{\lambda }\in L^r(\Omega ),\lambda \in \left( 0,1\right] \) and consider the following linear Dirichlet problem
Standard existence and regularity theory (see, for example, Struwe [20, p. 218]), implies that problem (10) has a unique solution \(v_{\lambda }(\cdot )\) such that
for some \(c_5>0\), all \(\lambda \in \left( 0,1\right] \), and with \(\alpha =1-\frac{N}{r}\in (0,1)\) (recall that \(r>N\)). Let \(\beta _{\lambda }(z)=Dv_{\lambda }(z)\). Then \(\beta _{\lambda }\in C^{0,\alpha }(\overline{\Omega })\) for every \(\lambda \in \left( 0,1\right] \). We have
Then Theorem 1 of Lieberman [13] (see also Corollary 1.1 of Guedda and Véron [8]) and (9), imply that we can find \(s\in (0,1)\) and \(c_6>0\) such that
Finally, the compact embedding of \(C^{1,s}_0(\overline{\Omega })\) into \(C^1_0(\overline{\Omega })\) and (8) imply that
This completes the proof. \(\square \)
4 Bifurcation-type theorem
Let
Proposition 6
If hypotheses H(f) hold, then \(\mathcal {L}\ne \emptyset \).
Proof
Using Proposition 5, we can find \(\lambda _0\in \left( 0,1\right] \) such that
Here, \(\delta _0>0\) is as postulated by hypothesis H(f)(iv).
We fix \(\lambda \in \left( 0,\lambda _0\right] \) and we consider the following truncation of the reaction in problem (\(P_{\lambda }\)):
(recall that \(\delta _0<c_0\leqslant w(z)\) for all \(z\in \overline{\Omega }\)). This is a Carathéodory function. We set \(\hat{K}_{\lambda }(z,x)=\int ^x_0\hat{k}_{\lambda }(z,s)ds\) and consider the function \(\hat{\varphi }_{\lambda }:W^{1,p}_0(\Omega )\rightarrow \mathbb R\) defined by
As before, we have \(\hat{\varphi }_{\lambda }\in C^1(W^{1,p}_0(\Omega ))\). Also, it follows from (12) that
In addition, we have that
Therefore, we can find \(\hat{u}_{\lambda }\in W^{1,p}_0(\Omega )\) such that
In (13) we choose \(h=(\tilde{u}_{\lambda }-\hat{u}_{\lambda })^+\in W^{1,p}_0(\Omega )\). Then
Next, we choose \(h=(\hat{u}_{\lambda }-w)^+\in W^{1,p}_0(\Omega )\) in (13). Then
(see hypothesis H(f)(iii) and use the nonlinear Green identity, see [3, p. 211])
So, we have proved that
Using (12) and (14), Eq. (13) becomes
From (14), (15) and Theorem 1 of Lieberman [13], we infer that
This completes the proof. \(\square \)
A byproduct of the above proof is the following corollary.
Corollary 7
If hypotheses H(f) hold, then \(S_{\lambda }\subseteq \mathrm{int}\, C_+\) for all \(\lambda >0\).
The next proposition shows that \(\mathcal {L}\) is an interval.
Proposition 8
If hypotheses H(f) hold, \(\lambda \in \mathcal {L}\) and \(\vartheta \in (0,\lambda )\), then \(\vartheta \in \mathcal {L}.\)
Proof
Since \(\lambda \in \mathcal {L}\), we can find \(u_{\lambda }\in S_{\lambda }\subseteq \mathrm{int}\, C_+\). Proposition 5 implies that we can find \(\tau \in [0,\lambda _0]\) (see (11)) such that
We introduce the Carathéodory function e(z, x) defined by
We set \(E_{\vartheta }(z,x)=\int ^x_0 e_{\vartheta }(z,s)ds\) and consider the functional \(\hat{\psi }_{\vartheta }:W^{1,p}_0(\Omega )\rightarrow \mathbb R\) defined by
We know that \(\hat{\psi }_{\vartheta }\in C^1(W^{1,p}_0(\Omega ))\). Moreover, \(\hat{\psi }_{\vartheta }\) is coercive (see (16)) and sequentially weakly lower semicontinuous. So, we can find \(u_{\vartheta }\in W^{1,p}_0(\Omega )\) such that
In (17) we first choose \(h=(\tilde{u}_{\tau }-u_{\vartheta })^+\in W^{1,p}_0(\Omega )\). Then
Next, in (17) we choose \(h=(u_{\vartheta }-u_{\lambda })^+\in W^{1,p}_0(\Omega )\). Then
So, we have proved that
It follows from (16), (17) and (18) that
The proof is now complete. \(\square \)
An interesting byproduct of the above proof is the following result.
Corollary 9
If hypotheses H(f) hold, \(\lambda \in \mathcal {L},u_{\lambda }\in S_{\lambda }\subseteq \mathrm{int}\, C_+\), and \(\vartheta <\lambda \), then \(\vartheta \in \mathcal {L}\) and we can find \(u_{\vartheta }\in S_{\vartheta }\subseteq \mathrm{int}\, C_+\) such that \(u_{\vartheta }\leqslant u_{\lambda }\).
In fact, we can improve the above result as follows.
Proposition 10
If hypotheses H(f) hold, \(\lambda \in \mathcal {L},u_{\lambda }\in S_{\lambda }\subseteq \mathrm{int}\, C_+\), and \(\vartheta <\lambda \), then \(\vartheta \in \mathcal {L}\) and we can find \(u_{\vartheta }\in S_{\vartheta }\subseteq \mathrm{int}\, C_+\) such that \(u_{\lambda }-u_{\vartheta }\in \mathrm{int}\, C_+\).
Proof
From Corollary 9 we know that \(\vartheta \in \mathcal {L}\) and we can find \(u_{\vartheta }\in S_{\vartheta }\subseteq \mathrm{int}\, C_+\) such that
Let \(\rho =||u_{\lambda }||_{\infty }\) and let \(\hat{\xi }_{\rho }>0\) be as postulated by hypothesis H(f)(v). Then
We set
We have
(see (19) and hypotheses H(f)(v)).
We can apply Proposition 2 and conclude that
The proof is now complete. \(\square \)
Denote \(\lambda ^*=\sup \mathcal {L}.\)
Proposition 11
If hypotheses h(f) hold, then \(\lambda ^*<+\infty \).
Proof
Let \(\epsilon >0\) be such that \(\hat{\lambda }_1+\epsilon <\eta \) (see hypothesis H(f)(ii)). We can find \(M>0\) such that
Also, hypothesis H(f)(i) implies that we can find large enough \(\tilde{\lambda }>0\) such that
It follows from (20) and (21) that
Let \(\lambda >\tilde{\lambda }\) and suppose that \(\lambda \in \mathcal {L}\). Then we can find \(u_{\lambda }\in S_{\lambda }\subseteq \mathrm{int}\, C_+\). We have
Since \(u_{\lambda }\in \mathrm{int}\, C_+\), we can find \(t\in (0,1)\) so small that
(see Proposition 2.1 of Marano and Papageorgiou [14]). We have
Using (24), we can define the Carathéodory function \(\beta (z,x)\) as follows
We set \(B(z,x)=\int ^x_0\beta (z,s)ds\) and consider the \(C^1\)-functional \(\sigma :W^{1,p}_0(\Omega )\rightarrow \mathbb R\) defined by
From (26) it is clear that \(\sigma (\cdot )\) is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find \(\bar{u}\in W^{1,p}_0(\Omega )\) such that
In (27) we first choose \(h=(\hat{y}_1-\bar{u})^+\in W^{1,p}_0(\Omega )\). Then
Also, in (27) we choose \(h=(\bar{u}-u_{\lambda })^+\in W^{1,p}_0(\Omega )\). Then
So, we have proved that
It follows from (26)–(28) that
Therefore we have \(\lambda ^*\leqslant \tilde{\lambda }<+\infty \). \(\square \)
Next, we show that the critical parameter \(\lambda ^*>0\) is admissible.
Proposition 12
If hypotheses H(f) hold, then \(\lambda ^*\in \mathcal {L}\).
Proof
Let \(\{\lambda _n\}_{n\geqslant 1}\subseteq (0,\lambda ^*)\) and assume that \(\lambda _n\rightarrow (\lambda ^*)^{-}\) as \(n\rightarrow \infty \). We can find \(u_n=u_{\lambda _n}\in S_{\lambda _n}\subseteq \mathrm{int}\,C_+\) for all \(n\in \mathbb N\). Then
Suppose that \(||u_n||\rightarrow \infty \). We set \(y_n=\frac{u_n}{||u_n||}\ n\in \mathbb N\). Then \(||y_n||=1, y_n\geqslant 0\) for all \(n\in \mathbb N\). So, we may assume that
From (29) we have
Hypotheses H(f)(i), (ii) imply that
This growth condition implies that
Then (32) and hypothesis H(f)(ii) imply that at least for a subsequence, we have
In (31) we choose \(h=y_n-y\in W^{1,p}_0(\Omega )\), pass to the limit as \(n\rightarrow \infty \), and use (30) and (32). Then
Therefore, if in (31) we pass to the limit as \(n\rightarrow \infty \) and use (34) and (33), then
Since \(\eta \leqslant \eta _0(z)\leqslant \hat{\eta }\) for almost all \(z\in \Omega \) (see (33)), using Proposition 4, we have
So, from (35) and since \(||y||=1\) (see (34)), it follows that y must be nodal, a contradiction (see (34)). Therefore
Hence, we may assume that
On account of Corollary 9, we may assume that \(\{u_n\}_{n\geqslant 1}\) is nondecreasing. Therefore \(u^*\ne 0\). Also, we have
From (36) and by passing to a subsequence if necessary, we can say that
From (37), (38) and Problem 1.19 of Gasinski and Papageorgiou [4], we have that
If in (29) we choose \(h=u_n-u^*\in W^{1,p}_0(\Omega )\), pass to the limit as \(n\rightarrow \infty \) and use (39) and the fact that \(\{N_f(u_n)\}_{n\geqslant 1}\subseteq L^{p'}(\Omega )\) is bounded, then
Finally, in (29) we pass to the limit as \(n\rightarrow \infty \) and use (39) and (40). We obtain
This completes the proof. \(\square \)
We have proved that
Proposition 13
If hypotheses H(f) hold and \(\lambda \in (0,\lambda ^*)\), then problem \((P_\lambda )\) admits at least two positive solutions
Proof
Let \(u^*\in S_{\lambda ^*}\subseteq \mathrm{int}\,C_+\) (see Proposition 12). Invoking Proposition 10, we can find \(u_\lambda \in S_\lambda \subseteq \mathrm{int}\,C_+\) such that
We consider the Carathéodory function \(\tau _\lambda (z,x)\) defined by
Recall that \(u_\lambda ^{-\gamma }\in L^{p'}(\Omega )\) (see the proof of Proposition 5). We set \(T_\lambda (z,x)=\int ^x_0\tau _\lambda (z,s)ds\) and consider the functional \(\tilde{\varphi }_\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb R\) defined by
We know that \(\tilde{\varphi }_\lambda \in C^1(W^{1,p}_0(\Omega ))\). Let \(K_{\tilde{\varphi }_\lambda }=\{u\in W^{1,p}_0(\Omega ):\tilde{\varphi }_\lambda '(u)=0\}\) (the critical set of \(\tilde{\varphi }_\lambda \)). Also, for \(u\in W^{1,p}_0(\Omega )\), we set
Claim 1
\(K_{\tilde{\varphi _\lambda }}\subseteq [u_\lambda )\cap \mathrm{int}\,C_+\).
Let \(u\in K_{\tilde{\varphi }_\lambda }\). We have
We choose \(h=(u_\lambda -u)^+\in W^{1,p}_0(\Omega )\). Then
This proves Claim 1.
Note that \(u_\lambda \in K_{\tilde{\varphi }_\lambda }\). We may assume that
or otherwise we already have a second positive smooth solution for problem \(P_{\lambda }\) (see (42)) and so we are done.
We introduce the following Carathéodory function
We set \(\hat{T_\lambda }(z,x)=\int ^x_0\hat{\tau }_\lambda (z,s)ds\) and consider the \(C^1\)-functional \(\hat{\varphi }_\lambda :W^{1,p}(\Omega )\rightarrow \mathbb R\) defined by
This functional is coercive (see (46)) and sequentially weakly lower semicontinuous. Hence we can find \(\tilde{u}_\lambda \in W^{1,p}_0(\Omega )\) such that
In (47) we choose \(h=(u_\lambda -\tilde{u}_\lambda )^+\in W^{1,p}_0(\Omega )\) and \(h=(\tilde{u}_\lambda -u^*)^+\in W^{1,p}_0(\Omega )\) and obtain that
From (42) and (46) it is clear that
Also, \(u_\lambda \) is a minimizer of \(\hat{\varphi }_\lambda \). Since \(u^*-u_\lambda \in \mathrm{int}\,C_+\) (see (41)), it follows that
We assume that \(K_{\tilde{\varphi }_\lambda }\) is finite or otherwise on account of Claim 1, we already have an infinity of positive smooth solutions for problem \(P_{\lambda }\) bigger than \(u_\lambda \) and so we are done. Because of (49), we can find \(\rho \in (0,1)\) small such that
Hypothesis H(f)(ii) implies that
Claim 2
\(\tilde{\varphi }_\lambda \) satisfies the C-condition.
Let \(\{u_n\}_{n\geqslant 1}\subseteq W^{1,p}_0(\Omega )\) such that \(\{\tilde{\varphi }_\lambda (u_n)\}_{n\geqslant 1}\subseteq \mathbb R\) is bounded and
We have
We choose \(h=-u^-_n\in W^{1,p}_0(\Omega )\) in (52) and also use (42). Then
Suppose that \(||u^+_n||\rightarrow \infty \) and let \(y_n=\frac{u^+_n}{||u^+_n||}\ n\in \mathbb N\). Then \(||y_n||=1,y_n\geqslant 0\) for all \(n\in \mathbb N\). So, we may assume that
From (42) and hypothesis H(f)(ii), we have
In (55) we choose \(h=y_n-y\in W^{1,p}_0(\Omega )\) and pass to the limit as \(n\rightarrow \infty \). Then
Then passing to the limit as \(n\rightarrow \infty \) in (55) and using (56) and (57), we obtain
As before, using Proposition 4, we have
This proves that \(\{u^+_n\}_{n\geqslant 1}\subseteq W^{1,p}_0(\Omega )\) is bounded. Hence
So, we may assume that
In (52) we choose \(h=u_n-u\in W^{1,p}_0(\Omega )\), pass to the limit as \(n\rightarrow \infty \) and use (59). Then
This proves Claim 2.
On account of (50), (51) and Claim 2 we can apply Theorem 1 (the mountain pass theorem) and find \(\hat{u}_\lambda \in W^{1,p}_0(\Omega )\) such that
Therefore \(\hat{u}_\lambda \in \mathrm{int}\,C_+\) is the second positive solution of \(P_{\lambda }\) and
The proof is now complete. \(\square \)
Therefore we have also proved Theorem A, which is the main result of this paper.
Remark 2
An interesting open problem is whether there is such a bifurcation-type theorem for resonant problems, that is,
or even for the nonuniformly nonresonant problems, that is,
with \(\eta \in L^\infty (\Omega )\) such that
In both cases it seems to be difficult to show that \(\lambda ^*<\infty \). Additional conditions on \(f(z,\cdot )\) might be needed.
References
Aizicovici, S., Papageorgiou, N.S., Staicu, V.: Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints. Mem. Am. Math. Soc. 196(915), 70 (2008)
Coclite, M., Palmieri, G.: On a singular nonlinear Dirichlet problem. Commun. Partial Differ. Equ. 14, 1315–1327 (1989)
Gasinski, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman and Hall/CRC, Boca Raton (2006)
Gasinski, L., Papageorgiou, N.S.: Exercises in Analysis, Part 2: Nonlinear Analysis. Springer, Cham (2016)
Ghergu, M., Rădulescu, V.D.: Singular elliptic problems with two parameters. J. Differ. Equ. 195, 520–536 (2003)
Ghergu, M., Rădulescu, V.D.: Singular Elliptic Problems: Bifurcation and Asymptotic Analysis. Clarendon Press, Oxford (2008)
Giacomoni, J., Schindler, I., Takač, P.: Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation. Ann. Sci. Norm. Super. Pisa Cl. Sci. 5(6), 117–158 (2007)
Guedda, M., Véron, L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13, 879–902 (1989)
Hewitt, E., Stromberg, K.: Real and Abstract Analysis. Springer, New York (1975)
Hirano, N., Saccon, C., Shioji, N.: Brezis–Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem. J. Differ. Equ. 245, 1997–2037 (2008)
Lair, A.V., Shaker, A.W.: Entire solution of a singular semilinear elliptic problem. J. Math. Anal. Appl. 200, 498–505 (1996)
Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary value problem. Proc. Am. Math. Soc. 11, 721–730 (1001)
Lieberman, G.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)
Marano, S.A., Papageorgiou, N.S.: Positive solutions to a Dirichlet problem with \(p\)-Laplacian and concave–convex nonlinearity depending on a parameter. Commun. Pure Appl. Anal. 12, 815–829 (2013)
Motreanu, D., Motreanu, V., Papageorgiou, N.S.: Topological and Variational Methods with Applications to Nonlinear Boundary Value Problems. Springer, New York (2014)
Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: Pairs of positive solutions for resonant singular equations with the \(p\)-Laplacian. Electr. J. Differ. Equ. 249, 22 (2017)
Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D. (2018) \((p,2)\)-equations asymmetric at both zero and infinity. Adv. Nonlinear Anal. 7(3) (2018). https://doi.org/10.1515/anona-2017-0195
Papageorgiou, N.S., Smyrlis, G.: A bifurcation-type theorem for singular nonlinear elliptic equations. Methods Appl. Anal. 22, 147–170 (2015)
Perera, K., Zhang, Z.: Multiple positive solutions of singular \(p\)-Laplacian problems by variational methods. Bound. Value Prob. 2005, 377–382 (2009)
Struwe, M.: Variational Methods. Springer, Berlin (1990)
Sun, Y., Wu, S., Long, Y.: Combined effects of singular and superlinear nonlinearities in some singular boundary value problems. J. Differ. Equ. 176, 511–531 (2001)
Acknowledgements
The authors wish to thank the referee for his/her remarks and suggestions. This research was supported by the Slovenian Research Agency Grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083. V.D. Rădulescu acknowledges the support through a Grant of the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, Project No. PN-III-P4-ID-PCE-2016-0130, within PNCDI III.
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Communicated by Ari Laptev.
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Papageorgiou, N.S., Rădulescu, V.D. & Repovš, D.D. Positive solutions for nonlinear parametric singular Dirichlet problems. Bull. Math. Sci. (2018). https://doi.org/10.1007/s13373-018-0127-z
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DOI: https://doi.org/10.1007/s13373-018-0127-z
Keywords
- Parametric singular term
- (\(p-1\))-Linear perturbation
- Uniform nonresonance
- Nonlinear regularity theory
- Truncation
- Strong comparison principle
- Bifurcation-type theorem