Abstract
We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearities. In this paper the concave term will be parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter \(\lambda \) varies.
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1 Introduction
Let \(\Omega \subseteq \mathbb R^{N} \) be a bounded domain with a \(C^2\)-boundary \(\partial \Omega \). In this paper we study the following nonlinear nonhomogeneous parametric singular problem:
The map \(a:\mathbb R^{N} \rightarrow \mathbb R^{N}\) involved in the differential operator of (\(P_{\lambda }\)) is strictly monotone, continuous (hence maximal monotone, too) and satisfies certain other regularity and growth conditions which are listed in hypotheses H(a) below (see Sect. 2). These conditions are general enough to incorporate in our framework many differential operators of interest such as the p-Laplacian and the (p, q)-Laplacian (that is, the sum of a p-Laplacian and a q-Laplacian). The operator \(u\mapsto \mathrm{div}\,a(Du)\) is not homogeneous and this is a source of difficulties in the analysis of problem (\(P_{\lambda }\)). The potential function \(\xi \in L^{\infty }(\Omega )\) is indefinite (that is, sign changing). So the operator \(u \mapsto -\mathrm{div}\, a(Du) + \xi (z) |u|^{p-2} u \) is not coercive and this is one more difficulty in the analysis of problem (\(P_{\lambda }\)). In the reaction (the right-hand side of (\(P_{\lambda }\))), the term \(\vartheta (\cdot )\) is singular at \(x=0\), while the perturbation contains the combined effects of a parametric concave term \(x\mapsto \lambda x^{q-1}\) (\(x\geqslant 0 \)) (recall that \(q<p\)), with \(\lambda >0\) being the parameter and of a Carathéodory function 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), which is assumed to exhibit \((p-1)\)-superlinear growth near \(+\infty \), but without satisfying the usual for superlinear problems Ambrosetti-Rabinowitz condition (the AR-condition for short). So in problem (\(P_{\lambda }\)) we have the competing effects of singular, concave and convex terms.
Using variational methods related to the critical point theory, combined with suitable truncation, perturbation and comparison techniques, we produce a critical parameter value \(\lambda ^* >0\) such that
- (i)
for all \(\lambda \in (0,\lambda ^*)\) problem (\(P_{\lambda }\)) has at least two positive solutions;
- (ii)
for \(\lambda = \lambda ^*\) problem (\(P_{\lambda }\)) has at least one positive solution;
- (iii)
for all \(\lambda > \lambda ^*\) problem (\(P_{\lambda }\)) has no positive solutions.
This work continues the recent paper by Papageorgiou et al. [16], where \(\xi \equiv 0 \) and in the reaction the parametric term is the singular one. It is also related to the works of Papageorgiou and Smyrlis [17] and Papageorgiou and Winkert [19], where the differential operator is the p-Laplacian, \(\xi \equiv 0\) and no concave terms are allowed. Singular p-Laplacian equations with no potential term and reactions of special form were considered by Chu et al. [2], Giacomoni et al. [5], Li and Gao [10], Mohammed [12], Perera and Zhang [20], and Papageorgiou et al. [14].
2 Mathematical background and hypotheses
In this section we present the main mathematical tools which we will use in the analysis of problem (\(P_{\lambda }\)). We also fix our notation and state the hypotheses on the data of the problem.
So, let X be a Banach space, \(X^*\) its topological dual, and let \(\varphi \in C^1(X).\) We say that \(\varphi (\cdot )\) satisfies the “C-condition”, if the following property holds:
This is a compactness-type condition on the functional \(\varphi (\cdot )\), which leads to the minimax theory of the critical values of \(\varphi (\cdot )\) (see, for example, Papageorgiou et al. [15]). We denote by \(K_\varphi \) the critical set of \(\varphi \), that is,
The main spaces in the analysis of problem (\(P_{\lambda }\)) are the Sobolev space \(W^{1,p}_0(\Omega )\)\((1<p<\infty )\) and the Banach space \(C_0^1(\overline{\Omega })=\{u\in C^1(\overline{\Omega }): u|_{\partial \Omega } = 0\}\). We denote by \(||\cdot ||\) the norm of \(W^{1,p}_0\). By the Poincaré inequality we have
The Banach space \(C_0^1(\Omega )\) is ordered with positive (order) cone
This cone has a nonempty interior given by
We will also use two additional ordered Banach spaces. The first one is
This cone is ordered with positive (order) cone
This cone has a nonempty interior given by
where \(\hat{d}(z) = d(z,\partial \Omega )\ \text{ for } \text{ all }\ z\in \overline{\Omega }\). On account of Lemma 14.16 of Gilbarg and Trudinger [6, p. 355], we have
with \(\hat{u}_1\) being the positive, \(L^p\)-normalized (that is, \(||\hat{u}_1||_p=1\)) eigenfunction corresponding to the principal eigenvalue \(\hat{\lambda }_1>0\) of the Dirichlet p-Laplacian. The nonlinear regularity theory and the nonlinear maximum principle (see, for example, Gasinski and Papageorgiou [4, pp. 737–738]), imply that \(\hat{u}_1\in \mathrm{int}\,C_+\).
The second ordered space is \(C^1(\overline{\Omega })\) with positive (order) cone
Clearly, this cone has a nonempty interior.
Concerning ordered Banach spaces with an order cone which has a nonempty interior (solid order cone), we have the following result which will be useful in our analysis (see Papageorgiou et al. [15, Proposition 4.1.22]).
Proposition 1
If X is an ordered Banach space with positive (order) cone K, int K \(\ne \emptyset \), and \(e \in int\,K\), then for every \(u\in X\) we can find \(\lambda _u >0\) such that \(\lambda _u e - u\in K.\)
Let \(l\in C^1(0,\infty )\) with \(l(t)>0\) for all \(t>0\). We assume that
Then the conditions on the map \(a(\cdot )\) are the following:
\(H(a): a(y) = a_0(|y|)y\) for all \(y\in \mathbb R^{N}\), with \(a_0(t)>0\) for all \(t>0\) and
- (i)
\(a_0\in C^1(0,+\infty ),\ t\mapsto a_0 (t)\) is strictly increasing on \((0,+\infty )\), \(a_0(t) t\rightarrow 0^+\) as \(t\rightarrow 0^+\) and
$$\begin{aligned} \lim _{t\rightarrow 0^+}\frac{a'_0(t)t}{a_0(t)} > -1; \end{aligned}$$ - (ii)
there exists \(c_3>0\) such that
$$\begin{aligned} |\nabla a(y)| \leqslant c_3 \frac{l(|y|)}{|y|}\ \text{ for } \text{ all }\ y\in \mathbb R^{N} \backslash \{0\}; \end{aligned}$$ - (iii)
\((\nabla a(y)\xi ,\xi )_{\mathbb R^{N}} \geqslant \frac{l(|y|)}{|y|}|\xi |^2\) for all \(y\in \mathbb R^{n} \backslash \{0\}\), \(\xi \in \mathbb R^{N}\);
- (iv)
if \(G_0(t) = \int _{0}^{t} a_0(s)s ds \), then there exists \(\tau \in (q,p]\) such that
$$\begin{aligned} \limsup _{t\rightarrow 0^+} \frac{\tau G_0(t)}{t^{\tau }} \leqslant c^* \end{aligned}$$
and \(0\leqslant p G_0(t) - a_0(t) t^2\) for all \(t>0\).
Remark 1
Hypotheses H(a)(i), (ii), (iii) are dictated by the nonlinear regularity theory of Lieberman [10] and the nonlinear maximum principle of Pucci and Serrin [21]. Hypothesis H(a)(iv) serves the needs of our problem, but in fact, it is a mild condition and it is satisfied in all cases of interest (see the examples below). These conditions were used by Papageorgiou and Rădulescu [13] and by Papageorgiou et al. [16].
Hypotheses H(a) imply that the primitive \(G_0(\cdot )\) is strictly increasing and strictly convex. We set \(G(y) = G_0(|y|)\) for all \(y\in \mathbb R^{N}.\) Evidently, \(G(\cdot )\) is convex, \(G(0) = 0\) and
Using hypotheses H(a)(i), (ii), (iii) and (2), we can easily obtain the following lemma, which summarizes the main properties of the map \(a(\cdot )\).
Lemma 2
If hypotheses H(a)(i), (ii), (iii) hold, then
- (a)
the map \(y\mapsto a(y)\) is continuous, strictly monotone (hence maximal monotone, too);
- (b)
\(|a(y)| \leqslant c_4 (|y|^{s-1} +|y|^{p-1})\) for some \(c_4>0\), and all \(y\in \mathbb R^{N}\);
- (c)
\((a(y),y)_{\mathbb R^{N}} \geqslant \frac{c_1}{p-1} |y|^p\) for all \(y\in \mathbb R^{N}\).
Using this lemma and (3), we obtain the following growth estimates for the primitive \(G(\cdot )\).
Corollary 3
If hypotheses H(a)(i), (ii), (iii) hold, then \(\frac{c_1}{p(p-1)} |y|^{p} \leqslant G(y) \leqslant c_5 (1+|y|^p) \) for some \(c_5>0\), and all \(y\in \mathbb R^{N}\).
The examples that follow confirm that the framework provided by hypotheses H(a) is broad and includes many differential operators of interest (see [13]).
Example 1
-
(a)
\(a(y) = |y|^{p-2}y\) with \(1<p<\infty \).
This map corresponds to the p-Laplace differential operator defined by
$$\begin{aligned} \Delta _p u= \mathrm{div}\, (|Du|^{p-2} Du)\ \text{ for } \text{ all }\ u\in W^{1,p}_0 (\Omega ). \end{aligned}$$ -
(b)
\(a(y)=|y|^{p-2} y + \mu |y|^{q-2} y\) with \(1<q<p<\infty ,\ \mu \geqslant 0\).
This map corresponds to the (p, q)-Laplace differential operator defined by
$$\begin{aligned} \Delta _p u+\Delta _q u~\text{ for } \text{ all }\ u\in W^{1,p}_0(\Omega ). \end{aligned}$$Such operators arise in models of physical processes. We mention the works of Cherfils and Ilyasov [1] (reaction-diffusion systems) and Zhikov [22] (homogenization of composites consisting of two materials with distinct hardening exponent in elasticity theory).
-
(c)
\(a(y) =(1+|y|^2)^{\frac{p-2}{2}}y\) with \(1<p<\infty \).
This map corresponds to the modified capillary operator.
-
(d)
\(a(y) = |y|^{p-2}y\left( 1+ \frac{1}{1+ |y|^p}\right) \) with \(1<p<\infty .\)
The hypotheses on the potential term \(\xi (\cdot )\) and on the singular part \(\vartheta (\cdot )\) of the reaction are the following:
\(H(\xi ): \xi \in L^\infty (\Omega ).\)
\(H(\vartheta ): \vartheta :(0,+\infty )\rightarrow (0,+\infty )\) is a locally Lipschitz function such that
(i) for some \(\gamma \in (0,1)\) we have
$$\begin{aligned} 0 < c_6\leqslant \liminf _{x\rightarrow 0^+}\vartheta (x)x^{\gamma } \leqslant \limsup _{x\rightarrow 0^+}\vartheta (x) x^\gamma \leqslant c_7; \end{aligned}$$(ii) \(\vartheta (\cdot )\) is nonincreasing.
Remark 2
In the literature we almost always encounter the following particular singular term
Of course, hypotheses \(H(\vartheta )\) provide a much more general framework and can accomodate also singularities like the ones that follow:
The following strong comparison principle can be found in Papageorgiou et al. [16, Proposition 6] (see also Papageorgiou and Smyrlis [17, Proposition 4]).
Proposition 4
If hypotheses \(H(a),H(\vartheta )\) hold, \(\hat{\xi }\in L^{\infty }(\Omega )\), \(\hat{\xi }(z) \geqslant 0\) for almost all \(z\in \Omega \), \(h_1,h_2 \in L^{\infty }(\Omega )\) satisfy
and \(u,v\in C^{1,\alpha }(\overline{\Omega })\) satisfy \(0<u(z)\leqslant v(z)\) for all \(z\in \Omega \) and for almost all \(z\in \Omega \) we have
\(-\mathrm{div}\, a(Du(z)) - \vartheta (u(z)) + \xi (z) u (z)^{p-1} = h_1(z)\)
\(- \mathrm{div}\, a(Dv(z) - \vartheta (v(z)) + \xi (z) v(z)^{p-1} = h_2(z)\),
then \(v-u\in \mathrm{int}\,\hat{C}_+.\)
In what follows, \(p^*\) is the critical Sobolev exponent corresponding to p, that is,
Now we introduce our hypotheses on the nonlinearity f(z, x).
\(H(f): f:\Omega \times \mathbb R\rightarrow \mathbb R_+\) is a Carathéodory function such that \(f(z,0)=0\) for almost all \(z\in \Omega \) and
- (i)
\(f(z,x) \leqslant a(z) (1+x^{r-1})\) for almost all \(z\in \Omega \), and all \(x\geqslant 0\), with \(a\in L^{\infty }(\Omega )\), \(p<r<p^*;\)
- (ii)
if \(F(z,x)=\int _0^x f(z,s) ds,\)
then \(\lim _{x\rightarrow +\infty }\frac{F(z,x)}{x^{p}} = +\infty \) uniformly for almost all \(z\in \Omega \);
- (iii)
there exists \(\sigma \in ((r-p) \max \{ \frac{N}{p}, 1 \},p^*),\ \sigma >q\) such that
$$\begin{aligned} 0<\hat{\beta }_0\leqslant \liminf _{x\rightarrow +\infty } \frac{f(z,x)x-pF(z,x)}{x^{\sigma }}\ \text{ uniformly } \text{ for } \text{ almost } \text{ all }\ z\in \Omega ; \end{aligned}$$ - (iv)
\(\limsup _{x\rightarrow 0^+} \frac{f(z,x)}{x^{r-1}} \leqslant \eta _0 \) uniformly for almost all \(z\in \Omega \);
- (v)
for every \(\rho >0\), there exists \(\hat{\xi }_\rho >0\) such that for almost all \(z\in \Omega \) the function
$$\begin{aligned} x\mapsto f(z,x) + \hat{\xi }_\rho x^{\rho -1} \end{aligned}$$is nondecreasing on \([0,\rho ]\).
Remark 3
Since our aim is to find positive solutions and the above hypotheses concern the positive semiaxis \(\mathbb R_+ = [0,+\infty )\), we may assume that
Hypotheses H(f)(ii), (iii) imply that
So, the nonlinearity \(f(z,\cdot )\) is \((p-1)\)-superlinear near \(+\infty \). However, this superlinearity of \(f(z,\cdot )\) is not formulated using the AR-condition. We recall that the AR-condition (unilateral version due to (4)), says that there exist \(\gamma >p\) and \(M>0\) such that
If we integrate (6a) and use (6b), we obtain the weaker condition
Therefore the AR-condition implies that \(f(z,\cdot )\) exhibits at least \((\gamma -1)\)-polynomial growth. Evidently, (7) implies the much weaker condition (5). In this work instead of the standard AR-condition, we employ the less restrictive hypothesis H(f)(iii). In this way we incorporate in our framework also \((p-1)\)-superlinear terms with “slower” growth near \(+\infty \), which fail to satisfy the AR-condition. The following function satisfies hypotheses H(f) but fails to satisfy the AR-condition (for the sake of simplicity we drop the z-dependence)
Finally, let us fix the notation which we will use throughout this work. For \(x\in \mathbb R\) we set \(x^{\pm } = \max \{ \pm x,0\}.\) Then for \(u\in W^{1,p}_0(\Omega )\) we define \(u^{\pm }(z) = u(z)^{\pm }\) for almost all \(z\in \Omega \). It follows that
If \(u,v\in W^{1,p}_0(\Omega )\) and \(u\leqslant v\), then we define.
Also, by \(\mathrm{int}_{c_0^1(\overline{\Omega })}[u,v]\) we denote the interior in the \(C_0^1(\overline{\Omega })\)-norm topology of the set \([u,v]\cap C_0^1(\overline{\Omega })\).
By \(A:W^{1,p}_0(\Omega )\rightarrow W^{-1,p'}(\Omega )=W^{1,p}_0(\Omega )^*\ (\frac{1}{p} + \frac{1}{p'} = 1)\) we denote the nonlinear operator defined by
We know (see Gasinski and Papageorgiou [4]), that \(A(\cdot )\) is continuous, strictly monotone (hence maximal monotone, too) and of type \((S)_+\), that is,
We introduce the following two sets related to problem (\(P_{\lambda }\)):
We let \(\lambda ^* = \sup \mathcal {L}. \)
3 Positive solutions
We start by considering the following purely singular problem:
From Papageorgiou et al. [16, Proposition 10], we have the following property.
Proposition 5
If hypotheses \(H(a),H(\xi ),H(\vartheta )\) hold, then problem (8) admits a unique positive solution \(v\in \mathrm{int}\,C_+\).
Let \(\beta > ||\xi ||_\infty \). Then hypotheses H(f)(i), (iv) and since \(1<q<p<r\), imply that we can find \(c_{10},c_{11}>0\) such that
Let \(k_\lambda (x) = \lambda c_{10} x^{q-1} + c_{11}x^{r-1} -\beta x^{p-1}\) for all \(x\geqslant 0\). With \(v\in \mathrm{int}\,C_+\) from Proposition 5, we consider the following auxiliary Dirichlet problem:
For this problem we prove the following result.
Proposition 6
If hypotheses \(H(a),H(\xi ),H(\vartheta )\) hold, then for all small enough \(\lambda >0\) problem \((10)_\lambda \) has a smallest positive solution
Proof
Recall that \(v\in \mathrm{int}\,C_+\) (see Proposition 5). Hence \(v\in \mathrm{int}\,K_+\) (see (1)). For \(s>N\) we consider the function \(\hat{u}^{1/s}_1\in K_+\). According to Proposition 1, we can find \(\mu >0\) such that
From the Lemma in Lazer and McKenna [9, p. 726], we have
Hypotheses \(H(\vartheta )\) imply that we can find \(c_{12}>0\) and \(\delta >0\) such that
It follows from (12), (13) that
Let \(\hat{k}_\lambda (x) = \lambda c_{10} x^{q-1} + c_{11}x^{r-1}\) for all \(x\geqslant 0\) and set \(\hat{K}_\lambda (x) = \int _{0}^{x} \hat{k}_\lambda (s)ds\). We consider the \(C^1\)-functional \(\psi _\lambda : W^{1,p}_0(\Omega )\rightarrow \mathbb R\) defined by
We introduce the function \(\mathfrak {I}_\lambda (t)=\lambda t^{1-p} + t^{r-p},t>0. \) Evidently, \(\mathfrak {I}_\lambda \in C^{1}(0,+\infty )\) and since \(1<p<r\), we see that
So, we can find \(t_0>0\) such that
Since \(\frac{p-1}{r-1}<1\), it follows that
So, we can find \(\lambda _0\in (0,1]\) such that
For \( \rho =t_0 \), we see from (14) that
where \(\overline{B}_{\rho } = \{u\in W^{1,p}_0(\Omega ):||u||\leqslant \rho \}\) and \(\partial \overline{B}_\rho = \{u\in W^{1,p}_0 (\Omega ):||u||=\rho \}\).
We fix \(\lambda \in (0,\lambda _0]\). Hypothesis H(a)(iv) implies that we can find \(c^*_0 > c^*\) and \(\delta >0\) such that
Let \(u\in \mathrm{int}\, C_+\) and choose small enough \(t\in (0,1)\) such that
Then we have
Since \(q<\tau \leqslant p\), choosing \(t\in (0,1)\) even smaller if necessary, we have
The functional \(\psi _\lambda (\cdot )\) is sequentially weakly lower semicontinuous and by the Eberlein-Smulian theorem and the reflexivity of \(W^{1,p}_0(\Omega )\), the set \(\overline{B}_\rho \) is sequentially weakly compact. So, by the Weierstrass-Tonelli theorem, we can find \(\overline{u}\in W^{1,p}_0(\Omega )\) such that
From (15), (16) and (17) it follows that
In (18) we choose \(h=-\overline{u}^{-}\in W^{1,p}_0(\Omega ).\) Using Lemma 2(c) and since \(\beta >||\xi ||_\infty \) we obtain
Then from (18) we have
From (19) and Theorem 7.1 of Ladyzhenskaya and Uraltseva [8, p. 286], we have \(\overline{u}\in L^{\infty }(\Omega )\). Hence \(k_\lambda (\overline{u}(\cdot ))\in L^\infty (\Omega )\). Recall that \(\vartheta (v(\cdot ))\in L^s(\Omega )\) with \(s>N\). From Theorem 9.15 of Gilbarg and Trudinger [6, p. 241], we know that there exists a unique solution \(y_0\in W^{2,s}(\Omega )\) to the following linear Dirichlet problem
By the Sobolev embedding theorem, we have
Let \(\eta _0(z)=Dy_0(z).\) Then \(\eta _0\in C^{\alpha }(\overline{\Omega },\mathbb R^{N})\) and we have
The regularity theory of Lieberman [11] implies that \(\overline{u}\in C_+ \backslash \{ 0 \}\). Moreover, from (19) we have
(from the nonlinear maximum principle, see Pucci and Serrin [21, pp. 111,120]).
Let \(\hat{S_\lambda }\) denote the set of positive solutions of problem \((10)_\lambda \). We have just seen that \(\emptyset \ne \hat{S_\lambda }\subseteq \mathrm{int}\,C_+\) for \(\lambda \in (0,\lambda _0].\) Moreover, from Papageorgiou et al. [16, Proposition 18], we know that \(\hat{S_\lambda }\) is downward directed (that is, if \(u_1,u_2,\in \hat{S_\lambda }\), then we can find \(u\in \hat{S_\lambda }\) such that \(u\leqslant u_1,u\leqslant u_2)\). So, by Lemma 3.10 of Hu and Papageorgiou [7, p. 178], we can find a decreasing sequence \(\{ \overline{u}_n \}_{n\geqslant 1} \subseteq \hat{S_\lambda }\) such that
For every \(n\in \mathbb N\) we have
Choosing \(h = \overline{u}_n \in W^{1,p}_0\) and since \(0\leqslant \overline{u}_n\leqslant \overline{u}_1\) for all \(n\in \mathbb N\), using Lemma 2(c), we see that \(\{\overline{u}_n\}_{n\geqslant 1} \subseteq W^{1,p}_0(\Omega )\) is bounded. So, we have
Next, in (20) we choose \(h=\overline{u}_n-\overline{u}\in W^{1,p}_0(\Omega )\), pass to the limit as \(n\rightarrow \infty \) and use (21). Then
We pass to the limit as \(n\rightarrow \infty \) in (20) and use (22). Then
Note that for all \(n\in \mathbb N\), we have
(by the weak comparison principle, see Damascelli [3, Theorem 1.2])
Therefore \(\overline{u}_\lambda \in \hat{S_\lambda }\subseteq \mathrm{int}\,C_+\) and \(\overline{u}_\lambda = \inf \hat{S_\lambda }\). \(\square \)
We will use \(\bar{u}_\lambda \in \mathrm{int}\,C_+\) from Proposition 6 to show the nonemptiness of \(\mathcal {L}\).
Proposition 7
If hypotheses \(H(a),H(\xi ),H(\vartheta ),H(f)\) hold, then \(\mathcal {L}\ne \emptyset \) and \(S_\lambda \subseteq \mathrm{int}\,C_+\).
Proof
From (9) we have
For \(\lambda \in \left( 0,\lambda _0\right] \) we have
With \(\beta >||\xi ||_\infty \) and \(\lambda \in \left( 0,\lambda _0\right] \), we consider the following truncation-perturbation of the reaction in problem (\(P_{\lambda }\)):
This is a Carathéodory function. We set \(\Gamma _\lambda (z,x)=\int ^x_0\gamma _\lambda (z,s)ds\) and consider the functional \(\hat{\sigma }_\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb R\) defined by
Using Proposition 3 of Papageorgiou and Smyrlis [17], we see that \(\hat{\sigma }_\lambda \in C^1(W^{1,p}_0(\Omega ))\). Also, from (26), Corollary 3 and since \(\beta >||\xi ||_\infty \), we see that \(\hat{\sigma }_\lambda (\cdot )\) is coercive. In addition, it is sequentially weakly lower semicontinuous. So, we can find \(u_\lambda \in W^{1,p}_0(\Omega )\) such that
In (27) first we choose \(h=(u_\lambda -\bar{u}_\lambda )^+\in W^{1,p}_0(\Omega )\). Then we have
Next, in (27) we choose \(h=(v-u_\lambda )^+\in W^{1,p}_0(\Omega )\). Then we have
So, we have proved that
It follows from (26), (27) and (28) that
Note that \(\vartheta _\lambda (u_\lambda )\leqslant \vartheta (v)\) (see (28) and hypothesis \(H(\vartheta )(ii)\)) and \(\vartheta (v)\in L^s(\Omega )\). So, as before (see the proof of Proposition 6), we infer that
Therefore we have seen that
The proof is now complete. \(\square \)
For \(\eta >0\), let \(\tilde{u}_{\eta }\in \mathrm{int}\,C_+\) be the unique solution of the following Dirichlet problem
By Proposition 9 of Papageorgiou et al. [16], we see that given \(u\in S_\lambda \subseteq \mathrm{int}\,C_+\) (that is, \(\lambda \in \mathcal {L}\)), we can find small \(\eta >0\) such that
We will use this to obtain a lower bound for the elements of \(S_\lambda \).
Proposition 8
If hypotheses \(H(a),H(\xi ),H(\vartheta ),H(f)\) hold and \(\lambda \in \mathcal {L}\), then \(v\leqslant u\) for all \(u\in S_\lambda \).
Proof
Let \(u\in S_\lambda \subseteq \mathrm{int}\,C_+\). Then on account of (29) we can define the following Carathéodory function
We set \(E(z,x)=\int ^x_0 e(z,s)ds\) and consider the functional \(\mu :W^{1,p}_0(\Omega )\rightarrow \mathbb R\) defined by
As before, Proposition 3 of Papageorgiou and Smyrlis [17] implies that \(\mu \in C^1(W^{1,p}_0(\Omega ))\). The coercivity of \(\mu (\cdot )\) (see (30)) and the sequential weak lower semicontinuity guarantee the existence of \(\tilde{v}\in W^{1,p}_0(\Omega )\) such that
In (31) we choose \(h=(\tilde{v}-u)^+\in W^{1,p}_0(\Omega )\). Then we have
Similarly, if in (31) we choose \(h=(\tilde{u}_\eta -\tilde{v})^+\in W^{1,p}_0(\Omega )\), then we have
So, we have proved that
It follows from (30), (31), (32) that \(\tilde{v}\) is a positive solution of (18). Then on account of Proposition 5, we have
The proof is now complete. \(\square \)
Next, we show a structural property of the set \(\mathcal {L}\), namely that \(\mathcal {L}\) is an interval. Moreover, we establish a kind of strong monotonicity property for the solution set \(S_\lambda \).
Proposition 9
If hypotheses \(H(a),H(\xi ),H(\vartheta ),H(f)\) hold, \(\lambda \in \mathcal {L},0<\mu <\lambda \) and \(u_\lambda \in S_\lambda \subseteq \mathrm{int}\,C_+\), then \(\mu \in \mathcal {L}\) and there exists \(u_\mu \in S_\mu \subseteq \mathrm{int}\,C_+\) such that \(u_\lambda -u_\mu \in \mathrm{int}\,C_+.\)
Proof
From Proposition 8 we know that \(v\leqslant u_\lambda \). Then with \(\beta >||\xi ||_\infty \) we can define the following truncation-perturbation of the reaction in problem (\(P_\mu \)):
Evidently, \(e_\mu (z,x)\) is a Carathéodory function. We set \(E_\mu (z,x)=\int ^x_0e_\mu (z,s)ds\) and consider the \(C^1\)-functional \(\hat{\psi }_\mu :W^{1,p}_0(\Omega )\rightarrow \mathbb R\) defined by
Clearly, \(\hat{\psi }_\mu (\cdot )\) is coercive (see (33) and recall that \(\beta >||\xi ||_\infty \)). It is also sequentially weakly lower semicontinuous. So, we can find \(u_\mu \in W^{1,p}_0(\Omega )\) such that
In (34) we first use \(h=(u_\mu -u_\lambda )^+\in W^{1,p}_0(\Omega )\). Then
Next, in (34) we use \(h=(v-u_\mu )^+\in W^{1,p}_0(\Omega )\). Then from Proposition 5 and since \(f\geqslant 0\), we obtain
We have proved that
It follows from (33), (34), (35) that \(u_\mu \in S_\mu \subseteq \mathrm{int}\,C_+\) and so \(\mu \in \mathcal {L}\).
Let \(\rho =||u_\lambda ||_\infty \) and let \(\hat{\xi }_\rho >0\) as postulated by hypothesis H(f)(v). We have
From (36) and Proposition 4 of Papageorgiou and Smyrlis [17], we obtain
The proof is now complete. \(\square \)
Proposition 10
If hypotheses \(H(a),H(\xi ),H(\vartheta ),H(f)\) hold, then \(\lambda ^*<+\infty \).
Proof
Recall that by hypotheses H(f)(ii), (iii), we have
So, we can find \(M>0\) such that
Hypotheses \(H(\vartheta )\) imply that we can find small \(\delta \in \left( 0,1\right] \) such that
Finally, hypotheses H(f)(i), (v) imply that we can find big \(\lambda _0>0\) such that
Combining (37), (38), (39) we have
Let \(\lambda >\lambda _0\) and assume that \(\lambda \in \mathcal {L}\). Then according to Proposition 7 we can find \(u_\lambda \in S_\lambda \subseteq \mathrm{int}\,C_+\). Let \(\Omega _0\subseteq \Omega \) be an open set with \(\overline{\Omega }_0\subseteq \Omega \) and \(C^2\)-boundary \(\partial \Omega _0\). We have
For \(\epsilon >0\), let \(m^{\epsilon }_{0}=m_0+\epsilon \) and with \(\rho =||u_\lambda ||_\infty \), let \(\hat{\xi }_\rho >0\) be as postulated by hypothesis H(f)(v). We can always take \(\hat{\xi }_\rho >||\xi ||_\infty \). We have
Then from (40) and Proposition 4, we see that for small enough \(\epsilon >0\) we have
which contradicts the definition of \(m_0\). Hence \(\lambda \notin \mathcal {L}\) and so \(\lambda ^*\leqslant \lambda _0<+\infty \). \(\square \)
By Propositions 9 and 10 it follows that
Proposition 11
If hypotheses \(H(a),H(\xi ),H(\vartheta ),H(f)\) hold and \(\lambda \in (0,\lambda ^*)\), then problem (\(P_{\lambda }\)) admits at least two positive solutions
Proof
Let \(0<\mu<\lambda<\eta <\lambda ^*\). We have \(\mu ,\eta \in \mathcal {L}\) (see (42)). On account of Proposition 9 we can find \(u_\mu \in S_\mu \subseteq \mathrm{int}\,C_+,\ u_0\in S_\lambda \subseteq \mathrm{int}\,C_+,\ u_\eta \in S_\eta \subseteq \mathrm{int}\,C_+\) such that
With \(\beta >||\xi ||_\infty \), we introduce the Carathéodory function \(d_\lambda (z,x)\) defined by
We set \(D_\lambda (z,x)=\int ^x_0 d_\lambda (z,s)ds\) and consider the functional \(\varphi _\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb R\) defined by
We know that \(\varphi _\lambda \in C^1(W^{1,p}_0(\Omega ))\) (see Papageorgiou and Smyrlis [17, Proposition 3]). Also, let
This is a Carathéodory function. We set \(\hat{D}_\lambda (z,x)=\int ^x_0\hat{d}_\lambda (z,s)ds\) and consider the \(C^1\)-functional \(\hat{\varphi }_\lambda :W^{1,p}_0(\Omega )\rightarrow \mathbb R\) defined by
Using (44) and (45) and the nonlinear regularity theory (see the proof of Proposition 7), we show that
From (47) we see that we can assume that
or otherwise we already have a second positive solution for (\(P_{\lambda }\)) (see (45)) and so we are done.
Clearly, \(\hat{\varphi }_\lambda (\cdot )\) is coercive (see (45)) and sequentially weakly lower semicontinuous. So, we can find \(\hat{u}_0\in W^{1,p}_0(\Omega )\) such that
But from (44) and (45) we see that
It follows from (43), (49), (50) that
On account of (44) and (46), we may assume that
Otherwise we already have an infinity of positive smooth solutions. From (51), (52) and Theorem 5.7.6 of Papageorgiou et al. [15], we see that we can find small \(\rho \in (0,1)\) such that
Hypothesis H(f)(ii) and Corollary 3 imply that if \(u\in \mathrm{int}\,C_+\), then
Claim 1
\(\varphi _\lambda \) satisfies the C-condition.
Consider a sequence \(\{u_n\}_{n\geqslant 1}\subseteq W^{1,p}_0(\Omega )\) such that
From (56) we have
In (57) we choose \(h=-u^-_n\in W^{1,p}_0(\Omega )\). From (44) and Lemma 2, we have
Next, in (57) we choose \(h=u^+_n\in W^{1,p}_0(\Omega )\). Then
Adding (59) and (60) and using hypothesis H(a)(iv), we obtain
From hypotheses H(f)(i), (iii) we see that we can find \(\hat{\beta }_1\in (0,\hat{\beta _0})\) and \(c_{20}>0\) such that
Using (62) in (61) and recalling that \(q<\sigma \) (see hypothesis H(f)(iii)) we obtain that
First, suppose that \(N\ne p\). It is clear from hypothesis H(f)(iii) that we may assume that \(\sigma<r<p^*\) (recall that \(p^*=+\infty \) if \(N\leqslant p\)). Let \(t\in (0,1)\) be such that
From the interpolation inequality (see, for example, Papageorgiou and Winkert [18, Proposition 2.3.17, p.116]), we have
From hypothesis H(f)(i), we have
In (57) we choose \(h=u^+_n\in W^{1,p}_0(\Omega )\) and use Lemma 2. Then
The hypothesis on \(\sigma \) (see H(f)(iii)) implies that \(tr<p\). Also we have \(q<p\). Therefore it follows from (66) that
If \(p=N\), then \(p^*=+\infty \) and by the Sobolev embedding theorem, we have that \(W^{1,p}_0(\Omega )\hookrightarrow L^s(\Omega )\) for all \(1\leqslant s<\infty \). So, we need to replace in the previous argument \(p^*\) by \(s>r>\sigma \) big enough. More precisely, as before, let \(t\in (0,1)\) be such that
Recall that \(r-\sigma <p\) (see hypothesis H(f)(iii)). Hence for large enough \(s>r\)
Then for such large \(s>r\), the previous argument is valid and we again obtain (67).
From (58) and (67) we have that \(\{u_n\}_{n\geqslant 1}\subseteq W^{1,p}_0(\Omega )\) is bounded. So, we may assume that
In (57) we choose \(h=u_n-u\in W^{1,p}_0(\Omega )\), pass to the limit as \(n\rightarrow \infty \), and use (68). Then
This proves Claim 1.
From (53), (54) and Claim 1, we see that we can apply the mountain pass theorem. So, we can find \(\hat{u}\in W^{1,p}_0(\Omega )\) such that
It follows rom (44) and (69) that
The proof is now complete. \(\square \)
Proposition 12
If hypotheses \(H(a),H(\xi ),H(\vartheta ),H(f)\) hold, then \(\lambda ^*\in \mathcal {L}\).
Proof
Let \(\{\lambda _n\}_{n\geqslant 1}\subseteq (0,\lambda ^*)\) be such that \(\lambda _n\uparrow \lambda ^*\). We know that \(\lambda _n\in \mathcal {L}\) for all \(n\in \mathbb N\) and so we can find \(u_n=u_{\lambda _n}\in S_{\lambda _n}\subseteq \mathrm{int}\, C_+\)\((n\in \mathbb N)\) increasing (see Proposition 9).
Let \(\hat{\varphi }_{\lambda _n}(\cdot )\) be the functional from the proof of Proposition 11, with \(u_\mu =u_{n-1},\ u_\mu =u_{n+1}(n\geqslant 2)\). Then we have
Also, we have
Using (70), (71) and reasoning as in the proof of Proposition 11 (see Claim 1), we obtain that
From this, as in the proof of Proposition 11, exploiting the \((S)_+\) property of \(A(\cdot )\), we obtain
Passing to the limit as \(n\rightarrow \infty \) in (71) and using (72), we have
The proof is now complete. \(\square \)
This proposition implies that
Summarizing the situation for problem (\(P_{\lambda }\)), we can state the following bifurcation-type result.
Theorem 13
If hypotheses \(H(a),H(\xi ),H(\vartheta ),H(f)\) hold, then there exists \(\lambda ^*>0\) such that
- (a)
for all \(\lambda \in (0,\lambda ^*)\) problem (\(P_{\lambda }\)) has at least two positive solutions
$$\begin{aligned} u_0,\hat{u}\in \mathrm{int}\,C_+,\ u_0\ne \hat{u}; \end{aligned}$$ - (b)
for \(\lambda =\lambda ^*\) problem (\(P_{\lambda }\)) has at least one positive solution \(u_*\in \mathrm{int}\, C_+\);
- (c)
for all \(\lambda >\lambda ^*\) problem (\(P_{\lambda }\)) has no positive solutions.
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Acknowledgements
This research was supported by the Slovenian Research Agency Grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083.
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Papageorgiou, N.S., Rădulescu, V.D. & Repovš, D.D. Nonlinear singular problems with indefinite potential term. Anal.Math.Phys. 9, 2237–2262 (2019). https://doi.org/10.1007/s13324-019-00333-7
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DOI: https://doi.org/10.1007/s13324-019-00333-7
Keywords
- Nonhomogeneous differential operator
- Indefinite potential
- Singular term
- Concave and convex nonlinearities
- Truncation
- Comparison principles
- Nonlinear regularity
- Nonlinear maximum principle