Abstract
We consider a Robin problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential term. The reaction is parametric and exhibits the competing effects of a concave (sublinear) and of a convex (superlinear) terms (“concave-convex” problem). The parameter multiplies the convex term. We prove an existence and multiplicity theorem which is global in parameter.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
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 parametric, nonlinear, nonhomogeneous Robin problem:
In this problem, the map \(a: {\mathbb {R}}^N \rightarrow {\mathbb {R}}^N\) involved in the differential operator of \((p_{\lambda })\), is strictly monotone and continuous (thus, maximal monotone too) and satisfies certain other regularity and growth conditions listed in hypotheses \({\widehat{H}}\) below. These conditions provide a general framework in which we can fit many differential operators of interest such as the p-Laplacian and the (p, q)-Laplacian.
In the boundary condition, \(\frac{\partial u}{\partial n_a}\) denotes the conormal derivative corresponding to the map a(.). If \(u\in C^{1}({\bar{\Omega }})\) then
with n(.) being the outward unit normal. The boundary coefficient \(\beta \in C^{0,\alpha }(\partial \Omega )\) with \(0<\alpha <1\) and \(\beta (z)\ge 0.\)
The reaction term f(z, x) is a Caratheodory function (that is, for all \(x\in {\mathbb {R}}\), \(z\mapsto f(z,x)\) is measurable, while for almost all \(z\in \Omega \), \(x\mapsto f(z,x)\) is continuous). The potential term \(\xi (z)(u(z))^{p-1}\) is indefinite, that is, \(\xi (.)\) is sign-changing and this makes the differential operator (left-hand side of the \((p_{\lambda })\)) non-coercive. In the reaction (right-hand side of \((p_{\lambda })\)), we have the competing effects of a “concave” (\((p-1)\)-sublinear) term \(c(u(z))^{\tau -1}\) (\(c>0, \ 1<\tau <p\)) and of a parametric perturbation which is “convex” (\((p-1)\)-superlinear). So, problem \((p_{\lambda })\) is a generalized version of the classical “concave-convex problem”. The study of such problems started with the seminal work of Ambrosetti–Brezis–Gerami [1], who considered semilinear Dirichlet equations driven by the Laplacian and with no potential term (that is, \(\xi =0\)). The concave and convex contributions in the reaction are of the power type, that is, the reaction in [1] has the form
Their work was extended to Dirichlet equations driven by the p-Laplacian, by Garcia Azorero–Peral Alonso–Manfredi [3] and by Guo–Zhang [6]. In these works the reaction is also of the power type as above. More general differential operators and reactions were considered by Papageorgiou–Radulescu–Repovs [12] (anisotropic p-Laplacian equations) and by Papageorgiou–Vetro [16] ((p, 2)-equations). In both works, the problem has Dirichlet boundary conditions, there is no potential term (thus the operator is coercive) and the parameter \(\lambda \) multiplies the concave term. Only Marano–Marino–Papageorgiou [8], deal with a Dirichlet p-Laplacian equation with no potential term and a parametric convex (superlinear) term. The work of [8] was extended recently by Gasiniski–Papageorgiou–Zhang [5] to problems driven by the Robin p-Laplacian and with a positive potential term (thus, the differential operator is coercive). Recently Bai–Papageorgiou–Zeng [2], studied nonparametric Robin problems driven by a similar nonhomogeneous differential operator as \((p_{\lambda })\) plus an indefinite potential term. The authors proved a multiplicity result producing solutions with sign information (positive, negative and nodal (sign-changing) solutions). Finally, we should also mention the recent relevant work of Papageorgiou–Radulescu–Zhang [15], which examines a nonlinear eigenvalue problem for the Robin p-Laplacian plus a positive potential term. They proved a bifurcation-type result but for large values of the parameter \(\lambda >0\).
Using variational tools combined with suitable truncation and comparison techniques, we prove an existence and multiplicity result for the positive solutions of \((p_{\lambda })\) which is global in \(\lambda >0\) (a bifurcation-type result but for small values of \(\lambda >0\)).
2 Mathematical Background and Hypotheses
The main spaces in the analysis of the problem \((p_{\lambda })\) are the Sobolev space \(W^{1,p}(\Omega )\) with \(1<p<\infty \), the Banach space \(C^1({\overline{\Omega }})\), and the boundary Lebesgue spaces \(L^s(\partial \Omega )\) with \(1\le s\le \infty \). By \(\Vert \cdot \Vert \), we denote the norm of \(W^{1,p}(\Omega )\), which is defined by
for all \(u\in W^{1,p}(\Omega )\).
The Banach space \(C^1({\overline{\Omega }})\) is an ordered with a positive (order) cone \(C_+\) which is defined by;
This cone has a nonempty interior given by
Also we use another cone in \(C^1({\overline{\Omega }})\), namely,
On \(\partial \Omega \) we consider the \((N-1)\)-dimensional Hausdorff (surface) measure \(\sigma (\cdot )\). Using this measure on \(\partial \Omega \), we can define in the usual way the boundary Lebesgue spaces \(L^s(\partial \Omega )\) with \(1\le s\le \infty \). From the theory of Sobolev spaces, we know that there exists a unique bounded linear operator \(\gamma _0: W^{1,p}(\Omega ) \rightarrow L^p(\partial \Omega )\), called the “trace operator”, such that
Using \(\gamma _0(.)\) we have a way to speak about the boundary values of a Sobolev function. The trace operator \(\gamma _0(\cdot )\) is compact into \(L^s(\partial \Omega )\) for all \(s\in [1, \frac{(N-1)p}{N-p})\) if \(p<N\), and into \(L^s(\partial \Omega )\) for all \(s\in [1,+\infty )\) if \(N\le p\). Moreover, we know that:
and
In the sequel, for the sake of notation simplicity, we drop the use of the operator \(\gamma _0(\cdot )\). All restrictions of Sobolev functions on \(\partial \Omega \) are understood in the sense of traces.
Given \(u:\Omega \rightarrow {\mathbb {R}}\) is measurable, then we define
These are measurable functions and \(u=u^+-u^-\), \(|u|=u^++u^-\). Moreover, if \(u\in W^{1,p}(\Omega )\), then \(u^\pm \in W^{1,p}(\Omega )\). Suppose \(u,v:\Omega \rightarrow {\mathbb {R}}\) are measurable functions with \(u(z)\le v(z)\) for a.a. \(z\in \Omega \). We define
Let X be a Banach space and \(\varphi \in C^1(X)\). We say that \(\varphi (.)\) satisfies the “C-condition”, if it has the following property:
“Every sequence \(\{u_n\}_{n\in {\mathbb {N}}}\subset X\) such that
-
\(\{\varphi (u_n)\}_{n\in {\mathbb {N}}}\subseteq {\mathbb {R}}\) is bounded,
-
\((1+\Vert u_n\Vert _X)\varphi '(u_n)\rightarrow 0 \ \text {in} \ X^*\),
admits a strongly convergent subsequence.”
This is a compactness-type condition on \(\varphi (.)\) which compensates for the fact that the ambient space X need not be locally compact (being in general infinite-dimensional). By \(K_\varphi \) we denote the critical set of \(\varphi (.)\), that is,
By \(p^*\) we denote the critical Sobolev exponent corresponding to \(p\in (1,\infty )\), defined by
Let \({\hat{l}}\in C^1(0,\infty )\) with \({\hat{l}}(t)>0\) for all \(t>0\). We assume that there exist constants \(c_1, \ c_2>0\) and \(1<s<p\) such that
for all \(t>0\).
We introduce the following conditions on the map \(a:{\mathbb {R}}^N\rightarrow {\mathbb {R}}^N\) (see also Bai-Papageorgiou–Zeng [2]):
\(\mathbf {{\hat{H}}:}\) \(a(y)=a_0(|y|)y\) for all \(y\in {\mathbb {R}}^N\) with \(a_0(t)>0\) for all \(t>0\) satisfies the following conditions:
-
(i)
\(a_0\in C^1(0,\infty )\), \(t\mapsto a_0(t)t\) is strictly increasing on \({\mathbb {R}}^+:= (0,\infty )\) and \(a_0(t)t\rightarrow 0^+\) as \(t\rightarrow 0^+\) such that \(\lim _{t\rightarrow 0^+} \frac{ta'_0(t)}{a_0(t)}>-1\);
-
(ii)
\(|\nabla a(y)|\le c_3\frac{{\hat{l}}(|y|)}{|y|}\) for all \(y\in {\mathbb {R}}^N\backslash \{0\}\) for some \(c_3>0\);
-
(iii)
\((\nabla a(y)\xi ,\xi )_{{\mathbb {R}}^N}\ge \frac{{\hat{l}}(|y|)}{|y|}|\xi |^2\) for all \(y\in {\mathbb {R}}^N\backslash \{0\}\) and all \(\xi \in {\mathbb {R}}^N\);
-
(iv)
if \(G_0(t)=\int _0^t a_0(s)sds\) for all \(t>0\), then for some \(q\in (\tau ,p]\) we have \(\lim _{t\rightarrow 0^+}\frac{ qG_0(t)}{t^q} \le c^*\) and \(t\mapsto G_0(t^{1/q})\) is convex.
Remark 2.1
Hypotheses \(\mathbf {{\hat{H}}}(i)\), (ii), (iii) are dictated by the nonlinear regularity theory of Lieberman [7] and the nonlinear maximum principle of Pucci and Serrin [17]. Hypothesis \(\mathbf {{\hat{H}}}(iv)\) is an extra condition for the needs of our problem, but it is mild and it is satisfied in all cases of interest as the examples which follow illustrate.
Example 2.2
The following maps satisfy hypotheses \(\mathbf {{\hat{H}}}\):
-
(a.)
\(a(y)=|y|^{p-2}y\) with \(1<p<\infty \).
-
(b.)
\(a(y)=|y|^{p-2}y+|y|^{q-2}y\) with \(1<q<p<\infty \).
-
(c.)
\(a(y)=\big (1+|y|^2\big )^{\frac{p-2}{2}}y\) with \(1<p<\infty \).
Note that the map in (a.) corresponds to the p-Laplacian while the map in (b.) corresponds to the (p, q)-Laplacian.
Using these hypotheses, we can prove the following properties of the map a(.) (see Papageorgiou–Radulescu [9]).
Lemma 2.3
If hypotheses \(\mathbf {{\hat{H}}}(i)\), (ii), (iii) hold, then
-
(a.)
\(y\rightarrow a(y)\) is continuous and strictly monotone, hence maximal monotone too;
-
(b.)
\(|a(y)|\le c_4(|y|^{s-1}+|y|^{p-1})\) for all \(y\in {\mathbb {R}}^N\), some \(c_4>0\);
-
(c.)
\((a(y),y)_{{\mathbb {R}}^N}\ge \frac{c_1}{p-1}|y|^p\) for all \(y\in {\mathbb {R}}^N\).
We set \(G(y)=G_0(|y|)\) for all \(y\in {\mathbb {R}}^N\). Then the map \(G(\cdot )\in C^1({\mathbb {R}}^N,{\mathbb {R}})\) and we have
So \(G(\cdot )\) is the primitive of the map \(a(\cdot )\) and on account of Lemma 2.3 (a), \(G(\cdot )\) is strictly convex. Since \(G(0)=0\), we have
Using Lemma 2.3 and (2.1), we deduce the following bilateral growth restrictions on G(.).
Corollary 2.4
If hypotheses \(\mathbf {{\hat{H}}}(i)\), (ii), (iii) hold, then
Let \(V:W^{1,p}(\Omega )\rightarrow W^{1,p}(\Omega )^*\) be the nonlinear operator defined by
This operator has the following properties (see Problem 2.192 of Gasinski–Papageorgiou [4]).
Proposition 2.5
If hypotheses \(\mathbf {{\hat{H}}}(i)\), (ii), (iii) hold, then the operator \(V(\cdot )\) is bounded (that is, maps bounded sets to bounded sets), continuous, monotone (hence, maximal monotone too) and of type \((S)_+\), that is, it has the following property, if \(u_n \overset{w}{\rightarrow }\ u\) in \(W^{1,p}(\Omega )\) and \(\limsup \nolimits _{n\rightarrow \infty }\langle V(u_n),u_n-u\rangle \le 0\), then \(u_n \rightarrow u\) in \(W^{1,p}(\Omega )\).
The hypotheses on the potential function \(\xi (.)\) and the boundary coefficient \(\beta (.)\) are the following:
\(\mathbf {H_0}\): \(\xi \in L^{\infty }(\Omega )\), \(\beta \in C^{0,\alpha }(\partial \Omega )\) with \(\alpha \in (0, 1]\), \(\beta (z) \ge 0\) for all \(z \in \partial \Omega \) and \(\xi \ne 0\) or \(\beta \ne 0\).
The above hypotheses imply that
is an equivalent norm on \(W^{1,p}(\Omega )\).
In the boundary condition, \(\frac{\partial u}{\partial n_a}\) denotes the conormal derivative of u corresponding to the map a(.) and it is interpreted using the nonlinear Green’s identity (see Paoageorgiou–Radulescu–Repovs [11], p.34).
The hypotheses on the perturbation f(z, x) are the following:
\({\textbf{H}}\): \(f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}\) be a Caratheodory function such that \(f(z, 0) = 0\) for almost every \(z \in \Omega \), and satisfies the following conditions:
-
(i)
\(0\le f(z, x) \le a(z)(1 + x^{r-1})\) for almost every \(z \in \Omega \) and all \(x\ge 0\), where \(a \in L^{\infty }(\Omega )\) and \(p<r<p^*\);
-
(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 every \(z \in \Omega \);
-
(iii)
there exists \(\theta \in \big (\max \{1,(r-p)\frac{N}{p}\},p^*\big )\) such that
$$\begin{aligned} 0<\beta \le \liminf _{x \rightarrow \infty } \frac{f(z,x)x-pF(z,x)}{x^\theta } \ \text { uniformly for almost every } \ z \in \Omega ; \end{aligned}$$ -
(iv)
\(\lim _{x\rightarrow 0^+}\frac{f(z,x)}{x^{q-1}}=0\) uniformly for almost every \(z\in \Omega \);
-
(v)
for every \(s>0\), there exists \(\eta _{s} > 0\) such that \(0<\eta _{s} \le f(z,x)\) for almost every \(z\in \Omega \), all \(x\ge s\) and for every \(\rho >0\), there exists \({\hat{\xi }}_{\rho }>0\) such that for almost every \(z\in \Omega \)
$$\begin{aligned} x\rightarrow f(z,x)+{\hat{\xi }}_{\rho }x^{p-1} \end{aligned}$$is nondecreasing on \([0,\rho ]\).
Remark 2.6
Since we look for positive solutions and all the above hypotheses concern the positive semiaxis, we may assume that \(f(z,x)=0\) for almost every \(z\in \Omega \), all \(x\le 0\). Hypotheses H(ii), (iii) imply that
So the perturbation f(z, x) is \((p-1)\)-superlinear. However, this superlinearity is not formulated in terms of Ambrosetti–Rabinowitz condition, which is common in the literature when dealing with superlinear problems (see Struwe [18], p.102). Instead, we use a less restrictive condition (see H(iii)), which incorporates in our framework also superlinear functions with “slower” growth as \(x\rightarrow \infty \). For example, the function
with \(q<s\le p\) and \(1<\eta \le p\), satisfies hypotheses H, but fails to satisfy the Ambrosetti–Rabinowitz condition.
In what follows \(\gamma _{p}:W^{1,p}(\Omega )\rightarrow {\mathbb {R}}\) is the \(C^1\)-functional defined by
Consider the following auxiliary Robin problem
Reasoning as in the proof of Proposition 3.3 of Bai–Papageorgiou–Zeng [2], we obtain the following result.
Proposition 2.7
If hypotheses \({\hat{H}}\), \(H_{0}\) hold, then problem (2) admits a unique positive solution
We introduce the following two sets:
In the next section, we establish the properties of these two sets.
3 Positive Solutions
First, we prove the nonemptiness of \({\mathfrak {L}}\) and the regularity of the elements of \(S_{\lambda }.\)
Fix \(\lambda >0\) and let \(k>\frac{1}{\lambda }\Vert \xi \Vert _{\infty }\). We introduce the Caratheodory function \(l_{\lambda }(z,x)\) defined by
with \({\bar{u}}\in int C_+\) being the unique positive solution of (2) (see Proposition 2.7).
We set \(L_{\lambda }(z,x)=\int _{0}^{x}l_{\lambda }(z,s)ds\) and consider the \(C^{1}\)-functional \(\psi _{\lambda }:W^{1,p}(\Omega )\rightarrow {\mathbb {R}}\) defined by
Proposition 3.1
If hypotheses \({\hat{H}}, H_{0},H\) hold, then \({\mathfrak {L}}\ne \emptyset \) and \(S_{\lambda }\subseteq int C_{+}\) for every \(\lambda >0 \).
Proof
For every \(u\in W^{1,p}(\Omega ) \), we have
where
for all \(x\ge 0\). Note that \(R_{\lambda }\ge 0\) and \(c{\bar{u}}(.)^{\tau -1}+\lambda f(.,{\bar{u}}(.) )+\lambda k{\bar{u}}(.)^{p-1}\in L^{\infty }(\Omega )\). Hence we have
Hypotheses H(i), (iv) imply that
Since \(1<q<p<r\) we have \(\Vert u\Vert ^p\le \Vert u\Vert ^q+\Vert u\Vert ^r.\) Therefore we can write that
Let \(\rho =\Vert u\Vert \). If \(\rho >1\) is large, then since \(1<\tau <p\), we have
We fix such a large \(\rho >1\) and choose \(\lambda _{0}>0\) small such that
Then from (3.2) it follows that
On account of hypotheses \({\hat{H}}(iv)\), we can find \(\delta >0\) such that
Let \(u\in C^1(\bar{\Omega })\) with \(u(z)>0\) for all \(z\in \Omega \). Since \({\bar{u}}\in int C_+\) (see Proposition 2.7), using Proposition 4.1.22, p.274 of Papageorgiou–Radulescu–Repovs [11], we can find \(t\in (0,1)\) small such that
Since \(q>1\), choosing \(t\in (0,1)\) even smaller if necessary, we see that
we consider the closed ball
Since \(W^{1,p}(\Omega )\) is reflexive, \({\bar{B}}_\rho \) is w-compact and by the Eberlein–Smulian theorem, it is sequently w-compact. Also, using the Sobolev embedding theorem, we see that \(\psi _{\lambda }(.)\) is sequentially weakly lower semicontinuous. So, by the Weierstrass–Tonelli theorem, we can find \({\hat{u}}_{\lambda }\in {\bar{B}}_\rho \) such that
From (3.3) and (3.5), we infer that
Then from (3.6) it follows that
In (3.7), we use the test function \(h=({\bar{u}}-{\hat{u}}_{\lambda })^+ \in W^{1,p}(\Omega ) \). Then we have
(by Proposition 2.7)
then
From (3.8), (3.1), (3.7) it follows that \({\hat{u}}_{\lambda }\in S_{\lambda }\) and hence \((0,\lambda _{0})\subset {\mathfrak {L}}\ne \emptyset \).
From Papageorgiou–Radulescu [10], we have that if \(u\in S_{\lambda }\), then \(u\in L^{\infty }(\Omega )\). Therefore nonlinear regularity theory of Lieberman [7] implies that \(u\in C_{+}\backslash \{0\}\).
Let \(k_{0}>\Vert \xi \Vert _\infty \). Then
with \(k_{0}+\xi (.)\in L^{\infty }(\Omega )\), \(k_{0}+\xi (z)>0\) for almost all \(z\in \Omega \). Then from Pucci and Serrin [17] (p.120) we infer that
\(\square \)
In the next proposition, we show that \({\mathfrak {L}}\) is a connected set (an interval).
Proposition 3.2
If hypotheses \({\hat{H}}, H_{0},H\) hold, \(\lambda \in {\mathfrak {L}}\) and \(0<\mu <\lambda \) then \(\mu \in {\mathfrak {L}}\).
Proof
Since \(\lambda \in {\mathfrak {L}}\) we can find \(u_{\lambda } \in S_{\lambda }\subseteq intC_+\). Let \(k_0>\Vert \xi \Vert _\infty \) and introduce the Caratheodory function \(\eta _\mu (z,x)\) defined by
We set \(H_{\mu }(z,x)=\int _{0}^{x}\eta _{\mu }(z,s)ds\) and consider the \(C^1\)-functional \({\hat{\psi }}_{\mu }:W^{1,p}(\Omega )\rightarrow {\mathbb {R}}\) defined by
Since \(k_{0}>\Vert \xi \Vert _{\infty }\), from (3.9) we see that \({\hat{\psi }}_{\mu }(.)\) is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find \(u_{\mu }\in W^{1,p}(\Omega )\) such that
Let \(u\in C^1({\bar{\Omega }})\) with \(u(z)>0\) for all \(z\in \Omega \). As before since \(u_{\lambda }\in int C_+\), we can find \(t\in (0,1)\) small such that
From (3.9) and (3.4) and since \(\tau <q\) (see hypotheses H(iv)) we see that for \(t\in (0,1)\) small
then
So, \(u_{\mu }\ne 0.\)
From (3.10) we have
In (3.11), first, we choose the test function \(h=-u_{\mu }^-\in W^{1,p}(\Omega ) \). Since \(k_0>\Vert \xi \Vert _{\infty }\), we obtain
then \(u_{\mu }\ge 0\), \(u_{\mu }\ne 0.\)
Next in (3.11), we choose the test function \(h=(u_{\mu }-{u}_{\lambda })^+ \in W^{1,p}(\Omega ) \). Using (3.9), we have
Since \(u_{\lambda }\in S_{\lambda }\)
then
So, we have proved that
From (3.12), (3.9), (3.11), we see that
then
\(\square \)
Hidden in the above proof is the following corollary.
Corollary 3.3
If hypotheses \({\hat{H}},H_{0},H\) hold, \(\lambda \in {\mathfrak {L}}\), \(u_{\lambda }\in S_{\lambda }\) and \(0<\mu <\lambda \), then \(\mu \in {\mathfrak {L}}\) and there exists \(u_{\mu }\in S_{\mu }\) such that \(u_{\mu }\le u_{\lambda }\).
We can improve this corollary.
Proposition 3.4
If hypotheses \({\hat{H}},H_{0},H\) hold, \(\lambda \in {\mathfrak {L}}\), \(u_{\lambda }\in S_{\lambda }\) and \(0<\mu <\lambda \), then \(\mu \in {\mathfrak {L}}\) and there exists \(u_{\mu }\in S_{\mu }\) such that
Proof
From Corollary (3.3) we already know that \(\mu \in {\mathfrak {L}}\) and there exists \(u_{\mu }\in S_{\mu }\subseteq intC_{+}\) such that
Let \(\rho =\Vert u_{\lambda }\Vert _{\infty }\) and let \({\hat{\xi }}_{\rho }>0\) be as postulated by hypotheses H(v). We can always take \({\hat{\xi }}_{\rho }>\frac{1}{\lambda }\Vert \xi \Vert _{\infty }\). We have
Since \(u_{\mu }\in intC_{+}\), we have \(u_{\mu }(z)\ge s>0\) for all \(z\in {\bar{\Omega }}\). Therefore
Then from (3.14) and Proposition 6 of Papageorgiou–Radulescu–Repovs [13], we have that
\(\square \)
Let \(\lambda ^{*}=\sup {\mathfrak {L}}.\)
Proposition 3.5
If hypotheses \({\hat{H}},H_{0},H\) hold, then \(\lambda ^{*}<\infty .\)
Proof
Hypotheses H and the fact that \(\xi \in L^{\infty }(\Omega )\), imply that we can find \(\lambda _{0}>0\) such that
Let \(\lambda > \lambda _{0}\) and suppose that \(\lambda \in {\mathfrak {L}}\). We can find \(u_{\lambda }\in S_{\lambda }\subseteq int C_+.\) Let \(m=\underset{{\bar{\Omega }}}{min} \ u_{\lambda }>0\) and for \(\delta >0\) let \(m_{\delta }=m+\delta \). For \(\rho =\Vert u_{\lambda }\Vert _{\infty }\), let \({\hat{\xi }}_{\rho }>0\) be as postulated by hypothesis H(v). We can always take \({\hat{\xi }}_{\rho }>\frac{1}{\lambda }\Vert \xi \Vert _{\infty }\). Then we have
Since \(0\le \eta _{m}\le f(z,m)\) (see hypothesis H(v)) as before using Proposition 6 of [13], from (3.16) we obtain that
a contradiction to the definition of m. Therefore \(\lambda \notin {\mathfrak {L}}\) and so \(\lambda ^*\le \lambda _{0}<\infty \). \(\square \)
Therefore we can say that \({\mathfrak {L}}\subset (0,\infty )\) is a bounded interval and \((0,\lambda ^{*})\subseteq {\mathfrak {L}}\subseteq (0,\lambda ^*]\).
Next, we show that for \(\lambda \in (0,\lambda ^*)\), we have a multiplicity of positive solutions for problem \((p_{\lambda })\).
Proposition 3.6
If hypotheses \({\hat{H}}\), \(H_{0}\), H hold, then problem \((p_{\lambda })\) has at least two positive solutions
Proof
Let \(0<\mu<\lambda <\theta \). From Proposition (3.4), we know that we can find \(u_{\theta }\in S_{\theta }\), \(u_{0}\in S_{\lambda }\) and \(u_{\mu }\in S_{\mu }\) such that
Let \(k>\frac{1}{\lambda }\Vert \xi \Vert _{\infty }\) and introduce the Caratheodory function \(j_{\lambda }(z,x)\) defined by
We set \(J_{\lambda }(z,x)=\int _{0}^{x}j_{\lambda }(z,s)ds\) and consider the \(C^{1}\)-functional \(\sigma _{\lambda }:W^{1,p}(\Omega )\rightarrow {\mathbb {R}}\) defined by
Using (3.18) and the nonlinear regularity theory, we obtain that
(see also the proof of Proposition (3.2)). From (3.18) and since \(k>\frac{1}{\lambda }\Vert \xi \Vert _{\infty }\), we see that \({\sigma }_{\lambda }(.)\) is coercive. Also, it is sequentially weakly lower semicontinuous. Therefore we can find \({\tilde{u}}_{0}\in W^{1,p}(\Omega )\) such that
If \({\tilde{u}}_{0}\ne u_0\), then on account of (3.19), (3.18), we see that \({\tilde{u}}_{0}\) is the second positive smooth solution of (\(p_{\lambda }\)) and so we are done. Therefore, we assume that \({\tilde{u}}_{0}=u_{0}\in intC_+\). Hence \(u_{0}\) is a minimizer of \({\sigma }_{\lambda }(.)\). We introduce another Caratheodory function \({\hat{j}}_{\lambda }(z,x)\) defined by
We set \({\hat{J}}_{\lambda }(z,x)=\int _{0}^{x}\hat{j}_{\lambda }(z,s)ds\) and consider the \(C^{1}\)-functional \({\hat{\sigma }}_{\lambda }:W^{1,p}(\Omega )\rightarrow {\mathbb {R}}\) defined by
Using (3.20) and the nonlinear regularity theory, we obtain that
From (3.18) and (3.20), we see that
Recall that \(u_0\) is a minimizer of \(\sigma _{\lambda }(.)\). Then from (3.22) and (3.17), we infer that
(see [10], Prpoposition 2.12). From (3.21) and (3.20), we see that we may assume that \(K_{\sigma _{\lambda }}\) is finite or otherwise we already have an infinity of positive smooth solutions and so we are done. Then (3.23) and Theorem 5.76, p.449 of [11], imply that we can find \(\rho \in (0,1)\) small such that
If \(u\in intC_+\), then on account of hypotheses H(ii), we have
Using hypotheses H(iii) and reasoning as in the proof of Proposition 4 in [14], we show that
From (3.24), (3.25) and (3.26) and the mountain pass theorem we can find \({\hat{u}}\in W^{1,p}(\Omega )\) such that
Therefore \({\hat{u}}\in intC_+\) is the second positive smooth solution of \((p_\lambda )\) (\(\lambda \in (0,\lambda ^*)\)), distinct from \(u_{0}\). \(\square \)
We need to determine the status of the critical parameter \(\lambda ^*>0\).
Proposition 3.7
If hypotheses \({\hat{H}},H_{0},H\) hold, then \(\lambda ^* \in {\mathfrak {L}}\).
Proof
Let \(\{\lambda _n\}_{n\in {\mathbb {N}}}\subseteq {\mathfrak {L}} \) be such that \(\lambda _n \rightarrow \lambda ^*\). From the proof of Proposition (3.8), we know that we can find \(u_n\in S_{\lambda _{n}}\subseteq intC_+, \ \ n\in {\mathbb {N}}\) such that
Note that in the definition of \({\hat{J}}_{\lambda _{n}}(.,.)\) (see (3.20)), we replace \(u_{\mu }\) by \(u_1\) and \(k>\frac{1}{\lambda _{1}}\Vert \xi \Vert _{\infty }\). We have
Also we have
From (3.29) and (3.30) and following the argument which shows that the functional \({\hat{\sigma }}_\lambda (.)\) satisfies the C-condition (see in [14] the proof of Proposition 4 and recall that \(\lambda _n\rightarrow \lambda ^*\)), we obtain that
Passing to the limit as \(n\rightarrow \infty \) in (3.30) and using (3.31), we obtain
\(\square \)
Summarizing, we can state the following existence and multiplicity theorem for the problem \((p_{\lambda })\), which is global in the parameter \(\lambda >0\) (a bifurcation-type theorem).
Theorem 3.8
If hypotheses \({\hat{H}},H_{0},H\) hold, then there exists \(\lambda ^*\) such that
-
(a)
for all \(\lambda \in (0,\lambda ^*)\), problem \((p_{\lambda })\) has at least two positive solutions
$$\begin{aligned} u_{0},{\hat{u}}\in intC_{+}; \end{aligned}$$ -
(b)
for \(\lambda =\lambda ^*\), problem \((p_{\lambda })\) has at least one positive solution
$$\begin{aligned} u^{*}\in intC_{+}; \end{aligned}$$ -
(c)
for all \(\lambda >\lambda ^*\), problem \((p_{\lambda })\) has no positive solutions.
Data Availability Statements
Data sharing not applicable to this article as no data sets were generated or analysed during the current study.
References
Ambrosetti, A., Brezis, H., Cerami, G.: Combined effects of concave and convex nonlinearities in some elliptic problems. J. Funct. Anal. 122, 519–543 (1994)
Bai, Y., Papageorgiou, N.S., Zeng, S.: Multiplicity results for nonlinear nonhomogeneous robin problems with indefinite potential term. Results Math. 78, 134 (2023). https://doi.org/10.1007/s00025-023-01907-5
Garcia Azorero, J., Peral Alonso, I., Manfredi, Juan: Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations. Commun. Contemp. Math. 02(03), 385–404 (2000)
Gasinski, L., Papageorgiou, N.S.: Exercises in Analysis. Part 2: Nonlinear Analysis. Springer, Cham (2016)
Gasinski, L., Papageorgiou, N.S., Zhang, Y.: Positive solutions for a class of nonlinear parametric Robin problems. Rend. Circ. Mat. Palermo II. Ser (2023). https://doi.org/10.1007/s12215-023-00918-z
Guo, Z., Zhang, Z.: \(W^{1, p}\) versus \(C^1\) local minimizers and multiplicity results for quasilinear elliptic equations. J. Math. Anal. Appl. 286(1), 32–50 (2003)
Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Commun. Part. Differ. Equ. 16, 311–361 (1991)
Marano, S.A., Marino, G., Papageorgiou, N.S.: On a Dirichlet problem with (p, q)-Laplacian and parametric concave-convex nonlinearity. J. Math. Anal. Appl. 475(2), 1093–1107 (2019)
Papageorgiou, N.S., Radulescu, V.D.: Coercive and noncoercive Neumann problems with indefinite potential. Forum Math. 28, 545–571 (2016)
Papageorgiou, N.S., Radulescu, V.D.: Nonlinear nonhomogeneous Robin problems with superlinear reaction term. Adv. Nonlinear Stud. 16, 737–764 (2016)
Papageorgiou, N.S., Radulescu, V.D., Repovs, D.D.: Nonlinear Analysis-Theory and Methods. Springer, Cham (2019)
Papageorgiou, N.S., Radulescu, V.D., Repovs, D.D.: Anisotropic equations with indefinite potential and competing nonlinearities. Nonlinear Anal. 201, 111861 (2020)
Papageorgiou, N.S., Radulescu, V.D., Repovs, D.D.: Nonlinear nonhomogeneous singular problems. Calc. Var. 59, 9 (2020)
Papageorgiou, N.S., Radulescu, V.D., Zhang, Y.: Anisotropic singular double phase Dirichlet problems. Discr. Cont. Dyn. Syst. 14, 4465–4502 (2021)
Papageorgiou, N.S., Radulescu, V.D., Zhang, W.: Global existence and multiplicity for nonlinear Robin eigenvalue problems. Results Math. 78, 133 (2023)
Papageorgiou, N.S., Vetro, C., Vetro, F.: Superlinear Robin problems with indefinite linear part. Bull. Malays. Math. Sci. Soc. 43, 537–562 (2020)
Pucci, P., Serrin, J.B.: The Maximum Principle. Springer, Basel (2007)
Struwe, M.: Variational Methods. Springer, Berlin (1990)
Funding
Open access funding provided by the Scientific and Technological Research Council of Türkiye (TÜBïTAK). The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
Author information
Authors and Affiliations
Contributions
All authors contributed to the study conception and design. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or non-financial interests to disclose. The authors declares that there are no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Öztürk, E., Papageorgiou, N.S. Global Multiplicity of Positive Solutions for Nonlinear Robin Problems with an Indefinite Potential Term. Results Math 79, 113 (2024). https://doi.org/10.1007/s00025-024-02142-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-024-02142-2
Keywords
- Nonlinear nonhomogeneous differential operator
- indefinite potential
- positive solutions
- truncations and comparisons
- nonlinear regularity