Abstract
We consider a parametric nonlinear Robin problem driven by the negative p-Laplacian plus an indefinite potential. The equation can be thought as a perturbation of the usual eigenvalue problem. We consider the case where the perturbation \(f(z,\cdot )\) is \((p-1)\)-sublinear and then the case where it is \((p-1)\)-superlinear but without satisfying the Ambrosetti–Rabinowitz condition. We establish existence and uniqueness or multiplicity of positive solutions for certain admissible range for the parameter \(\lambda \in {\mathbb {R}}\) which we specify exactly in terms of principal eigenvalue of the differential operator.
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1 Introduction
Let \(\varOmega \subseteq {\mathbb {R}}^N\) be a bounded domain with a \(C^2\)-boundary \(\partial \varOmega \). In this paper we study the following nonlinear parametric Robin problem
In this problem \(\varDelta _p\) denotes the p-Laplace differential operator defined by
Also \(\xi (\cdot ) \in L^\infty (\varOmega )\) is an indefinite (that is, sign changing) potential function, \(\lambda \in {\mathbb {R}}\) is a parameter and f(z, x) is a Carathéodory perturbation function (that is, for all \(x \in {\mathbb {R}}\), \(z \rightarrow f(z,x)\) is measurable and for a.a. \(z \in \varOmega \), \(x \rightarrow f(z,x)\) is continuous). In the boundary condition \(\dfrac{\partial u}{\partial n_p}\) denotes the generalized normal derivative defined by
with \(n(\cdot )\) being the outward unit normal on \(\partial \varOmega \). This kind of generalized normal derivative is dictated by the nonlinear Green’s identity (see, for example, Gasiński–Papageorgiou [8] (p. 211)). The boundary weight term \(\beta \in C^{0,\alpha }(\partial \varOmega )\) (\(0<\alpha <1\)) and \(\beta (z) \ge 0\) for all \(z \in \partial \varOmega \).
Problem \((P_{\lambda })\) can be viewed as a perturbation of the usual eigenvalue problem for the Robin p-Laplacian plus an indefinite potential. We look for positive solutions and we consider two distinct cases depending on the growth of the perturbation \(f(z,\cdot )\) near \(+\infty \):
-
\(f(z,\cdot )\) is \((p-1)\)-sublinear.
-
\(f(z,\cdot )\) is \((p-1)\)-superlinear.
Let \({\widehat{\lambda }}_1 \in {\mathbb {R}}\) be the principal eigenvalue of the differential operator \(u \rightarrow - \varDelta _p u + \xi (z)|u|^{p-2}u\) with Robin boundary condition. In the first case (\((p-1)\)-sublinear perturbation), we show that for all \(\lambda \ge {\widehat{\lambda }}_1\), problem \((P_{\lambda })\) has no positive solution, while for \(\lambda < {\widehat{\lambda }}_1\), problem \((P_{\lambda })\) has at least one positive solution. Moreover, this positive solution is unique, if we impose a monotonicity condition on the quotient \(x \rightarrow \dfrac{f(z,x)}{x^{p-1}}\) for \(x>0\). In the second case (\((p-1)\)-superlinear perturbation), the situation changes and uniqueness of the positive solution fails. In fact the problem exhibits a kind of bifurcation phenomenon. Namely, for \(\lambda \ge {\widehat{\lambda }}_1\) problem \((P_{\lambda })\) has no positive solution, while for \(\lambda < {\widehat{\lambda }}_1\) problem \((P_{\lambda })\) has at least two positive solutions. Finally for both situations, we establish the existence of minimal positive solutions. Our work here extends to the p-Laplacian that of Papageorgiou–Rǎdulescu–Repovš [20]. Eigenvalue problems for the p-Laplacian plus an indefinite potential were studied by Papageorgiou–Rǎdulescu [18] (semilinear problems (that is, \(p=2\)) with Robin boundary condition) and by Mugnai–Papageorgiou [16] (nonlinear problems with Neumann boundary condition (that is, \(\beta \equiv 0\))). Both works deal with nonparametric problems and prove existence and multiplicity results under resonance conditions. We also mention the works of Hu–Papageorgiou [10,11,12]. In [11] the authors treat superdiffusive logistic equation with Robin boundary condition, while in [10, 12], they deal with equations driven by a nonhomogeneous differential operator.
2 Auxiliary results
In this section we present some auxiliary results and notions which we will need in the sequel.
First we deal with the following eigenvalue problem:
Our hypotheses on the functions \(\xi (\cdot )\) and \(\beta (\cdot )\) are the following:
- \(H(\xi )\)::
-
\(\xi \in L^\infty (\varOmega )\).
- \(H(\beta )_1\)::
-
\(\beta \in C^{0,\alpha }(\partial \varOmega )\) with \(\alpha \in (0,1)\) and \(\beta (z) \ge 0\) for all \(z \in \partial \varOmega \).
In addition to the Sobolev space \(W^{1,p}(\varOmega )\), we will also use the Banach space \(C^1({\overline{\varOmega }})\) which is an ordered Banach space with positive cone \(C_+=\{u \in C^1({\overline{\varOmega }}) \, : \, u(z) \ge 0 \text{ for } \text{ all } z \in {\overline{\varOmega }}\}\). This cone has a nonempty interior given by
Also on \(\partial \varOmega \) we consider the \((N-1)\)-dimensional Hausdorff (surface) measure \(\sigma (\cdot )\). With this measure on \(\partial \varOmega \), we can define the Lebesgue spaces \(L^\tau (\partial \varOmega )\) \(1 \le \tau \le +\infty \). We know that there exists a unique continuous linear map \(\gamma _0 : W^{1,p}(\varOmega ) \rightarrow L^p(\partial \varOmega )\) known as the “trace map” s.t. \(\gamma _0(u)=u |_{\partial \varOmega }\) for all \(u \in W^{1,p}(\varOmega ) \cap C({\overline{\varOmega }})\). So, we understand the trace map as representing the “boundary values” of a Sobolev function \(u \in W^{1,p}(\varOmega )\). We know that \(\gamma _0\) is compact into \(L^\tau (\partial \varOmega )\) for all \(\tau \in \left[ 1, \dfrac{(N-1)p}{N-p}\right) \) when \(p<N\) and into \(L^\tau (\partial \varOmega )\) for all \(\tau \in [1,+\infty )\) when \(p \ge N\). Moreover, we have
In the sequel for the sake of notational simplicity we drop the use of the trace map \(\gamma _0\). It is understood that all restrictions of Sobolev functions on \(\partial \varOmega \) are taken in the sense of traces.
In what follows by \(\vartheta : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}\) we denote the \(C^1\)-functional defined by
From Fragnelli–Mugnai–Papageorgiou [7], we have the following proposition concerning problem (1) (see also Mugnai–Papageorgiou [16] and Papageorgiou–Rǎdulescu [18] where special cases of (1) are investigated).
Proposition 1
If hypotheses \(H(\xi )\), \(H(\beta )_1\) hold, then problem (1) admits a smallest eigenvalue \({\widehat{\lambda }}_1 \in {\mathbb {R}}\) s.t.
-
$$\begin{aligned} {\widehat{\lambda }}_1=\inf \left[ \frac{\vartheta (u)}{\Vert u\Vert _p^p} \, : \, u \in W^{1,p}(\varOmega ), \, u \ne 0 \right] . \end{aligned}$$(2)
-
\({{\widehat{\lambda }}}_1\) is isolated and simple.
-
The infimum in (2) is realized on the one-dimensional eigenspace of \({\widehat{\lambda }}_1\); the elements of this eigenspace do not change sign and if \({\widehat{u}}_1\) denotes the positive, \(L^p\)-normalized (that is, \(\Vert {\widehat{u}}_1\Vert _p=1\)) eigenfunction, then \({\widehat{u}}_1 \in D_+\).
-
If \({\widehat{\lambda }}>{\widehat{\lambda }}_1\) is another eigenvalue and \({\widehat{u}} \in W^{1,p}(\varOmega )\) a corresponding eigenfunction, then \({\widehat{u}} \in C^1({\overline{\varOmega }})\) is nodal (that is, sign changing).
-
As a consequence of these properties, we have the following useful lemma.
Lemma 1
If hypotheses \(H(\xi )\), \(H(\beta )_1\) hold, \(\eta \in L^\infty (\varOmega )\), \(\eta (z) \le {\widehat{\lambda }}_1\) for a.a. \(z \in \varOmega \) and the inequality is strict on a set of positive measure, then there exists \({\widehat{c}}>0\) s.t.
Proof
Let \(\zeta : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}\) be the \(C^1\)-functional defined by
From (2) we have \(\zeta \ge 0\). Suppose that the claim of the lemma is not true. Then we can find \(\{u_n\}_{n \ge 1} \subseteq W^{1,p}(\varOmega )\) s.t.
The p-homogeneity of \(\zeta (\cdot )\) implies that we may assume that \(\Vert u_n\Vert _p=1\) for all \(n \in {\mathbb {N}}\). Then clearly \(\{u_n\}_{n \ge 1} \subseteq W^{1,p}(\varOmega )\) is bounded (see hypotheses \(H(\xi )\), \(H(\beta )_1\)) and so we may assume that
To fix things we assume that \(\mu >0\) (the reasoning is the same if \(\mu <0\)). Then from (5) and since \(u=\mu {\widehat{u}}_1 \in D_+\), we have
wich contradicts (2). \(\square \)
Let \(A:W^{1,p}(\varOmega ) \rightarrow W^{1,p}(\varOmega )^*\) be the nonlinear map defined by
From Motreanu–Motreanu–Papageorgiou [14] (p. 40), we have the following result concerning this map.
Proposition 2
The map \(A(\cdot )\) is bounded (that is, maps bounded sets to bounded sets), monotone, continuous (hence maximal monotone too) and of type \((S)_+\), that is, if \( u_n \xrightarrow {w} u\) in \(W^{1,p}(\varOmega )\) and \(\limsup _{n \rightarrow + \infty } \langle A(u_n),u_n-u\rangle \le 0\), then \(u_n \rightarrow u\) in \(W^{1,p}(\varOmega )\).
Recall that if X is a Banach space and \(\varphi \in C^1(X, {\mathbb {R}})\), then we say that \(\varphi \) satisfies the Cerami condition (the C-condition for short), if the following is true:
“Every sequence \(\{u_n\}_{n \ge 1} \subseteq X\) s.t. \(\{\varphi (u_n)\}_{n \ge 1} \subseteq {\mathbb {R}} \) is bounded and \((1 + \Vert u_n\Vert ) \varphi '(u_n) \rightarrow 0 \) in \(X^*\) as \(n \rightarrow +\infty \), admits a strongly convergent subsequence”.
Let \(f_0: \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) be a Carathéodory function s.t.
with \(a \in L^\infty (\varOmega )_+\) and \(p^*= {\left\{ \begin{array}{ll} \frac{Np}{N-p} &{} \text{ if } p<N\\ +\infty &{}\text{ if } N \le p\end{array}\right. }\) (the critical Sobolev exponent). Let \(F_0(z,x)= \int _0^x f_0(z,s)ds\) and consider the \(C^1\)-functional \(\varphi _0:W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}\) defined by
From Papageorgiou–Rǎdulescu [17], we have the following result relating local minimizers of \(\varphi _0\) and which is an outgrowth of the nonlinear regularity theory. The first such result was proved by Brezis-Nirenberg [4] for \(p=2\) and the space \(H_0^1(\varOmega )\).
Proposition 3
If \(u_0 \in W^{1,p}(\varOmega )\) is a local \(C^1({\overline{\varOmega }})\)-minimizer of \(\varphi _0\), that is, there exists \(\delta _1>0\) s.t.
then \(u_0 \in C^{1,\tau }({\overline{\varOmega }})\) with \(\tau \in (0,1)\) and it is also a local \(W^{1,p}(\varOmega )\)-minimizer of \(\varphi _0\), that is, there exists \(\delta _2>0\) s.t.
To make good use of this result, we need a strong comparison principle. In this direction we have the following proposition which is a special case of a more general result due to Fragnelli–Mugnai–Papageorgiou [6]. Given \(h_1,h_2 \in L^\infty (\varOmega )\), we say that \(h_1 \prec h_2\) if and only if for every \(K \subseteq \varOmega \) compact, there exists \(\varepsilon =\varepsilon (K)>0\) s.t.
Note that if \(h_1,h_2 \in C(\varOmega )\) and \(h_1(z)<h_2(z)\) for all \(z \in \varOmega \), then \(h_1 \prec h_2\).
Proposition 4
If \(\xi , h_1,h_2 \in L^\infty (\varOmega )\), \(h_1 \prec h_2\), \(u \in C^1({\overline{\varOmega }}) {\setminus } \{0\}\), \( v \in D_+\) and they satisfy
then \((v-u)(z)>0\), for all \(z \in \varOmega \) and \(\dfrac{\partial (v-u)}{\partial n}\Big |_{D_0}<0\) where \(D_0=\{z \in \partial \varOmega : v(z)=u(z)\}\).
Remark 1
If in \(C^1({\overline{\varOmega }})\) we introduce the order cone
then the above proposition says that \(v-u \in {\mathrm{int }} \, {\widehat{C}}_+\). If \(D_0 = \emptyset \), then \({\widehat{C}}_+=C_+\).
For problem \((P_{\lambda })\), we introduce the following two sets:
For the set \(S(\lambda )\) we have the following general result.
Proposition 5
If hypotheses \(H(\xi )\), \(H(\beta )_1\) hold and \(f: \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a Carathéodory function s.t. for a.a. \(z \in \varOmega \), \(f(z,0)=0\), \(f(z,x) \ge 0\) for all \(x > 0\), \(f(z,x)=0\) for all \(x < 0\) and \(f(z,x) \le a(z)(1+x^{p^*-1})\) for a.a. \(z \in \varOmega \), all \(x \ge 0\), with \(a \in L^\infty (\varOmega )_+\), then \(S(\lambda ) \subseteq D_+\) (possibly empty).
Proof
Suppose that \(u \in S(\lambda )\). Then
(see Papageorgiou–Rǎdulescu [17]). From (6) and Papageorgiou–Rǎdulescu [19] we have \(u \in L^\infty (\varOmega )\). Then Theorem 2 of Lieberman [13] implies that \(u \in C_+ {\setminus } \{0\}\). From (6) and since \(f \ge 0\), we have
\(\square \)
Proposition 6
If hypotheses \(H(\xi )\), \(H(\beta )_1\) hold, \(f: \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a Carathéodory function s.t. for a.a. \(z \in \varOmega \) \(f(z,0)=0\), \(f(z,x)>0\) for all \(x>0\), \(f(z,x) \le a(z)(1+x^{p^*-1})\) for a.a. \(z \in \varOmega \), all \(x\ge 0\), with \(a \in L^\infty (\varOmega )_+\) and \(\lambda \ge {\widehat{\lambda }}_1\), then \(S(\lambda )=\emptyset \).
Proof
Arguing by contradiction, suppose that \(S(\lambda ) \ne \emptyset \) and let \(u \in S(\lambda )\). From Proposition 5 we know that \(u \in D_+\). Also, let \({\widehat{u}}_1 \in D_+\) be the principal eigenfunction from Proposition 5. Consider the function
From the nonlinear Picone’s identity of Allegretto-Huang [2] (see also Motreanu–Motreanu–Papageorgiou [14] (p. 255)), we have
Then we have
a contradiction. Therefore \(S(\lambda )=\emptyset \) for all \(\lambda \ge {\widehat{\lambda }}_1\). \(\square \)
3 \((p-1)\)-sublinear perturbation
In this section, we deal with the case of a \((p-1)\)-sublinear perturbation \(f(z,\cdot )\).
\(H_1\): \(f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a Carathéodory function s.t. for a.a. \(z \in \varOmega \), \(f(z,0)=0\), \(f(z,x)>0\) for all \(x >0\) and
-
(i)
for every \(\rho >0\), there exists \(a_\rho \in L^\infty (\varOmega )_+\) s.t. \(f(z,x) \le a_\rho (z)\) for a.a. \(z \in \varOmega \), all \(x \in [0,\rho ]\);
-
(ii)
\(\lim \nolimits _{x \rightarrow + \infty }\dfrac{f(z,x)}{x^{p-1}}=0\) uniformly for a.a. \(z \in \varOmega \);
-
(iii)
there exist \(\delta >0\), \(q \in (1,p)\) and \(c_1>0\) s.t.
$$\begin{aligned} c_1x^{q-1} \le f(z,x) \quad \text{ for } \text{ a.a. } z \in \varOmega , \text{ all } x \in [0,\delta ]. \end{aligned}$$
Remark 2
Since we are looking for positive solutions and the above hypotheses concern the positive semiaxis \({\mathbb {R}}_+=[0,+\infty )\), without any loss of generality we may assume that \(f(z,x)=0\) for a.a. \(z \in \varOmega \), all \(x < 0\). Hypothesis \(H_1\)(ii) says that for a.a. \(z \in \varOmega \) the perturbation \(f (z,\cdot )\) is \((p-1)\)-sublinear near \(+\infty \). Finally hypothesis \(H_1\)(iii) implies the presence of a concave term near \(0^+\).
Example 1
The following functions satisfy hypotheses \(H_1\). For the sake of simplicity we drop the z-dependence.
Proposition 7
If hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_1\) hold and \(\lambda < {\widehat{\lambda }}_1\), then \(S(\lambda ) \ne \emptyset \) and so \({\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)\).
Proof
Let \(\eta > \Vert \xi \Vert _\infty \) and consider the following Carathéodory function
We set \(G_\lambda (z,x)=\int _0^x g_\lambda (z,s)ds\) and consider the \(C^1\)-functional \(\varphi _\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}\) defined by
Hypotheses \(H_1\)(i), (ii) imply that given \(\varepsilon >0\), we can find \(c_2=c_2(\varepsilon )>0\) s.t.
Then for all \(u \in W^{1,p}(\varOmega )\) we have
Here by \(|\cdot |_N\) we denote the Lebesgue measure on \({\mathbb {R}}^N\). Choosing \(\varepsilon \in (0, {\widehat{\lambda }}_1 - \lambda )\) (recall that \(\lambda < {\widehat{\lambda }}_1\)), from (9) and Lemma 1, we have
Using the Sobolev embedding theorem and the compactness of the trace operator, we see that
Then invoking the Weierstrass-Tonelli theorem, we can find \(u_\lambda \in W^{1,p}(\varOmega )\) s.t.
Let \(t \in (0,1)\) be small s.t.
Here \(\delta >0\) is as in hypothesis \(H_1\)(iii). Then we have
Since \(t \in (0,1)\) and \(q<p\), by choosing \(t \in (0,1)\) even smaller if necessary, we have
From (10), we have
In (11) we choose \(h=-u_\lambda ^- \in W^{1,p}(\varOmega )\). Then
Then equation (11) becomes
\(\square \)
In fact we can show that problem \((P_{\lambda })\) for \(\lambda < {\widehat{\lambda }}_1\) has a smallest positive solution.
Fix \(\lambda < {\widehat{\lambda }}_1\) and \(r \in (p,p^*)\). Hypotheses \(H_1\)(i), (ii), (iii) imply that we can find \(c_7(\lambda )>0\) with \(\lambda \rightarrow c_7(\lambda )\) bounded on bounded subsets of \({\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)\) s.t.
This unilateral growth estimate for the reaction term of problem \((P_{\lambda })\) leads to the following auxiliary nonlinear Robin problem:
For this problem we have the following existence and uniqueness result.
Proposition 8
If hypotheses \(H(\xi )\), \(H(\beta )_1\) hold, then for every \(\lambda \in {\mathbb {R}}\) problem (\(Au_\lambda \)) admits a unique positive solution \(u_*^\lambda \in D_+.\)
Proof
First we show the existence of a positive solution for problem (\(Au_\lambda \)). To this end, we consider the \(C^1\)-functional \(\psi _\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}\) defined by
We have
Using (14) in (13) and recalling that \(q<p<r\), we infer that \(\psi _\lambda (\cdot )\) is coercive. Also, it is sequentially weakly lower semicontinuous (use the Sobolev embedding theorem and the compactness of the trace map). So, by the Weierstrass-Tonelli theorem, we can find \(u_*^\lambda \in W^{1,p}(\varOmega )\) s.t.
Since \(q<p<r\), as before (see the proof of Proposition 7), we can show that
From (15) we have
In (16) we choose \(h=-u^{\lambda ^-}_*\in W^{1,p}(\varOmega )\). Then
Therefore Eq. (16) becomes
As before, the nonlinear regularity theory (see [13]) implies \(u_*^\lambda \in C_+ {\setminus } \{0\}\).
Moreover, from (17) we have
Next we show the uniqueness of this positive solution. To this end suppose that \(v_*^\lambda \in W^{1,p}(\varOmega )\) is another positive solution of (\(Au_\lambda \)). As above we can show that \(v^\lambda _* \in D_+\).
We have
Interchanging the roles of \(u_*^\lambda \) and \(v_*^\lambda \) in the above argument, we also have
Adding (18) and (19) and using the nonlinear Picone’s identity, we have
Since the function \(x \rightarrow \dfrac{c_1}{x^{p-q}}-c_7(\lambda )x^{r-p}\) is strictly decreasing on \((0,+\infty )\), from (20) we infer that
This proves the uniqueness of the positive solution \(u_*^\lambda \in D_+\) of problem (\(Au_\lambda \)). \(\square \)
Remark 3
There is an alternative approach to the uniqueness of the positive solution \(u_*^\lambda \in D_+\) of problem (\(Au_\lambda \)) which does not use the nonlinear Picone’s identity. For this we need to assume that \(\beta (z)>0\) for all \(z \in \partial \varOmega \). First note that, if \(\rho =\Vert u_*^\lambda \Vert _\infty \), then we can find \({\widehat{\xi }}_\rho >0\) s.t. for a.a. \(z \in \varOmega \), the function \(x \rightarrow c_1 x^{q-1}-c_7(\lambda )x^{r-1}+ {\widehat{\xi }}_\rho x^{p-1}\) is nondecreasing on \([0,\rho ]\). As before let \(v_*^\lambda \in D_+\) be another positive solution of (\(Au_\lambda \)) and let \(t>0\) be the biggest real s.t.
We assume that \(t \in (0,1)\). We have
Invoking Proposition 4 (recall \(\beta >0\)), we have
where we recall that \({\widehat{C}}_+=\left\{ y \in C^1({\overline{\varOmega }}): y(z) \ge 0 \text{ for } \text{ all } z \in {\overline{\varOmega }}, \, \dfrac{\partial u}{\partial n}\Big |_{\partial \varOmega }\le 0 \right\} \).
Evidently (22) contradicts the maximality of \(t>0\). Therefore we must have \(t \ge 1\) and so
Interchanging the roles of \(u_*^\lambda \in D_+\) and \(v_*^\lambda \in D_+\) in the above argument we also have
So, again we have proved uniqueness of the positive solution of problem (\(Au_\lambda \)). Recall that \(\lambda \rightarrow c_7(\lambda )\) is bounded on bounded sets of \(\lambda \in {\mathbb {R}}\). So, if \(B \subseteq {\mathbb {R}}\) is bounded, \({\widehat{c}}_7 \ge c_7(\lambda )\) for all \(\lambda \in B\) and \({\widehat{u}} \in D_+\) is the unique positive solution of the auxiliary problem
(see Proposition 8), then \({\widehat{u}} \le u_*^\lambda \) for all \(\lambda \in B\).
Next using \(u_*^\lambda \in D_+\), we can have a lower bound for the elements of the set \(S(\lambda )\). This fact will be used to produce the smallest positive solution for problem \((P_{\lambda })\) when \(\lambda < {\widehat{\lambda }}_1\).
So, we have the following result.
Proposition 9
If hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_1\) hold and \(\lambda < {\widehat{\lambda }}_1\), then \(u_*^\lambda \le u\) for all \(u \in S(\lambda )\).
Proof
As before let \(\eta > \Vert \xi \Vert _\infty \). For \(u \in S(\lambda )\) we consider the following Carathéodory function
We set \({\widehat{G}}_\lambda (z,x)= \int _0^x {\widehat{g}}_\lambda (z,s)ds\) and consider the \(C^1\)-functional \({\widehat{\psi }}_\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}\) defined by
From (23) and since \(\eta > \Vert \xi \Vert _\infty \), we see that the functional \({\widehat{\psi }}_\lambda \) is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find \({\widehat{u}}_*^\lambda \in W^{1,p}(\varOmega )\) s.t.
As before, since \(q<p<r\), we have that
From (24) we have
In (25) first we choose \(h=-{\widehat{u}}^{\lambda ^-}_* \in W^{1,p}(\varOmega )\). We obtain
Next in (25) we choose \(({\widehat{u}}^\lambda _*-u)^+ \in W^{1,p}(\varOmega )\). We have
Therefore, we have proved that
Finally we have
\(\square \)
Proposition 10
If hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_1\) hold and \(\lambda < {\widehat{\lambda }}_1\), then problem \((P_{\lambda })\) admits a smallest positive solution \({\overline{u}}_\lambda \in D_+\).
Proof
As in Filippakis–Papageorgiou [5], we have that \(S(\lambda )\) is downward directed, that is, if \(u_1, u_2 \in S(\lambda )\), there is \(u \in S(\lambda )\) s.t. \(u \le u_1\), \(u \le u_2\). Invoking Lemma 3.10 of Hu–Papageorgiou [9] (p. 178), we can find \(\{u_n\}_{n \ge 1} \subseteq S(\lambda )\) decreasing s.t.
We have
for all \(h \in W^{1,p}(\varOmega )\). Since \(u_n \le u_1 \in S(\lambda ) \subseteq D_+\), from (26) and hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_1\)(i) it follows that
So, we may assume that
In (26) we choose \(h=u_n - {\overline{u}}_\lambda \in W^{1,p}(\varOmega )\), pass to the limit as \(n \rightarrow +\infty \) and use (27). Then we have
If in (26) we pass to the limit as \(n \rightarrow +\infty \) and use (28), then
From Proposition 9 we have
Hence \({\overline{u}}_\lambda \ne 0\) and so we conclude that
\(\square \)
Next we examine the map \(\lambda \rightarrow {\overline{u}}_\lambda \) from \((-\infty , {\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\).
Proposition 11
If hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_1\) hold, then the map \(\lambda \rightarrow {\overline{u}}_\lambda \) from \({\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\) is nondecreasing (that is, if \(\lambda < \mu \), then \({\overline{u}}_\lambda \le {\overline{u}}_\mu \)) and left continuous.
Proof
Suppose that \(\lambda , \mu \in {\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\) and \(\lambda < \mu \). Let \({\overline{u}}_\mu \in S(\mu )\) be the minimal positive solution of problem \((P_\mu )\) (see Proposition 10). For \(\eta > \Vert \xi \Vert _\infty \) we introduce the following Carathéodory function
We set \(E_\lambda (z,x)= \int _0^x e_\lambda (z,s)ds\) and consider the \(C^1\)-functional \({\widetilde{\psi }}_\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}\) defined by
From (29) and since \(\eta > \Vert \xi \Vert _\infty \), we see that \({\widetilde{\psi }}_\lambda \) is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find \(u_\lambda \in W^{1,p}(\varOmega )\) s.t.
Let \(m_\mu =\min _{{\overline{\varOmega }}} {\overline{u}}_\mu >0\) (recall that \({\overline{u}}_\mu \in D_+\)) and choose \(t \in (0,1)\) small s.t. \(t {\widehat{u}}_1(z) \le \min \{m_\mu ,\delta \}\) for all \(z \in {\overline{\varOmega }}\) (here \(\delta >0\) is as in hypothesis \(H_1\)(iii)). Because \(q<p\) and by choosing \(t \in (0,1)\) even smaller if necessary, we have that
From (30) we have
As in the proof of Proposition 9, using this time (31) and (29), we show that
This proves that \(\lambda \rightarrow {\overline{u}}_\lambda \) is nondecreasing.
Next we show the left continuity of this map. So, let \(\{\lambda _n, \lambda \}_{n \ge 1} \subseteq {\mathcal {L}}\) and suppose that \(\lambda _n \rightarrow \lambda ^-\). From the first part of the proof we have \({\overline{u}}_{\lambda _n} \le {\overline{u}}_\lambda \) for all \(n \in {\mathbb {N}}\) and so we infer that \(\{{\overline{u}}_{\lambda _n}\}_{n \ge 1} \subseteq W^{1,p}(\varOmega )\) is bounded. So, we may assume that
We have
for all \(h \in W^{1,p}(\varOmega )\), all \(n \in {\mathbb {N}}\). In (33) we choose \(h= {\overline{u}}_{\lambda _n}-{\widetilde{u}} \in W^{1,p}(\varOmega )\), pass to the limit as \(n \rightarrow +\infty \) and use (32). Then
So, if in (33) we pass to the limit as \(n \rightarrow +\infty \) and use (34), then
for all \(h \in W^{1,p}(\varOmega )\).
Set \(B=\{\lambda _n\}_{n \ge 1}\) and let \({\widehat{c}}_7 \ge c_7({\widetilde{\lambda }})\) for all \({\widetilde{\lambda }} \in B\) (recall that \(\lambda \rightarrow c_7(\lambda )\) is bounded on bounded sets). Consider \({\widehat{u}} \in D_+\) the unique positive solution of
(see Proposition 8 and the Remark following it).
We know that
Then from (35) we infer that \({\widetilde{u}} \in S(\lambda )\).
Suppose that \({\widetilde{u}} \ne {\overline{u}}_\lambda \). Then we can find \(z_0 \in {\overline{\varOmega }}\) s.t.
From Theorem 2 of Lieberman [13], we know that there exist \(M>0\) and \(\tau \in (0,1)\) s.t.
Exploiting the compact embedding of \(C^{1,\tau }({\overline{\varOmega }})\) into \(C^{1}({\overline{\varOmega }})\) and using (34), from (37) we have
which contradicts the monotonicity of \(\lambda \rightarrow {\overline{u}}_{\lambda }\) (recall \(\lambda _n < \lambda \) for all \(n \in {\mathbb {N}}\)). Therefore \({\widetilde{u}}={\overline{u}}_{\lambda }\) and so from (38) we conclude that the map \(\lambda \rightarrow {\overline{u}}_{\lambda }\) is left continuous from \({\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\). \(\square \)
If we strengthen the conditions on the perturbation \(f(z,\cdot )\), we can have uniqueness of the positive solution for problem \((P_{\lambda })\), \(\lambda < {\widehat{\lambda }}_1\).
The new hypotheses on f(z, x) are the following:
\(H_2\): \(f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a Carathéodory function s.t. for a.a. \(z \in \varOmega \), \(f(z,0)=0\), \(f(z,x)>0\) for all \(x >0\), hypotheses \(H_2\)(i), (ii), (iii) are the same as the corresponding hypotheses \(H_1\)(i), (ii), (iii) and
-
(iv)
for a.a. \(z \in \varOmega \) the function \(x \rightarrow \dfrac{f(z,x)}{x^{p-1}}\) is strictly decreasing on \((0,+\infty )\).
Example 2
The function \(f_1(x)=x^{q-1}\) for all \(x \ge 0\) with \(1<q<p\) satisfies hypotheses \(H_2\). On the other hand the function
need not satisfy hypotheses \(H_2\) unless additional restrictions are imposed on the exponents \(q, \tau \).
Proposition 12
If hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_2\) hold and \(\lambda < {\widehat{\lambda }}_1\), then problem \((P_{\lambda })\) has a unique positive solution \(u_\lambda \in D_+\).
Proof
Existence follows from Proposition 7. The uniqueness is proved as in the proof of Proposition 8 using the nonlinear Picone’s identity (for an alternative approach, see the Remark following the proof of Proposition 8). \(\square \)
In this case, because of the uniqueness of the positive solution, Proposition 11 takes the following form:
Proposition 13
If hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_2\) hold, then the map \(\lambda \rightarrow u_\lambda \) is nondecreasing and continuous from \({\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\).
In fact, by strengthening hypothesis \(H(\beta )_1\) (since we will use Proposition 4) and with an additional condition on the perturbation \(f(z,\cdot )\) we can improve the monotonicity property of the maps \(\lambda \rightarrow {\overline{u}}_\lambda \) in Proposition 11 and of the map \(\lambda \rightarrow u_\lambda \) in Proposition 12.
So, we introduce the following conditions on the functions \(\beta (z)\) and f(z, x):
\(H(\beta )_2\): \(\beta \in C^{0,\alpha }(\partial \varOmega )\) with \(\alpha \in (0,1)\) and \(\beta (z)>0\) for all \(z \in \partial \varOmega \).
\(H_3\): \(f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a Carathéodory function s.t. for a.a. \(z \in \varOmega \), \(f(z,0)=0\), \(f(z,x)>0\) for all \(x >0\), hypotheses \(H_3\)(i), (ii), (iii) are the same as the corresponding hypotheses \(H_1\)(i), (ii), (iii) and
-
(iv)
for every \(\rho >0\), there exists \({\widehat{\xi }}_\rho >0\) s.t. for a.a. \(z \in \varOmega \) the function \(x \rightarrow f(z,x)+ {\widehat{\xi }}_\rho x^{p-1}\) is nondecreasing on \([0,\rho ]\).
We also introduce a corresponding strengthening of hypotheses \(H_2\).
\(H_4\): \(f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a Carathéodory function s.t. for a.a. \(z \in \varOmega \), \(f(z,0)=0\), \(f(z,x)>0\) for all \(x >0\), hypotheses \(H_4\)(i), (ii), (iii), (iv) are the same as the corresponding hypotheses \(H_2\)(i), (ii), (iii), (iv) and
-
(v)
for every \(\rho >0\), there exists \({\widehat{\xi }}_\rho >0\) s.t. for a.a. \(z \in \varOmega \), the function \(x \rightarrow f(z,x)+ {\widehat{\xi }}_\rho x^{p-1}\) is nondecreasing on \([0,\rho ]\).
Proposition 14
If hypotheses \(H(\xi )\), \(H(\beta )_2\), \(H_3\) hold, then the map \(\lambda \rightarrow {\overline{u}}_\lambda \) is strictly increasing from \({\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\) in the sense that \(\lambda < \mu \, \Rightarrow \, {\overline{u}}_\mu - {\overline{u}}_\lambda \in {\mathrm{int }}\, {\widehat{C}}_+\) with \(D_0=\{z \in \partial \varOmega : {\overline{u}}_\mu (z)= {\overline{u}}_\lambda (z)\}\).
Proof
Let \(\lambda ,\mu \in {\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\) with \(\lambda < \mu \). From Proposition 11 we know that
Let \(\rho =\Vert {\overline{u}}_\mu \Vert _\infty \) and let \({\widehat{\xi }}_\rho >0\) be as postulated by hypothesis \(H_3\)(iv). We set
For \(\delta >0\) we define \({\overline{u}}_\lambda ^\delta = {\overline{u}}_\lambda + \delta \in D_+\). We have
Note that if \(\mu <0\), then \({\widetilde{\xi }}_\rho ={\widehat{\xi }}_\rho +|\mu |\) and we have
If \(\mu \ge 0\), then \({\widetilde{\xi }}_\rho ={\widehat{\xi }}_\rho \) and using hypothesis \(H_3\)(iv) we have
Returning to (39), we have
Since \(\mu >\lambda \) and \({\overline{u}}_\lambda \in D_+\), we have
Then since \(\chi (\delta ) \rightarrow 0^+\) as \(\delta \rightarrow 0^+\), for \(\delta >0\) small we have
Using this in (40) we have
In this case in the definition of \({\widehat{C}}_+\), \(D_0=\{z \in \partial \varOmega : {\overline{u}}_\mu (z)={\overline{u}}_\lambda (z)\}\). \(\square \)
Similarly we have:
Proposition 15
If hypotheses \(H(\xi )\), \(H(\beta )_2\), \(H_4\) hold, then the map \(\lambda \rightarrow {\overline{u}}_\lambda \) is strictly increasing from \({\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\).
The next theorem summarizes the situation for problem \((P_{\lambda })\) when the perturbation \(f(z,\cdot )\) is \((p-1)\)-sublinear.
Theorem 1
We have:
-
1.
If hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_1\) hold, then
-
(a)
for all \(\lambda \ge {\widehat{\lambda }}_1\) problem \((P_{\lambda })\) has no positive solution;
-
(b)
for all \(\lambda < {\widehat{\lambda }}_1\) problem \((P_{\lambda })\) has at least one positive solution and it admits a smallest positive solution \({\overline{u}}_\lambda \in D_+\);
-
(c)
the map \(\lambda \rightarrow {\overline{u}}_\lambda \) from \({\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\) is nondecreasing (that is, if \(\lambda \le \mu \), then \({\overline{u}}_\lambda \le {\overline{u}}_\mu \)) and left continuous.
-
(a)
-
2.
If hypotheses \(H(\xi )\), \(H(\beta )_2\), \(H_3\) hold, then the map \(\lambda \rightarrow {\overline{u}}_\lambda \) from \({\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\) is strictly increasing as in Proposition 14.
-
3.
If hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_2\) hold and \(\lambda < {\widehat{\lambda }}_1\), then problem \((P_{\lambda })\) has a unique solution \(u_\lambda \in D_+\) and the map \(\lambda \rightarrow u_\lambda \) from \({\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\) is nondecreasing and continuous.
-
4.
If hypotheses \(H(\xi )\), \(H(\beta )_2\), \(H_4\) hold, then the solution map \(\lambda \rightarrow {\overline{u}}_\lambda \) from \({\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\) is strictly increasing.
4 \((p-1)\)-superlinear perturbation
In this section we consider the case where the perturbation \(f(z,\cdot )\) is \((p-1)\)-superlinear. In this case uniqueness of the solution fails and the problem exhibits a bifurcation-type behaviour, namely there are no positive solutions for all \(\lambda \ge {\widehat{\lambda }}_1\) and there are at least two positive solutions for \(\lambda < {\widehat{\lambda }}_1\).
The new hypotheses on the perturbation term f(z, x) are the following:
\(H_5\): \(f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a Carathéodory function s.t. for a.a. \(z \in \varOmega \) \(f(z,0)=0\), \(f(z,x)\ge 0\) for all \(x > 0\), there exist \(\varOmega _0 \subseteq \varOmega \) with \(|\varOmega _0|_N>0\) s.t. \(f(z,x)>0\) for all \(z \in \varOmega _0\), all \(x>0\) and
-
(i)
\(f(z,x) \le a(z)(1+x^{r-1})\) for a.a. \(z \in \varOmega \), all \(x \ge 0\), with \(a \in L^\infty (\varOmega )_+\), \(r \in (p,p^*)\);
-
(ii)
if \(F(z,x)=\int _0^xf(z,s)ds\), then
$$\begin{aligned} \lim _{x \rightarrow + \infty }\dfrac{F(z,x)}{x^{p}}=+\infty \quad \text{ uniformly } \text{ for } \text{ a.a. } z\in \varOmega \end{aligned}$$and there exists \(\tau \in (\max \{1,(r-p)\frac{N}{p}\},p^*)\) s.t.
$$\begin{aligned} 0< {\widetilde{\xi }} \le \liminf _{x \rightarrow +\infty } \dfrac{f(z,x)x-pF(z,x)}{x^\tau } \quad \text{ uniformly } \text{ for } \text{ a.a. } z \in \varOmega ; \end{aligned}$$ -
(iii)
\(\lim _{x \rightarrow 0^+}\dfrac{f(z,x)}{x^{p-1}}=0\) uniformly for a.a. \(z \in \varOmega \).
Remark 4
As we did for the “sublinear” case, since we are looking for positive solutions and the above hypotheses concern the positive semiaxis, without any loss of generality, we may assume that \(f(z,x)=0\) for a.a. \(z \in \varOmega \), all \(x < 0\). Hypothesis \(H_5\)(ii) implies that for a.a. \(z \in \varOmega \) \(f(z,\cdot )\) is \((p-1)\)-superlinear. However, note that we do not use the usual in such cases “Ambrosetti–Rabinowitz condition” (the AR-condition for short, unilateral version since we are looking for positive solutions), which says that there exist \(q>p\) and \(M>0\) s.t.
(see Ambrosetti–Rabinowitz [3] and Mugnai [15]). Integrating (41) and using (42) we obtain
Hence from (41) and (43) we infer that near \(+\infty \), \(f(z,\cdot )\) exhibits at least \((q-1)\)-polynomial growth. Our hypothesis \(H_5\)(ii) is more general. Indeed, suppose that the AR-condition holds. We may assume that \(q > \max \{1, (r-p) \frac{N}{p}\}\). We have
The function
satisfies hypotheses \(H_5\), but not the AR-condition (see (41)).
From Propositions 5 and 6 we have
It follows that \({\mathcal {L}} \subseteq (-\infty , {\widehat{\lambda }}_1)\). In the next proposition we show that equality holds.
Proposition 16
If hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_5\) hold, then \({\mathcal {L}} = (-\infty , {\widehat{\lambda }}_1)\).
Proof
We fix \(\lambda \in (-\infty , {\widehat{\lambda }}_1)\) and consider the Carathéodory function \(k_\lambda (z,x)\) defined by
We set \(K_\lambda (z,x)= \int _0^x k_\lambda (z,x)ds\) and consider the \(C^1\)-functional \(w_\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}\) defined by
As before \(\eta > \Vert \xi \Vert _\infty \). Hypotheses \(H_5\)(i), (iii) imply that given \(\varepsilon >0\), we can find \(c_{14} =c_{14}(\varepsilon )>0\) s.t.
Choosing \(\varepsilon \in (0, {\widehat{\lambda }}_1-\lambda )\) (recall \(\lambda < {\widehat{\lambda }}_1\)), for every \(u \in W^{1,p}(\varOmega )\) we have
Since \(p<r\), from (46) we infer that \(u=0\) is a strict local minimizer of \(w_\lambda \). So, we can find \(\rho \in (0,1)\) small s.t.
(see Aizicovici–Papageorgiou–Staicu [1], proof of Proposition 29).
Hypothesis \(H_5\)(ii) implies that
Claim: \(w_\lambda \) satisfies the C-condition.
Let \(\{u_n\}_{n \ge 1} \subseteq W^{1,p}(\varOmega )\) be a sequence s.t.
From (50) we have
In (51) we choose \(h=-u_n^- \in W^{1,p}(\varOmega )\). Using (44) we obtain
From (49), (52) and (44) it follows that
In (51) we choose \(h=u_n^+ \in W^{1,p}(\varOmega )\). Then
Adding (53) and (54) we obtain
Hypotheses \(H_5\)(i), (ii) imply that we can find \({\widetilde{\xi }}_0 \in (0, {\widetilde{\xi }})\) and \(c_{18}>0\) s.t.
Using (56) in (55), we infer that
First suppose that \(N > p\). Clearly in hypothesis \(H_5\)(ii), we can always assume that \(\tau<r<p^*\) (recall that \(p^*=+\infty \) if \(p \ge N\)). Let \(t \in (0,1)\) be such that
From the interpolation inequality (see, for example, Gasiński–Papageorgiou [8] (p. 905)), we have
In (51) we choose \(h=u_n^+ \in W^{1,p}(\varOmega )\). Then
From hypothesis \(H_5\)(i) we see that we can always take \(r \in (p,p^*)\) close to \(p^*\) and as \(r \rightarrow (p^*)^-\), we have \(\tau >p\). So, there is no loss of generality in assuming that \(\tau >p\). Then from (60) and (57), we have
From hypothesis \(H_5\)(ii) and (58) we see that
If \(N\le p\), then \(p^*=+\infty \), while the Sobolev embedding theorem says that \(W^{1,p}(\varOmega ) \hookrightarrow L^q(\varOmega )\) for all \(q \in [1,+\infty )\). Let \(q>r>\tau \) and choose \(t \in (0,1)\) s.t.
Note that
Since by hypothesis \(H_5\)(ii) we have \(r-\tau <p\) (recall \(N \le p\)), for the previous argument (case \(N \le p\)) to work, we use \(q>r\) big s.t. \(tr<p\) (see (63), (64)). Then again we conclude that (62) holds. Because of (62) we may assume that
In (51) we choose \(h=u_n-u \in W^{1,p}(\varOmega )\), pass to the limit as \(n \rightarrow +\infty \) and use (65). Then we have
This proves the Claim.
Then (47), (48) and the Claim permit the use of the mountain pass theorem (see, for example, Gasiński–Papageorgiou [8] (p. 648)). So, we can find \(u_\lambda \in W^{1,p}(\varOmega )\) s.t.
From (66) it follows that \(u_\lambda \ne 0\) and \(u_\lambda \in S(\lambda ) \subseteq D_+\) (see Proposition 5). Therefore \({\mathcal {L}}=(-\infty ,{\widehat{\lambda }}_1)\). \(\square \)
In fact as we did in the “sublinear” case, we can produce the minimal positive solution for problem \((P_{\lambda })\), \(\lambda < {\widehat{\lambda }}_1\).
Proposition 17
If hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_5\) hold and \(\lambda \in {\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)\), then problem (\(P_\lambda \)) has a smallest positive solution \({\overline{u}}_\lambda \in D_+\).
Proof
We argue as in the proof of Proposition 10. Recall that \(S(\lambda )\) is downward directed (see Filippakis–Papageorgiou [5]). Using Lemma 3.10 of Hu–Papageorgiou [9] (p. 178), we can find a decreasing sequence \(\{u_n\}_{n \ge 1} \subseteq S(\lambda )\) s.t.
We have
In (67) we choose \(h= u_n \in W^{1,p}(\varOmega )\), we obtain
Recall that
From (68) to (69) it follows that
So, we may assume that
In (67) we choose \(h=u_n -{\overline{u}}_\lambda \in W^{1,p}(\varOmega )\), pass to the limit as \(n \rightarrow + \infty \) and use (70). Then
Passing to the limit as \(n \rightarrow + \infty \) in (67) and using (71), we obtain
If we can show that \({\overline{u}}_\lambda \ne 0\), then \({\overline{u}}_\lambda \in S(\lambda ) \subseteq D_+\). Arguing by contradiction, suppose that \({\overline{u}}_\lambda = 0\). Then
We set \(y_n=\dfrac{u_n}{\Vert u_n\Vert }\), \(n \in {\mathbb {N}}\). Then for all \(n \in {\mathbb {N}}\) we have \(\Vert y_n\Vert =1\), \(y_n \ge 0\). So, we may assume that
From (67) we have
Here \(N_f(y)(\cdot )= f(\cdot , y(\cdot ))\) for all \(y \in W^{1,p}(\varOmega )\). We set \(\rho =\Vert u_1\Vert _\infty \). Hypotheses \(H_5\)(i), (iii) imply that
Then by passing to a suitable subsequence if necessary and using hypothesis \(H_5\)(iii), we have
(see Aizicovici–Papageorgiou–Staicu [1], proof of Proposition 14).
In (73) we choose \(h = y_n -y \in W^{1,p}(\varOmega )\), pass to the limit as \(n \rightarrow + \infty \) and use (72) and (74). Then
So, if in (73) we pass to the limit as \(n \rightarrow + \infty \) and use (74) and (75), then
Choosing \(h=y \in W^{1,p}(\varOmega )\), we obtain
a contradiction to Proposition 1. Therefore
\(\square \)
As in the “sublinear” case, we have:
Proposition 18
If hypotheses \(H(\xi )\), \(H(\beta )_1\), \(H_5\) hold, then the map \(\lambda \rightarrow {\overline{u}}_\lambda \) from \({\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\) is nondecreasing and left continuous.
Again by strengthening the conditions on the functions \(\beta (\cdot )\) and \(f(z, \cdot )\) we can improve the monotonicity of the map \(\lambda \rightarrow {\overline{u}}_\lambda \).
The new hypotheses on the perturbation f(z, x) are the following:
\(H_6\): \(f : \varOmega \times {\mathbb {R}} \rightarrow {\mathbb {R}}\) is a Carathéodory function s.t. for a.a. \(z \in \varOmega \), \(f(z,0)=0\), \(f(z,x) \ge 0\) for all \(x > 0\), there exists \(\varOmega _0 \subseteq \varOmega \) with \(f(z,x)>0\) for all \(z \in \varOmega _0\), all \(x>0\), hypotheses \(H_6\)(i), (ii), (iii) are the same as the corresponding hypotheses \(H_5\)(i), (ii), (iii) and
-
(iv)
for every \(\rho >0\), there exists \({\widehat{\xi }}_\rho >0\) s.t. for a.a. \(z \in \varOmega \), the function
$$\begin{aligned} x \rightarrow f(z,x)+ {\widehat{\xi }}_\rho x^{p-1} \end{aligned}$$is nondecreasing on \([0,\rho ]\).
Proposition 19
If hypotheses \(H(\xi )\), \(H(\beta )_2\), \(H_6\) hold, then the map \(\lambda \rightarrow {\overline{u}}_\lambda \) from \({\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\) is strictly decreasing.
In fact under these stronger conditions on \(\beta (z)\) and f(z, x), we can produce a second positive solution for problem \((P_{\lambda })\), when \(\lambda \in {\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)\).
Proposition 20
If hypotheses \(H(\xi )\), \(H(\beta )_2\), \(H_6\) hold and \(\lambda \in {\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)\), then problem (\(P_\lambda \)) admits at least two positive solutions
Proof
From Proposition 16 we already have a positive solution \(u_\lambda \in D_+\). We may assume that \(u_\lambda \) is the minimal positive solution, that is, \(u_\lambda = {\overline{u}}_\lambda \) (see Proposition 17). We introduce the following Carathéodory function
with \(\eta > \Vert \xi \Vert _\infty \) as always. We set \(Z_\lambda (z,x)= \int _0^x \zeta _\lambda (z,s)ds\) and consider the \(C^1\)-functional \(j_\lambda : W^{1,p}(\varOmega ) \rightarrow {\mathbb {R}}\) defined by
From (76) it is clear that
with \(w_\lambda \in C^1(W^{1,p}(\varOmega ))\) as in the proof of Proposition 16. From (77) and the Claim in the proof of Proposition 16, it follows that
Claim: We may assume that \(u_\lambda \in D_+\) is a local minimizer of \(j_\lambda \).
Let \(\lambda<\mu <{\widehat{\lambda }}_1\) and let \(u_\mu \in S(\mu ) \subseteq D_+\) (see Proposition 16). We consider the following truncation of \(\zeta _\lambda (z, \cdot )\)
Evidently this is a Carathéodory function. We set \({\widehat{Z}}_\lambda (z,x)= \int _0^x {\widehat{\zeta }}_\lambda (z,s)ds\) and consider the \(C^1\)-functional \({\widehat{j}}_\lambda : W^{1,p} (\varOmega ) \rightarrow {\mathbb {R}}\) defined by
If \(K_{{\widehat{j}}_\lambda }= \{u \in W^{1,p}(\varOmega ): {\widehat{j}}_\lambda ^\prime (u)=0\}\), then we will show that
So, let \( u \in K_{{\widehat{j}}_\lambda }\). Then
In (80) we choose \(h=(u_\lambda -u)^+ \in W^{1,p}(\varOmega )\). Then
Also in (80), we choose \(h=(u-u_\mu )^+ \in W^{1,p}(\varOmega )\). Then
So, we have proved that
Since \(\eta > \Vert \xi \Vert _\infty \), from (76) and (79) it follows that \({\widehat{j}}_\lambda \) is coercive. Also, the Sobolev embedding theorem and the compactness of the trace map imply that \({\widehat{j}}_\lambda \) is sequentially weakly lower semicontinuous. So, from the Weierstrass-Tonelli theorem, we can find \({\widetilde{u}}_\lambda \in W^{1,p}(\varOmega )\) s.t.
If \({\widetilde{u}}_\lambda \ne u_\lambda \), then from (76), (79) and (82), we see that
So, this is the desired second solution of \((P_{\lambda })\) and we are done.
Therefore, we assume that \({\widetilde{u}}_\lambda = u_\lambda \). From Proposition 19, we have
From (76) and (79) it is clear that
From this equality and (83) we infer that
This proves the Claim.
In proving (81), we established that
We assume that \(K_{j_\lambda }\) is finite or otherwise (84) implies that we already have a whole sequence of distinct positive solutions of \((P_{\lambda })\), all bigger that \(u_\lambda \), hence we are done. Then we can find \(\rho \in (0,1)\) small s.t.
(see Aizicovici–Papageorgiou–Staicu [1], proof of Proposition 29). Note that hypothesis \(H_6\)(ii) and (76) imply that
From (78), (85) and (86) we see that we can apply the mountain pass theorem and find \({\widehat{u}}_\lambda \in W^{1,p}(\varOmega )\) s.t.
From (76), (84), (85) and (87), we infer that
\(\square \)
So, summarizing the situation for problem \((P_{\lambda })\) when the perturbation \(f(z, \cdot )\) is \((p-1)\)-superlinear, we have the following theorem
Theorem 2
If hypotheses \(H(\xi )\), \(H(\beta )_2\), \(H_6\) hold, then
-
(a)
for all \(\lambda \ge {\widehat{\lambda }}_1\) problem \((P_{\lambda })\) has no positive solution;
-
(b)
for all \(\lambda < {\widehat{\lambda }}_1\) problem \((P_{\lambda })\) has at least two positive solutions
$$\begin{aligned} u_\lambda , {\widehat{u}}_\lambda \in D_+, \quad u_\lambda \le {\widehat{u}}_\lambda , \quad {\widehat{u}}_\lambda \ne u_\lambda ; \end{aligned}$$ -
(c)
for all \(\lambda < {\widehat{\lambda }}_1\) problem \((P_{\lambda })\) has a smallest positive solution \({\overline{u}}_\lambda \in D_+\) and the map \(\lambda \rightarrow {\overline{u}}_\lambda \) from \({\mathcal {L}}=(-\infty , {\widehat{\lambda }}_1)\) into \(C^1({\overline{\varOmega }})\) is strictly increasing and left continuous.
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Vetro, C. Perturbed eigenvalue problems for the Robin p-Laplacian plus an indefinite potential. Anal.Math.Phys. 10, 69 (2020). https://doi.org/10.1007/s13324-020-00416-w
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DOI: https://doi.org/10.1007/s13324-020-00416-w
Keywords
- Positive solutions
- Sublinear and superlinear perturbation
- Nonlinear Picone’s identity
- Nonlinear maximum principle
- Nonlinear regularity
- Indefinite potential
- Minimal positive solution
- Uniqueness