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The Fučik spectrum of the p-Laplace equations with different weights and its resonance problems
Boundary Value Problems volume 2017, Article number: 31 (2017)
Abstract
In the present paper, our purpose is to construct the Fučik spectrum \(C_{l}^{\pm}\) (cf. (2.13) and (2.17) below) with different weights for the p-Laplacian. As an application, we will discuss the existence of nontrivial solutions to the p-Laplace equations with resonance on the Fučik spectrum by making use of variational methods and Morse theory.
1 Introduction
In this paper, we consider the existence of nontrivial solutions to the Dirichlet boundary value problem
where Ω is a bounded domain in \(\mathbb{R}^{N}\) with smooth boundary ∂Ω, \(\Delta_{p}\) denotes the p-Laplace operator, that is, \(\Delta_{p}u=\nabla(|\nabla u|^{p-2}\nabla u)\), \(u^{\pm} = \max\{\pm u(x), 0\}\), and \(a(x), b(x) \in L^{r}(\Omega)\), where \(r > \frac{N}{p}\) if \(1 < p \leq N\) and \(r =1\) if \(p > N\). We assume that \(g: \Omega\times\mathbb{R} \rightarrow \mathbb{R}\) is a Caratheodory function.
It is well known that the value of \((\lambda, \mu)\) plays an important role in the study of the solvability of (1.1). The approach here of course requires the preliminary study of weighted asymmetric eigenvalue problems of the form
The set \(\Sigma_{p}(a, b)\), the Fučik spectrum of the p-Laplacian with the weights \(a(x)\) and \(b(x)\) on \(W_{0}^{1, p}\), is defined by those \((\lambda,\mu)\in\mathbb{R}^{2}\) such that (1.2) has a nontrivial solution u.
The generalized notion of spectrum was introduced for \(p=2\) (i.e. in the linear case) in the 1970s by Fučik [1] and Dancer [2] in connection with the study of the so-called jumping non-linearities. From then on, further work has been done in the study of \(\Sigma_{2}(1,1)\); cf. [3–5]. The quasilinear case of \(p \neq2\) and \(N = 1\), in the situation where \(a(x)\) and \(b(x)\) are nonconstant and different, was investigated in [6–9] (for \(a(x)\) and \(b(x) > 0\)) and [10] (for \(a(x)\) and \(b(x)\) are indefinite). From the above papers, \(\Sigma_{p}(a, b)\) has the same general shape as in the linear ODE case.
In the quasilinear PDE case, when \(\lambda a = \mu b\),
It is well known that problem (1.3) has an unbound sequence of variational eigenvalues \(\lambda_{l}(a)\) satisfying a standard min-max characterization, \(\lambda_{l}(a) \rightarrow+\infty\) (\(l\rightarrow\infty\)). The first eigenvalue \(\lambda_{1}(a)\) of \(-\Delta_{p} u\) is positive, simple, and admits a positive eigenfunction \(\varphi_{a}\) (see Lindqvist [11] for the case \(a(x)=1\) and Cuesta [12] for the case where \(a(x)\) is an indefinite weights), where \(\lambda_{1}(a) := \min\{\int_{\Omega}| \nabla u|^{p}: u \in W_{0}^{1, p}(\Omega) \mbox{ and } \int_{\Omega} a|u|^{p} =1\}\) < \(\lambda_{2}(a) := \min\{ \lambda\in \mathbb{R}: \lambda \mbox{ is the eigenvalue and } \lambda>\lambda_{1}(a )\}\). We denote \(e_{a}(x) := \frac{\varphi_{a}}{\|\varphi_{a}\|_{1, p}}\). According to Arias et al. [13], we know that \(\Sigma_{p}(a, b)\) contains the two lines \(\lambda_{1}(a)\times\mathbb {R}\) and \(\mathbb{R} \times\lambda_{1}(b)\).
As in [13], the first nontrivial eigenvalue which admits a sign-changing eigenfunction of the eigenvalue problem with weights \(a(x)\) and \(b(x)\) is defined as \(c(a,b) := \inf_{\gamma\in\Gamma}\max_{t \in[-1, 1]} \) \(A(\gamma(t))\), where \(A(u) := \int_{\Omega}|\nabla u|^{p}\,dx\), \(\Gamma:= \{\gamma\in C([-1,1], \mathscr{M}_{a, b}): \gamma(-1) = \varphi_{a} \mbox{ and } \gamma(1) = \varphi_{b} \}\), and \(\mathscr{M}_{a, b} := \{u\in W_{0}^{1, p}(\Omega): \int_{\Omega}a(x)(u^{+})^{p} + b(x)(u^{-})^{p}\,dx =1\}\). Hence, the first nontrivial curve in \(\Sigma_{p}(a, b)\) through \((c(a, b), c(a, b))\) asymptotic to \(\lambda_{1}(a)\times\mathbb{R}\) and \(\mathbb{R} \times\lambda_{1}(b)\) at infinity was constructed by Cuesta et al. [14] for the case \(a = b = 1\), and by Arias et al. [13] for the case with two different nonconstant weights. More recently, unbounded curves \(\{C_{l}^{\pm}\}\) in \(\Sigma_{p}(a, b)\) have been constructed and variationally characterized by min-max procedures by Micheletti and Pistoia [15] for \(p > 2\), \(a = b = 1\), and by Perera [16] for \(p > 1\), \(a = b = 1\) and [17] for the fractional p-Laplace operator.
Related studies can also be found in [18–20].
It is our purpose in this paper to initiate the study of (1.2) and its relation with the solvability of (1.1) in the general case: \(a(x), b(x)\geq0\) are possibly nonconstant and different. Hence we will discuss the existence of the Fučik spectrum for (1.2) besides \(\lambda_{1}(a)\times\mathbb{R}\), \(\mathbb {R}\times \lambda_{1}(b)\) and \(\mathcal {C} := \{(\alpha(s), s\alpha(s)): \alpha(s) = c(a, sb), s > 0\}\) (in [13]), which is an open question raised in [13].
As an application, we will concern ourselves with the resonance type of the problem (1.1). The resonance problem has been studied by Perera [16] and Guo and Liu [21] for the case \((\lambda,\mu)\in\Sigma_{p}(a,b)\), with \(a = b = 1\). The purpose of this paper is to discuss the general resonance case \((\lambda, \mu) \in\Sigma_{p}(a, b)\), with nonconstant and different weights \(a(x)\) and \(b(x)\). When \((\lambda, \mu) \in\lambda_{1}(a) \times \mathbb{R}\) (\((\lambda, \mu) \in\mathbb{R} \times\lambda_{1}(b)\)), we obtain the existence of nontrivial solution for (1.1) by the Morse theorem. However, the usual Morse theory supposes that the functional \(\Phi(u)\) satisfies the Palais-Smale condition ((P.S.) condition for short), which is not clear in the case where the nonlinear term \(g(x, u)\) is asymptotic to \(|u|^{p-2}u\) at infinity. To overcome this difficulty, Cerami introduced a weaker compactness condition (see [22]) that allows for more general results; see for example [23]. Fortunately, replacing the usual (P.S.) condition by Cerami’s weaker compactness condition ((\(C_{c}\)) condition for short), we still have the deformation lemma (see [24]).
2 Fučik spectrum with different weights
In this section, we consider the eigenvalues problem of asymmetric p-Laplacian with weights (1.2). It is always assumed that \(a(x), b(x)\in L^{r}(\Omega)\) with r as in the introduction, i.e. \(r > \frac{N}{p}\) if \(1 < p \leq N\) and \(r = 1\) if \(p > N\).
As is well known, the associated functional of the problem (1.2) is
We introduce two new functionals as follows:
where \(u \in W_{0}^{1, p}(\Omega)\), \(s_{0}\) is a positive constant to be determined later.
The Finsler manifold we need is defined as follows:
\(S(b)\) can be considered in a similar way. We will consider the functionals \(J_{s, t, s_{0}}^{+}\) on the manifold \(S(a)\) and \(J_{s, t, s_{0}}^{-}\) on the manifold \(S(b)\), respectively. Let us just deduce the case \(J_{s, t, s_{0}}^{+}\) on \(S(a)\), since the other case can be done similarly.
For each integer \(l\geq1\), we define maps family \({\mathcal{F}}_{l}(a)\) as
where \(S^{l-1}\) is the unit sphere in \(\mathbb{R}^{l}\).
As observed by Drábek and Robinson [25], we can define an unbounded sequence of functionals \(J(u)\) by
where \(J(u) = \int_{\Omega}| \nabla u| \,dx = J_{0, 0, 0}(u)\). It is shown that \(\lambda_{l}(a)\) is the lth eigenvalue of \(-\Delta_{p}\) with weight \(a(x)\), although it is not known whether this gives a complete list of eigenvalues. Furthermore,
In the sequel we fix some \(l \geq2\) and suppose that
For any \(\varepsilon\in (0, \lambda_{l}(a) - \lambda_{l-1}(a) )\), we can choose an \(A^{l - 2} \in{\mathcal{F}}_{l - 1}(a)\), such that
Let \(h_{l - 2}: S^{l - 2} \to A^{l - 2}\) be any fixed continuous surjection and denote
where \(S_{+}^{l - 1}\) is the upper hemisphere of \(S^{l - 1}\) with boundary \(S^{l - 2}\). Then \({\mathcal{F}}_{l}^{+}(a)\) is a homotopy-stable family of compact subsets of \(S(a)\) with closed boundary \(A^{l - 2}\), that is,
-
(i)
every set \(A_{+} \in{\mathcal{F}}_{l}^{+}(a)\) contains \(A^{l - 2}\);
-
(ii)
for any set \(A_{+} \in{\mathcal{F}}_{l}^{+}(a)\) and any deformation \(\eta\in C([0, 1]\times S(a); S(a))\) satisfying \(\eta(t, u) = u\) for all \((t, u) \in(\{ 0 \} \times S(a)) \cup([0, 1] \times A^{l - 2})\) we have \(\eta(\{ 1 \} \times A) \in{\mathcal{F}}_{l}^{+}(a)\).
Now, we define
Similarly, we can define
We can make sure that \(c_{l}^{\pm}(s, t, s_{0})\) is the critical value of \(J_{s, t, s_{0}}^{\pm}(u)\), under some assumptions. Furthermore, the critical point of \(J_{s, t, s_{0}}^{\pm}(u)\) corresponding to \(c_{l}^{\pm}(s, t, s_{0})\) in which \(s_{0}\) is determined in a special way, is just the solution to (1.2), that is, we will establish the following result.
Theorem 2.1
Suppose that the weights \(a(x)\) and \(b(x)\) are nonnegative and, for some fixed constants \(k' \geq1 \geq k > 0\),
where k and \(k'\) satisfy
where \(\delta\in(0, 1)\). Suppose that s and t are in some intervals as follows:
Then, for any \(s_{0}\) in \([\delta(\lambda_{l}(a)-\lambda_{l-1}(a)), \lambda_{l}(a)-\lambda _{l-1}(a) )\), \(c_{l}^{+}(s, t, s_{0})\) is a critical value of \(J_{s, t, s_{0}}^{+}(u)\) on \(S(a)\). In particular, the set of the Fučik spectrum of \(-\Delta_{p}\) with weights \(a(x)\) and \(b(x)\) greater than \(c(a(x),b(x))\) includes \((s+c_{l}^{+}(s, t, s_{0}), t+c_{l}^{+}(s, t, s_{0}))\) for s and t in the above intervals, and some well-defined \(s_{0} \in [\delta(\lambda_{l}(a)-\lambda_{l-1}(a)),\lambda _{l}(a)-\lambda_{l-1}(a) )\) satisfying
Lemma 2.1
\(J_{s, t, s_{0}}^{+}(u)\) satisfies the (P.S.) condition on \(S(a)\).
Proof
Let \(\{u_{n}\}\subset S(a)\) and \(t_{n}\in\mathbb{R}\) be sequence, such that, for some constant k,
and
for all \(w\in W_{0}^{1, p}(\Omega)\), where \(\varepsilon_{n}\rightarrow0\), as \(n \rightarrow\infty\) and \(\|\cdot\|_{1, p}\) represents the norm of \(W_{0}^{1, p}(\Omega)\). From (2.7) it follows that \(\{ u_{n}\}\) remains bounded in \(W_{0}^{1, p}(\Omega)\); consequently, for a subsequence, \(u_{n}\rightarrow u\) weakly in \(W_{0}^{1, p}(\Omega)\), strongly in \(L^{p}(\Omega)\) and almost everywhere in Ω. Putting \(w=u_{n} - u\) in equation (2.8), we also see that \(t_{n}\) remains bounded. We then have
where the right-hand side goes to zero as \(n\rightarrow\infty\), i.e.
It is sufficient to obtain \(u_{n} \rightarrow u\) strongly in \(W_{0}^{1, p}\). □
Lemma 2.2
As a function of s, t, \(s_{0}\), the min-max value
is continuous on \(s_{0}\).
Proof
Let \(\{s_{n}\}\subset [\delta(\lambda_{l}(a)-\lambda_{l-1}(a)), \lambda_{l}(a)-\lambda_{l -1}(a)]\) be a sequence such that
Taking \(A_{+} \in\mathcal{F}_{l}^{+}(a)\) in the following equation:
Then we can see that
Noting that \(k a(x) \leq b(x) \leq k' a(x)\), we have
where \(\varepsilon_{n}\rightarrow0\), as \(n\rightarrow\infty\). By (2.9) we have
Letting \(n\rightarrow\infty\), we have
which shows that \(\lim_{n\rightarrow\infty} c_{l}^{+}(s, t,s_{n}) = c_{l}^{+}(s, t, s_{0})\). The proof is completed. □
Proof of Theorem 2.1
We first claim that
If it is not the case, that is, \(c_{l}^{+}(s, t, s_{0}) < \lambda_{l}(a) - s\), then we can choose some \(A_{+}\in{\mathcal{F}}^{+}_{l}(a)\), such that
By setting \(A=A_{+}\cup(-A_{+})\), we get an odd symmetric set A. The set A is thus in \({\mathcal{F}}_{l}(a)\). Since the functional \(J(u) = J_{0,0, 0}(u)\) is even, from (2.4) it follows that
which leads to a contradiction. Thus, (2.10) follows.
The inequality (2.10) is crucial in proving the existence of critical value of \(J_{s, t, s_{0}}^{+}(u)\) on the manifold \(S(a)\). Since the assumption in the theorem implies that, for \(\varepsilon\in(0, \lambda_{l}(a)-\lambda_{l-1}(a)-s_{0})\), we have
By Lemma 2.1, we know that \(J_{s, t, s_{0}}^{+}\) satisfies the (P.S.) condition on \(S(a)\). Then it follows from Theorem 3.2 of Ghoussoub [26] that \(c^{+}_{l}(s, t, s_{0})\) is a critical value of \(J_{s, t, s_{0}}^{+}(u)\), that is, according to the Lagrange multiple rule, there is some \(u\in W_{0}^{1,p}(\Omega)\), such that
The min-max value \(c^{+}_{l}(s, t, s_{0})\) is also a function of \(s_{0}\), and from Lemma 2.2, we know that \(c^{+}_{l}(s, t, s_{0})\) is continuous on \(s_{0}\). Set
Next we claim that
In fact, for every \(A_{+}\in{\mathcal{F}}^{+}_{l}(a)\), since \(A=A_{+}\cup(-A_{+})\in{\mathcal{F}}_{l}(a)\), on A, we have
Hence, we can estimate the maximum of \(J_{s, t, \delta(\lambda_{l}(a)-\lambda_{l-1}(a))}(u)\) on \(A_{+}\) as follows:
By the conditions (2.5) and (2.6), we have
Then we see that \(\psi(\delta(\lambda_{l}(a)-\lambda_{l-1}(a))) \geq0\).
Moreover, for any \(\varepsilon> 0\), we pick up an \(A \in{\mathcal{F}}_{l}(a)\), such that
and
Denote \(A_{+}=h(S^{l-1}_{+}(a))\), and then \(A_{+}\in{\mathcal{F}}^{+}_{l}(a)\). For every \(u \in A_{+}\),
We can deduce that \(\psi(\lambda_{l}(a)-\lambda_{l-1}(a))\leq0\). Hence there exists some \(s_{0} \in[ \delta(\lambda_{l}(a)-\lambda_{l-1}(a)),\lambda _{l}(a)-\lambda_{l-1}(a) )\), such that
which means that if we take \(s_{0}\) in \(J_{s, t, s_{0}}(u)\) as above, then
□
We see that there exists some u which is also the solution, in the weak sense, to the following equation:
Clearly,
where s, t, \(s_{0}\) and \(c^{+}_{l}(s, t, s_{0})\) are as in Theorem 2.1.
In a similar way, suppose \(\lambda_{l-1}(b)<\lambda_{l}(b)\) for some fix \(l\geq2\), and then we can get a corresponding conclusion as follows.
Theorem 2.2
Suppose that the weights \(a(x)\) and \(b(x)\) are nonnegative and, for some fixed constants \(k'\geq1 \geq k>0\),
where k and \(k'\) satisfy
where \(\delta\in(0,1)\). Suppose that s and t are in some intervals as follows:
Then, for any \(s_{0}\) in \([ \delta(\lambda_{l}(b)-\lambda_{l-1}(b)),\lambda_{l}(b)-\lambda _{l-1}(b) )\), \(c_{l}^{-}(s, t, s_{0})\) is a critical value of \(J_{s, t, s_{0}}^{-}(u)\) on \(S(b)\). In particular, the set of the Fučik spectrum of \(-\Delta_{p}\) with weights \(a(x)\) and \(b(x)\) greater than \(c(a(x),b(x))\) includes \((t+c_{l}^{-}(s, t, s_{0}),s+c_{l}^{-}(s, t, s_{0}))\) for s and t in the above intervals, and some well-defined \(s_{0} \in [\delta(\lambda_{l}(b)-\lambda_{l-1}(b)),\lambda _{l}(b)-\lambda_{l-1}(b) )\) satisfies
Hence,
where s, t, \(s_{0}\) and \(c^{-}_{l}(s, t, s_{0})\) are as in Theorem 2.2.
Remark 2.1
Theorem 2.1 and Theorem 2.2 contain the results in [16], in which \(a(x)=b(x)=1\); so our results must be much more generic than those in [16].
Remark 2.2
Theorem 2.1 and Theorem 2.2 answer partly the open question raised in [13]: whether there exists any other Fučik spectrum for (1.2) besides \(\lambda_{1}(a)\times\mathbb{R}\), \(\mathbb{R}\times\lambda_{1}(b)\) and \(\mathcal{C}:=\{(\alpha(s),s\alpha(s)):\alpha(s)=c(a,sb),s>0\}\).
3 Resonance problems with respect to the Fučik spectrum
As an application, we consider the resonance problems for the Dirichlet boundary problem (1.1) of the Fučik spectrum type. The first case: \((\lambda, \mu)\) is in the part of the two lines \(\lambda _{1}(a) \times R\) and \(R \times\lambda_{1}(b)\), which has been defined in [13]. The second case: \((\lambda, \mu) \in C_{l}^{\pm}\), which is defined in Section 2; see equations (2.13) and (2.17). Similar to Section 2, we always assumed that \(a(x), b(x)\in L^{r}\) with r as in the introduction. We assume that \(g: \Omega\times\mathbb{R}\rightarrow\mathbb{R}\) is a Caratheodory function. In order to state our main results, we assume that \(g(x,u)\) satisfies the following conditions:
- (f1):
-
\(g(x,u)\in C(\bar{\Omega}\times \mathbb{R},\mathbb{R})\) with \(g(x,0)=0\) and \(\lim_{|s|\rightarrow \infty}\frac{g(x,s)}{|s|^{p-1}}=0\) uniformly;
- (f2):
-
\(pG(x,s)-g(x,s)s > 0\) a.e. \(x \in\Omega\), \(\forall s \neq0\);
- (f3):
-
\(\lim_{|s|\rightarrow\infty}(g(x,s)s - pG(x,s))= -\infty\);
- (f4):
-
for some \(v\in(1,p)\), there are constants \(m, a_{m}>0\) s.t. \(G(x,u)\geq a_{m} |u|^{v}\) for \(|u|\leq m\);
- (F1):
-
\(|g(x,u)|\leq V(x)^{p-q}|u|^{q-1}+W(x)^{p-1}\) with \(q \leq p\) and nonnegative \(V,W\in L^{p}(\Omega)\);
- (F2):
-
\(\exists\theta\in(0, 1)\) and \(M > 0\) such that \(G(x,u) - \theta^{p} G(x, \frac{u}{\theta}) \leqslant M\), for \(|u| \geqslant M\).
As is well known, the weak solution to the problem (1.1) is a critical point of
where \(G(x, t)=\int_{0}^{t} g(x, s)\,ds\), and then we have the following.
Theorem 3.1
Suppose that conditions (f1)-(f4) hold, \(a(x), b(x)\geq \epsilon_{0} > 0\), a.e. in Ω. Then (1.1) has at least one nontrivial weak solution in \(W_{0}^{1,p}(\Omega)\) if one of the following conditions holds:
- (f5):
-
\(\lambda=\lambda_{1}(a)\); and μ satisfies
$$c(a,a)a(x)>\mu b(x)\quad \textit{and}\quad \mu \int_{\Omega}b(x)e_{a}^{p}(x)\,dx\geq 1; $$ - (f6):
-
\(\mu=\lambda_{1}(b)\); and λ satisfies
$$c(b,b)b(x)>\lambda a(x)\quad \textit{and}\quad \lambda \int_{\Omega}a(x)e_{b}^{p}(x)\,dx\geq1. $$
Theorem 3.2
Suppose that (F1), (f3) and (f5) hold, \(a(x), b(x)\geq \epsilon_{0} > 0\), a.e. in Ω and satisfy the conditions mentioned in Theorem 2.1 (or in Theorem 2.2), and then the problem (1.1) has a nontrivial solution if \((\lambda,\mu)\in C_{l}^{+}\) (or \((\lambda,\mu)\in C_{l}^{-}\)).
In order to prove Theorem 3.1, we need to state the following conclusions about critical groups. Firstly, we introduce some notations. Let \(E:= W_{0}^{1, p}(\Omega)\), \(E^{*}\) represent the dual space to E, and J be a \(C^{1}(E,\mathbb{R})\). Then the critical set of J on E is defined as
For \(c\in\mathbb{R}\cup\{+\infty\}\), we use
standing for the closed and open sublevel sets of functional J on E. Recall that the critical groups of J at its isolated point u with a critical value \(c=J(u)\) are defined by
where U is an isolated neighborhood of u, and \(H_{q}(\cdot,\cdot)\) is the qth singular relative homology groups with the integer coefficients \(\mathbb{Z}\). Moreover, it is well known that \(C_{q}(J,u)\) is independent of the choice of U and due to the excision property of singular homology theory. Suppose that \(-\infty<\inf J(K)\), and we choose \(c<\inf J(K)\). The critical groups of J at infinity are defined by
From the deformation lemma, we see that \(C_{q}(J,\infty)\) is independent of the choice of \(c<\inf J(K)\) while J satisfies the (P.S.) condition. For details of the topological notation mentioned here and throughout this paper refer to [27].
Lemma 3.1
Suppose that conditions (f1)-(f3) hold and \(\lambda =\lambda_{1}(a)\) (or \(\mu=\lambda_{1}(b)\)); then the functional Φ satisfies (\(C_{c}\)) condition for all \(c \in \mathbb{R}\).
Proof
We just give the proof of the case when \(\lambda=\lambda_{1}(a)\), and the other case of \(\mu=\lambda_{1}(b)\) can be treated similarly.
Suppose that \(\{u_{n}\}\in W_{0}^{1, p}\) is a sequence such that
It is sufficient to obtain that \(\{u_{n}\}\) is bounded. Assume the contrary, that is, \(\|u_{n}\|\rightarrow\infty\). Let \(v_{n}=\frac{u_{n}}{\|u_{n}\|_{1,p}}\). Then, up to a subsequence, we still denote the subsequence by \(\{v_{n}\}\). There is a \(v_{0}\) such that \(\{v_{n}\}\) converges weakly to \(v_{0}\) in \(W_{0}^{1,p}(\Omega)\), \(v_{n}\rightarrow v_{0}\) strongly in \(L^{p}(\Omega)\), and \(v_{n}(x)\rightarrow v_{0}(x)\) almost everywhere in Ω. In (3.3) divided by \(\|u_{n}\|_{1,p}^{p-1}\), and then for every \(w \in W_{0}^{1,p}\), we have
In (3.4), first we claim that \(\|u_{n}\|_{1,p}^{1-p}\int_{\Omega}g(x,u_{n})w\,dx \rightarrow0\). Indeed, it follows from (f1) that, for any \(\epsilon > 0\), there exists \(C_{\epsilon} > 0\) such that
By Hölder’s inequality and Sobolev’s inequality, we get
Hence from (3.4) we have
Putting \(w=v_{n}-v_{0}\), it is sufficient to obtain
And consequently
Using the inequality
where \(\xi, \eta\in\mathbb{R}^{N}\), \(c=c(p)>0\), and \(s=2\) if \(p\geq 2\), \(s=p\) if \(1< p<2\), one easily obtains \(v_{n} \rightarrow v_{0}\) in \(W_{0}^{1, p}\) with \(\|v_{0}\|_{1, p}=1\).
Hence from (3.5) we see that
Taking \(w=v_{0}^{\pm}\), respectively, we get
and
respectively. From the above equality (3.7), we know either \(v_{0}^{+} = 0\) or \(v_{0} = e_{a}\), where \(e_{a}\) is the eigenfunction corresponding to \(\lambda= \lambda_{1}(a)\). If \(v_{0} = e_{a}\), then \(u_{n}(x) = \|u_{n}\|_{1, p} e_{a} \rightarrow+\infty\) for almost everywhere in Ω. If \(v_{0}^{+} = 0\), then we must have \(v_{0} = -e_{b} < 0\) where \(e_{b}\) is the eigenfunction corresponding to \(\mu= \lambda_{1}(b)\), since \(v_{0} \neq0\) and equality (3.8), then \(u_{n}(x) =- \|u_{n}\|_{1, p} e_{b} \rightarrow-\infty\) for almost everywhere in Ω.
From assumption (f3), we have
However, by (3.2) and (3.3), one has
which implies that
This contradicts (3.9). Thus \(\{u_{n}\}\) is bounded, and the proof completed. □
Lemma 3.2
Suppose that conditions (f1)-(f3) and (f5) (or (f6)) hold, \(a(x), b(x)\geq\epsilon_{0} > 0\), a.e. in Ω. Then we have
Proof
We just give the proof of the case (f5), and the other case (f6) can be treated in a similar way.
By (f1), we know that Φ is of class \(C^{1}\) in \(W_{0}^{1,p}(\Omega)\). We consider the decomposition of \(W_{0}^{1,p}=V \oplus W\), where \(V=\operatorname{span} \{e_{a}\}\) is the one-dimensional eigenspace associated with \(\lambda_{1}(a)\), with \(e_{a} > 0\) in Ω and \(\|e_{a}\|_{1,p}=1\), and \(W := \{w\in W_{0}^{1,p}(\Omega): \int_{\Omega}a(x)we_{a}^{p-1}\,dx=0\}\) is a complementary subspace of V in \(W_{0}^{1,p}(\Omega)\). Therefore (see [28]), there exists \(\bar{\lambda} >\lambda_{1}(a)\) such that
We claim that \(\bar{\lambda}=c(a,a)\), if it is not the case, that is, there exists \(u\in W\), such that
Hence there exist \(\lambda_{1}(a)<\theta<c(a,a)\) and \(u\in W\), such that
Hence, θ is the eigenvalue of problem (1.3), which contradicts Theorem 11 in [13] stating that problem (1.3) does not admit any eigenvalue in the open interval \(( \lambda_{1}(a), c(a, a) )\). Then we have
On the other hand, (f1) implies that
And from (f3), we obtain: for any \(M > 0\), there is some \(R > 0\) such that
Integrating the equality
over the interval \([t,T]\subset[R,+\infty)\), we have
Noting that \(\lim_{T\rightarrow\infty}T^{-p}G(x,T)=0\), and letting \(T\rightarrow\infty\) in (3.11), we have
Similarly, it is shown that \(G(x,t)\geq\frac{M}{p}\), \(\forall t \leq-R\), a.e. \(x\in\Omega\). Hence
Now, letting \(v=\xi e_{a}\in V\) (\(\xi\in\mathbb{R}\)), from the assumption (f5) we have
It follows from Lebesgue’s theorem that
Now we turn to proving
Indeed, by (f1),we obtain that, for any \(\epsilon > 0\), there exists \(C_{\epsilon} > 0\) such that
By (3.10) and (3.14), we have, for any \(w \in W\),
With the condition (f5), we can choose ϵ small enough such that \((c(a,a)-\lambda_{1}(a))a(x)-p\epsilon\geq0\), \(c(a,a)a(x)-\mu b(x)-p\epsilon\geq0\). Then fix ϵ, and we have \(\Phi(w)\geq-pC_{\epsilon}|\Omega|\) where \(-pC_{\epsilon}|\Omega|\) is a constant. Obviously (3.13) holds true.
We suppose that Φ has only a finite number of critical points. Then, for sufficiently small number c, we have by [24], that
□
Lemma 3.3
Theorem 2.1 in [21]
Suppose that conditions (f1)-(f4) hold, and let zero be an isolated critical point of Φ. Then the Morse critical groups for Φ at zero are trivial, that is,
Proof of Theorem 3.1
In view of Lemma 3.1 and Lemma 3.2, we have \(C_{1}(\Phi,\infty)\neq0\), and according to the Morse equality, the functional Φ has at least one critical point u such that \(C_{1}(\Phi,u)\neq0\). Since \(C_{1}(\Phi,0)= 0\) by Lemma 3.3, we conclude that \(u\neq0\). The proof is complete. □
Proof of Theorem 3.2
Here we only consider the case \((\lambda, \mu)\in C_{l}^{+}\), since the case \((\lambda, \mu)\in C_{l}^{-}\) can be done similarly. Let \((\lambda, \mu)=(s+c_{l}^{+}(s, t, s_{0}), t+c_{l}^{+}(s, t, s_{0}))\), where \(s_{0} = c_{l}^{+}(s, t, s_{0})\). We introduce a new functional as follows:
First we will show that, for sufficiently large j, there is some \(u_{j}\in W_{0}^{1,p}(\Omega)\) such that
Let \(A^{l-2} \in{\mathcal{F}}_{l-1}^{+}(a)\) be as in Section 2. By (2.12), there exists \(j_{0}\), for any \(j \geqslant j_{0}\), and we have
For such \(j, u \in A^{l-2}\) and \(R>0\), by (F1) and \(a(x) \geqslant \epsilon_{0}\) (a.e. in Ω) we will have
Then we see that
Let
For \(u \in F\), by (3.18) and (F1), we can deduce the following from (3.15):
Denote \(r := (\int_{\Omega}a |u|^{p})^{\frac{1}{p}}\), \(k(r) := \frac {r^{p}}{j} - C (r^{q}+1)\). It is sufficient to deduce that \(k(r)\) (\(r \in[0, +\infty) \)) takes its minimum at point \(r_{m} = (\frac {Cqj}{p})^{\frac{1}{p-q}}\), and its minimum is
Thus, we have
Now using (3.17), for any \(j \geqslant j_{0}\), we can choose sufficiently high \(R_{j} > 0\), thus
where \(B=\{R_{j}u: u \in A^{l-2}\}\). In fact, (3.20) holds only if \(R_{j}\) is larger than the positive solution of the following equation:
It is sufficient to find that (3.20) holds as \(R_{j} \geqslant C j^{\frac{1}{p - q}}\). As a matter of convenience, and without loss of generality, we suppose \(R_{j_{0}}\) is large enough and satisfies
Then we will consider the homotopy-stable family of compact subsets of \(W_{0}^{1, p}(\Omega)\) with boundary B given by
where \(h_{l-2}:S^{l-2}\to A^{l-2}\) is as in Section 2. We claim that the set F is dual to the class \(\mathcal{F}_{j}^{+}\), i.e.,
It is clear from (3.19) and (3.20) that \(F\cap B=\emptyset\). Let \(A \in\mathcal{F}_{j}^{+}\). If \(0\in A\), then we are done. Otherwise, we denote by π the radial projection onto S, \(\pi(A)\in\mathcal{F}_{l}^{+}\), and we have
So \(F\cap\pi(A)\neq\emptyset\), and we obtain \(F\cap A\neq\emptyset\).
Now let us define a minimax sequence on the class \(\mathcal{F}^{+}\):
Noting the truth that the set F is dual to the class \(\mathcal {F}_{j}^{+}\), and by (3.19), we have
Hence it follows from a deformation argument of Cerami [22] that, for the above j, there exists a sequence \({u_{j}}\) such that
It is easy to prove that \(\{c_{j}\}_{j_{0}}^{\infty}\) is a non-decreasing sequence. According to (3.24) and (3.25), for \(j \geqslant j_{0}\) we can get
We claim that the sequence \(\{c_{j}\}_{j_{0}}^{\infty}\) is bounded. Assume the contrary, that is, \(c_{j} \rightarrow-\infty\) as \(j \rightarrow \infty\), and then we will prove that \(\{c_{j}\}_{j_{0}}^{\infty}\) has some fixed growth rate, which contradicts (3.26).
In fact, for a fixed number \(j \geqslant j_{0}\) and \(\forall\epsilon> 0\), there exists \(A \in\mathcal{F}_{j}^{+}\) and a continuous surjection h, such that \(A = h(S_{+}^{l - 1})\) and
Noting that \(H(\cdot) := \frac{R_{j + 1}}{R_{j}} h(\cdot) = \theta ^{-1}h(\cdot) \in\mathcal{F}_{j + 1}^{+}\). Therefore, for all \(u \in S_{+}^{l - 1}\), we have
where \(C_{M}\) is a constant and only related to the condition (F2). Noting the arbitrariness of \(u \in S_{+}^{l - 1}\), we have
According to the arbitrariness of ϵ, we have
From the assumption that \(c_{j} \rightarrow- \infty\) as \(j \rightarrow \infty\), there will exist \(j_{1} \geqslant j_{0}\), such that \(c_{j_{1}} < 0\), and then we can get
Suppose that \(|c_{j} |\) is large enough, and then we can translate the last formula into the following:
Then we have
contradicting (3.26). Thus, \(\{c_{j}\}_{j_{0}}^{\infty}\) is bounded. According to (3.25), we obtain (3.16).
We complete the proof by showing that a subsequence of \(\{u_{j}\}\) mentioned in (3.16) converges to a solution to (1.1). It is sufficient to prove that \(\{u_{j}\}\) is bounded. Assume the contrary, that is, \(\|u_{j}\|\rightarrow\infty\). Setting \(v_{j}:=\frac{u_{j}}{\|u_{j}\|}\), then up to a subsequence, we still denote the subsequence by \(\{v_{j}\}\). There is a \(v_{0}\) such that \(v_{j}\rightharpoonup v_{0}\) weakly in \(W_{0}^{1,p}(\Omega)\), strongly in \(L^{p}(\Omega)\), and almost everywhere in Ω. We have
It is sufficient to obtain
Hence, we can deduce that \(v_{j}\rightarrow v_{0}\) strongly in \(W_{0}^{1,p}\), and \(\|v_{0}\|=1\). For each \(w \in W_{0}^{1, p}\), we have
Going to the limit as \(j \rightarrow\infty\), we get
So \(v_{0}\in E_{\lambda a,\mu b}\). Thus,
contradicting (3.16). Hence, \(\{u_{j}\}\) is bounded. There exists a subsequence of \(\{u_{j}\}\) converging to a weak solution to (1.1). □
References
Fučik, S: Solvability of Nonlinear Equations and Boundary Value Problems. Springer, Dordrecht (1980)
Dancer, N: On the Dirichlet problem for weakly nonlinear elliptic partial differential equation. Proc. R. Soc. Edinb. 76, 283-300 (1977)
Arias, M, Campos, J: Radial Fučik spectrum of the Laplace operator. J. Math. Anal. Appl. 156, 654-666 (1995)
Dancer, N: Generic domain dependance for nonsmooth equations and the open set problem for jumping nonlinearities. Topol. Methods Nonlinear Anal. 1, 139-150 (1993)
Micheletti, AM: A remark on the resonance set for a semilinear elliptic equation. Proc. R. Soc. Edinb. 124, 803-809 (1994)
Drábek, P: Solvability and Bifurcations of Nonlinear Equations. Pitman Research Notes in Mathematics, vol. 264. Wiley, New York (1992)
Arias, J, Campos, M: Fučik spectrum of a singular Sturm-Liouville problem. Nonlinear Anal. TMA 27, 679-697 (1996)
Rynne, B: The Fučik spectrum of general Sturm-Liouville problems. J. Differ. Equ. 161, 87-109 (2000)
Reichel, W, Walter, W: Sturm-Liouville type problems for the p-Laplacian under asymptotic nonresonance conditions. J. Differ. Equ. 156, 50-70 (1999)
Alif, M, Gossez, J-P: On the Fučik spectrum with indefinite weights. Differ. Integral Equ. 14, 1511-1530 (2001)
Lindqvist, P: On the equation \(\operatorname{div}(|\nabla u|^{p-2}\nabla u)+\lambda|u|^{p-2}u=0\). Proc. Am. Math. Soc. 109(1), 157-164 (1990); Addendum: Proc. Am. Math. Soc. 116(2), 583-584 (1992)
Cuesta, M: Eigenvalue problems for the p-Laplacian with indefinite weight. Electron. J. Differ. Equ. 2001, 33 (2001)
Arias, M, Campos, J, Cuesta, M, Gossez, J-P: Asymmetric elliptic problems with indefinite weights. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 19, 581-616 (2002)
Cuesta, M, De Figueiredo, D, Gossez, JP: The beginning of the Fučik spectrum of the p-Laplacian. J. Differ. Equ. 159, 212-238 (1999)
Micheletti, A, Pistoia, A: On the Fučik spectrum for the p-Laplacian. Differ. Integral Equ. 14, 867-882 (2001)
Perera, K: Resonance problems with respect to the Fučik spectrum of the p-Laplacian. Electron. J. Differ. Equ. 2002, 36 (2002)
Perera, K, Squassina, M, Yang, Y: A note on the Dancer-Fučik spectra of the fractional p-Laplacian and Laplacian operators. Adv. Nonlinear Anal. 4(1), 13-23 (2015)
Pucci, P, Radulescu, V: The impact of the mountain pass theory in nonlinear analysis: a mathematical survey. Boll. Unione Mat. Ital. 9(3), 543-582 (2010)
Pucci, P, Radulescu, V: Combined effects in quasilinear elliptic problems with lack of compactness. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 22(2), 189-205 (2011)
Barbu, L, Enache, C: Maximum principles, Liouville-type theorems and symmetry results for a general class of quasilinear anisotropic equations. Adv. Nonlinear Anal. 5(4), 395-405 (2016)
Guo, Y, Liu, J: Solutions of p-sublinear p-Laplacian equation via Morse theory. J. Lond. Math. Soc. 72, 632-644 (2005)
Cerami, G: An existence criterion for the critical points on unbounded manifolds. Istit. Lomb. Accad. Sci. Lett. Rend. A. 112(2), 332-336 (1979)
Bartolo, P, Benci, V, Fortunato, D: Abstract critical point theorems and applications to nonlinear problems with strong resonance at infinity. Nonlinear Anal. 7, 981-1012 (1983)
Bartsch, T, Li, SJ: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. 28, 419-441 (1997)
Drábek, P, Robinson, S: Resonance problems for the p-Laplacian. J. Funct. Anal. 169, 189-200 (1999)
Ghoussoub, M: Duality and Perturbation Methods in Critical Point Theory. Cambridge University Press, Cambridge (1993)
Chang, KC: Infinite Dimensional Morse Theory and Multiple Solutions of Nonlinear Differential Equations. Birkhäuser, Boston (1993)
Anane, A, Tsouli, N: On the second eigenvalue of the p-Laplacian. In: Benkirane, A, Gossez, J-P (eds.) Nonlinear Partial Differential Equations. Pitman Research, Notes in Math., vol. 343, pp. 1-9 (1996)
Acknowledgements
This research was supported in part by the Natural Science Foundation of China under grants 11501106 and 11226303, the Natural Science Foundation of Guangdong Province of China under grant 2016A030313835, the Excellent Teacher Scheme in Guangdong Higher Education Institutions under grant YQ2014332 and the Training Programme Foundation for Excellent Young Scholar of Guangdong Province under grant YQ2015164.
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Gan, W., Geng, D. & Chen, J. The Fučik spectrum of the p-Laplace equations with different weights and its resonance problems. Bound Value Probl 2017, 31 (2017). https://doi.org/10.1186/s13661-017-0763-3
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DOI: https://doi.org/10.1186/s13661-017-0763-3