Abstract
We consider a perturbed version of the Robin eigenvalue problem for the p-Laplacian. The perturbation is \((p - 1)\)-superlinear. Using the Nehari manifold method, we show that for all parameters \(\lambda < {\hat{\lambda }}_1\) (= the principal eigenvalue of the differential operator), there exists a ground-state nodal solution of the problem.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Suppose that \(\Omega \subseteq {\mathbb {R}}^N\) is a bounded domain with a \(C^2\)-boundary \(\partial \Omega \). In this paper, we study the following nonlinear parametric Robin problem:
In this problem, \(\Delta _p\) denotes the p-Laplace differential operator defined by
On the set \([Du=0]\) of critical points, this operator is degenerate for \(p>2\) and is singular if \(1<p<2\). The analysis developed in this paper includes the borderline case \(p=N\). In this situation, the Dirichlet energy \(\int \limits _\Omega |Du|^N\mathrm{d}x\) is conformally invariant. The borderline case is important in the theory of quasi-conformal mappings.
Problem (\(P_\lambda \)) contains the perturbation \(u \mapsto \xi (z)|u|^{p - 2} u\) with the potential function \(\xi \in L^\infty (\Omega )\), \(\xi (z) \geqslant 0\) for a.a. \(z \in \Omega \). In the reaction (right-hand side of problem (\(P_\lambda \))), we have the combined effects of a parametric term \(u \mapsto \lambda |u|^{p-2} u\) and of a Carathéodory perturbation f(z, x). (That is, for all \(x \in {\mathbb {R}}\) the mapping \(z \mapsto f(z,x)\) is measurable and for a.a. \(z \in \Omega \) the function \(x \mapsto f(z,x)\) is continuous.) We assume that f(z, x) exhibits \((p -1)\)-superlinear growth as \(x \rightarrow \pm \infty \).
We can view problem (\(P_\lambda \)) as a superlinear perturbation of the Robin eigenvalue problem for the operator \(u \mapsto - \Delta _p u + \xi (z) |u|^{p-2} u\). In the boundary condition, \(\frac{\partial u}{\partial n_p}\) denotes the conormal derivative of u corresponding to the p-Laplace differential operator. This directional derivative is interpreted using the nonlinear Green’s identity (see Papageorgiou et al. [13, p. 35]), and if \(u \in C^1({\overline{\Omega }})\), then
with \(n(\cdot )\) being the outward unit normal on \(\partial \Omega \). The boundary coefficient \(\beta \in C^{0,\alpha }(\partial \Omega )\) with \(0< \alpha < 1\) satisfies \(\beta (z) \geqslant 0\) for all \(z \in \partial \Omega \).
The nonlinear Robin boundary condition in problem (\(P_\lambda \)) is motivated by certain nonlinear patterns in which the flux across the boundary is not linearly proportional to the density function. A typical example is Boltzmann’s fourth power law in heat transfer problems, where
where \(h_0\) is the surrounding temperature; see Özisik [10]. Another example is based on the Michaelis–Menten hypothesis in some biochemical reaction problems where the substrate concentration satisfies the boundary condition
see Ross [15].
We are looking for ground-state (that is, least energy) nodal (sign-changing) solutions of problem (\(P_\lambda \)). Using the Nehari manifold method, we show that if \(\lambda < {\hat{\lambda }}_1\) (here \({\hat{\lambda }}_1\) is the principal eigenvalue of the differential operator \(u \mapsto - \Delta _p u + \xi (z) |u|^{p-2}u\) with Robin boundary condition), then problem (\(P_\lambda \)) has a ground-state nodal solution. We prove this result by relaxing the usual Nehari monotonicity hypothesis which is the following:
-
(N)
“For a.e. \(z \in \Omega \), the quotient function \(x \mapsto \frac{f(z,x)}{|x|^{p -1}}\) is strictly increasing on \(\mathring{{\mathbb {R}}}_- \cup \mathring{{\mathbb {R}}}_+\) with \(\mathring{{\mathbb {R}}}_- =(-\infty , 0)\) and \(\mathring{{\mathbb {R}}}_+ =(0,+\infty )\).”
This condition was used by Szulkin and Weth [16] to have uniqueness of the projection on the Nehari manifold. Instead, in the present paper, we assume that the quotient is simply increasing.
In the past, the problem of the existence of ground-state solutions for such parametric problems was investigated only in the context of semilinear Dirichlet problems driven by the Laplace differential operator. We mention the work of Szulkin and Weth [16], who produce a ground-state solution using the stronger monotonicity condition (N), but they do not show that their ground-state solution is nodal. Later, Tang [17] obtained a ground-state solution using the relaxed monotonicity condition, but the ground-state solution need not be nodal. Ground-state nodal solutions under the relaxed monotonicity hypothesis were obtained recently by Lin and Tang [7]. All the aforementioned works deal with semilinear equations (that is, \(p=2\)), and the boundary condition is Dirichlet. Ground-state nodal solutions under the strong Nehari monotonicity condition (see hypothesis (N) above) were obtained by Liu and Dai [8] (Dirichlet problems) and Gasiński and Papageorgiou [4] (problems with a nonlinear boundary condition). In both these works, the reaction is nonparametric and has a different structure. We also mention the work of Papageorgiou, Rădulescu and Repovš [12], who studied problem (\(P_\lambda \)) when \(p=2\) (semilinear equation) looking for positive solutions and proved a bifurcation-type result with critical parameter being \({\hat{\lambda }}_1\). Finally, we point out that eigenvalue problems with nonlinear Robin boundary condition naturally arise in the study of reaction–diffusion equation where a distributed absorption competes with a boundary source; see Lacey et al. [5] for details.
Our main result in this paper is the following theorem. Hypotheses \({H}_0\) and \({H}_1\) on the data of the problem can be found in Sect. 2.
Theorem 1
If hypotheses \({H_0},\) \({H_1}\) are fulfilled and \(\lambda < {\hat{\lambda }}_1,\) then the following properties hold true.
-
(a)
Problem (\(P_\lambda \)) has a ground-state nodal solution \(u_* \in C^1({\overline{\Omega }});\)
-
(b)
If, in addition, \(e(z,x) > 0\) for a.a. \(z \in \Omega ,\) all \(x \ne 0,\) then \(u_*\) has two nodal domains; here, \(e(z,x)=f(z,x)x-pF(Z,x)\)
2 Mathematical preliminaries and hypotheses
The main space in the analysis of problem (\(P_\lambda \)) is the Sobolev space \(W^{1,p}(\Omega )\). By \(\Vert \cdot \Vert \), we denote the norm of \(W^{1,p}(\Omega )\) defined by
Also, we will use the boundary Lebesgue spaces \(L^p(\partial \Omega )\). On \(\partial \Omega \), we consider the \((N-1)\)-dimensional Hausdorff (surface) measure \(\sigma (\cdot )\). Using this measure, we can define in the usual way the “boundary” Lebesgue spaces \(L^q(\partial \Omega )\), \(1 \leqslant q \leqslant + \infty \). From the theory of Sobolev spaces, we know that there exists a unique continuous linear operator \({\hat{\gamma }}_0 : W^{1,p}(\Omega )\mapsto L^p(\partial \Omega )\), known as the “trace operator,” such that
So, the trace operator extends the notion of boundary values to all Sobolev functions. We know that this operator is compact into \(L^r(\partial \Omega )\) for \(r < \frac{(N-1)p}{N-p}\) if \(p<N\) and into \(L^r(\partial \Omega )\) for \(1 \leqslant r < + \infty \) if \(N \leqslant p\). The trace operator is not surjective, and we have
We introduce our hypotheses on the potential function \(\xi (\cdot )\) and on the boundary coefficient \(\beta (\cdot )\).
\(H_0\): \(\xi \in L^\infty (\Omega )\), \(\xi (z) \geqslant 0\) for a.a. \(z \in \Omega \), \(\beta \in C^{0,\alpha }(\partial \Omega )\) with \(0< \alpha < 1\), \(\beta (z) \geqslant 0\) for all \(z \in \partial \Omega \) and \(\xi \not \equiv 0\) or \(\beta \not \equiv 0\).
Remark 1
We see that these hypotheses incorporate also the Neumann problem \((\beta \equiv 0).\)
In what follows, by \(\gamma _p : W^{1,p}(\Omega )\mapsto {\mathbb {R}}\) we denote the \(C^1\)-functional defined by
for all \(u \in W^{1,p}(\Omega )\).
Using Lemma 4.11 of Mugnai and Papageorgiou [9] and Proposition 2.4 of Gasiński and Papageorgiou [3], we have
Another way to see this is via a simple contradiction argument. So, suppose that we could find \(\{u_n\}_{n\in {\mathbb {N}}}\subseteq W^{1,p}(\Omega )\) such that for all \(n\in {\mathbb {N}}\)
By homogeneity, we may assume that \(\Vert u_n\Vert =1\) for all \(n\in {\mathbb {N}}\). So, we may assume that
Then, in the limit as \(n\rightarrow \infty \) and since the norm in a Banach space is weakly lowers semicontinuous, we obtain
We have
But then we have
a contradiction to the fact that \(\Vert u_n\Vert =1\) for all \(n\in {\mathbb {N}}\).
We consider the nonlinear eigenvalue problem
We say that \({\hat{\lambda }} \in {\mathbb {R}}\) is an eigenvalue of the operator \(u \mapsto - \Delta _p u + \xi (z) |u|^{p- 2} u\) with Robin boundary condition, if problem (2) admits a nontrivial solution \({\hat{u}} \in W^{1,p}(\Omega )\), known as an eigenfunction corresponding to the eigenvalue \({\hat{\lambda }}\).
By using the Lagrange multiplier rule, we see that problem (2) has a smallest eigenvalue \({\hat{\lambda }}_1\), which is characterized variationally by
On account of (1), we see that \({\hat{\lambda }}_1 > 0\). Also, this eigenvalue is isolated in the spectrum and simple. The infimum in (3) is realized on the corresponding one dimensional eigenspace and so the eigenfunctions corresponding to \({\hat{\lambda }}_1 > 0\) have fixed sign. By \({\hat{u}}_1\), we denote the positive, \(L^p\)-normalized (that is, \(\Vert {\hat{u}}_1\Vert _p = 1\)) eigenfunction corresponding to \({\hat{\lambda }}_1\). The nonlinear regularity theory (see Lieberman [6]) and the nonlinear maximum principle (see Pucci and Serrin [14]) imply that \({\hat{u}}_1 \in \mathrm{int\,} C_+ = \left\{ u \in C^1({\overline{\Omega }}) : u(z) > 0 \hbox { for all } z \in {\overline{\Omega }}\right\} \) (the interior of positive (order) cone \(C_+ = \left\{ u \in C^1({\overline{\Omega }}) : u(z) \geqslant 0 \hbox { for all } z \in {\overline{\Omega }}\right\} \) of \(C^1({\overline{\Omega }})\)). The Ljusternik–Schnirelmann minimax scheme implies the existence of a whole strictly increasing sequence \(\left\{ {\hat{\lambda }}_k\right\} _{k \in {\mathbb {N}}}\) of eigenvalues of problem (2) such that \({\hat{\lambda }}_k \rightarrow + \infty \), known as “variational eigenvalues.” We do not know if they exhaust the spectrum of the operator. We know that if \({\hat{\lambda }} \ne {\hat{\lambda }}_1\) is a nonprincipal eigenvalue, then the corresponding eigenfunctions \({\hat{u}} \in C^1({\overline{\Omega }})\) (regularity theory) are nodal functions. Details can be found in Fragnelli et al. [2].
Let \(A_p : W^{1,p}(\Omega )\mapsto W^{1,p}(\Omega )^*\) be the nonlinear map defined by
It is well known that this map is bounded (maps bounded sets to bounded sets), continuous, monotone (thus maximal monotone too) and of type \((S)_+\), that is
We have
for all u, \(h \in W^{1,p}(\Omega )\).
Now, we introduce our hypotheses on the perturbation f(z, x). Recall that
\({H_1}\): \(f : \Omega \times {\mathbb {R}}\mapsto {\mathbb {R}}\) is a Carathéodory function such that \(f(z,0) = 0\) for a.a. \(z \in \Omega \) and
-
(i)
\(| f(z,x) | \leqslant a(z) \left( 1+|x|^{r-1}\right) \) for a.a. \(z \in \Omega \), all \(x \in {\mathbb {R}}\), with \(a \in L^\infty (\Omega )\) and \(p<r<p^*\);
-
(ii)
if \(F(z,x)=\int \limits _0^x f(z,s) \mathrm{d}s\), then \(\lim _{x \rightarrow \pm \infty }\frac{F(z,x)}{|x|^p} = +\infty \) uniformly for a.a. \(z \in \Omega \);
-
(iii)
\(\lim _{x \rightarrow 0} \frac{f(z,x)}{|x|^{p-2}x} = 0\) uniformly for a.a. \(z \in \Omega \);
-
(v)
for a.a. \(z \in \Omega \), the quotient function \(x \mapsto \frac{f(z,x)}{|x|^{p-1}}\) is increasing on \(\mathring{{\mathbb {R}}}_- \cup \mathring{{\mathbb {R}}}_+\).
Remark 2
If \(f(z,\cdot )\) is \((p-1)\)-superlinear as \(x \rightarrow \pm \infty ,\) then hypothesis \({H_1}\)(ii) is satisfied. Note that we use the relaxed Nehari monotonicity condition (see hypothesis \({H_1}\)(iv)).
We will prove our existence theorem first using the strong Nehari monotonicity condition (see (N)), and then via approximations of the perturbation, we will establish the result for the general case. For this reason, we introduce the following set of hypotheses:
\({H_1'}\): \(f : \Omega \times {\mathbb {R}}\mapsto {\mathbb {R}}\) is a Carathéodory function such that \(f(z,0) = 0\) for a.a. \(z \in \Omega \), hypotheses \({H_1'}\)(i), (ii), (iii) are the same as the corresponding hypotheses in \({H_1}\) and
-
(iv)
hypothesis (N) holds.
If \(u \in W^{1,p}(\Omega )\), then we define \(u^\pm = \max \left\{ \pm u,0\right\} \) and we have
We denote by \(|\cdot |_N\) the Lebesgue measure on \({\mathbb {R}}^N\).
Let \(\varphi _\lambda : W^{1,p}(\Omega )\mapsto {\mathbb {R}}\) be the energy (Euler) functional defined by
Evidently, \(\varphi _\lambda \in C^1(W^{1,p}(\Omega ))\). We introduce the following two sets:
We see that \({N_0} \subseteq {N}\). The set N is known as the “Nehari manifold” for the functional \(\varphi _\lambda (\cdot )\). Note that every nontrivial solution of problem (\(P_\lambda \)) belongs to the Nehari manifold. Since we look for nodal solutions, we introduce the Nehari submanifold \({N_0}\). Hypotheses \({H_0}\), \({H_1'}\) imply that \(\emptyset \ne {N_0} \subseteq {N}\) (see also Proposition 5 and Papageorgiou et al. [11]).
3 Ground-state nodal solutions
We define
We look for an element of \({N_0}\) which realizes the infimum \({\hat{m}}_\lambda ^0\) and which is a critical point of \(\varphi _\lambda \). Such a function will be a ground-state nodal solution of problem (\(P_\lambda \)).
Proposition 2
If hypotheses \({H_0},\) \({H_1}\) hold, then for all \(\tau ,\) \(t \geqslant 0\) and all \(u \in W^{1,p}(\Omega )\) we have
Proof
We have
Using the fact that \(\left\{ u^+> 0\right\} \cap \left\{ u^- > 0\right\} = \emptyset \), we have
Similarly, we have
Finally, we have
Let \(x \ne 0\) and \(\mu \geqslant 0\). Then,
Returning to (7) and using (8), we obtain
Finally, we use (5), (6) and (9) in (4) and obtain
This proof is now complete. \(\square \)
From this proposition, we infer at once the following two useful corollaries.
Corollary 3
If hypotheses \({H_0},\) \({H_1}\) hold and \(u \in {N_0},\) then \(\varphi _\lambda (u) = \max _{\tau ,t \geqslant 0} \varphi _\lambda (\tau u^+ - t u^-).\)
Corollary 4
If hypotheses \({H_0},\) \({H_1}\) hold and \(u \in {N},\) then \(\varphi _\lambda (u) = \max _{\tau \geqslant 0} \varphi _\lambda (\tau u).\)
Evidently, Corollary 3 implies \(\emptyset \ne {N_0} \subseteq {N}\).
Next, we relate nodal elements of \(W^{1,p}(\Omega )\) with the Nehari submanifold \({N_0}\). In particular, we infer that \({N_0} \ne \emptyset \).
Proposition 5
If hypotheses \({H_0},\) \({H_1'}\) hold, \(\lambda < {\hat{\lambda }}_1\) and \(u \in W^{1,p}(\Omega )\) with \(u^\pm \ne 0,\) then we can find a unique pair \((\tau _u,t_u) \in \mathring{{\mathbb {R}}}_+ \times \mathring{{\mathbb {R}}}_+\) such that \(\tau _u u^+ - t_u u^- \in {N_0}.\)
Proof
Let \(u \in W^{1,p}(\Omega )\) with \(u^\pm \ne 0\) (nodal function) and consider the corresponding fibering function
Using the chain rule, we see that for all \(t > 0\) the following equivalence holds:
On account of hypothesis \(H_1'\mathrm{(iv)}=(N)\), the integral in the right-hand side of (10) is strictly increasing in \(t > 0\).
Hypotheses \(H_1'\mathrm{(iv)},\) (iii) imply that given \(\varepsilon > 0\), we can find \(c_1 = c_1(\varepsilon ) > 0\) such that
Then, we have
Choosing \(\varepsilon \in (0, {\hat{\lambda }}_1 - \lambda )\) (recall that \(\lambda < {\hat{\lambda }}_1\)), we obtain
On the other hypotheses \(H_1'\mathrm{(i)}\), (ii) imply that given \(\eta > 0\), we can find \(c_4 = c_4(\eta ) > 0\) such that
Then, we have
for some \(c_5 > 0\) (see (12)).
Recall that \(\eta > 0\) is arbitrary. So, choosing \(\eta > 0\) large we have
We conclude that there exists unique \(\tau _u > 0\) (see (10)) such that
In a similar fashion, working this time with the fibering function
we produce a unique \(t_u > 0\) such that
We conclude that
This ends the proof of the proposition. \(\square \)
Using the previous proposition, we can have a minimax characterization of \({\hat{m}}_\lambda ^0 = \inf _{N_0} \varphi _\lambda \). Let \(W_n = \left\{ u \in W^{1,p}(\Omega ): u^\pm \ne 0\right\} \) (the nodal elements of the Sobolev space \(W^{1,p}(\Omega )\)).
Proposition 6
If hypotheses \({H_0}, {H_1'}\) hold and \(\lambda < {\hat{\lambda }}_1,\) then \({\hat{m}}_\lambda ^0 = \inf _{u \in W_n} \max _{\tau , t \geqslant 0} \varphi _\lambda (\tau u^+ - t u^-).\)
Proof
Let \(\xi _\lambda = \inf _{u \in W_n} \max _{\tau , t \geqslant 0} \varphi _\lambda (\tau u^+ - t u^-) < + \infty \) (since \({N_0} \subseteq W_n\)). We have
On the other hand, we have
From (13) and (14), we conclude that \(\xi _\lambda = {\hat{m}}_\lambda ^0.\) \(\square \)
Next, we show that \({\hat{m}}_\lambda ^0\) is realized on \({N_0}\).
Proposition 7
If hypotheses \({H_0},\) \({H_1'}\) hold and \(\lambda < {\hat{\lambda }}_1,\) then there exists \({\hat{u}} \in {N_0}\) such that \(\varphi _\lambda ({\hat{u}}) = {\hat{m}}_\lambda ^0 > 0.\)
Proof
Let \(\left\{ u_n\right\} _{n \in {\mathbb {N}}} \subseteq {N_0}\) be a minimizing sequence. We show that this sequence is bounded in \(W^{1,p}(\Omega )\). Arguing by contradiction, suppose that up to a subsequence, we have \(\Vert u_n\Vert \rightarrow + \infty \). Let \(v_n = \frac{u_n}{\Vert u_n\Vert }\), \(n \in {\mathbb {N}}\). We have that \(\Vert v_n\Vert = 1\) for all \(n \in {\mathbb {N}}\) and so we may assume that
Suppose that \(v = 0\). Using (11), we see that for every \(\rho > 0\) we have
With \(\varepsilon _n \rightarrow 0^+\) and \(t_n = \frac{\rho }{\Vert v_n\Vert }\) (\(n \in {\mathbb {N}}\)), we have
But \(\rho > 0\) is arbitrary. Let \(\rho \rightarrow + \infty \) to reach a contradiction. Therefore, \(v \ne 0\).
Let \(\Omega _* = \left\{ z \in \Omega : v(z) \ne 0\right\} \). We have \(|\Omega _*|_N > 0\) and
Hypotheses \(H_1'\mathrm{(i)},\) (ii) imply that there exists \(c_8 > 0\) such that
We have
a contradiction. So, the minimizing sequence \(\left\{ u_n\right\} _{n \in {\mathbb {N}}} \subseteq {N_0}\) is bounded in \(W^{1,p}(\Omega )\). We may assume that
From (18), it follows that
Since \(u_n \in {N_0}\) (\(n \in {\mathbb {N}}\)), we have
From (18), (19) and the sequential weak lower semicontinuity of \(\gamma _p(\cdot )\), we have
Similarly, we show that
We have
On account of hypotheses \(H_1'\mathrm{(i)}\), (iii), given \(\varepsilon > 0\), we can find \(c_{10} = c_{10}(\varepsilon ) > 0\) such that
Since \(u_n \in {N_0}\) (\(n \in {\mathbb {N}}\)), we have
Choose \(\varepsilon \in \left( 0,\frac{c_{11}}{c_{12}}\right) \). Then,
Then, from (23), (24) and (25), we have
In a similar fashion, we show that \({\hat{u}}^- \ne 0\). Then, from (22), it follows that
It remains to show that \({\hat{m}}_\lambda ^0 > 0\). We have
We define \(e(z,x) = f(z,x) x - p F(z,x)\).
Claim: For a.a. \(z \in \Omega \), \(e(z,\cdot )\) is strictly increasing on \({\mathbb {R}}_+ = [0,+ \infty )\) and strictly decreasing on \({\mathbb {R}}_- = (- \infty , 0]\).
First, we show the claim under the extra condition that for a.a. \(z \in \Omega \), \(f(z,\cdot )\) is differentiable. Then, for \(x > 0\), we have
In a similar fashion, we show that
Therefore, the claim is true if \(f(z.\cdot )\) is differentiable.
Next, we drop the differentiability hypothesis on \(f(z,\cdot )\). To this end, we consider a mollifier \(\theta \in C_c^\infty ({\mathbb {R}})\) such that
We set \(\theta _\varepsilon (t) = \frac{1}{\varepsilon } \theta \left( \frac{t}{\varepsilon }\right) \). Then, \(\theta _\varepsilon \in C_c^\infty ({\mathbb {R}})\), \(\mathrm{supp \,} \theta _\varepsilon \subseteq [0,\varepsilon ]\), \(\int \limits ^{+\infty }_{-\infty } \theta _\varepsilon (s) \mathrm{d}s =1\). We define
From Evans and Gariepy [1, pp. 122–123], we know that for a.a. \(z \in \Omega \)
For \(x> u > 0\), we have
Since \(f_\varepsilon (z,\cdot )\) is differentiable, from the first part of the proof of the claim we have that
where \(e_\varepsilon (z,x) = f_\varepsilon (z,x)x - p F_\varepsilon (z,x)\), \(F_\varepsilon (z,x) = \int \limits ^x_0 f_\varepsilon (z,s) \mathrm{d}s\). Passing to the limit as \(\varepsilon \rightarrow 0^+\), we obtain
Similarly we show that
This proves the claim.
Returning to (27) and recalling that \(e(z,0) = 0\) for a.a. \(z \in \Omega \) we infer that
Therefore, finally we have
This proof is now complete. \(\square \)
Remark 3
We indicate an alternative way show that \({\hat{m}}_\lambda ^0 > 0.\) Using a contradiction argument as in the proof Proposition 7, we can show that \(\varphi _\lambda |_{N}\) is coercive. Since \(\varphi _\lambda \) is sequentially weakly lower semicontinuous, invoking the Weierstrass–Tonelli theorem, we can find \({\hat{u}}_0 \in {N}\) such that \(\varphi _\lambda ({\hat{u}}_0) = \inf _{N} \varphi _\lambda = {\hat{m}}_\lambda .\) Since N is a natural constraint for \(\varphi _\lambda \) (see [13]), we have that \({\hat{u}}_0 \in K_{\varphi _\lambda } \subseteq C^1({\overline{\Omega }})\) (nonlinear regularity theory). From Corollary 4, we know that \(\varphi _\lambda ({\hat{u}}_0) = \max _{\tau \geqslant 0} \varphi _\lambda (\tau {\hat{u}}_0)\) and on account of hypothesis \(H_1'\mathrm{(iii)}\) and since \(\lambda < {\hat{\lambda }}_1,\) for \(\tau \in (0,1)\) small we have \(\varphi _\lambda (\tau {\hat{u}}_0) > 0,\) hence \(0 < {\hat{m}}_\lambda = \varphi _\lambda ({\hat{u}}_0).\) But clearly \({\hat{m}}_\lambda \leqslant {\hat{m}}_\lambda ^0\) (since \({N_0 \subseteq N}).\)
Next, following the arguments of Willem [18, p. 74] and of Szulkin and Weth [16, p. 612], we show that the Nehari submanifold \({N_0}\) is a natural constraint (see [13, p. 425]).
Proposition 8
If hypotheses \({H_0}, {H_1'}\) hold, \(\lambda < {\hat{\lambda }}_1\) and \({\hat{u}} \in {N_0}\) is as in Proposition 7, then \({\hat{u}} \in K_{\varphi _\lambda } = \left\{ u \in W^{1,p}(\Omega ): \varphi _\lambda '(u) = 0\right\} \) (the critical set of \(\varphi _\lambda ).\)
Proof
Since \({\hat{u}} \in {N_0}\), we have
For \(\tau ,\) \(t \in \mathring{{\mathbb {R}}} {\setminus } \{1\}\), we have
Arguing by contradiction, suppose that \(\varphi _\lambda '({\hat{u}}) \ne 0\). Then, we can find \(\delta > 0\) and \(\eta > 0\) such that
Consider the parallelogram \(D=\left( \frac{1}{2},\frac{3}{2}\right) ^2\) and the function \(\mu (\tau ,t) = \tau u^+ - t u^-\), \(\tau \), \(t \geqslant 0\). From (29), we see that
Using Lemma 2.3 of Willem [18, p. 38], with \(\varepsilon = \min \left\{ \frac{{\hat{m}}_\lambda ^0 - \ell }{4}, \frac{\eta \delta }{8}\right\} \), \(S = {\overline{B}}_\delta ({\hat{u}})= \left\{ u \in W^{1,p}(\Omega ): \Vert u - {\hat{u}} \Vert { \leqslant } \delta \right\} \), we can find a deformation \({\hat{h}}(t,u)\) such that
From these properties of the deformation, we infer that
Let \(\beta (\tau ,t) = {\hat{h}}(1,\mu (\tau ,t))\) and set
By \({\hat{d}}_B\), we denote the Brouwer degree. From the proof Proposition 5, we see that
Note that \(\mu |_{\partial D} = \beta |_{\partial D}\) (see (30)), the definition of \(\ell \) and the choice of \(\varepsilon > 0\)). So, from the properties of the Brouwer degree (see [13, p. 178]), we have
which contradicts (30).
Therefore, we conclude that \({\hat{u}} \in K_{\varphi _\lambda }\). \(\square \)
So, under the stronger monotonicity hypothesis \((N) = H_1'\mathrm{(iv)}\), we have proved the existence of a ground-state nodal solution for problem (\(P_\lambda \)), when \(\lambda < {\hat{\lambda }}_1\).
Next, we replace the strong monotonicity condition by the relaxed one \(H_1'\mathrm{(iv)}\). To be able to treat this more general situation, let \(\theta > 0\) and consider the following perturbation of f(z, x):
Then, \(f_\theta (z,x)\) is a Carathéodory function which satisfies hypothesis \({H_1'}\). We set \(F_\theta (z,x) = \int \limits ^x_0 f_\theta (z,s) \mathrm{d}s\) and consider the \(C^1\)-functional \(\varphi _\lambda ^\theta : W^{1,p}(\Omega )\mapsto {\mathbb {R}}\) defined by
for all \(u \in W^{1,p}(\Omega )\).
We see that
For this functional, we introduce the Nehari manifold
and the Nehari submanifold
Proposition 9
If hypotheses \({H_0}, {H_1}\) hold and \(\lambda < {\hat{\lambda }}_1,\) then we can find \(\nu _0 > 0\) such that
Proof
Let \(u \in {N}^\theta \). We have
Since \(r > p\), for \(t \in (0,1)\) small we have
The proof of the proposition is now complete. \(\square \)
Now, we are ready to state and prove the main result of this paper, Theorem 1, which establishes the existence of a ground-state nodal solution under the relaxed monotonicity condition \(H_1\mathrm{(iv)}\).
3.1 Proof of Theorem 1
(a) Let \({\overline{u}} \in {N}_0\). We have
Now let \(\theta _n \rightarrow 0^+\). Using Propositions 7, 8, 9 and (32), we see that we can \(u_n= u_{\theta _n} \in {N}^{\theta _n}\) (\(n \in {\mathbb {N}}\)) such that
Claim: The sequence \(\left\{ u_n\right\} _{n \in {\mathbb {N}}} \subseteq W^{1,p}(\Omega )\) is bounded.
We argue indirectly. So, suppose that the claim is not true. Then, for at least a subsequence, we have
Let \(v_n = \frac{u_n}{\Vert u_n\Vert }\), \(n \in {\mathbb {N}}\). Then, \(\Vert v_n\Vert = 1\) for all \(n \in {\mathbb {N}}\) and so we may assume that
Suppose that \(v =0\). Let \(k > 1\). From (33), we have
We pass to the limit as \(n \rightarrow \infty \). Since \(v = 0\), from (34) we obtain
But \(k > 1\) is arbitrary. So, let \(k \rightarrow +\infty \) to have a contradiction.
Next, we assume that \(v \ne 0\). We set \({\hat{\Omega }} = \left\{ z \in \Omega : v(z) \ne 0\right\} \). Then, \(|{\hat{\Omega }}|_N > 0\) and we have \(|u_n(z)| \rightarrow + \infty \) for a.a. \(z \in {\hat{\Omega }}\). We have
Since \(|{\hat{\Omega }}|_N > 0\) and \(|u_n(z)| \rightarrow + \infty \) for a.a. \(z \in {\hat{\Omega }}\), using hypothesis \(H_1\mathrm{(ii)}\) and reasoning as in the proof Proposition 7, we show that
So, if in (35) we pass to the limit as \(n \rightarrow \infty \), we have a contradiction. This proves the claim.
On account of the claim, we can say that
From (33), we have
for all \(h \in W^{1,p}(\Omega )\), all \(n \in {\mathbb {N}}\).
In (37) we use the test function \(h = u_n - u_* \in W^{1,p}(\Omega )\). Passing to the limit as \(n \rightarrow \infty \) and using (36), we obtain
Since \(u_n^\pm \in {N}^{\theta _n}\) (\(n \in {\mathbb {N}}\)), using Proposition 9 and (38), we have
Then, we have
We will show that in fact equality holds. Given \(\varepsilon > 0\) let \(y_\varepsilon \in {N}_0\) such that
For \(\tau \), \(t > 0\) we have
On account of hypothesis \(H_1\mathrm{(ii)}\), we see that we can find \(M > 1\) such that
From Proposition 5, we know that there exist \(\tau _n\), \(t_n > 0\) (\(n \in {\mathbb {N}}\)) unique such that
Since \({\hat{m}}_\lambda ^{\theta _n} > 0\) (see Proposition 7) and \(\varphi _\lambda ^{\theta _n}\left( \tau _n y_\varepsilon ^+ - t_n y_\varepsilon ^- \right) \geqslant {\hat{m}}_\lambda ^{\theta _n} > 0\) (see (40)). Hence, from (39) we infer that \(\tau _n\), \(t_n < M\) for all \(n \in {\mathbb {N}}\). We have
We pass to the limit as \(n \rightarrow \infty \) and obtain
Since \(\varepsilon > 0\) is arbitrary, we let \(\varepsilon \downarrow 0\) and obtain
The regularity theory of Lieberman [6] implies that \(u_* \in C^1({\overline{\Omega }})\).
(b) With the additional assumption that \(e(z,x) > 0\) for a.a. \(z \in \Omega \), all \(x \ne 0\), we will show that \(u_*\) has two nodal domains.
We argue by contradiction. So, suppose that
and \(\Omega _1 = \left\{ {\hat{u}}_1 > 0\right\} \), \(\Omega _2 = \left\{ {\hat{u}}_2 < 0\right\} \) are connected open subsets of \(\Omega \), \(\Omega _1 \cap \Omega _2 = \emptyset \) and
Let \(y = {\hat{u}}_1 + {\hat{u}}_2\). Then, \(y^+ = {\hat{u}}_1\), \(y^- = - {\hat{u}}_2\). We have
Then,
which is a contradiction. Hence, \({\hat{u}}_3 = 0\) and we conclude that \(u_*\) has two nodal domains.
The proof of Theorem 1 is now complete. \(\square \)
References
Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)
Fragnelli, G., Mugnai, D., Papageorgiou, N.S.: The Brezis–Oswald result for quasilinear Robin problems. Adv. Nonlinear Stud. 16, 603–622 (2016)
Gasiński, L., Papageorgiou, N.S.: Positive solutions for the Robin \(p\)-Laplacian problem with competing nonlinearties. Adv. Calc. Var. 12, 31–56 (2019)
Gasiński, L., Winkert, P.: Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold. J. Differ. Equ. 274, 1037–1066 (2021)
Lacey, A.A., Ockendon, J.R., Sabina, J.: Multidimensional reaction-diffusion equations with nonlinear boundary conditions. SIAM J. Appl. Math. 58(5), 1622–1647 (1998)
Lieberman, G.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)
Lin, X., Tang, X.: Ground state sign-changing solutions for semilinear Dirichlet problems. Bound. Value Prob. 2018, 52 (2018)
Liu, W., Dai, G.: Existence and multiplicity results for double phase problem. J. Differ. Equ. 265(9), 4311–4334 (2018)
Mugnai, D., Papageorgiou, N.S.: Resonant nonlinear Neumann problems with indefinite weight. Ann. Sc. Norm. Super. Pisa Cl. Sci. 11, 729–788 (2012)
Özisik, M.N.: Boundary Value Problems of Heat Conduction. Dover, New York (1989)
Papageorgiou, N.S., Qin, D., Rădulescu, V.D.: Anisotropic double phase problems with indefinite potential: multiplicity solutions. Anal. Math. Phys. 10, 63 (2020)
Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: Positive solutions for perturbations of the Robin eigenvalue problem plus an indefinite potential. Discrete Contin. Dyn. Syst. 37, 2589–2618 (2017)
Papageorgiou, N.S., Rădulescu, V.D., Repovš, D.D.: Nonlinear Analysis—Theory and Methods. Springer Monographs in Mathematics. Springer, Cham (2019)
Pucci, P., Serrin, J.: The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, vol. 73. Birkhäuser Verlag, Basel (2007)
Ross, L.W.: Perturbation analysis of diffusion-coupled biochemical reaction kinetics. SIAM J. Appl. Math. 19, 323–329 (1970)
Szulkin, A., Weth, T.: The method of Nehari manifold. In: Gao, D., Motreanu, D. (eds.) Handbook of Nonconvex Analysis and Application, pp. 597–632. International Press, Somerville, MA (2010)
Tang, X.: Non-Nehari manifold for superlinear Schrödinger equation. Taiwan. J. Math. 18, 1957–1979 (2014)
Willem, M.: Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24. Birkhäuser Boston Inc., Boston, MA (1996)
Acknowledgements
The authors wish to thank three anonymous reviewers for their corrections and remarks. The research of Vicenţiu D. Rădulescu and Youpei Zhang was supported by a grant of the Romanian Ministry of Research, Innovation and Digitization, CNCS/CCCDI-UEFISCDI, project number PCE 137/2021, within PNCDI III. Youpei Zhang was partially supported by the Fundamental Research Funds for the Central Universities of Central South University (No. 2019zzts211). This paper has been completed, while Youpei Zhang was visiting University of Craiova (Romania) with the financial support of China Scholarship Council (No. 201906370079). Youpei Zhang would like to thank China Scholarship Council and Embassy of the People’s Republic of China in Romania.
Author information
Authors and Affiliations
Corresponding author
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
Papageorgiou, N.S., Rădulescu, V.D. & Zhang, Y. Ground-state nodal solutions for superlinear perturbations of the Robin eigenvalue problem. Z. Angew. Math. Phys. 73, 49 (2022). https://doi.org/10.1007/s00033-022-01679-y
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00033-022-01679-y
Keywords
- Nehari manifold
- Nehari submanifold
- Natural constraint
- Robin boundary condition
- Ground-state nodal solution