Abstract
We consider two types of nonlinear eigenvalue problems involving Laplace and p-Laplace operators \((p>2)\). The main result establishes the existence of at least two nontrivial weak solutions in the case of the perturbed equation and the existence of a continuous spectrum in the case of the (p, 2)-equation. In both cases, variational methods play a central role in our arguments.
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1 Introduction
The Laplace non-homogeneous equation can be found in many areas of physics. Problems of this type have been studied by Barile and Figueiredo [2], Motreanu and Tanaka [15], Papageorgiou and Rădulescu [17], Papageorgiou et al. [18], etc. Our main purpose in the present paper is to study nonlinear Laplace equations and (p, 2)-Laplace equations, which are the sum of the p-Laplace \((p>2)\) and the Laplace operator.
Let \(\Omega \subset \mathbb {R}^{N} \ (N \ge 2)\) be an open bounded set with smooth boundary. A classical result establishes
has a discrete spectrum in \(H^{1}_{0}(\Omega )\). In other words, there exists an increasing sequence of eigenvalues \(0< \lambda _{1} < \lambda _{2} \le \lambda _{3} \le \cdots \rightarrow \infty \). This is a consequence of the Riesz–Fredholm theorem for compact self-adjoint operators. For the proof we refer to [6, Ch. VI].
The lowest eigenvalue of problem (1) can be characterized from a variational point of view as the minimum of the Rayleigh quotient, namely
The similar problem
with weight function V can be found in pioneering papers of Bocher [4], Pleijel [20], Hess and Kato [12]. Minakshisundaram and Pleijel [14] proved that problem (3) admits an unbounded sequence \((\lambda _{n})\) of eigenvalues, provided that V is nonnegative, \(V \in L^{\infty }(\Omega )\) and \(V>0\) on some set of positive measure. We should also mention here the case where the weight function V may change the sign and may have singular points. This type of equations was studied by Szulkin and Willem [23].
Let us consider the following problem
where f is defined by
for some fixed \(c\in (0,\lambda _1)\).
We assume that g satisfies the following hypotheses:
-
(H1)
there exists a positive constant \(C \in (0,1)\) such that \(|g(x,t)| \le Ct\) for every \(t \ge 0\) and a.a. \(x \in \Omega \);
-
(H2)
there exists \(t_{0}>0\) such that \(G(x,t_{0}):=\int _{0}^{t_{0}}g(x,s)ds>0\) for a.a. \(x \in \Omega \);
-
(H3)
\(\lim _{u \rightarrow \infty }\frac{g(x,u)}{u}=0\) uniformly for a.a. \(x \in \Omega \).
In other words, the function g has a superlinear growth near the origin but it increases at most linearly at infinity.
Theorem 1
Assume that f is given by relation (5) and conditions (H1), (H2) and (H3) are fulfilled. Then \(\lambda _{1}\) defined in (2) is an isolated eigenvalue of problem (4) and the corresponding set of eigenvectors is a cone. Moreover, any \(\lambda \in (0,\lambda _{1})\) is not an eigenvalue of problem (4) but there exists \(\mu _{1}>\lambda _{1}\) such that every \(\lambda \in (\mu _{1},\infty )\) is an eigenvalue of problem (4).
The proof of the above result can be found in [13].
2 Main Results
This work is mainly inspired by papers of Chorfi and Rădulescu [9], Mihăilescu and Rădulescu [13], Onete [16] and Rădulescu [22].
The main purpose of this paper is to study the following (p, 2)-eigenvalue problem
with \(p>2\) and f as in (5). We are also concerned with problem (4) under the effect of a perturbation with an unbalanced growth at zero and the infinity
where \(u \in H^{1}_{0}(\Omega )\). This type of equations describes phenomena in mathematical physics. Many mathematicians have dealt with this type of problems (see, e.g., Benci et al. [3] and Cherfils and Ilyasov [8]).
We assume that \(\lambda \) is a real parameter and the reaction r satisfies the following conditions:
-
(R1)
r is continuous and there exists \(\varepsilon >0\) such that r is nonnegative and nontrivial on \([0,\varepsilon ]\);
-
(R2)
there exists a real number \(p_1 \in (0,1)\) such that for some \(c>0\)
$$|r(t)|\le c(1+|t|^{p_1})\ \text{ for } \text{ all } t\in {\mathbb {R}}$$and
$$\lim _{t\rightarrow 0}\frac{r(t)}{t}=0.$$
Hypothesis (R2) says that r should have a sublinear growth near the origin, since \(|t|^s<|t|\) for \(s>1\) and \(|t|\le 1\).
In our case \(\lambda \) is an eigenvalue in context of nonlinear operator. For more details we refer the reader to [10, p. 117].
If we denote
where \(F(x,t)=\int _0^tf(x,s)ds\) and \(R(t)=\int _{0}^{t}r(s)ds\), then \(\lambda \) is an eigenvalue for the pair (A, B) of nonlinear operators (as in [10]) if and only if there is a corresponding eigenfunction \(u \in H^{1}_{0}(\Omega ) \backslash \{0\}\), which is a solution of problem (7).
Let us introduce some notation. We say that \(u \in W^{1,p}_{0}(\Omega )\backslash \{0\}\) is a solution of problem (6) if
for every \(v \in W^{1,p}_{0}(\Omega )\). For all \(u \in W^{1,p}_{0}(\Omega )\) we denote
In what follows, we denote
and similar for strong inequalities
Then \(u_{+},u_{-} \in W^{1,p}_{0}(\Omega )\) and
(see, e.g., [11, Theorem 7.6]).
The main results in this paper are the following.
Theorem 2
Assume that f is given by relation (5) and conditions (H1), (H2) and (H3) are fulfilled. Then for some \(M \in (0,1)\), every \(\lambda \ge \frac{\lambda _{1}}{M}\) large enough is an eigenvalue of problem (6).
We refer to [9] for related concentration properties of the spectrum.
Theorem 3
Assume that hypotheses (H1)-(H3), (R1) and (R2) are fulfilled. Then there is a real number \(\Lambda \) such that problem (7) has at least two solutions for all \(\lambda > \Lambda \).
We refer to the recent monograph [19] for the main abstract methods used in this paper.
3 Proof of Theorem 2
First we show that all positive eigenvalues of problem (6) are bigger than \(\lambda _{1}\). Due to [9, Theorem 2] we can use \(g(x,u^{+})\) instead of f(x, u). Substituting (10) and (11) into (9) we obtain
whenever \(v \in W^{1,p}_{0}(\Omega )\). Now we set \(v=u_{+}\). We obtain
Using (H1) in (13) we obtain
From the above inequality we conclude that \(\lambda > \lambda _{1}\) since \(\lambda _{1}\) cannot be an eigenvalue.
Next, we have to show that problem (6) has a solution for all \(\lambda \) large enough.
Let us consider the following problem
The energy functional \(E:W^{1,p}_{0}(\Omega ) \rightarrow \mathbb {R}\) associated to problem (15) is defined by
Fix \(\lambda > \lambda _{1}\). Hypothesis (H3) implies that there is a positive constant \(C=C(\lambda )\) such that
It follows that
hence E is coercive.
In the next step we will show that there exists \(\lambda >0\) large enough such that
In this case we will use our assumption (H2) and fix \(t_{0} \in \mathbb {R}\) such that
Let \(K \subset \Omega \) be a compact subset, sufficiently large, and \(w \in W^{1,p}_{0}(\Omega )\) such that \(w=t_{0}\) in K and \(0 \le w \le t_{0}\) in \(\Omega \).
From (H2) it follows that
We conclude that
provided that \(\lambda >0\) is large enough. For these value of \(\lambda \), the energy functional E has a negative global minimum, hence problem (6) admits a solution. The proof is complete.\(\square \)
4 Proof of Theorem 3
Let us define the energy functional \(J:H^{1}_{0}(\Omega ) \rightarrow \mathbb {R}\) associated to problem (7) by
where
and
We first establish that J is well-defined. Using (R2) we know that there is a positive constant \(C_{1}\) such that for all \(t \in \mathbb {R}\)
Thus, by the \(L^{p}\) spaces embedding and Sobolev embedding theorem [6, Corollary IX.14], the functional \(J:H^{1}_{0}(\Omega ) \rightarrow \mathbb {R}\) is well-defined. The functional J is of class \(C^{1}\) and, for every \(\zeta \in H^{1}_{0}(\Omega )\),
The operator R is not constant since \(\nabla R(u)=r(u) \nabla u \ne 0\).
In the following step we will prove that the functional J is coercive.
Indeed, from the fact that \(F(x,u) \le \frac{c}{2}u^{2}+C\) (see hypothesis (H3) and relation (5)), there exists \(\gamma >0\) such that
Since Poincaré’s inequality holds, we obtain
By (26), we get
Combining (29) with (30) we obtain
hence J is coercive.
Next, we show that the functional J satisfies the Palais–Smale compactness condition. Let \((u_{j})\) be a sequence in \(H^{1}_{0}(\Omega )\) such that
The coercivity of J implies that \((u_{n})\) is bounded and from the Eberlein-Šmulian theorem we know that \(u_{n} \xrightarrow {w} u\) in \(H^{1}_{0}(\Omega ).\) Hence
The last implication follows from Kadec-Klee property which stand that if for any sequence \((u_{n})\) in Banach space X and \(u \in X\) such that \(\lim _{n \rightarrow \infty }\Vert u_{n}\Vert =\Vert u\Vert \), we have \(\Vert u_{n}-u\Vert \rightarrow 0\) provided \(u_{n} \rightarrow u\) weakly as \(n \rightarrow \infty \). This shows that J satisfies the Palais–Smale condition. To finish the proof we will use a version of the Pucci–Serrin theorem (three critical points theorem, cf. [21, Corollary 1]) in the generalized version established by Bonanno (see [5, Theorem 2.3 and Remark 2.2]). This theorem says that if J is a real-valued \(C^{1}\)-functional defined on a real Banach space having two local minima and satisfying the Palais–Smale condition, then J has at least three critical points. The three critical points theorem of Pucci and Serrin and the Palais–Smale compactness condition must be regarded in relationship with the mountain pass theorem of Ambrosetti and Rabinowitz.
Therefore we have to show that \(J:H^{1}_{0}(\Omega ) \rightarrow \mathbb {R}\) has at least two local minima for large values of \(\lambda \).
Since J is coercive and sequentially weakly lower semicontinuous, then it has a minimizer \(u_{0}\) such that
It follows that \(u_{0} \ne 0\). For \(\hat{p} >1\), we have
So \(r(t)\le \varepsilon t^{\hat{p}}\) for \(t \in [-\delta ,\delta ]\). Then for \(u \in C^{1}_{0}(\Omega )\) with \(\Vert u\Vert _{C^{1}_{0}(\overline{\Omega })} \le \delta \), we have
Since \(\hat{p}+1>2\), for \(\varepsilon \in (0,1)\) small
It follows that \(u=0\) is a local minimizer of J. Now we can apply the Pucci–Serrin theorem.
Now we can give the following alternative argument, without using the Pucci–Serrin theorem (in the version established by Bonanno). Suppose that \(u_0\) and \(u_1\) are two local minimizers of J. We may assume that
We also assume that the critical set \(K_J\) of J is finite (otherwise we already have infinitely many solutions). Then, by Theorem 5.7.6 in [19, p.449], we can find \(0<\rho <\Vert u_0-u_1\Vert \) such that
Since J satisfies the Palais–Smale condition (being coercive) we can apply the mountain pass theorem of Ambrosetti and Rabinowitz (see [1]) and obtain a third critical point distinct from \(u_0\) and \(u_1\).
The proof is now complete.\(\square \)
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Change history
13 July 2024
Incorrect author query comments removed from Keywords and Proof of Theorem 3 section.
13 July 2024
A Correction to this paper has been published: https://doi.org/10.1007/s00025-024-02223-2
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The author would like to thank Professor N.S. Papageorgiou and Professor V.D. Radulescu for fruitful discussions and numerous useful comments.
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The original online version of this article was revised:, In the Introduction section of this article, the term ‘\(\Omega \in \)’ should have read “\(\Omega \subset \)”. In the sentence beginning ‘This theorem says that.....’ in this article, the text ‘Theorem 1.3’ should have read ‘Theorem 2.3.’ In the sentence beginning ‘First we show that all positive ...’ in this article, the text ‘[9, Theorem 1]’ should have read ‘[9, Theorem 2]’.
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Bień, K. Nonlinear Eigenvalue Problems for (p, 2)-Equations and Laplace Equations with Perturbations. Results Math 79, 148 (2024). https://doi.org/10.1007/s00025-024-02176-6
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DOI: https://doi.org/10.1007/s00025-024-02176-6
Keywords
- Nonlinear eigenvalue problem
- Palais–Smale condition
- nonlinear operator
- (p, 2)-equation
- continuous spectrum
- Pucci–Serrin three critical point theorem
- perturbation