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 \(\Omega \in \mathbb {R}^{N} \ (N \ge 2)\Omega \subset \) be an open bounded set with smooth boundary. A classical result establishes

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda u &{} \text {in} \; \Omega , \\ u=0 &{} \text {on} \; \partial \Omega . \end{array} \right. \end{aligned}$$
(1)

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

$$\begin{aligned} \lambda _{1}=\inf _{u \in H^{1}_{0}(\Omega )}\frac{\int _{\Omega }|\nabla u|^{2}dx}{\int _{\Omega }u^{2}dx}. \end{aligned}$$
(2)

The similar problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda V(x) u &{} \text {in} \; \Omega , \\ u=0 &{} \text {on} \; \partial \Omega \end{array} \right. \end{aligned}$$
(3)

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

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\lambda f(x,u) &{} \text {in} \; \Omega , \\ u=0 &{} \text {on} \; \partial \Omega , \end{array} \right. \end{aligned}$$
(4)

where f is defined by

$$\begin{aligned} f(x,u)= \left\{ \begin{array}{ll} g(x,u), &{} u \ge 0, \\ cu, &{} u<0 , \end{array} \right. \end{aligned}$$
(5)

for some fixed \(c\in (0,\lambda _1)\).

We assume that g satisfies the following hypotheses:

  1. (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. \(x \in \Omega \);

  2. (H2)

    there exists \(t_{0}>0\) such that \(G(x,t_{0}):=\int _{0}^{t_{0}}g(x,s)ds>0\) for a.a. \(x \in \Omega \);

  3. (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

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u-\Delta _{p}u=\lambda f(x,u) &{} \text {in} \; \Omega , \\ u=0 &{} \text {on} \; \partial \Omega , \end{array} \right. \end{aligned}$$
(6)

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

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u= f(x,u)+\lambda r(u) &{} \text {in} \; \Omega , \\ u=0 &{} \text {on} \; \partial \Omega , \end{array} \right. \end{aligned}$$
(7)

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:

  1. (R1)

    r is continuous and there exists \(\varepsilon >0\) such that r is nonnegative and nontrivial on \([0,\varepsilon ]\);

  2. (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

$$\begin{aligned} A(u):=\frac{1}{2} \int _{\Omega } |\nabla u|^{2}dx- \int _{\Omega }F(x,u)dx \ \text { and } B(u):=\int _{\Omega }R(u)dx, \end{aligned}$$
(8)

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

$$\begin{aligned} \int _{\Omega }\nabla u \nabla v dx + \int _{\Omega }|\nabla u|^{p-2}\nabla u \nabla v dx = \lambda \int _{\Omega }f(x,u)vdx \end{aligned}$$
(9)

for every \(v \in W^{1,p}_{0}(\Omega )\). For all \(u \in W^{1,p}_{0}(\Omega )\) we denote

$$\begin{aligned} u_{\pm }(x):=\max \{\pm u(x),0\}, \ \text {for } x \in \Omega . \end{aligned}$$
(10)

In what follows, we denote

$$[u \ge 0]=\{x \in \Omega : \, u(x) \ge 0\},$$
$$ [u \le 0]=\{x \in \Omega : \, u(x) \le 0\}$$

and similar for strong inequalities

$$[u> 0]=\{x \in \Omega : \, u(x) > 0\},$$
$$ [u< 0]=\{x \in \Omega : \, u(x) < 0\}.$$

Then \(u_{+},u_{-} \in W^{1,p}_{0}(\Omega )\) and

$$ \begin{aligned} \nabla u_{+}= \left\{ \begin{array}{ll} \nabla u &{} \text {on } [u>0] \\ 0 &{} \text {on } [u \le 0] \end{array} \right. \& \nabla u_{-}= \left\{ \begin{array}{ll} \nabla u &{} \text {on } [u<0] \\ 0 &{} \text {on } [u \ge 0] \end{array} \right. \end{aligned}$$
(11)

(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 1] we can use \(g(x,u^{+})\) instead of f(x, u). Substituting (10) and (11) into (9) we obtain [9, Theorem 2]

$$\begin{aligned} \int _{\Omega }\nabla u_{+} \nabla v dx + \int _{\Omega }|\nabla u|^{p-2} \nabla u_{+} \nabla v dx = \lambda \int _{\Omega }g(x,u_{+})vdx \end{aligned}$$
(12)

whenever \(v \in W^{1,p}_{0}(\Omega )\). Now we set \(v=u_{+}\). We obtain

$$\begin{aligned} \int _{\Omega }|\nabla u_{+}|^{2}dx+\int _{\Omega }|\nabla u|^{p-2}|\nabla u_{+}|^{2}dx= \lambda \int _{\Omega }g(x,u_{+})u_{+}dx. \end{aligned}$$
(13)

Using (H1) in (13) we obtain

$$\begin{aligned} \lambda _{1} \int _{\Omega }u_{+}^{2} dx \le \int _{\Omega } |\nabla u_{+}|^{2}dx \le \lambda \int _{\Omega } g(x,u_{+})u_{+}dx \le \lambda C \int _{\Omega } u_{+}^{2}dx. \end{aligned}$$
(14)

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

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u-\Delta _{p}u=\lambda g(x,u_{+}) &{} \text {in} \; \Omega , \\ u=0 &{} \text {on} \; \partial \Omega . \end{array} \right. \end{aligned}$$
(15)

The energy functional \(E:W^{1,p}_{0}(\Omega ) \rightarrow \mathbb {R}\) associated to problem (15) is defined by

$$\begin{aligned} E(u)=\frac{1}{2} \int _{\Omega }|\nabla u|^{2}dx+\frac{1}{p} \int _{\Omega } |\nabla u|^{p}dx - \lambda \int _{\Omega } G(x,u_{+})dx. \end{aligned}$$
(16)

Fix \(\lambda > \lambda _{1}\). Hypothesis (H3) implies that there is a positive constant \(C=C(\lambda )\) such that

$$\begin{aligned} \lambda G(x,u) \le \frac{\lambda _{1}}{2}u^{2}+C \ \text {for all } (x,u) \in \Omega \times \mathbb {R}. \end{aligned}$$
(17)

It follows that

$$\begin{aligned} \begin{aligned} E(u)&\ge \frac{1}{2} \int _{\Omega } | \nabla u|^{2} dx + \frac{1}{p} \int _{\Omega } | \nabla u |^{p} dx - \frac{\lambda _{1}}{2} \int _{\Omega } u^{2}dx - C |\Omega | \\ {}&\ge \frac{1}{p}\Vert u\Vert ^{p}_{W^{1,p}_{0}}-C|\Omega | , \end{aligned} \end{aligned}$$
(18)

hence E is coercive.

In the next step we will show that there exists \(\lambda >0\) large enough such that

$$\begin{aligned} \inf \{E(u): \ u \in W^{1,p}_{0}(\Omega )\}<0. \end{aligned}$$
(19)

In this case we will use our assumption (H2) and fix \(t_{0} \in \mathbb {R}\) such that

$$\begin{aligned} G(x,t_{0})>0 \ \text {for all } x \in \Omega . \end{aligned}$$
(20)

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

$$\begin{aligned} \begin{aligned} \int _{\Omega }G(x,w)dx&= \int _{K}G(x,w)dx+\int _{\Omega \backslash K}G(x,w)dx \\&\ge \int _{K}G(x,t_{0})dx-\frac{C}{2} \int _{\Omega \backslash K}w^{2}dx \\&\ge \int _{K} G(x,t_{0})dx-\frac{Ct_{0}^{2}}{2} |\Omega \backslash K |>0. \end{aligned} \end{aligned}$$
(21)

We conclude that

$$\begin{aligned} E(w)<0, \end{aligned}$$
(22)

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

$$\begin{aligned} J(u):=C(u)+\lambda D(u), \end{aligned}$$
(23)

where

$$\begin{aligned} C(u):=\frac{1}{2} \int _{\Omega }|\nabla u|^{2}dx - \int _{\Omega }F(x,u)dx \ \text { for all } u \in H^{1}_{0}(\Omega ) \end{aligned}$$
(24)

and

$$\begin{aligned} D(u):=-\int _{\Omega }R(u)dx \ \text { for all } u \in H^{1}_{0}(\Omega ). \end{aligned}$$
(25)

We first establish that J is well-defined. Using (R2) we know that there is a positive constant \(C_{1}\) such that for all \(t \in \mathbb {R}\)

$$\begin{aligned} |R(t)| \le C_{1} (|t|+|t|^{p_{1}+1}). \end{aligned}$$
(26)

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 \(\zeta \in H^{1}_{0}(\Omega )\),

$$\begin{aligned} <J'(u),\zeta >= \int _{\Omega }\nabla u \nabla \zeta dx - \int _{\Omega }f(x,u)\zeta dx - \lambda \int _{\Omega }r(u)\zeta dx. \end{aligned}$$
(27)

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

$$\begin{aligned} C(u) \ge \frac{1}{2} \int _{\Omega } |\nabla u|^{2}dx -\frac{1}{2} \int _{\Omega }u^{2}dx-C|\Omega |_N \ge \gamma \int _{\Omega }|\nabla u|^{2}dx-C|\Omega |_N. \end{aligned}$$
(28)

Since Poincaré’s inequality holds, we obtain

$$\begin{aligned} C(u) \ge \gamma \Vert u\Vert _{H^{1}_{0}}^{2}. \end{aligned}$$
(29)

By (26), we get

$$\begin{aligned} D(u) \ge -C_{2} (\Vert u\Vert +\Vert u\Vert ^{p_{1}+1}). \end{aligned}$$
(30)

Combining (29) with (30) we obtain

$$\begin{aligned} J(u) \ge \gamma \Vert u\Vert ^{2}-C_{2}\lambda (\Vert u\Vert +\Vert u\Vert ^{p_{1}+1}), \end{aligned}$$
(31)

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

$$\begin{aligned}{} & {} |J(u_{j})| \le M, \ M>0,\end{aligned}$$
(32)
$$\begin{aligned}{} & {} \Vert J'(u_{j})\Vert _{H^{-1}} \rightarrow 0. \end{aligned}$$
(33)

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

$$\begin{aligned} \lim _{n \rightarrow \infty } \int _{\Omega }\nabla u_{n} \nabla (u_{n}-u)dx=0 \Rightarrow \Vert \nabla u_{n}\Vert _{2} \rightarrow \Vert \nabla u\Vert _{2} \Rightarrow \Vert u_{n}-u\Vert _{H^{1}_{0}(\Omega )}. \end{aligned}$$
(34)

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 1.3 and Remark 2.2]). This theorem says that if J is a real-valued \(C^{1}\)-functional [5, Theorem 2.3 and Remark 2.2]). 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

$$ J(u_{0})<0=J(0) \; \text {for } \lambda >0 \; \text {large}. $$

It follows that \(u_{0} \ne 0\). For \(\hat{p} >1\), we have

$$ \lim _{t \rightarrow 0} \frac{r(t)}{t^{\hat{p}}}=0. $$

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

$$ J(u) \ge \gamma \Vert u\Vert ^{2}-C\lambda \varepsilon \Vert u\Vert ^{\hat{p}+1}. $$

Since \(\hat{p}+1>2\), for \(\varepsilon \in (0,1)\) small

$$ J(u) \ge J(0)=0 \; \text {for } \Vert u\Vert _{C^{1}_{0}(\overline{\Omega })} \le \delta . $$

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

$$J(u_0)\le J(u_1).$$

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

$$J(u_0)\le J(u_1)<\inf \{J(u):\ \Vert u-u_1\Vert =\rho \}.$$

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.\(\square \)