Multiplicity of Solutions for an Elliptic Kirchhoff Equation

Abstract

In this paper we study the existence of positive solution to the Kirchhoff elliptic problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\left( 1+\gamma G'\left( \Vert \nabla u\Vert ^2_{L^2(\Omega )}\right) \right) \Delta u = \lambda f(u) &{} \text{ in } \; \Omega ,\\ u = 0 &{} \text{ on } \; \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

where $\Omega$ is an open, bounded subset of $\mathbb {R}^N$ ($N\ge 3$), f is a locally Lipschitz continuous real function, $f(0)\ge 0$, $G'\in C(\mathbb {R}^+)$ and $G'\ge 0$. We prove the existence of at least two solutions with $L^\infty (\Omega )$ norm between two consecutive zeroes of f for large $\lambda$.

1 Introduction

We consider the following Kirchhoff elliptic boundary value problem

where $\Omega$ is an open, bounded subset of $\mathbb {R}^N$ ($N\ge 3$), $G\in C^1(\mathbb {R}^+)$ with $G, G'\ge 0$, $\gamma \ge 0$ and f is a locally Lipschitz continuous real function defined in $[0,\infty )$ with $f(0)\ge 0$. We extend f to $(-\infty ,0)$ by $f(s)=f(0)\ge 0$ for every $s< 0$ to guarantee that any solution of ($P_{\lambda ,\gamma }$) is nonnegative. In the semilinear local case, $\gamma =0$, it is well known (see [2]) that maximum principle implies the nonexistence of solution u with $\Vert u\Vert _{L^\infty (\Omega )}=\alpha$ provided that $f(\alpha )\le 0$. K. J. Brown and H. Budin proved in [4], by using variational methods, how changes in the sign of f lead to multiple positive solutions of the equation for sufficiently large $\lambda$.

In [6] P. Hess proved for $\lambda$ big enough the existence of $w_{1,\lambda }, \ldots , w_{m,\lambda }$, $u_{1,\lambda }, \ldots , u_{m-1,\lambda }$, $2m-1$ positive solutions of the nonlinear elliptic eigenvalue problem $(P_{\lambda ,0})$, if $f(0)>0$ and the graph of f has m positive humps and $m-1$ negative humps, each positive hump having greater area than the previous negative hump. More precisely, if $\alpha _k$ denotes the right end point of the k-th positive hump and $F(s):=\int _0^s f(t)dt$, then it is assumed that $f(0)>0$, $f(\alpha _1), \ldots , f(\alpha _m)=0$ and

$$\begin{aligned} \max \{F(s):0\le s \le \alpha _{k-1}\}<F(\alpha _k), \, k=2,\ldots , m. \end{aligned}$$

He showed that $w_{1.\lambda }\le \alpha _1$ and $\Vert w_{k+1,\lambda }\Vert _{L^\infty (\Omega )}, \Vert u_{k,\lambda }\Vert _{L^\infty (\Omega )}\in (\alpha _k, \alpha _{k+1})$ for every $k=1,\ldots , m-1$. Moreover, $w_{1,\lambda }\le w_{2,\lambda } \le \ldots \le w_{m,\lambda }$ and $u_{k,\lambda }\le w_{k+1,\lambda }$ for every $k=1,\ldots , m-1$ since $w_{k,\lambda }$ is the maximal solution of $(P_{\lambda ,0})$ in $[0,\alpha _k]$ with $k=1,\ldots , m$.

Z. Liu proved, in [9], assuming that $f(0)>0$ and under the hypotheses in [6] that all those solutions are ordered, is that to say, $w_{1,\lambda }< u_{1,\lambda }< w_{2,\lambda }< u_{2,\lambda }<w_{3,\lambda }<\cdots<u_{m-1,\lambda }<w_{m,\lambda }$.

E. N. Dancer and K. Schmitt have shown in [5] that this relation between the area of the positive hump and the previous negative one is necessary for the existence of a positive solution with norm in the interval corresponding to the positive hump. Indeed, they showed that if $0<\alpha <\beta$ are two consecutive zeroes of f such that $f(s)>0$ for every $s\in (\alpha ,\beta )$ then a necessary condition for the existence of solution u with $\Vert u\Vert _{L^\infty (\Omega )}=r\in (\alpha ,\beta )$ is

$$\begin{aligned} \int _{s_0}^r f(s)ds>0,\, \forall s_0\in (0, \alpha ). \end{aligned}$$

(1.1)

In [7] J. García-Melián and L. Iturriaga have studied problem $(P_{\lambda ,0})$ for a nonnegative function f with r positive zeros, $\alpha _1,\ldots , \alpha _r$. They showed that, for large enough $\lambda$, there exist at least two solutions $w_{k,\lambda },u_{k,\lambda }$ with $\Vert w_{k,\lambda }\Vert _{L^\infty (\Omega )}<\alpha _k<\Vert u_{k,\lambda }\Vert _{L^\infty (\Omega )}$ if f verifies a suitable non-integrability condition near each of its zeros. Specifically this condition is: there exists $\delta >0$ such that

$$\begin{aligned} \int _{\alpha _k}^{\alpha _k+\delta }\frac{f(t)}{(t-\alpha _k)^{\frac{2(N-1)}{N-2}}}dt=+\infty \end{aligned}$$

(1.2)

for $k = 1,...,r$. In addition, they obtained that

$$\begin{aligned} \lim _{\lambda \rightarrow +\infty } \Vert w_{k,\lambda }\Vert _{L^\infty (\Omega )}=\lim _{\lambda \rightarrow +\infty } \Vert u_{k,\lambda }\Vert _{L^\infty (\Omega )}=\alpha _k. \end{aligned}$$

They used the sub- and supersolutions method and topological degree arguments combined with the use of a suitable Liouville theorem.

The above condition (1.2) was improved in [3] by B. Barrios, J. García-Melián and L. Iturriaga under the assumption that f has an isolated positive zero $\alpha$ such that

$$\begin{aligned} \frac{f(t)}{(t-\alpha )^{\frac{N+2}{N-2}}} \text{ is } \text{ decreasing } \text{ in } (\alpha ,\alpha +\delta ) \text{ for } \text{ some } \text{ small } \delta >0. \end{aligned}$$

(1.3)

They proved, still for nonnegative f, that for large enough $\lambda$ there exist at least two ordered positive solutions $w_{\alpha ,\lambda }<u_{\alpha ,\lambda }$ verifying the properties $\Vert w_{\alpha ,\lambda }\Vert _{L^\infty (\Omega )}<\alpha <\Vert u_{\alpha ,\lambda }\Vert _{L^\infty (\Omega )}$ and $w_{\alpha ,\lambda }(x), u_{\alpha ,\lambda }(x)\rightarrow \alpha$ uniformly on compact subsets of $\Omega$ as $\lambda \rightarrow +\infty$. The existence of these solutions holds independently of the behavior of f near zero or infinity.

In this paper we are interested in the multiplicity results for the Kirchhoff problem $(P_{\lambda ,\gamma })$ (in the case $\gamma > 0$) providing a description of the behavior of solutions for $\lambda \rightarrow +\infty$. Thus, in addition to $f(0)\ge 0$ we only assume

(H):

There exists $0<\alpha < \beta$ such that $f(\alpha )=f(\beta )=0$, $f(s)>0$ for every $s\in (\alpha ,\beta )$ and

$$\begin{aligned} r_\alpha =\inf \left\{ r\in (\alpha ,\beta ): \int _{s_0}^r f(s)ds>0,\, \forall s_0\in (0, \alpha )\right\} \text { is finite.} \end{aligned}$$

We observe that f may change sign and that hypothesis (H) implies that (1.1) is satisfied for every $r\in (r_\alpha ,\beta )$.

In order to state the main result we use the following notation:

$$\begin{aligned} \mathcal {S}=\{(\mu ,u): u\in H_0^1(\Omega )\cap L^\infty (\Omega ) \text { solves } (P_{\mu ,\gamma }), \mu >0\} \end{aligned}$$

and for any $B\subset \mathcal {S}$, $\mu >0$ we define $B_\mu =\{u: (\mu ,u)\in \mathcal {S}\}$.

Theorem 1.1

Assume that f is a locally Lipschitz continuous real function with $f(0)\ge 0$ and satisfies (H). Assume also that $G\in C^1(\mathbb {R}^+)$ with $G, G'\ge 0$ and $\gamma \ge 0$. Then there exists $\overline{\lambda }>0$ such that, for every $\lambda >\overline{\lambda }$, problem $(P_{\lambda ,\gamma })$ admits at least two solutions $u_\lambda , w_\lambda \in H_0^1(\Omega )\cap L^\infty (\Omega )$ with $u_\lambda \ne w_\lambda$ and $\Vert u_\lambda \Vert _{L^\infty (\Omega )}, \Vert w_\lambda \Vert _{L^\infty (\Omega )} \in (r_\alpha ,\beta )$. Moreover, one of the following alternatives holds:

• 1. There exists an unbounded continuum $\Sigma \subset \mathcal {S}$ such that $\Sigma _\lambda$ has at least two elements.

• 2. $\mathcal {S}_\lambda$ is infinite.

Even more $\lim _{\lambda \rightarrow +\infty } \Vert w_\lambda \Vert _{L^\infty (\Omega )}=\beta$ and, if we assume that f is nonnegative, and it satisfies (1.3), then $\lim _{n\rightarrow \infty } \Vert u_{\lambda _n}\Vert _{L^\infty (\Omega )}=\alpha$ for some sequence $\lambda _n\rightarrow +\infty$.

We observe that in contrast with the semilinear case, $\gamma =0$, due to the lack of a convenient comparison result for the Kirchhoff problem $(P_{\lambda ,\gamma })$ we have no information on whether the solutions are ordered or not.

With respect to the plan of the paper, in the second section we show the main nonexistence results for $(P_{\lambda ,\gamma })$, we establish the existence of $\overline{\lambda }$ and main properties of the solution $w_\lambda$ as a minimum of a truncated functional. The third section is devoted to the existence of the second solution $u_\lambda$ using some Leray-Schauder degree computations which leads to the continuum alternative. Finally, in the last section we collect all the results leading to the proof of the main result.

2 Qualitative Properties of Solutions

In this section we collect the main results concerning with the existence, nonexistence and main properties of solutions to the problem ($P_{\lambda ,\gamma }$).

Let us remark that we are mainly interested in positive bounded solutions and the interaction between its $L^\infty (\Omega )$ norm and the positive zeroes of f. Thus, for every $k>0$, we may take

$$\begin{aligned} f_k(s) = f(\max \{0,\min \{s,k\}\}), \, s\in \mathbb {R}. \end{aligned}$$

(2.1)

and consider the truncated problem

This truncated problem provides us with bounded solutions to ($P_{\lambda ,\gamma }$) as it is stated in the following lemma.

Lemma 2.1

Let $u\in H^1_0(\Omega )$ be a weak solution to $(P_k)$, then $u\in L^\infty (\Omega )$ and u is a solution to ($P_{\lambda ,\gamma }$) if $\Vert u\Vert _{L^\infty (\Omega )}\le k$. $\square$

The next result is related to the nonexistence of bounded solutions with a specific $L^\infty (\Omega )$ norm.

Lemma 2.2

There is no bounded solution to ($P_{\lambda ,\gamma }$) with $L^\infty (\Omega )$ norm equal to $r>0$ in the following cases:

• 1. $f(r)\le 0$.

• 2. $f(s)>0$ for every $s\in (\alpha , r]$, $f(\alpha )=0$ and $\int _{s_0}^r f(s)ds \le 0$ for some $s_0\in (0,\alpha )$.

Proof

Observe that if $u\in H_0^1(\Omega )\cap L^\infty (\Omega )$ is a solution of ($P_{\lambda ,\gamma }$) with $\Vert u\Vert _{L^\infty (\Omega )}=r$ and $\Vert u\Vert _{H_0^1(\Omega )}=r_0$ then

$$\begin{aligned} -\Delta u = \frac{\lambda }{1+\gamma G'(r_0)} f(u) \equiv \tilde{f}(u). \end{aligned}$$

Thus, the first item is a direct consequence of the strong maximum principle ($\tilde{f}(r)\le 0$, see [2]) and the second is proved in [5] ($\int _{s_0}^r \tilde{f}(s)ds \le 0$).

Remark 2.3

Combining the results in the previous two lemmas, in the case $f(k)=0$, any solution u to $(P_k)$ it is also a solution to ($P_{\lambda ,\gamma }$) satisfying $\Vert u\Vert _{L^\infty (\Omega )} < k$. In addition, we may consider the functional $I_k: H_0^1(\Omega )\rightarrow \mathbb {R}$ given by

$$\begin{aligned} I_k(u)=\frac{1}{2}\int _\Omega |\nabla u|^2+\frac{1}{2}\gamma G\left( \int _\Omega |\nabla u|^2\right) - \lambda \int _\Omega \int _0^u f_k(s)ds, \, u\in H_0^1(\Omega ), \end{aligned}$$

and observe that $I_k\in C^1(H_0^1(\Omega ))$ and critical points of $I_k$ are solutions to $(P_k)$.

The existence of solution to $(P_k)$ when $f(k)=0$ is then standard as it is stated in the following lemma:

Lemma 2.4

Assume that $f(k)=0$ for some $k>0$. Then there exists a solution u to ($P_{\lambda ,\gamma }$) with $\Vert u\Vert _{L^\infty (\Omega )} < k$.

Proof

Observe that, since $f_k$ is bounded (which also implies that any solution of $(P_k)$ belongs to $L^\infty (\Omega )$), the functional $I_k$ is coercive. In addition, $I_k$ is weak lower semicontinuous which implies that $I_k$ has a global minimum $w_\lambda$ which, taking into account Remark 2.3, is a solution of ($P_{\lambda ,\gamma }$).

The solution given by Lemma 2.4 may be the trivial one and in the next result we give sufficient conditions to prove that it is not the trivial one. The proof follows closely that of [6] and we include it here for the reader convenience.

Lemma 2.5

Assume that $f(k)=0$ for some $k>0$ and that for some $0<\varepsilon <k$ there exists $0< s_0< k-\varepsilon< s_1 < k$ such that

$$\begin{aligned} \max _{0\le s\le k-\varepsilon }\left\{ \int _0^{s} f_k(t)dt\right\} = \int _0^{s_0} f_k(t)dt, \quad \int _{s_0}^{s_1} f_k(t)dt >0. \end{aligned}$$

Then there exists $\lambda _\varepsilon >0$ such that $\Vert w_\lambda \Vert _{L^\infty (\Omega )}\ge k-\varepsilon$ for every $\lambda >\lambda _\varepsilon$, where $w_\lambda$ is the minimum of $I_k$ established in Lemma 2.4.

Proof

Assume that $\Vert w_{\lambda }\Vert _{L^{\infty }(\Omega )}<k-\varepsilon$ and observe that, in this case

$$\begin{aligned} \int _\Omega \int _0^{w_\lambda } f_k(t)dt&\le \int _\Omega \max _{0\le s\le k-\varepsilon }\left\{ \int _0^{s} f_k(t)dt\right\} \nonumber \\&= |\Omega | \int _0^{s_1} f(t)dt - |\Omega | \int _{s_0}^{s_1} f(t)dt. \end{aligned}$$

(2.2)

Now we take $v \in C^\infty _0(\Omega )$ with $\Vert v\Vert _{L^\infty (\Omega )}\le k$ and $v(x) = s_1 > k-\varepsilon$ when $x\not \in \Omega _\delta \equiv \{x\in \Omega :\, \text {dist}(x,\partial \Omega ) < \delta \}$ for some $\delta$.

Therefore, using (2.2) we have:

$$\begin{aligned} \int _\Omega \int _0^{v} f_k(t)dt&= \int _{\Omega \setminus \Omega _\delta } \int _0^{s_1} f_k(t)dt + \int _{\Omega _\delta } \int _0^{v} f_k(t)dt \\&= |\Omega | \int _0^{s_1} f(t)dt + \int _{\Omega _\delta } \int _{s_1}^{v} f_k(t)dt \\&\ge \int _\Omega \int _0^{w_\lambda } f_k(t)dt +|\Omega | \int _{s_0}^{s_1} f(t)dt - C |\Omega _\delta |. \end{aligned}$$

Now we fix $\delta >0$ such that $\eta \equiv |\Omega | \int _{s_0}^{s_1} f(t)dt - C |\Omega _\delta | > 0$ and we have that

$$\begin{aligned} I_k(v) - I_k (w_\lambda )&\le \frac{1}{2}\int _\Omega |\nabla v|^2+\frac{1}{2}\gamma G\left( \int _\Omega |\nabla v|^2\right) \\&\quad - \lambda \int _\Omega \int _0^{v} f_k(s)ds + \lambda \int _\Omega \int _0^{w_\lambda } f_k(s)ds \\&\le \frac{1}{2}\int _\Omega |\nabla v|^2+\frac{1}{2}\gamma G\left( \int _\Omega |\nabla v|^2\right) - \lambda \eta . \end{aligned}$$

On the other hand for $\lambda > \lambda _\varepsilon \equiv \frac{1}{\eta }\left( \frac{1}{2}\int _\Omega |\nabla v|^2+\frac{1}{2}\gamma G\left( \int _\Omega |\nabla v|^2\right) \right)$ (since v is fixed) we have that

$$\begin{aligned} I_k (v) < I_k (w_\lambda ). \end{aligned}$$

This implies that $\Vert w_{\lambda }\Vert _{L^{\infty }(\Omega )}>k-\varepsilon$ for every $\lambda >\lambda _\varepsilon$ and we finish the proof.

The following result is concerned with the existence of solution having $L^\infty (\Omega )$-norm between two consecutive zeros $0<\alpha < \beta$ when (H) is satisfied.

Lemma 2.6

Assume (H) satisfied. Then there exist $\underline{\lambda },\overline{\lambda }>0$ such that

$$\begin{aligned} \Lambda = \{\lambda \ge 0: (P_{\lambda ,\gamma }) \text { admits solution }u \text { with } \alpha \le r_\alpha<\Vert u\Vert _{L^\infty (\Omega )}<\beta \} \end{aligned}$$

is a nonempty closed set with $[\overline{\lambda },+\infty )\subset \Lambda \subset [\underline{\lambda },+\infty )$.

Proof

Using Lemma 2.4 we have the existence of a global minimum $w_\lambda$ of $I_\beta$ which is a solution to problem ($P_{\lambda ,\gamma }$). This solution may be the trivial one and we will prove now that, for large enough $\lambda$, $\Vert w_\lambda \Vert _{L^\infty (\Omega )}\in (r_\alpha , \beta )$. This is a consequence of Lemma 2.5 with $k=\beta$, $\varepsilon =\beta -\alpha$ and $s_1>r_\alpha$. (Moreover, as in the proof of Lemma 2.5 we also have that $\lim _{\lambda \rightarrow +\infty } I_\beta (w_\lambda ) = -\infty$).

As a consequence $[\overline{\lambda }, +\infty )\subset \Lambda$ for any $\overline{\lambda }>\lambda _\epsilon$. In particular $\Lambda \ne \emptyset$ and we can take $\underline{\lambda }= \inf \Lambda$. Observe that, since $(P_{0,\gamma })$ admits only the trivial solution, $0\not \in \Lambda$.

Now we prove that $\Lambda$ is closed and as a consequence $\underline{\lambda }>0$. Let $\lambda _n$ be a convergent sequence in $\Lambda$ and denote $\lambda =\lim _{n\rightarrow \infty } \lambda _n$.

Let take $u_n \in H_0^1(\Omega )\cap L^\infty (\Omega )$ with $r_\alpha< \Vert u_n\Vert _{L^\infty (\Omega )}<\beta$ satisfying

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\left( 1+\gamma G'\left( \Vert \nabla u_n\Vert ^2_{L^2(\Omega )}\right) \right) \Delta u_n = \lambda _n f(u_n) &{} \text{ in } \; \Omega ,\\ u_n = 0 &{} \text{ on } \; \partial \Omega .\\ \end{array}\right. } \end{aligned}$$

Observe that the sequence $\lambda _n f(u_n)\left( 1+\gamma G'\left( \Vert \nabla u_n\Vert ^2_{L^2(\Omega )}\right) \right) ^{-1}$ is bounded in $L^\infty (\Omega )$. In addition, applying [8, Theorem 6.1] we deduce that the sequence $u_n$ is bounded in $C^{0,\gamma }(\overline{\Omega })$. Consequently, Ascolí-Arzelá Theorem assures that $u_n$ possesses a subsequence converging strongly in $C(\overline{\Omega })$ to $u\in L^\infty (\Omega )$ with $r_\alpha \le \Vert u\Vert _{L^\infty (\Omega )}\le \beta$.

Moreover, taking $u_n$ as test function and using that f is continuous we get that $u_n$ is bounded in $H_0^1(\Omega )$. This implies, in particular, that $u\in H_0^1(\Omega )$ and, up to a subsequence, $u_n\rightarrow u$ weakly in $H_0^1(\Omega )$. In addition, taking $u_n-u$ as test function, we can also assure that $u_n$ strongly converges to u in $H_0^1(\Omega )$ and we can pass to the limit and it follows that

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\left( 1+\gamma G'\left( \Vert \nabla u\Vert ^2_{L^2(\Omega )}\right) \right) \Delta u = \lambda f(u) &{} \text{ in } \; \Omega ,\\ u = 0 &{} \text{ on } \; \partial \Omega .\\ \end{array}\right. } \end{aligned}$$

Since u solves ($P_{\lambda ,\gamma }$) the strong maximum principle allows us to assure that $r_\alpha \ne \Vert u\Vert _{L^\infty (\Omega )}\ne \beta$ (see Lemma 2.2). Therefore we have proved that $\lambda \in \Lambda$, i.e. $\Lambda$ is closed and we conclude the proof.

Remark 2.7

Assume that (H) is satisfied and denote by $w_\lambda$ the minimum of $I_\beta$ established in Lemma 2.6 for every $\lambda \ge \overline{\lambda }$. Then, as a consequence of Lemma 2.5, for every $\varepsilon >0$ there exists $\lambda _\varepsilon >0$ such that $\Vert w_\lambda \Vert _{L^\infty (\Omega )}\ge \beta -\varepsilon$ for every $\lambda >\lambda _\varepsilon$.

Remark 2.8

Observe that when $f(k)\ne 0$, Lemma 2.4 also gives a bounded solution to $(P_k)$ which is not necessarily a solution to ($P_{\lambda ,\gamma }$) unless it is less than k. Moreover, Lemma 2.5 is also true for solutions to $(P_k)$. Thus, if $\alpha<k<\beta$, with $\alpha ,\beta$ as in Remark 2.7 we have that the norm of this bounded solution to $(P_k)$ has to be greater than k.

3 Leray-Schauder Degree Computations

In this section we perform the Leray Schauder degree computations to obtain the multiplicity result, again we follow closely [6].

Observe that u solves ($P_{\lambda ,\gamma }$) and $\Vert u\Vert _{L^\infty (\Omega )}\le k$ if and only if

$$\begin{aligned} u = (-\Delta )^{-1}\left( \lambda f_k(u)\left( 1+\gamma G'\left( \Vert \nabla u\Vert ^2_{L^2(\Omega )}\right) \right) ^{-1}\right) \equiv \Phi _{\lambda ,k}(u). \end{aligned}$$

Moreover, $\Phi _{\lambda ,k}: H_0^1(\Omega )\rightarrow \mathbb {R}$ is compact and we can use Leray-Schauder degree theory.

Lemma 3.1

For every $\lambda , k > 0$, there exists $R>0$ such that $\text {deg} (I-\Phi _{\lambda ,k}, B_{R_\lambda }(0),0)=1$.

Proof

Let us consider $H(t,u)=u-t \Phi _{\lambda ,k}(u)$, for every $t\in [0,1]$ and $u\in H_0^1(\Omega )$. Since solutions to $H(t,u)=0$ are bounded in $L^\infty (\Omega )$ we have that they are also bounded in $H_0^1(\Omega )$ and, for big enough $R>0$, H is a valid homotopy and the result follows.

Observe that R depends on $\lambda$ and k but it can be chosen constant in compact sets.

Now we denote by $S_k$ to the set of solutions to ($P_{\lambda ,\gamma }$) with $L^\infty (\Omega )$-norm less that k. Recall that, when $f(k)=0$, $S_k$ is the set of solutions to $(P_k)$. Let us denote

$$\begin{aligned} U_{\varepsilon ,k} = \left\{ v \in H_0^1(\Omega ): \text {dist} (v,S_k) < \varepsilon \right\} . \end{aligned}$$

Lemma 3.2

Let $0<\alpha <\beta$ be two consecutive zeroes of f and $\lambda >0$. Then there exists $\overline{\varepsilon }>0$ such that $\text {deg} (I-\Phi _{\lambda ,\beta }, U_{\varepsilon ,\alpha },0)=1$ for every $0< \varepsilon < \overline{\varepsilon }$.

Proof

It is clear that $\text {deg} (I-\Phi _{\lambda ,\alpha }, U_{\varepsilon ,\alpha },0)=1$ since there is no solution of $u=\Phi _{\lambda ,\alpha }(u)$ in $B_R(0)\setminus U_{\varepsilon ,\alpha }$.

Let us consider $H(t,u)=u-t \Phi _{\lambda ,\alpha }(u)+(1-t)\Phi _{\lambda ,\beta }(u)$, for every $t\in [0,1]$ and $u\in H_0^1(\Omega )$. We claim that H is a valid homotopy for small $\varepsilon$ which implies the final result. Indeed, assume on the contrary that for some $\varepsilon _n\rightarrow 0$, $t_n\rightarrow t\in [0,1]$ and $u_n\in \partial U_{\varepsilon _n,\alpha }$ we have that

$$\begin{aligned} u_n=t_n \Phi _{\lambda ,\alpha }(u_n)+(1-t_n)\Phi _{\lambda ,\beta }(u_n), \end{aligned}$$

or equivalently

$$\begin{aligned} -\left( 1+\gamma G'\left( \Vert \nabla u_n\Vert ^2_{L^2(\Omega )}\right) \right) \Delta u_n = \lambda (t_n f_\alpha (u_n)+ (1-t_n)f_\beta (u_n)). \end{aligned}$$

Whenever $\Vert u_n\Vert _{L^\infty (\Omega )}\le \alpha$, since $f_\beta (u_n)=f_\alpha (u_n)$ we have that $u_n\in S_\alpha$ which is a contradiction with the fact that $u_n\in \partial U_{\varepsilon _n,\alpha }$, in particular $\Vert u_n\Vert _{L^\infty (\Omega )} > \alpha$. Moreover, since $f_\beta (s)=f_\alpha (s)=0$ for every $s\ge \beta$, using Lemma 2.2 we have that $\Vert u_n\Vert _{L^\infty (\Omega )}<\beta$.

Observe that $\lambda (t_n f_\alpha (u_n)+ (1-t_n)f_\beta (u_n))\left( 1+\gamma G'\left( \Vert \nabla u_n\Vert ^2_{L^2(\Omega )}\right) \right) ^{-1}$ is bounded in $L^\infty (\Omega )$ and, applying [8, Theorem 6.1] we deduce that the sequence $u_n$ is bounded in $C^{0,\gamma }(\overline{\Omega })$. Consequently, Ascolí-Arzelá Theorem assures that $u_n$ possesses a subsequence converging strongly in $C(\overline{\Omega })$ to $u\in L^\infty (\Omega )$ with $\alpha \le \Vert u\Vert _{L^\infty (\Omega )}\le \beta$.

Moreover, taking $u_n$ as test function we get that $u_n$ is bounded in $H_0^1(\Omega )$. This implies, in particular, that $u\in H_0^1(\Omega )$ and, up to a subsequence, $u_n\rightarrow u$ weakly in $H_0^1(\Omega )$. In addition, taking $u_n-u$ as test function, we can also assure that $u_n$ strongly converges to u in $H_0^1(\Omega )$ and we can pass to the limit and it follows that

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\left( 1+\gamma G'\left( \Vert \nabla u\Vert ^2_{L^2(\Omega )}\right) \right) \Delta u = \lambda (t f_\alpha (u)+(1-t)f_\beta (u) )&{} \text{ in } \; \Omega ,\\ u = 0 &{} \text{ on } \; \partial \Omega .\\ \end{array}\right. } \end{aligned}$$

Even more, since $\varepsilon _n\rightarrow 0$ and $u_n\in U_{\varepsilon _n,\alpha }$ we have that $u\in S_\alpha$ which also implies that $\Vert u\Vert _{L^\infty (\Omega )}<\alpha$ and this is a contradiction.

With the notation of Lemma 2.6, let us take now $\lambda \in \Lambda$ and denote $w_\lambda$ the solution of ($P_{\lambda ,\gamma }$) with $r_\alpha<\Vert w_\lambda \Vert _{L^\infty (\Omega )}<\beta$. Let us denote $\Lambda _1\subset \Lambda$ such that $w_\lambda$ is the minimum of $I_\beta$ and let us recall that $[\overline{\lambda },+\infty )\subset \Lambda _1$. Observe that, by definition of $\Lambda$, $w_\lambda \not \in S_\alpha$, in particular we can take $0<\underline{\varepsilon }<\overline{\varepsilon }$ such that $U_{\varepsilon ,\alpha }\cap B_\varepsilon (w_\lambda )=\emptyset$ for every $0<\varepsilon < \underline{\varepsilon }$. In the following lemma we prove multiplicity of solution for every $\lambda \in \Lambda _1$.

Theorem 3.3

Assume that (H) is satsified. For every $\lambda _0 \in \Lambda _1$ one of the following multiplicity results holds:

• 1. $w_{\lambda _0}$ is not an isolated solution to $(P_{\lambda _0,\gamma })$.

• 2. There exists un unbounded continuum $\Sigma \subset \{(\lambda ,u): u \text { solves } (P_{\lambda ,\gamma })\}$ such that $(\lambda _0,w_{\lambda _0})\in \Sigma$ and there exists a second solution $v_{\lambda _0} \ne w_{\lambda _0}$ with $(\lambda _0,v_{\lambda _0})\in \Sigma$.

Proof

Observe that whenever $w_{\lambda _0}$ is isolated, taking into account that it is a minimum for $I_\beta$, it is proved in [10, Theorem 1.1] that

$$\begin{aligned} \text {deg} (I-\Phi _{\lambda _0,\beta }, B_\varepsilon (w_{\lambda _0}), 0)=1, \, \varepsilon <\underline{\varepsilon }. \end{aligned}$$

Hence we may use [1, Theorem 4.4.1] to deduce the existence of the unbounded continuum $\Sigma$ with $(\lambda _0,w_{\lambda _0})\in \Sigma$. For the existence of the second solution $v_{\lambda _0}$ with $(\lambda _0,v_{\lambda _0})\in \Sigma$ we use the ideas of [1, Theorem 4.4.2] in the interval [a, b] with $a<\underline{\lambda }$ and $b=\lambda _0$, $U_1=B_\varepsilon (w_{\lambda _0})$ and taking into account that U may depend continuously on $\lambda$. Therefore we take

• $\Sigma _\alpha =\{(\lambda ,u): \, \lambda \in [a,b], u \text { solves } (P_{\lambda ,\gamma }), \Vert u\Vert _{L^\infty (\Omega )}\le \alpha \}$,

• $V_{\varepsilon ,\alpha }=\{(\lambda ,v)\in \mathbb {R}\times H_0^1(\Omega ): \text {dist}((\lambda ,v),\Sigma _\alpha )<\varepsilon \}$,

• $V_{\varepsilon ,\alpha }(\lambda )=\{u\in H_0^1(\Omega ): (\lambda ,u)\in V_{\varepsilon ,\alpha }\}$ and

• $U(\lambda )=B_R(0)\setminus \overline{V}_{\varepsilon ,\alpha }(\lambda )$, for every $\lambda \in [a,b]$.

By construction we have that $(P_{\lambda ,\gamma })$ has no solution on $\partial V_{\varepsilon ,\alpha }(\lambda )$ for every $\lambda \in [a,b]$ and $(P_{a,\gamma })$ has no solutions in $\overline{V}_{\varepsilon ,\alpha }(a)$ and the result follows as in [1, Theorem 4.4.2].

Remark 3.4

Observe that $\text {deg} (I-\Phi _{\lambda _0,\beta }, B_\varepsilon (w_{\lambda _0}), 0)=1$ also implies, by the excision property combined with Lemma 3.1 and Lemma 3.2, that $\text {deg} (I-\Phi _{\lambda _0,\beta }, B_R(0)\setminus (\overline{U}_{\varepsilon ,\alpha } \cup \overline{B}_\varepsilon (w_{\lambda _0})), 0)=-1$ which also gives the multiplicity result.

4 Proof of Theorem 1.1 Completed

We can take $\overline{\lambda }$ given by Lemma 2.6. Thus, for every $\lambda >\overline{\lambda }$, Lemma 2.6 provides us with the solution $w_\lambda$ with $\Vert w_\lambda \Vert _{L^\infty (\Omega )}\in (r_\alpha ,\beta )$ as the minimum of $I_\beta$.

The existence of the second solution $u_\lambda$ is consequence of Theorem 3.3 which also provides the multiplicity alternatives. Observe that, since no solution exist with $L^\infty (\Omega )$ norm in $[\alpha , r_\alpha ]$ (see Lemma 2.2), then $u_\lambda \in \Sigma _\lambda$ implies $\Vert u_\lambda \Vert _{L^\infty (\Omega )}\in (r_\alpha ,\beta )$.

As a consequence of Lemma 2.5 (see also Remark 2.7) we have that

$$\begin{aligned} \lim _{\lambda \rightarrow +\infty } \Vert w_\lambda \Vert _{L^\infty (\Omega )}=\beta . \end{aligned}$$

Thus, only remains to prove that, if we assume that f is nonnegative and it satisfies (1.3) then $\lim \Vert u_{\mu _n}\Vert _{L^\infty (\Omega )}=\alpha$ for some divergent sequence $\mu _n$. Indeed, in this case $r_\alpha =\alpha$ and this is a consequence of the results in [3] where it is proved the existence of $v_\lambda$ solution to

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

(4.1)

with $\lim _{\lambda \rightarrow +\infty } \Vert v_\lambda \Vert _{L^\infty (\Omega )}=\alpha$ and $\Vert v_\lambda \Vert _{L^\infty (\Omega )}>\alpha$. Thus, given $\lambda _n\rightarrow +\infty$ we can take $\mu _n = \lambda _n \left( 1+\gamma G'\left( \Vert \nabla v_{\lambda _n} \Vert ^2_{L^2(\Omega )}\right) \right)$ and $u_{\mu _n} = v_{\lambda _n}$. Since $G'\ge 0$ we have that $\mu _n\rightarrow +\infty$ and the proof is completed. $\square$

Remark 4.1

Let us observe that the multiplicity result cannot be deduced directly from the know results for the semilinear case. More precisely, given u a solution to the semilinear problem (4.1) then u is also a solution to

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle -\left( 1+\gamma G'\left( \Vert \nabla u\Vert ^2_{L^2(\Omega )}\right) \right) \Delta u = \mu f(u) &{} \text{ in } \; \Omega ,\\ u = 0 &{} \text{ on } \; \partial \Omega ,\\ \end{array}\right. } \end{aligned}$$

(4.2)

with $\mu = \left( 1+\gamma G'\left( \Vert \nabla u\Vert ^2_{L^2(\Omega )}\right) \right) \lambda$. This means that, for fixed $\lambda$, multiple solutions to (4.1) may lead to solutions to (4.2) for different values of the parameter $\mu$.

This is not enough to describe the global behavior in the case $\gamma \ne 0$ since the reverse depend on G and $\Vert u\Vert _{H_0^1(\Omega )}$:

$$\begin{aligned} \lambda = \frac{\mu }{\left( 1+\gamma G'\left( \Vert \nabla u_\mu \Vert ^2_{L^2(\Omega )}\right) \right) }, \end{aligned}$$

and we can assure that $u_\mu$ is a solution to (4.1) for this value of $\lambda$ but we cannot assure that it is precisely $v_\lambda$.

Remark 4.2

As in the semilinear case, depending on the behavior of f below $\alpha$ and above $\beta$, the results proved in this paper may lead to the existence of solutions with $L^\infty (\Omega )$ norm less that $\alpha$ or greater than $\beta$. Even more, the existence of multiple zeroes of f may lead to a more general multiplicity result.