1 Introduction

In this paper we study the classical positive solutions to the Dirichlet problem for a class of semilinear elliptic equations whose nonlinear term is of subcritical nature in a generalized sense and involves indefinite nonlinearities. More precisely, given \(\Omega \subset {\mathbb {R}}^N \), \(N> 2,\) a bounded, connected open subset, with \(C^{2}\) boundary \(\partial \Omega \), we look for positive solutions to:

$$\begin{aligned} -\Delta u =\lambda u+a(x)f(u), \quad \text{ in } \Omega , \qquad u= 0, \quad \text{ on } \partial \Omega , \end{aligned}$$
(1.1)

where \(\lambda \in {\mathbb {R}}\) is a real parameter, \(a\in C^1({\bar{\Omega }})\) changes sign in \(\Omega \),

$$\begin{aligned} f(s):=g(s)+h(s),\qquad \text{ with } \quad h(s):=\dfrac{\vert s\vert ^{2^* -2}s}{[\ln (e+\vert s \vert )]^\alpha }, \end{aligned}$$
(1.2)

\(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent, \(\alpha > 0\) is a fixed exponent, and \(f,g \in C^1({\mathbb {R}})\) satisfy

$$\begin{aligned} \mathrm {(H)}\, {\left\{ \begin{array}{ll} \mathrm {(H)}_0 &{}\lim _{s\rightarrow 0} \frac{f(s)}{\vert s\vert ^{p-2}s}=L_1,\qquad \text {for some}\ L_1>0,\ \text {and}\ p\in \left( 2,\frac{2N}{N-2}\right] \\ \mathrm {(H)}_\infty &{}\lim _{s\rightarrow \infty } \frac{g(s)}{\vert s\vert ^{q-2}s}=L_2,\quad \ \ \text {for some}\ L_2\ge 0,\ \text {and}\ q\in \left( 2,\frac{2N}{N-2}\right) \\ \mathrm {(H)}_{g'} &{} \displaystyle \vert g'(s)\vert \le C(1+\vert s\vert ^{q-2}),\quad \ \text {for}\ s\in {\mathbb {R}}. \end{array}\right. } \end{aligned}$$

We will say that f satisfies hypothesis (H) whenever (H)\(_0\), (H)\(_\infty \), and (H)\(_{g'}\) are satisfied. Since we are interested in positive solutions, we

$$\begin{aligned} \text {redefine }f\text { to be zero on }(-\infty , 0], \end{aligned}$$
(1.3)

note that, since (H)\(_0\), \(f(0)=0\) and that

$$\begin{aligned} \lim _{s\rightarrow 0^+} \left( \frac{f(s)}{s}-L_1\vert s\vert ^{p-2}\right) =0. \end{aligned}$$
(1.4)

When \(\lambda =0\), \(a(x)\equiv 1\) and \(g(s)\equiv 0\), this kind of nonlinearity has been studied in [5,6,7, 16], and in [11] for the case of the \(p-\)laplacian operator, with \(\alpha > \frac{p}{N-p}\). It is known the existence of uniform \(L^\infty \) a-priori bounds for any positive classical solution, and as a consequence, the existence of positive solutions. When \(\alpha \rightarrow 0,\) there is a positive solution blowing up at a non-degenerate point of the Robin function as \(\alpha \rightarrow 0,\) see [9] for details.

Let \((\lambda _1,\varphi _1)\) stands for the first eigen-pair of the Dirichlet eigenvalue problem \(-\Delta \varphi =\lambda \varphi \text{ in } \Omega ,\ \varphi = 0 \text{ on } \partial \Omega \,.\) From [10] it is known that \((\lambda _1,0)\) is a bifurcation point of positive solutions \((\lambda ,u_\lambda )\) to the equation (1.1). If f behaves like \(\vert u\vert ^{p-2}u\) at zero with \(2 \le p \le 2^*\), the influence of the negative part of the weight a is displayed under the sign of \(\int _{\Omega } a(x)\varphi _1(x)^{p}\, dx\), where \(\varphi _1\) is the first positive eigenfunction for \(-\Delta \) in \(H_0^1(\Omega )\). Specifically, whenever

$$\begin{aligned} \int _{\Omega } a(x)\varphi _1(x)^{p}\, dx < 0 \end{aligned}$$
(1.5)

the bifurcation of positive solutions from the trivial solution set is ’on the right’ of the first eigenvalue, in other words, for values of \(\lambda >\lambda _1\). When

$$\begin{aligned} \int _{\Omega } a(x)\varphi _1(x)^{p}\, dx >0 \end{aligned}$$

the bifurcation from the trivial solution set is ’on the left’ of the first eigenvalue, in other words, for values of \(\lambda < \lambda _1\).

Inspired by the work of Alama and Tarantello in [1], we will focus our attention to the case of a(x) changing sign and (1.5) is being satisfied, and, among other things, we will prove the existence of a turning point for a value of the parameter \(\Lambda >\lambda _1\), and in particular the existence of solutions when \(\lambda = \lambda _1\). We will use local bifurcation and variational techniques.

All throughout the paper, for \(v:\Omega \rightarrow {\mathbb {R}}\), \(v=v^+-v^-\) where

$$\begin{aligned} v^+(x):=\max \{v(x),0\} \qquad \text{ and }\quad v^-(x):=\max \{-v(x),0\}. \end{aligned}$$

Let us also define

$$\begin{aligned} \Omega ^\pm :=\{x\in \Omega : \ \pm a(x)>0\},\qquad \Omega ^0 :=\{x\in \Omega : \ a(x) = 0\}, \end{aligned}$$

and assume that both \(\Omega ^+,\ \Omega ^-\) are non empty sets.

For this nonlinearity the Palais–Smale condition of the energy functional becomes a delicate issue, needing Orlicz spaces and a Orlicz–Sobolev embedding theorem.

In order to prove (PS) condition, Alama and Tarantello ([1]) assume that the zero set \(\Omega ^0\) has a non empty interior. This is also a common hypothesis for other authors when dealing with changing sign superlinear nonlinearities [8, 20, 23]. But this is a technical hypothesis. (PS)-condition will be proved in Proposition 3.1 without assuming that hypothesis. We neither use Ambrosetti-Rabinowitz condition.

Let us now denote

$$\begin{aligned} C_0=\inf \{C\ge 0:\, f'(s)+C \ge 0 \hbox { for all } s\ge 0\}, \end{aligned}$$
(1.6)

and remark that hypothesis (H) implies that \(C_0<+\infty \). Observe also that

$$\begin{aligned} f(s)+C_0s \ge 0,\hbox { for all } s\ge 0;\quad f(s)s+C_0s^2 \ge 0,\hbox { for all } s\in {\mathbb {R}}. \end{aligned}$$
(1.7)

Let u be a weak solution to (1.1). By a regularity result, see Lemma 2.1, \(u\in C^{2}(\Omega )\cap C^{1,\mu }({\overline{\Omega }}).\) So by a solution, we mean a classical solution.

Assume that u is a non-negative nontrivial solution. It is easy to see that the solution is strictly positive. Indeed, adding \(\pm C_0a(x)u\) to the r.h.s. of the equation, splitting \(a=a^+-a^-\), taking into account (1.4) and (1.7), and letting in each side the nonnegative terms, we can write

$$\begin{aligned}&\left( -\Delta +a^-(x)\left[ \frac{f(u)}{u}+C_0\right] +C_0a(x)^+ \right) u\nonumber \\&\quad =\lambda u+a(x)^+\big [f(u)+C_0u\big ]+C_0a(x)^-u, \quad \text{ in } \Omega . \end{aligned}$$
(1.8)

Now, the strong Maximum Principle implies that \(u>0\) in \(\Omega ,\) and \(\frac{\partial u}{\partial \nu }<0\) on \(\partial \Omega \).

Our main result is the following theorem.

Theorem 1.1

Assume that \(g \in C^1({\mathbb {R}})\) satisfies hypothesis (H). Let \(C_0>0\) be defined by (1.6). If a changes sign in \(\Omega \), and (1.5) holds, then there exists a \(\Lambda \in {\mathbb {R}},\)

$$\begin{aligned} \lambda _1< \Lambda < \min \Big \{\lambda _1\big (\mathrm{int}\,(\Omega ^0)\big )\, ,\quad \lambda _1\big (\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\big ) +C_0\sup a^+ \Big \} \end{aligned}$$

and such that (1.1) has a classical positive solution if and only if \(\lambda \le \Lambda \).

Moreover, there exists a continuum (a closed and connected set) \( {\mathscr {C}}\) of classical positive solutions to (1.1) emanating from the trivial solution set at the bifurcation point \((\lambda ,u)=(\lambda _1,0)\) which is unbounded. Furthermore,

  1. (a)

    For every, \(\lambda \in \big (\lambda _1, \Lambda )\), (1.1) admits at least two classical ordered positive solutions.

  2. (b)

    For \(\lambda =\Lambda \), problem (1.1) admits at least one classical positive solution.

  3. (c)

    For every \(\lambda \le \lambda _1\), problem (1.1) admits at least one classical positive solution.

The paper is organized in the following way. Section 2 contains a regularity result and a non existence result. (PS)-condition and an existence of solutions result for \(\lambda <\lambda _1\) based in the Mountain Pass Theorem will be proved in Sect. 3. A bifurcation result for \(\lambda >\lambda _1\) is developed in Sect. 4. The main result is proved in Sect. 5. Appendix A contains some useful estimates. Orlicz spaces, and a Orlicz–Sobolev embeddings theorems, will be treated in Appendix B.

2 A Regularity Result and a Non Existence Result

Next, we recall a regularity Lemma stating that any weak solution is in fact a classical solution.

Lemma 2.1

If \(u\in H_0^1(\Omega )\) weakly solves (1.1) with a continuous function f with polynomial critical growth

$$\begin{aligned} \vert f(x,s)\vert \le C(1+\vert s\vert ^{2^*-1}), \end{aligned}$$

then, \(u\in C^{2}(\Omega )\cap C^{1,\mu }({\overline{\Omega }})\) and

$$\begin{aligned} \Vert u\Vert _{C^{1,\mu }({\overline{\Omega }})}\le C\Big (1+\Vert u\Vert _{L^{(2^*-1)r}({\overline{\Omega }})}^{2^*-1}\Big ), \end{aligned}$$

for any \(r>N\) and \(\mu =1-N/r.\) Moreover, if \(\partial \Omega \in C^{2,\mu }\), then \(u\in C^{2,\mu }({\overline{\Omega }})\).

Proof

Due to an estimate of Brézis-Kato [3], based on Moser’s iteration technique [17], \(u\in L^{r}(\Omega )\) for any \(r>1\); and by elliptic regularity \(u\in W^{2,r}(\Omega ),\) for any \(r>1\) (see [22, Lemma B.3] and comments below).

Moreover, by Sobolev embeddings for \(r>N\) and interior elliptic regularity \(u\in C^{1,\alpha }({\overline{\Omega }})\cap C^{2}(\Omega )\). Furthermore, if \(\partial \Omega \in C^{2,\alpha }\), then \(u\in C^{2,\alpha }({\overline{\Omega }})\). \(\square \)

Proposition 2.2

Let f satisfy hypothesis (H) and let \(C_0\) be defined in (1.6). Assume that a changes sign in \(\Omega \).

  1. 1.

    Problem (1.1) does not admit a positive solution \(u\in H_0^1 (\Omega )\) for any

    $$\begin{aligned} \lambda \ge \lambda _1\big (\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\big )+C_0\sup a^+. \end{aligned}$$
  2. 2.

    If \(\mathrm{int}\,(\Omega ^0)\ne \emptyset ,\) then \(\lambda _1\big (\mathrm{int}\,(\Omega ^0)\big )<+\infty \) and (1.1) does not admit a positive solution for any

    $$\begin{aligned} \lambda \ge \lambda _1\big (\mathrm{int}\,(\Omega ^0)\big ). \end{aligned}$$

Proof

1. Let \(\lambda \ge \lambda _1\big (\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\big ) + C_0\sup a^+ ,\) and assume by contradiction that there exists a non-negative non-trivial solution \(u\in H_0^1 (\Omega )\) to (1.1) for the parameter \(\lambda \). Since the Maximum Principle \(u>0\) in \(\Omega \), see (1.8).

Let \({\hat{\varphi }} \) be the positive eigenfunction of \(\big ( -\Delta , H_0^1 (\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\big )\big )\) of \(L^2\)-norm equal to 1. For simplicity, we will also denote by \({\hat{\varphi }}\) the extension by 0 of \({\hat{\varphi }}\) in all \(\Omega \). By Hopf’s maximum principle, we have \(\frac{\partial {\hat{\varphi }} }{\partial \nu } <0\) on \(\partial \big (\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\big )\), where \(\nu \) is the outward normal.

Again, if we multiply the equation (1.1) by \({\hat{\varphi }}\) and integrate along \(\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\) we find, after integrating by parts,

$$\begin{aligned} 0&>\int _{\partial (\mathrm{int}\,(\Omega ^+\cup \Omega ^0))} u\dfrac{\partial {\hat{\varphi }}}{\partial \nu }d\sigma \\&\quad + \int _{\mathrm{int}(\Omega ^+\cup \Omega ^0)} \Big [\lambda _1 \big (\mathrm{int}\,(\Omega ^+\cup \Omega ^0)\big )-\lambda +C_0a^+(x)\Big ] u{\hat{\varphi }} \,dx\\&= \int _{\Omega ^+} a^+(x) \big [f(u)+C_0u\big ]{\hat{\varphi }} \, dx> 0, \end{aligned}$$

a contradiction.

2. Let \(\lambda \ge \lambda _1\big (\mathrm{int}\,(\Omega ^0)\big )\) and, by contradiction, assume the existence of a positive solution \(u\in H_0^1 (\Omega )\) of problem (1.1) for the parameter \(\lambda \). Let \({\tilde{\varphi }}\) be a positive eigenfunction associated to \(\lambda _1 \big (\mathrm{int}\,(\Omega ^0)\big )<+\infty \). For simplicity, we will also denote by \({\tilde{\varphi }}\) the extension by 0 in all \(\Omega \). If we multiply equation (1.1) by \({\tilde{\varphi }}\) and integrate along \(\Omega ^0\) we find, after integrating by parts,

$$\begin{aligned} \int _{\mathrm{int}\,(\Omega ^0)} \nabla u \cdot \nabla {\tilde{\varphi }} \, dx&= \lambda \int _{\mathrm{int}\,(\Omega ^0)} u{\tilde{\varphi }}\, dx . \end{aligned}$$

On the other hand

$$\begin{aligned} \int _{\mathrm{int}\,(\Omega ^0)} \nabla u \cdot \nabla {\tilde{\varphi }} \,dx = \lambda _1 (\mathrm{int}\,(\Omega ^0)) \int _{\mathrm{int}\,(\Omega ^0)} {\tilde{\varphi }} u\, dx + \int _{\partial (\mathrm{int}\,(\Omega ^0))} u\dfrac{\partial {\tilde{\varphi }}}{\partial \nu } d\sigma . \end{aligned}$$

Hence

$$\begin{aligned} 0&>\int _{\partial (\mathrm{int}\,(\Omega ^0))} u\dfrac{\partial {\tilde{\varphi }} }{\partial \nu }d\sigma = \Big ( \lambda -\lambda _1 \big (\mathrm{int}\,(\Omega ^0)\big )\Big ) \int _{\mathrm{int}\,(\Omega ^0)}u{\tilde{\varphi }}\, dx \ge 0, \end{aligned}$$

a contradiction. \(\square \)

3 An Existence Result for \(\lambda <\lambda _1\)

In this section, we prove the existence of a nontrivial solution to equation (1.1) for \(\lambda <\lambda _1\), through the Mountain Pass Theorem.

3.1 On Palais–Smale Sequences

In this subsection, we define the framework for the functional \(J_\lambda \) associated to the problem (1.1)\(_\lambda \). Hereafter, we denote by \(\Vert \cdot \Vert \) the usual norm of \(H_0^1 (\Omega )\):

$$\begin{aligned} \Vert u\Vert =\left( \int _{\Omega } \vert \nabla u \vert ^2 \, dx \right) ^{1/2}. \end{aligned}$$

Given \(f(s)=h(s)+g(s)\) defined by (1.2), let us denote by \(F(s):= \int _0^s f(t) \, dt.\) Observe that (1.7) implies the following

$$\begin{aligned} F(s)+\frac{1}{2} C_0s^2 \ge 0,\hbox { for all } s\ge 0. \end{aligned}$$
(3.1)

Consider the functional \(J_\lambda : H_0^{1}(\Omega )\rightarrow {\mathbb {R}}\) given by

$$\begin{aligned} J_\lambda [v]&:= \frac{1}{2}\int _{\Omega } \vert \nabla v \vert ^2\, \, dx - \frac{\lambda }{2}\int _{\Omega }(v^+)^2\, dx -\int _{\Omega }a(x)F(v^+)\, dx . \end{aligned}$$

Take note that for all \(v\in H_0^1(\Omega )\), \( J_\lambda \big [v^+\big ]\le J_\lambda [v]. \)

The functional \(J_\lambda \) is well defined and belongs to the class \(C^1\) with

$$\begin{aligned} J_\lambda '[v]\, \psi&\,=\int _{\Omega } \nabla v\nabla \psi \,dx - \lambda \int _{\Omega }v^+\psi \, dx -\int _{\Omega }a(x)f(v^+)\psi \, dx , \end{aligned}$$

for all \(\psi \in H_0^{1}(\Omega ).\) As a result, non-negative critical points of the functional \(J_\lambda \) correspond to non-negative weak solutions to (1.1).

The next Proposition establishes that Palais–Smale sequences are bounded whenever \(\lambda <\lambda _1 (\mathrm{int}\,\Omega ^0)\), where \(\lambda _1 (\mathrm{int}\,\Omega ^0)\) may be infinite.

Proposition 3.1

Assume that \(g \in C^1({\mathbb {R}})\) fulfills hypothesis (H) and that \(\lambda <\lambda _1 (\mathrm{int}\,\Omega ^0)\le +\infty \).

Then any (PS) sequence, that is, a sequence satisfying the conditions

\((J_1)\):

\(J_\lambda [u_n] \le C\),

\((J_2)\):

\(\vert J_\lambda ^{'} [u_n]\, \psi \vert \le \varepsilon _n\, \Vert \psi \Vert \), where \(\varepsilon _n \rightarrow 0\) as \(n \rightarrow +\infty \)

is a bounded sequence.

Proof

1. Let \(\{u_n\}_{n\in {\mathbb {N}}} \) be a (PS) sequence in \(H_0^{1}(\Omega )\) and, in contradiction, assume that \(\Vert u_n\Vert \rightarrow +\infty \). Let us first prove the following claim:

Claim. Let \(v\in H_0^1(\Omega )\) be the weak limit of \(v_n=\frac{u_n}{\Vert u_n\Vert }\) and assume that \(v_n\rightarrow v\), strongly in \(L^{2^*-1} (\Omega )\) and a.e. Then \(v= 0\) a.e. in \(\Omega \).

Assume that \(v\not \equiv 0\) and write \(\gamma _n =\Vert u_n\Vert \). Let \(\omega _n:=\{x\in \Omega : v_n^+(x)>1\} \), then for any \(\psi \in C_0^1 (\Omega ) \),

$$\begin{aligned} \left| \frac{\ln (e+\gamma _n)^\alpha }{\gamma _n^{2^* -1}} \frac{(u_n^+ (x))^{2^*-1}}{[\ln (e+\gamma _n\,v_n^+ (x))]^\alpha }\vert \psi \vert \right| \le \vert v_n^+(x)\vert ^{2^*-1}\Vert \psi \Vert _\infty ,\qquad \forall x\in \omega _n. \end{aligned}$$

Let \(x\in \Omega \setminus \omega _n,\) based on the estimates (A.1),

$$\begin{aligned} \left| \frac{\ln (e+\gamma _n)^\alpha }{\gamma _n^{2^* -1}} \frac{(u_n^+ (x))^{2^*-1}}{[\ln (e+\gamma _n\,v_n^+ (x))]^\alpha }\vert \psi \vert \right| \le \Big (\vert v_n^+ (x)\vert ^{2^*-2}\Big )\Vert \psi \Vert _\infty \le \Vert \psi \Vert _\infty \end{aligned}$$

Besides, by the reverse of the Lebesgue dominated convergence theorem, see for instance [2, Theorem 4.9, p. 94] , there exists \(h_i\in L^1(\Omega ) \), \(1\le i\le 3\) such that, up to a subsequence,

$$\begin{aligned} \vert v_n^+\vert ^{2^*-1}\le h_1,\; \vert v_n^+\vert ^{p-1}\le h_2, \; \vert v_n^+\vert ^{2^*-2}\le h_3 ,\; \ a.e.\ x\in \Omega , \end{aligned}$$

for all \(n\in {\mathbb {N}}\), and therefore

$$\begin{aligned} \left| \frac{\ln (e+\gamma _n)^\alpha }{\gamma _n^{2^* -1}} f(u_n^+)\psi \right| \le C\left( h_1+ h_2 +h_3 +1)\right) \Vert \psi \Vert _\infty \in L^1 (\Omega ). \end{aligned}$$

By Lebesgue’s dominated convergent theorem, we have

$$\begin{aligned} \frac{\ln (e+\gamma _n)^\alpha }{\gamma _n^{2^* -1}} a(\cdot ) f(u_n^+)\psi \rightarrow a(\cdot ) (v^+)^{2^* -1} \psi \qquad \hbox {strongly in } L^1 (\Omega ). \end{aligned}$$

We have used here that if \(v^+(x)\not =0,\) then

$$\begin{aligned} \lim _{n\rightarrow +\infty }\frac{\ln (e+\gamma _n)}{\ln (e+\gamma _n\,v_n^+(x))}=1, \end{aligned}$$

and if \(v^+(x)=0,\) then

$$\begin{aligned} \lim _{n\rightarrow +\infty }\left( \frac{\ln (e+\gamma _n)}{\ln (e+\gamma _n\,v_n^+(x))}\right) ^\alpha \vert v_n^+(x)\vert ^{2^*-1}\le \lim _{n\rightarrow +\infty } \vert v_n^+(x)\vert ^{2^*-2}= 0. \end{aligned}$$

On the other hand

$$\begin{aligned} \frac{\ln (e+\gamma _n)^\alpha }{\gamma _n^{2^*-1}}\int _\Omega \nabla u_n \cdot \nabla \psi \, dx \rightarrow 0. \end{aligned}$$

Hence, using \((J_2)\) for an arbitrary test function \(\psi \), multiplying by \(\frac{\ln (e+\gamma _n)^\alpha }{\gamma _n^{2^* -1}}\) and passing to the limit we find

$$\begin{aligned} \int _\Omega a(x)(v^+)^{2^*-1}\psi \, dx=0 \quad \forall \psi \in C_0^1 (\Omega ). \end{aligned}$$

In particular \(v^+= 0\) a.e. in \(\Omega \setminus \Omega ^0\).

Assume that \(\ \mathrm{int}\,\Omega ^0\ne \emptyset \), and that \(\lambda <\lambda _1 (\mathrm{int}\,\Omega ^0) \). Thus, for any \(\psi \in C_0^1 (\mathrm{int}\,\Omega ^0)\) we have from \((J_2)\)

$$\begin{aligned} \int _{\mathrm{int}\,\Omega ^0} \nabla u_n \cdot \nabla \psi \, dx -\lambda \int _{\mathrm{int}\,\Omega ^0} u_n^+ \psi \, dx =o(1). \end{aligned}$$

Dividing by \(\Vert u_n\Vert \) and passing to the limit we have

$$\begin{aligned} \int _{\mathrm{int}\,\Omega ^0}\nabla v \cdot \nabla \psi \, dx =\lambda \int _{\mathrm{int}\,\Omega ^0}v^+ \psi \, dx. \end{aligned}$$

From the Maximum Principle, \(v\ge 0\) in \(\mathrm{int}\,\Omega ^0.\) Since \(\lambda <\lambda _1(\mathrm{int}\,\Omega ^0)\) then it must be \(v^+\equiv 0\) in \(\mathrm{int}\,\Omega ^0\). Hence \(v^+\equiv 0\) in \(\Omega \).

On the other hand, taking \(u_n^-\) as a test function in the condition \((J_2),\)

$$\begin{aligned} \left| -\int _\Omega \vert \nabla u_n^-\vert ^2 dx - \int _\Omega a(x)f (u_n^+)u_n^- dx \right| = \int _\Omega \vert \nabla u_n^-\vert ^2 dx \le \epsilon _n \Vert u_n^-\Vert \end{aligned}$$

so \(\Vert u_n^-\Vert \rightarrow 0\) and then \(v^-\equiv 0,\) and we conclude the proof of the claim.

2. In order to achieve a contradiction, we use a Hölder inequality, and properties on convergence into an Orlicz space, cf. Appendix B.

To this end, the analysis of Lemma A.2 gives us the existence of \(\alpha ^* >0\) such that the function \(s\rightarrow \frac{s^{2^*-1}}{[\ln (e+s)]^\alpha } \) is increasing along \([0,+\infty [\) if \(\alpha \le \alpha ^*\). In this case, we will denote

$$\begin{aligned} m(s)=\frac{s^{2^*-1}}{[\ln (e+s)]^\alpha } \end{aligned}$$
(3.2)

If \(\alpha > \alpha ^*\) the function \(s\rightarrow \frac{s^{2^*-1}}{[\ln (e+s)]^\alpha } \) possesses a local maximum \(s_1\) in \([0,+\infty [\). Let us denote by \({\overline{s}}_1\) the unique solution \(s>s_1\) such that

$$\begin{aligned} \frac{s_1^{2^*-1}}{[\ln (e+s_1)]^\alpha }=\frac{s^{2^*-1}}{[\ln (e+s)]^\alpha } \end{aligned}$$

and define the non-decreasing function

$$\begin{aligned} m(s) := \left\{ \begin{array}{ll} \frac{s^{2^*-1}}{[\ln (e+s)]^\alpha } &{} \hbox { if } s\not \in [s_1,{\overline{s}}_1],\\ \frac{s_1^{2^*-1}}{[\ln (e+s_1)]^\alpha } &{}\hbox { if } s\in [s_1, {\overline{s}}_1]. \end{array} \right. \end{aligned}$$
(3.3)

It follows that

$$\begin{aligned} s\rightarrow M (s)=\int _0^{s} m (t)\, dt \qquad \text{ is } \text{ a }\quad N-\text {function in}\ [0,+\infty [. \end{aligned}$$
(3.4)

By using

$$\begin{aligned} \lim _{s\rightarrow +\infty } \frac{\ln (e+s)}{\ln (e+2s)}=1 \qquad \text{ and }\quad \lim _{s\rightarrow 0} \frac{\ln (e+s)}{\ln (e+2s)}=1, \end{aligned}$$

we get that

$$\begin{aligned} \lim _{s\rightarrow +\infty } \frac{m(2s)}{m(s)}<+\infty \qquad \text{ and }\quad \lim _{s\rightarrow 0^+} \frac{m(2s)}{m(s)}<+\infty , \end{aligned}$$

which implies that there exists \(K>0\) such that \(m(2s)\le Km(s)\) for all \(s\ge 0\) and consequently M satisfies the \(\Delta _2\)-condition (B.1).

Since \(v_n\rightharpoonup 0\) in \(H_0^1(\Omega )\) and strongly in \(L^2(\Omega )\), it follows from \((J_2) \) applied to \(\psi =u_n\) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _\Omega a(x)\frac{f(u_n^+)u_n}{\Vert u_n\Vert ^2}\, dx =\lim _{n\rightarrow \infty }\int _\Omega a(x)\frac{f(u_n^+)}{\Vert u_n\Vert }\,v_n^+\, dx =1. \end{aligned}$$
(3.5)

Since the Hölder inequality into Orlicz spaces, see Proposition B.11.(ii),

$$\begin{aligned} \int _{\Omega }\left| a(x)\frac{f(u_n^+)}{\Vert u_n\Vert }\, v_n^+\right| \, dx\le \frac{\Vert a\Vert _\infty }{\Vert u_n\Vert }\, \big \Vert f(u_n^+)\big \Vert _{M^*}\, \big \Vert v_n^+\big \Vert _{M} \end{aligned}$$
(3.6)

By Theorem  B.3 and Theorem  B.12 we have

$$\begin{aligned} \Vert v_n-v\Vert _M\rightarrow 0. \end{aligned}$$
(3.7)

Moreover, since there exists \(C>0\) such that \(m(s)\le C s^{2^*-1}\), \(M(s)\le Cs^{2^*}\) for all \(s\ge 0,\) and the sequence \(\{u_n\}_{n\in {\mathbb {N}}}\subset H_0^1(\Omega )\), then, for each \(n\in {\mathbb {N}}\), there exists a \(C_n\) such that

$$\begin{aligned} \int _\Omega \vert u_n^+ \vert \, m\big (\vert u_n^+\vert \big ) \le C_n, \quad \int _\Omega M\big (\vert u_n^+\vert \big ) \le C_n. \end{aligned}$$

By using definition B.8 of \(M^*\) and identities of Proposition B.9 we have

$$\begin{aligned} M^*\left( m\big (\vert u_n^+\vert \big )\right) =\vert u_n^+ \vert \, m\big (\vert u_n^+\vert \big )- M\big (\vert u_n^+\vert \big ) \end{aligned}$$

then, for each \(n\in {\mathbb {N}}\),

$$\begin{aligned} \int _\Omega M^*\left( m\big (\vert u_n^+\vert \big )\right) dx\le 2C_n. \end{aligned}$$

Observe that \(\ \vert f(s)\vert \le C(1+m(s))\), so then

$$\begin{aligned} \Vert f(u_n^+)\Vert _{M^*} \le C\left\| 1+m\big (u_n^+\big )\right\| _{M^*} \le C\left[ 1+\int _\Omega M^*\Big (m\big (\vert u_n^+\vert \big )\Big )\right] \le C'_n, \end{aligned}$$

see Proposition B.11.(iii) and (i), concluding that the l.h.s. is bounded for each n.

Consequently, \(a(x)\frac{f(u_n^+)}{\Vert u_n\Vert } \in L_{M^*} (\Omega )\), which is the dual of \(L_{M} (\Omega )\) (see [15], Theorem 14.2).

On the other hand, from \(J_2\), for all \(\psi \in C_c^\infty (\Omega ),\)

$$\begin{aligned} \left| \int _{\Omega } \nabla v_{n} \nabla \psi \, dx - \lambda _n\int _{\Omega }v_{n}\psi \, dx -\int _{\Omega }a(x)\frac{f(u_n^+)}{\Vert u_n\Vert }\psi \, dx\right| \le \frac{\varepsilon _n}{\Vert u_n\Vert }\Vert \psi \Vert . \end{aligned}$$
(3.8)

Taking the limit, and since \(C_c^\infty (\Omega )\) is dense in \( L_M(\Omega )\) (see [13]),

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\Omega }a(x)\frac{f(u_n^+)}{\Vert u_n\Vert }\ \psi \, dx=0,\qquad \text{ for } \text{ all }\quad \psi \in L_M(\Omega ). \end{aligned}$$
(3.9)

Moreover, since (3.7), \(v_n \rightarrow v=0\) in \(L_M(\Omega )\), [2, Proposition 3.13 (iv)], and (3.9) imply

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _\Omega a(x)\frac{f(u_n^+)}{\Vert u_n\Vert }\, v_n\, dx =0, \end{aligned}$$

which contradicts (3.5). This concludes the proof. \(\square \)

Theorem 3.2

Assume the hypothesis of Proposition 3.1 and let \(\{u_n\}_{n\in {\mathbb {N}}} \) be a (PS) sequence in \(H_0^{1}(\Omega )\).

Then, there exists a subsequence, denoted by \(\{u_n\}_{n\in {\mathbb {N}}}\), such that

$$\begin{aligned} u_n\rightarrow u\qquad \text{ in }\quad H_0^{1}(\Omega ). \end{aligned}$$

Proof

From Proposition 3.1 we know that the sequence is bounded. Consequently, there exists a subsequence, denoted by \(\{u_{n}\}_{n\in {\mathbb {N}}}\), and some \(u\in H_0^1(\Omega )\) such that

$$\begin{aligned}&u_{n} \rightharpoonup u\ \ \text {weakly in}\ H_0^1(\Omega ), \end{aligned}$$
(3.10)
$$\begin{aligned}&\int _\Omega a(x)g(u_n)\vert u_n-u\vert \, dx \rightarrow 0, \end{aligned}$$
(3.11)
$$\begin{aligned}&u_n\rightarrow u \ \text { a.e.} \end{aligned}$$
(3.12)

By testing \((J_2)\) against \(\psi =u_n-u\) and using (3.10), and (3.11) we get

$$\begin{aligned} \Vert u_n-u\Vert ^2= & {} \int _\Omega \nabla u_n \cdot \nabla (u_n-u)\, dx +o(1)\\\le & {} \Vert a\Vert _\infty \int _\Omega \frac{\vert u_n\vert ^{2^*-1}}{[\ln (e+\vert u_n\vert )]^\alpha } \vert u_n-u\vert \, dx +o(1). \end{aligned}$$

Claim.

$$\begin{aligned} \int _\Omega \frac{\vert u_n\vert ^{2^*-1}}{[\ln (e+\vert u_n\vert )]^\alpha } \vert u_n-u\vert dx =o(1), \end{aligned}$$

In order to prove this claim, we use, as in the above proposition, a Hölder inequality and a compact embedding into some Orlicz space, c.f. Appendix B.

By Theorem  B.3 and Theorem  B.12 we have

$$\begin{aligned} \Vert u_n-u\Vert _M\rightarrow 0, \end{aligned}$$
(3.13)

where m, and M are defined by (3.2)–(3.4), as in the above proposition. On the other hand, because there exists \(C>0\) such that \(m(s)\le C s^{2^*-1}\) and \(M(s)\le Cs^{2^*}\) for all \(s\ge 0,\) and the sequence \(\{u_n\}_{n\in {\mathbb {N}}}\) is bounded in \(H_0^1(\Omega )\), then

$$\begin{aligned} \Vert u_n\, m(\vert u_n\vert )\Vert _{L^1 (\Omega )} \le C, \quad \Vert M(\vert u_n \vert )\Vert _{L^1(\Omega )}\le C\qquad \text{ for } \text{ all }\quad n\in {\mathbb {N}}\end{aligned}$$

By using definition B.8 of \(M^*\) and identities of Proposition B.9 we have

$$\begin{aligned} M^*\left( m (\vert u_n\vert )\right) =\vert u_n\vert \, m(\vert u_n\vert )- M(\vert u_n\vert ) \end{aligned}$$

then

$$\begin{aligned} \int _\Omega M^*\left( m(\vert u_n\vert )\right) dx\le C \end{aligned}$$

for all \(n\in {\mathbb {N}}\). Finally, by inequality (B.5) of Proposition B.12 we get

$$\begin{aligned} \sup \Big \{\Vert m(\vert u_n\vert )\Vert _{M^*},\ n\in {\mathbb {N}}\Big \}\le C+1 . \end{aligned}$$

Now, using Holder’s inequality (B.6) and that \(\frac{s^{2^*-1}}{[\ln (e+s)]^\alpha } \le m(s)\) for all \(s\ge 0\), we get

$$\begin{aligned} \int _\Omega \frac{\vert u_n\vert ^{2^*-1}}{[\ln (e+\vert u_n\vert )]^\alpha } \vert u_n-u\vert dx \le \Vert u_n-u\Vert _M\, \Vert m(\vert u_n \vert )\Vert _{M^*} \le (C+1)\Vert u_n-u\Vert _M \end{aligned}$$

and it follows from (3.13) that \(\Vert u_n-u\Vert \rightarrow 0\). \(\square \)

3.2 An Existence Result for \(\lambda <\lambda _1\)

The next theorem provides a solution to (1.1) for \(\lambda <\lambda _1\) based on the Mountain Pass Theorem.

Theorem 3.3

Assume that \(\Omega \subset {\mathbb {R}}^N \) is a bounded domain with \(C^{2}\) boundary. Assume that the nonlinearity f defined by (1.2) satisfies (H), and that the weight \(a\in C^1({\overline{\Omega }})\). Then, the boundary value problem (1.1)\(_\lambda \) has at least one classical positive solution for any \(\lambda <\lambda _1\).

Proof

We verify the hypothesis of the Mountain Pass Theorem, see [14, Theorem 2, Section 8.5]. Observe that the derivative of the functional \(J_\lambda ':H_0^{1}(\Omega )\rightarrow H_0^{1}(\Omega )\) is Lipschitz continuous on bounded sets of \(H_0^{1}(\Omega )\); also the (PS) condition is satisfied, see Proposition 3.1. Clearly \(J_\lambda [0]=0\).

  1. 1.

    Let now \(u\in H_0^{1}(\Omega )\) with \(\Vert u\Vert =r\), for \(r>0\) to be chosen below. Then,

    $$\begin{aligned} J_\lambda [u]=\frac{r^2}{2}-\frac{\lambda }{2}\int _{\Omega } (u^+)^{2}\, dx-\int _{\Omega }\,a(x)F(u^+) \,\, \, dx. \end{aligned}$$
    (3.14)

    From hypothesis (H) we have

    $$\begin{aligned} \left| \int _{\Omega }\,a(x)G(u^+) \,\, \, dx\right|&\le C\int _{\Omega } \big (\vert u\vert ^{p}+ \vert u\vert ^{q}\big )\, dx\le C\left( r^{p}+r^{q}\right) . \end{aligned}$$

    where \(G(s):= \int _0^s g(t) \, dt.\) Now, definition (1.2) implies that

    $$\begin{aligned} \left| \int _{\Omega }\,a(x)F(u^+) \,\, \, dx\right| \le C\left( r^{p}+r^{q}+r^{2^*}\right) . \end{aligned}$$

    In view of (3.14), and as a result of the Poincaré inequality, we get

    $$\begin{aligned} J_\lambda [u]\ge \frac{1}{2}\left( 1-\frac{\vert \lambda \vert }{\lambda _1} \right) \, r^2 -C\left( r^{p}+r^{q}+r^{2^*}\right) \ge C_1 r^2, \end{aligned}$$

    taking \(\vert \lambda \vert <\lambda _1\), \(r>0\) small enough, and using that \(p,\ q,\ 2^*>2\).

  2. 2.

    Now, fix some element \(0\le u_0\in H_{0}^{1}(\Omega )\), \( u_0> 0\) in \(\Omega ^+\), \( u_0\equiv 0\) in \(\Omega ^-\). Let \(v=tu_0\) for a certain \(t=t_0>0\) to be selected a posteriori. Since

    $$\begin{aligned} f(tu_0)=\vert t\vert ^{2^* -2}t\, f(u_0) \left( \dfrac{\ln (e+\vert u_0\vert )}{\ln (e+\vert tu_0\vert )}\right) ^\alpha +g(tu_0), \end{aligned}$$
    (3.15)

    then \(f(tu_0)/t\rightarrow +\infty \) as \(t\rightarrow +\infty \) in \(\Omega ^+\).

From definition, and integrating by parts,

$$\begin{aligned} F(s)= & {} \int _0^s\left( \dfrac{t^{2^*-1}}{\ln (e+t)^\alpha }+g(t)\right) \,dt\\= & {} \frac{1}{2^* }\, sh(s) +G(s) +\frac{\alpha }{2^* } \int _0^s\left( \dfrac{1}{\ln (e+t)}\right) ^{\alpha +1}\, \dfrac{t^{2^*}}{e+t}\, dt. \end{aligned}$$

It can be easily seen that \(\lim _{s \rightarrow +\infty } \frac{G(s)}{sf(s)}=0.\)

Therefore, using l’Hôpital’s rule we can write

$$\begin{aligned} \lim _{s \rightarrow +\infty } \frac{F(s)}{sf(s)} =\frac{1}{2^* }\in \left( 0,\frac{1}{2} \right) , \end{aligned}$$
(3.16)

hence

$$\begin{aligned} \lim _{t \rightarrow +\infty } \frac{F(tu_0)}{tu_0f(tu_0)} =\frac{1}{2^* }\in \left( 0,\frac{1}{2} \right) \qquad \text{ in }\quad \Omega ^+. \end{aligned}$$
(3.17)

Let \(C_0\ge 0\) be such that \(F(s)+\frac{1}{2} C_0 s^2\ge 0\) for all \(s\ge 0\) (see (1.7)), and let

$$\begin{aligned} {\widetilde{\Omega }}_{\delta }^+:=\{x\in \Omega ^+\ : \ a(x)=a^+(x)>\delta \}. \end{aligned}$$
(3.18)

By definition, \(u_0\equiv 0\) in \(\Omega ^-\), so, introducing \(\pm \frac{1}{2} C_0 (tu_0)^2\), splitting the integral, and using (3.17)–(3.18) we obtain

$$\begin{aligned}&-\int _{\Omega } a(x) F(tu_0)\, dx =-\int _{\Omega ^+} a^+(x) F(tu_0)\, dx \\&\quad \le \frac{C_0 t^2}{2} \int _{\Omega ^+}a^+(x)u_0^2 \, dx -\int _{{\widetilde{\Omega }}_{\delta }^+} a^+(x) \left[ \frac{1}{2} C_0 (tu_0)^2 +F(tu_0)\right] \, dx \\&\quad \le C+\frac{C_0 t^2}{2} \int _{\Omega ^+}a^+(x)u_0^2 \, dx -\frac{\delta t^2}{2}\int _{{\widetilde{\Omega }}_{\delta }^+} \left[ C_0 u_0^2 + \frac{u_0 f(tu_0)}{2^*t}\right] \, dx . \end{aligned}$$

Hence, there exists a positive constant \(C>0\) such that

$$\begin{aligned} J_\lambda [tu_0]&= \frac{t^2}{2}\Vert u_0\Vert ^2 -t^2\frac{\lambda }{2}\Vert u_0\Vert _{L^2(\Omega )}^2- \int _{\Omega ^+} a^+(x) F(tu_0)\\&\le C(1+t^2) -\frac{\delta \, t^2}{2}\int _{{\widetilde{\Omega }}_{\delta }^+} \left[ C_0 (u_0)^2 + \frac{u_0 f(tu_0)}{2^*t}\right] \, dx<0 \end{aligned}$$

for \(t=t_0>0\) big enough.

Step 3. We have at last checked that all the hypothesis of the Mountain Pass Theorem are accomplished. Let

$$\begin{aligned} \Gamma :=\{ {\mathbf {g}}\in C\big ([0,1];H_{0}^{1}(\Omega )\big ) : \,{\mathbf {g}} (0)=0,\ {\mathbf {g}}(1)=t_0u_0\}, \end{aligned}$$

then, there exists \(c\ge C_1\,r^2>0\) such that

$$\begin{aligned} c:=\inf _{{\mathbf {g}}\in \Gamma } \max _{0\le t\le 1} J_\lambda [{\mathbf {g}}(t)] \end{aligned}$$

is a critical value of \(J_\lambda \), that is, the set \({\mathscr {K}}_c:=\{v\in H_{0}^{1}(\Omega ): \, J_\lambda [v]=c,\ J_\lambda '[v]=0\}\ne \emptyset \). Thus there exists \(u\in H_0^1 (\Omega )\), \(u\ge 0\), \(u\ne 0\) such that for each \(\psi \in H_0^{1}(\Omega )\), we have

$$\begin{aligned} \int _{\Omega }\nabla u\cdot \nabla \psi \,dx= \int _{\Omega }\big [\lambda u^+ +a(x)f(u^+)\big ] \psi \, dx . \end{aligned}$$
(3.19)

and thereby, u is a nontrivial weak solution to (3.19). By Lemma 2.1, u is a classical solution, and by (1.8), \(u>0\) in \(\Omega .\) \(\square \)

4 A Bifurcation Result for \(\lambda >\lambda _1\)

Next Proposition uses Crandall-Rabinowitz’s local bifurcation theory, see [10], and Rabinowitz’s global bifurcation theory, see [19].

Proposition 4.1

Let us define

$$\begin{aligned} \Lambda :=\sup \{\lambda >0 : \, (1.1)_{\lambda }\ \text {admits a positive solution} \}. \end{aligned}$$

If (1.5) holds then,

$$\begin{aligned} \lambda _1<\Lambda < \min \Big \{\lambda _1\big (\mathrm{int}\,(\Omega ^0)\big )\, ,\quad \lambda _1\big (\mathrm{int}\,\big (\Omega ^+\cup \Omega ^0\big )\big ) +C_0\sup a^+ \Big \} \end{aligned}$$

where \(C_0>0\) is such that \(f(s)+C_0s \ge 0\) for all \(s\ge 0,\) (see definition (1.6)).

Moreover, there exists an unbounded continuum (a closed and connected set) \( {\mathscr {C}}\) of classical positive solutions to (1.1) emanating from the trivial solution set at the bifurcation point \((\lambda ,u)=(\lambda _1,0)\).

Proof

Proposition 2.2 establish the upper bounds for \(\Lambda \). Next, we concentrate our attention in proving that \(\Lambda >\lambda _1.\) Choosing \(\lambda \) as the bifurcation parameter, we check that the conditions of Crandall - Rabinowitz’s Theorem [10] are satisfied. For \(r>N\), we define the set \(W^{2,r}_+:=\{ u\in W^{2,r}(\Omega ): \, u>0\ \text {in}\ \Omega \}\), and consider \(W^{2,r}_+(\Omega )\cap W^{1,r}_0(\Omega )\) endowed with the topology of \(W^{2,r}(\Omega )\). If \(r>N\), we have that \(W^{2,r}_+(\Omega )\cap W^{1,r}_0(\Omega )\hookrightarrow C_0^{1,\mu }(\Omega )\) for \(\mu =1-\frac{N}{r}\in (0,1)\). Moreover, from Hopf’s lemma, we know that if \({\tilde{u}}\) is a positive solution to (1.1) then \({\tilde{u}}\) lies in the interior of \(W^{2,r}_+(\Omega )\cap W^{1,r}_0(\Omega )\).

We consider the map \({\mathscr {F}}:{\mathbb {R}}\times W^{2,r}_+(\Omega ) \cap W^{1,r}_{0}(\Omega )\rightarrow L^{r}(\Omega )\) for \(r>N\),

$$\begin{aligned} {\mathscr {F}}: (\lambda ,u)\rightarrow -\Delta u-\lambda u- a(x)f(u) \end{aligned}$$

The map \({\mathscr {F}}\) is a continuously differentiable map. Since hypothesis (i), \(g(0)=0\), and so \(a(x)F(0)=0\), \({\mathscr {F}} (\lambda ,0)=0\) for all \(\lambda \in {\mathbb {R}}\), and since \(F_u(x,0)=0\),

$$\begin{aligned} D_{u}{\mathscr {F}} (\lambda _1,0) w&:=-\Delta w - \lambda _1 w ,\\ D_{\lambda ,u}{\mathscr {F}}(\lambda _1,0) w&:= -w. \end{aligned}$$

Observe that

$$\begin{aligned} N\big (D_{u}{\mathscr {F}} (\lambda _1,0)\big )&=span[\varphi _1], \qquad \text {codim}\, R\big (D_{u}{\mathscr {F}} (\lambda _1,0)\big )=1,\\ D_{\lambda ,u}{\mathscr {F}}(\lambda _1,0) \varphi _1&=-\varphi _1\not \in R\big (D_{u}{\mathscr {F}} (\lambda _1,0)\big ), \end{aligned}$$

where \(N(\cdot )\) is the kernel, and \(R(\cdot )\) denotes the range of a linear operator.

Hence, the hypotheses of Crandall-Rabinowitz’s Theorem are satisfied and \((\lambda _1,0)\) is a bifurcation point. Thus, decomposing

$$\begin{aligned} C_0^{1,\mu } ({\overline{\Omega }})=span[\varphi _1] \oplus Z, \end{aligned}$$

where \(Z=span[\varphi _1]^{\bot }\), there exists a neighborhood \({\mathscr {U}}\) of \((\lambda _1,0)\) in \({\mathbb {R}} \times C_0^{1,\mu } ({\overline{\Omega }})\), and continuous functions \(\lambda (s), {\tilde{w}}(s),\) \( s\in (-\varepsilon ,\varepsilon ),\) \(\lambda : (-\varepsilon ,\varepsilon )\rightarrow {\mathbb {R}},\) \({\tilde{w}}: (-\varepsilon ,\varepsilon )\rightarrow Z\) such that \(\lambda (0)=\lambda _1,\) \({\tilde{w}}(0)=0,\) with \(\int _{\Omega } {\tilde{w}} \varphi _1\, dx =0,\) and the only nontrivial solutions to (1.1) in \({\mathscr {U}}\), are

$$\begin{aligned} \big \{\big (\lambda (s),s\varphi _1+s\,{\tilde{w}}(s)\big ):\ s\in (-\varepsilon ,\varepsilon )\big \}. \end{aligned}$$
(4.1)

Set \(u=u(s)=s\varphi _1 +s\,{\tilde{w}}(s)\). Note that by continuity \({\tilde{w}}(s)\rightarrow 0\) as \(s\rightarrow 0\), which guarantees that \(u(s)>0\) in \(\Omega \) for all \(s\in (0,\varepsilon )\) small enough.

Next, we show that \(\lambda (s)>\lambda _1\) for all s small enough. Since (3.15), and hypothesis (H)\(_0\) on f, note that \(\frac{a(x)f(su)}{s^{p-1}u^{p-1}}\rightarrow L_1a(x)\) as \(s\rightarrow 0\). In fact, as \({\tilde{w}}(s)\rightarrow 0\) uniformly as \(s\rightarrow 0\), hypothesis (H)\(_0\) yields

$$\begin{aligned} \frac{a(x)f\big (s\varphi _1+s\,{\tilde{w}}(s)\big )}{s^{p-1}\big (\varphi _1+{\tilde{w}}(s)\big )^{p-1}}\longrightarrow L_1a(x)\ \text {uniformly in}\ \Omega \qquad \text{ as }\quad s\rightarrow 0. \end{aligned}$$

Hence, multiplying and dividing by \(\big (\varphi _1+{\tilde{w}}(s)\big )^{p-1}\), we deduce

$$\begin{aligned} \frac{1}{s^{p-1}}\int _{\Omega }a(x)f\big (u(s)\big )\varphi _1&\underset{ s\rightarrow 0}{\rightarrow } L_1\int _{\Omega }a(x)\varphi _1^p. \end{aligned}$$

Now we prove that \(\lambda (s)>\lambda _1\) arguing by contradiction. Assume that there is a sequence \((\lambda _n,u_n)=\big (\lambda (s_n),u(s_n)\big )\) of bifurcated solutions to (1.1) in \({\mathscr {U}}\), with \(\lambda (s_n)\le \lambda _1\). Multiplying (1.1)\(_{\lambda _n}\) by \(\varphi _1\) and integrating by parts

$$\begin{aligned} 0\le \frac{\big (\lambda _1-\lambda (s_n)\big )}{s_n^{p-1}}\int _{\Omega } u(s_n)\varphi _1=\frac{1}{s_n^{p-1}}\int _{\Omega }a(x) f\big (u(s_n)\big )\varphi _1\rightarrow L_1\int _{\Omega }a(x)\varphi _1^p<0 \end{aligned}$$

which yields a contradiction, and consequently, \(\Lambda >\lambda _1\).

Finally, Rabinowitz’s global bifurcation Theorem [19] states that, in fact, the set \( {\mathscr {C}}\) of positive solutions to (1.1) emanating from \((\lambda _1,0)\) is a continuum (a closed and connected set) which is either unbounded, or contains another bifurcation point, or contains a pair of points \((\lambda ,u)\), \((\lambda ,-u)\) with \(u\ne 0\). Since (1.8), any non-negative non-trivial solution is strictly positive, and moreover \((\lambda _1,0)\) is the only bifurcation point to positive solutions, so \( {\mathscr {C}}\) can not reach another bifurcation point. Since (1.3), neither \( {\mathscr {C}}\) contains a pair of points \((\lambda ,u)\), \((\lambda ,-u)\) with \(u\ne 0\), which states that \( {\mathscr {C}}\) is unbounded, ending the proof. \(\square \)

5 Proof of Theorem 1.1

First we prove an auxiliary result.

Proposition 5.1

For each \(\lambda \in (\lambda _1,\Lambda )\), the following holds:

  1. (i)

    Problem (1.1)\(_\lambda \) admits a positive solution

    $$\begin{aligned} u_\lambda =\inf \big \{u(x): \, u>0 \text { solving } (1.1) _\lambda \big \}, \end{aligned}$$

    in other words \(u_\lambda \) is minimal.

  2. (ii)

    Moreover, the map \(\lambda \rightarrow u_\lambda \) is strictly monotone increasing, that is, if \(\lambda<\mu <\Lambda \), then \(u_\lambda (x)<u_\mu (x)\) for all \(x\in \Omega \), and \(\frac{\partial u_\lambda }{\partial \nu }(x)>\frac{\partial u_\mu }{\partial \nu }(x)\) for all \(x\in \partial \Omega \).

  3. (iii)

    Furthermore, \(u_\lambda \) is a local minimum of the functional \(J_\lambda \).

Proof

(i.a) Step 1. Existence of positive solutions for any \(\lambda \in (\lambda _1,\Lambda )\).

Let \(\lambda \in (\lambda _1,\Lambda )\) be fixed. By definition of \(\Lambda \), there exists a \(\lambda _0\in (\lambda ,\Lambda )\) such that the problem (1.1)\(_{\lambda _0}\) admits a positive solution \(u_0\). It is easy to verify that \(u_0>0\) is a supersolution to (1.1)\(_\lambda \). Indeed, for any \(\psi \in H_0^1(\Omega )\) with \(\psi \ge 0\) in \(\Omega \)

$$\begin{aligned} \int _\Omega \nabla u_0\cdot \nabla \psi \, dx - \lambda \int _\Omega u_0\psi \, dx -\int _\Omega a(x)f(u_0)\psi \, dx= (\lambda _0-\lambda ) \int _\Omega u_0\psi \, dx \ge 0. \end{aligned}$$

Moreover, for every \(\delta >0\) satisfying

$$\begin{aligned} 0<\delta < \left( \frac{\lambda -\lambda _1}{2L_1\,\Vert a^-\Vert _\infty } \right) ^\frac{1}{p-2}\frac{1}{\Vert \varphi _1\Vert _\infty } \end{aligned}$$
(5.1)

the function \({\underline{u}}=\delta \varphi _1\) is a subsolution for (1.1)\(_\lambda \) whenever \(\lambda >\lambda _1\). Let \(\delta >0\) satisfying (5.1) and such that \(g(s)\ge 0\) for any \(s\in [0,\delta \Vert \varphi _1\Vert _{L^\infty (\Omega )}]\). For any \(\psi \in H_0^1(\Omega )\), \(\psi >0\) with in \(\Omega \) we deduce

$$\begin{aligned}&\delta \int _\Omega \nabla \varphi _1\cdot \nabla \psi \, dx - \lambda \delta \int _\Omega \varphi _1 \psi \, dx -\int _\Omega a(x)f(\delta \varphi _1)\psi \, dx\\&\quad = -(\lambda -\lambda _1)\delta \int _\Omega \varphi _1\psi \, dx -\int _\Omega a(x)f(\delta \varphi _1)\psi \, dx\\&\quad = -(\lambda -\lambda _1)\delta \int _\Omega \varphi _1 \psi \, dx - \int _\Omega a(x)\left[ \dfrac{(\delta \varphi _1)^{2^* -1}}{[\ln (e+\delta \varphi _1)]^\alpha }+g(\delta \varphi _1)\right] \psi \, dx \\&\quad \le -(\lambda -\lambda _1)\delta \int _\Omega \varphi _1 \psi \, dx +\Vert a^-\Vert _\infty \int _\Omega \big [h(\delta \varphi _1) +g(\delta \varphi _1)\big ]\psi \, dx < 0. \end{aligned}$$

This allows us to take \({\underline{u}}=\delta \varphi _1\) as a subsolution for (1.1)\(_\lambda \) with \({\underline{u}}<u_0\). The sub- and supersolution method now guarantees a positive solution u to (1.1)\(_\lambda \), with \({\underline{u}}\le u\le u_0\).

(i.b) Step 2. Existence of a minimal positive solution \(u_\lambda \) for any \(\lambda \in (\lambda _1,\Lambda )\).

To show that there is in fact a minimal solution, for each \(x\in \Omega \) we define

$$\begin{aligned} {\underline{u}}_\lambda (x):=\inf \big \{u(x) : \, u>0 \text { solving } (1.1) _\lambda \big \}. \end{aligned}$$

Firstly, we claim that \({\underline{u}}_\lambda \ge 0,\) \({\underline{u}}_\lambda \not \equiv 0\). Assume that \({\underline{u}}_\lambda \equiv 0\) by contradiction. This would yield a sequence \(u_n\) of positive solutions to (1.1)\(_\lambda \) such that \(\Vert u_{n}\Vert _{C({\overline{\Omega }})}\rightarrow 0\) as \(n\rightarrow \infty \), or in other words, \((\lambda ,0)\) is a bifurcation point from the trivial solution set to positive solutions. Set \(v_n:=\frac{u_{n}}{\Vert u_{n}\Vert _{C({\overline{\Omega }})}}\). Observe that \(v_{n}\) is a weak solution to the problem

$$\begin{aligned} -\Delta v_{n} = \lambda v_{n} + a(x)f(u_{n})/\Vert u_{n}\Vert _{C({\overline{\Omega }})} \text { in }\; \Omega \,;\qquad v_{n}= 0 \text { on }\; \partial \Omega \,. \end{aligned}$$
(5.2)

It follows from \(\mathrm {(H)}_0\) that \( \frac{a(x)f(u_{n})}{\Vert u_{n}\Vert _{C({\overline{\Omega }})}}\rightarrow 0\) in \(C({\overline{\Omega }}) \) as \(n\rightarrow \infty \). Therefore, the right-hand side of (5.2) is bounded in \(C({{\overline{\Omega }}})\). Hence, by the elliptic regularity, \(v_n\in W^{2,r}(\Omega )\) for any \(r>1\), in particular for \(r>N\). Then, the Sobolev embedding theorem implies that \(||v_n||_{C^{1,\alpha }({\overline{\Omega }})}\) is bounded by a constant C that is independent of n. Then, the compact embedding of \(C^{1,\mu }({\overline{\Omega }})\) into \(C^{1,\beta }({\overline{\Omega }})\) for \(0<\beta <\mu \) yields, up to a subsequence, \(v_{n}\rightarrow \Phi \ge 0\) in \(C^{1,\beta }({\overline{\Omega }})\). Since \(\Vert v_{n}\Vert _{C({\overline{\Omega }})}=1\), we have that \(\Vert \Phi \Vert _{C({\overline{\Omega }})}=1\). Hence, \(\Phi \ge 0,\) \(\Phi \not \equiv 0.\)

Using the weak formulation of equation (5.2), passing to the limit, and taking into account that \(\lambda \) is fixed and \(v_{n}\rightarrow \Phi \), we obtain that \(\Phi \ge 0,\) \(\Phi \not \equiv 0,\) is a weak solution to the equation

$$\begin{aligned} -\Delta \Phi =\lambda \Phi \text { in }\; \Omega \, ,\qquad \Phi = 0 \text { on }\; \partial \Omega . \end{aligned}$$

Then, by the maximum principle, it follows that \(\Phi =\varphi _1>0\), the first eigenfunction, and \(\lambda =\lambda _1\) is its corresponding eigenvalue, which contradicts that \(\lambda >\lambda _1\).

Secondly, we show that \({\underline{u}}_\lambda \) solves (1.1)\(_\lambda \). We argue on the contrary. Observe that the minimum of any two positive solutions to (1.1)\(_\lambda \) furnishes a supersolution to (1.1)\(_\lambda \). Assume that there are a finite number of solutions to (1.1)\(_\lambda \), then \({\underline{u}}_\lambda (x):=\min \big \{u(x): u>0 \text { solves } (1.1) _\lambda \big \}\) and \({\underline{u}}_\lambda \) is a supersolution. Choosing \(\varepsilon _0\) small enough so that \(\varepsilon _0\varphi _1 <{\underline{u}}_\lambda \), the sub- supersolution method provides a solution \(\varepsilon _0\varphi _1 \le v\le {\underline{u}}_\lambda \). Since v is a solution and \({\underline{u}}_\lambda \) is not, then \( v\le {\underline{u}}_\lambda \), \(v\ne u\), contradicting the definition of \({\underline{u}}_\lambda \), and achieving this part of the proof.

Assume now that there is a sequence \(u_n\) of positive solutions to (1.1)\(_\lambda \) such that, for each \(x\in \Omega \), \(\inf u_n(x)={\underline{u}}_\lambda (x)\ge 0,\) \({\underline{u}}_\lambda \not \equiv 0.\) Let \({\underline{u}}_1:=\min \{u_1,u_2\}\). Choosing \(\varepsilon _1\) small enough so that \(\varepsilon _1\varphi _1 <{\underline{u}}_1\), the sub- supersolution method provides a solution \(\varepsilon _1\varphi _1 \le v_1\le {\underline{u}}_1 \). We reason by induction.

Let \({\underline{u}}_n:=\min \{v_{n-1},u_{n+1}\}\). Choosing \(\varepsilon _n\) small enough so that \(\varepsilon _n\varphi _1 <{\underline{u}}_n\), the sub- supersolution method provides a solution \(\varepsilon _n\varphi _1 \le v_n\le {\underline{u}}_n\le v_{n-1}\). With this induction procedure, we build a monotone sequence of solutions \(v_n\), such that

$$\begin{aligned} 0< v_n\le {\underline{u}}_n\le v_{n-1}\le {\underline{u}}_{n-1}\le \cdots \le v_{1}. \end{aligned}$$
(5.3)

Since monotonicity and Lemma 2.1, \(\Vert v_n\Vert _{C({\overline{\Omega }})}\le \Vert v_1\Vert _{C({\overline{\Omega }})}\), by elliptic regularity, \(\Vert v_n\Vert _{C^{1,\mu }({\overline{\Omega }})}\le C\) for any \(\mu <1\), and by compact embedding \(v_n\rightarrow v\) in \(C^{1,\beta }({\overline{\Omega }})\) for any \(\beta <\alpha \). Using the weak formulation of equation (1.1)\(_\lambda \), passing to the limit, and taking into account that \(\lambda \) is fixed, we obtain that v is a weak solution to the equation (1.1)\(_\lambda \). Hence \(v(x)\ge {\underline{u}}_\lambda >0\). Moreover, since (5.3), \(v_n(x)\downarrow v(x)\) pointwise for \(x\in \Omega \), so \(\inf v_n(x)=v(x)\). Also, and due to (5.3), \({\underline{u}}_n(x)\downarrow v(x)\) pointwise for \(x\in \Omega \), and \(\inf {\underline{u}}_n(x)=v(x)\).

On the other hand, by construction \({\underline{u}}_n\le u_{n+1}\), so, for each \(x\in \Omega \), \(v(x)=\inf {\underline{u}}_n(x)\le \inf u_n(x)={\underline{u}}_\lambda (x)\). Therefore, and by definition of \({\underline{u}}_\lambda \), necessarily \(v= {\underline{u}}_\lambda \), proving that \({\underline{u}}_\lambda \) solves (1.1)\(_\lambda \), and achieving the proof of step 2.

(ii) The monotonicity of the minimal solutions is concluded from a sub- supersolution method. Reasoning as in step 1, \(u_\mu \) is a strict supersolution to (1.1)\(_\lambda \), so \(w:= u_\mu (x)- u_\lambda (x)\ge 0\), \(w\not \equiv 0\). Moreover, \(w= 0\) on \(\partial \Omega ,\) and we can always choose \(c_0:=C_0\Vert a\Vert _{\infty }>0\) where \(C_0\) is defined by (1.6), so that \(a^-(x)f'(s)+ c_0\ge 0\) and \(a^+(x)f'(s)+ c_0\ge 0\) for all \(s\ge 0\), then

$$\begin{aligned}&\Big (-\Delta +a^-(x) f'\big (\theta u_\mu +(1-\theta ) u_\lambda \big )+ c_0\Big ) w = (\mu -\lambda )u_\mu + \lambda w \\&\quad + \big [a^+(x)f'\big (\theta u_\mu +(1-\theta ) u_\lambda \big ) +c_0\big ]w>0\ \text{ in } \Omega , \end{aligned}$$

finally, the Maximum Principle implies that \(w>0\) in \(\Omega ,\) and \(\frac{\partial w}{\partial \nu }<0\) on \(\partial \Omega ,\) ending the proof of step 3.

(iii) Since [4, Theorem 2] if there exists an ordered pair of \(L^\infty \) bounded sub and supersolution \({\underline{u}}\le {\overline{u}}\) to (1.1)\(_\lambda \), and neither \({\underline{u}}\) nor \({\overline{u}}\) is a solution to (1.1)\(_\lambda ,\) then there exist a solution \({\underline{u}}<u<{\overline{u}}\) to (1.1)\(_\lambda \) such that u is a local minimum of \(J_\lambda \) at \(H_0^1(\Omega )\).

Reasoning as in (i), \({\overline{u}}:=u_\mu \) with \(\mu >\lambda \) is a strict supersolution to (1.1)\(_\lambda \), and \({\underline{u}}:=\delta \varphi _1\) is a strict sub-solution for \(\delta >0\) small enough, such that \({\underline{u}}(x)<{\overline{u}}(x)\) for each \(x\in \Omega \). This achieves the proof. \(\square \)

Proof of Theorem 1.1

Theorem 3.3 provides the existence of positive solutions for \(\lambda <\lambda _1\), and Proposition 5.1 provide the existence of minimal positive solutions for \(\lambda \in (\lambda _1,\Lambda ).\)

(a) Step 1. Existence of a second positive solution for \(\lambda \in (\lambda _1,\Lambda ).\)

Fix an arbitrary \(\lambda \in (\lambda _1,\Lambda )\), and let \(u_{\lambda }\) be the minimal solution to (1.1)\(_{\lambda }\) given by Proposition 5.1, minimizing \(J_\lambda \). A second solution follows seeking a solution through variational arguments [12, Theorem 5.10] and the Mountain Pass procedure shown below.

First, reasoning as in Proposition 5.1(iii), we get a local minimum \({\tilde{u}}_\lambda >0\) of \(J_\lambda \). If \({\tilde{u}}_\lambda \ne u_\lambda \), then \({\tilde{u}}_\lambda \) is the second positive solution, ending the proof. Assume that \({\tilde{u}}_\lambda = u_\lambda \).

Now we reason as in [12, Theorem 5.10] on the nature of local minima. Thus, either

  1. (i)

    there exists \(\varepsilon _0>0\), such that \(\inf \big \{J_\lambda (u) : \Vert u-{\tilde{u}}_\lambda \Vert = \varepsilon _0 \big \} >J_\lambda ({\tilde{u}}_\lambda ),\) in other words, \({\tilde{u}}_\lambda \) is a strict local minimum, or

  2. (ii)

    for each \(\varepsilon >0,\) there exists \(u_\varepsilon \in H_0^1(\Omega )\) such that \(J_\lambda \) has a local minimum at a point \(u_\varepsilon \) with \(\Vert u_\varepsilon -{\tilde{u}}_\lambda \Vert =\varepsilon \) and \(J_\lambda (u_\varepsilon )=J_\lambda ({\tilde{u}}_\lambda ).\)

Let us assume that (i) holds, since otherwise case (ii) implies the existence of a second solution.

Consider now the functional \(I_\lambda : H_0^{1} (\Omega ) \rightarrow {\mathbb {R}}\) given by \(I_\lambda [v]=J_\lambda [u_\lambda +v]-J_\lambda [u_\lambda ]\), more specifically

$$\begin{aligned} I_\lambda [v]&:=\frac{1}{2} \int _{\Omega }\, \vert \nabla v\vert ^2\, dx-\frac{\lambda }{2} \int _{\Omega }\, (v^+)^2 \, dx -\int _{\Omega }\, {\tilde{G}}_\lambda (x,v^+) \, dx. \end{aligned}$$

where

$$\begin{aligned} {\tilde{G}}_\lambda (x,s)&:= a(x)\big [F(u_\lambda (x)+s)-F(u_\lambda (x))-f(u_\lambda (x))s\big ]\\&\ = a(x)\left[ \frac{1}{2}f'(u_\lambda (x))s^2+o(s^2)\right] . \end{aligned}$$

Obviously \(I_\lambda [v^+]\le I_\lambda [v],\) and observe that \(I_\lambda '[v]=0\iff J_\lambda '[u_\lambda +v]=0.\)

Fix now some element \(0\le v_0\in H_{0}^{1}(\Omega )\cap L^\infty (\Omega )\), \( v_0> 0\) in \(\Omega ^+\), \( v_0\equiv 0\) in \(\Omega ^-\). Let \(v=tv_0\) for a certain \(t=t_0>0\) to be selected a posteriori, and evaluate

$$\begin{aligned} I_\lambda [tv_0]=\frac{1}{2}t^2\,\left( \Vert \nabla v_0\Vert _{L^2(\Omega )}^2-\lambda \,\Vert v_0\Vert _{L^2(\Omega )}^2\right) -\int _{\Omega }\, {\tilde{G}}_\lambda (x,tv_0) \, \, dx. \end{aligned}$$

Reasoning as in the proof of Theorem 3.3 for large positive t, since \(F(t)/t^2\rightarrow \infty \) as \(t\rightarrow \infty ,\) and using also (3.1) we obtain that

$$\begin{aligned} I_\lambda [tv_0]&\le C(1+t+t^2) -\int _{\Omega ^+} a^+(x) \Big [F(u_\lambda +tv_0)+\frac{1}{2}C_0(u_\lambda +tv_0)^2\Big ]\nonumber \\&\le C(1+t+t^2) -\delta \int _{{\widetilde{\Omega }}_{\delta }^+}\Big [F(u_\lambda +tv_0)+\frac{1}{2}C_0(u_\lambda +tv_0)^2\Big ]\, dx, \end{aligned}$$

so

$$\begin{aligned} I_\lambda [tv_0]<0 \end{aligned}$$

for \(t=t_0\) big enough, and where \({\widetilde{\Omega }}_{\delta }^+\) is defined by (3.18). Thus, the Mountain Pass Theorem implies that if

$$\begin{aligned} \Gamma :=\{ {\mathbf {g}}\in C\big ([0,1];H_{0}^{1}(\Omega )\big ) : \,{\mathbf {g}} (0)=0,\ I_\lambda [{\mathbf {g}}(1)]<0\}, \end{aligned}$$

then, there exists \(c>0\) such that

$$\begin{aligned} c:=\inf _{{\mathbf {g}}\in \Gamma } \max _{0\le t\le 1} I_\lambda [{\mathbf {g}}(t)] \end{aligned}$$

is a critical value of \(I_\lambda \), and thereby \({\mathscr {K}}_c:=\{v\in H_{0}^{1}(\Omega ) : \, I_\lambda [v]=c,\ I_\lambda '[v]=0\}\) is non empty.

Since for any \({\mathbf {g}}\in \Gamma \) we have \(I_\lambda [{\mathbf {g}}^+(t)] \le I_\lambda [{\mathbf {g}}(t)]\) for all \(t\in [0,1]\), it follows that \({\mathbf {g}}^+\in \Gamma \), and we derive the existence of a sequence \(v_n\) such that

$$\begin{aligned} I_\lambda [v_n]\rightarrow c,\qquad \Vert I_\lambda '[v_n]\Vert \rightarrow 0,\qquad v_n\ge 0. \end{aligned}$$

On the other hand, \(w_n:=u_\lambda +v_n\) is a (PS) sequence for the original functional \(J_\lambda .\) Since Theorem 3.2, if \(\lambda <\lambda _1( \mathrm{int}\,\Omega ^0),\) \(v_n\rightarrow v_\lambda \) en \(H_0^1(\Omega ),\) so \(I_\lambda '[v]=0\) and \(I_\lambda [v]=c>0,\) hence \(v_\lambda \ge 0\) is a nontrivial critical point of \(I_\lambda \). Consequently, \(w_\lambda :=u_\lambda +v_\lambda \) is a positive critical point of \(J_\lambda ,\) such that, for each \(\psi \in H_0^{1}(\Omega )\), we have

$$\begin{aligned} \int _{\Omega }\nabla w_\lambda \cdot \nabla \psi \,dx= \int _{\Omega }\Big (\lambda w_\lambda +a(x)f(w_\lambda )\Big ) \psi \, dx , \end{aligned}$$

and thereby \(w_\lambda :=u_\lambda +v_\lambda \ge u_\lambda \), \(w_\lambda \ne u_\lambda \) is a second positive solution to (1.1)\(_\lambda \).

(b) Step 2. Existence of a classical positive solution for \(\lambda =\Lambda \).

We prove the existence of a solution for \(\lambda =\Lambda \). For each \(\lambda \in (\lambda _1,\Lambda )\), problem (1.1) admits a minimal positive weak solution \(u_\lambda \) and \(\lambda \rightarrow u_\lambda \) is increasing, see Proposition 5.1. Taking the monotone pointwise limit, let us define

$$\begin{aligned} u_{\, \Lambda }(x):=\lim _{\lambda \uparrow \Lambda } u_{\, {\lambda }}(x). \end{aligned}$$

We next see that \(\Vert u_{\, \Lambda }\Vert <+\infty \), reasoning on the contrary. Assume that there exists a sequence of solutions \(u_n:=u_{\, {\lambda _n}}\) such that \(\Vert u_{\, \lambda _n}\Vert \rightarrow +\infty \) as \(\lambda _n\rightarrow \Lambda \). Set \(v_n:=u_n/\Vert u_{n}\Vert ,\) then there exists a subsequence, again denoted by \(v_n\) such that \(v_n \rightharpoonup v\) in \(H_0^1(\Omega )\), and \(v_n \rightarrow v\) in \(L^p(\Omega )\) for any \(p<2^*\) and a.e. Arguing as in the claim of Proposition 3.1, \(v\equiv 0\). Moreover

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _\Omega a(x)\frac{f(u_n)}{\Vert u_n\Vert }\, v_n\, dx =1. \end{aligned}$$
(5.4)

On the other hand, from the weak formulation, for all \(\psi \in C_c^\infty (\Omega ),\)

$$\begin{aligned} \int _{\Omega } \nabla v_{n} \cdot \nabla \psi \, dx = \lambda _n\int _{\Omega }v_{n}\psi \, dx +\int _{\Omega }a(x)\frac{f(u_n)}{\Vert u_n\Vert }\psi \, dx. \end{aligned}$$
(5.5)

Taking the limit, and since \(C_c^\infty (\Omega )\) is dense in \( L^2(\Omega )\)

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\Omega }a(x)\frac{f(u_n)}{\Vert u_n\Vert }\ \psi \, dx=0,\qquad \text{ for } \text{ all }\quad \psi \in L^2(\Omega ). \end{aligned}$$
(5.6)

Since Lemma 2.1, \(u\in C^{2}(\Omega )\cap C^{1,\mu }({\overline{\Omega }})\) and so \(a(x)\frac{f(u_n)}{\Vert u_n\Vert } \in L^2 (\Omega )\). Moreover \(v_n \rightarrow v=0\) in \(L^2(\Omega )\). Hence [2, Proposition 3.13 (iv)], and (5.6) imply

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _\Omega a(x)\frac{f(u_n)}{\Vert u_n\Vert }\, v_n\, dx =0, \end{aligned}$$

which contradicts (5.4) and yields \(\Vert u_{\, \Lambda }\Vert <+\infty \).

By Sobolev embedding and the Lebesgue dominated convergence theorem, \(u_{n}\rightarrow u_{\, \Lambda }\) in \(L^{2^*}(\Omega )\).

Now, by substituting \(\psi =u_n\) in (5.5), using Hölder inequality and Sobolev embeddings we obtain

$$\begin{aligned} \Big [\Vert u_n\Vert \le \Lambda \Vert v_n\Vert _{L^2(\Omega )}\Vert u_n\Vert + C,\quad \text {with } \Vert v_n\Vert _{L^2(\Omega )}\rightarrow 0\Big ]\Rightarrow \Vert u_n\Vert \le C. \end{aligned}$$

By compactness, for a subsequence again denoted by \(u_n\), \(u_n \rightharpoonup u^*\) in \(H_0^1(\Omega )\), \(u_n \rightarrow u^*\) in \(L^p(\Omega )\) for any \(p<2^*\) and a.e. By uniqueness of the limit, \(u_\Lambda =u^*.\) Finally, by taking limits in the weak formulation of \(u_{n}\) as \(\lambda _n \rightarrow \Lambda \), we get

$$\begin{aligned} \int _{\Omega } \nabla u_{\, \Lambda } \cdot \nabla \psi = \Lambda \int _{\Omega }u_{\, \Lambda }\psi +\int _{\Omega }a(x)f(u_{\, \Lambda })\psi \,. \end{aligned}$$

Hence \(u_{\, \Lambda }\) is a positive weak solution to (1.1)\(_{\Lambda }\). Lemma 2.1 yields that \(u_\Lambda \in C^{2}(\Omega )\cap C^{1,\mu }({\overline{\Omega }})\) is a classical solution.

(c) Step 3. Existence of a classical positive solution for \(\lambda \le \lambda _1\).

The existence of a classical positive solution for \(\lambda <\lambda _1\) is done in Theorem 3.3. Let’s look for a solution when \(\lambda =\lambda _1.\)

Since step 1, for any \(\lambda \in (\lambda _1,\Lambda )\) there exists a second positive solution to (1.1)\(_\lambda \). Let’s denote it by \({\tilde{u}}_\lambda \ne u_\lambda .\) Now, define the pointwise limit

$$\begin{aligned} {\tilde{u}}_{\, \lambda _1}(x):=\limsup _{\lambda \rightarrow \lambda _1}\ {\tilde{u}}_{\, {\lambda }}(x). \end{aligned}$$
(5.7)

Reasoning as in step 2, \(\Vert {\tilde{u}}_{\, \lambda _1}\Vert <+\infty \) and \({\tilde{u}}_{\, \lambda _1}\in C^{2}(\Omega )\cap C^{1,\mu }({\overline{\Omega }})\) is a classical solution to (1.1)\(_{\lambda _1}\).

Moreover, \({\tilde{u}}_{\, \lambda _1}> 0.\) Assume on the contrary that \({\tilde{u}}_{\, \lambda _1}= 0.\) By the Crandall-Rabinowitz’s Theorem [10], the only nontrivial solutions to (1.1) in a neighbourhood of the bifurcation point \((\lambda _1,0)\) are given by (4.1)). Since Proposition 5.1, those are the minimal solutions \(u_\lambda \), and due to \({\tilde{u}}_\lambda \ne u_\lambda ,\) \({\tilde{u}}_\lambda \) are not in a neighbourhood of \((\lambda _1,0)\), contradicting the definition of \({\tilde{u}}_{\, \lambda _1}(x)\), (5.7)

Hence, \({\tilde{u}}_{\, \lambda _1}\ge 0\), and reasoning as in (1.8), the Maximum Principle implies that \({\tilde{u}}_{\, \lambda _1}> 0.\) \(\square \)