On a Dirichlet problem with $(p,q)$-Laplacian and parametric concave-convex nonlinearity

TL;DR

This paper investigates a Dirichlet problem involving a combined $(p,q)$-Laplacian operator with parametric nonlinearities, establishing bifurcation results and analyzing how the smallest positive solution varies with the parameter.

Contribution

It provides new bifurcation analysis and studies the monotonicity and continuity of the smallest positive solutions in a $(p,q)$-Laplacian problem with parametric nonlinearities.

Findings

Bifurcation-type results describing solution set changes with parameter.

Monotonicity of the smallest positive solution map.

Continuity of the smallest positive solution with respect to the parameter.

Abstract

A homogeneous Dirichlet problem with $(p,q)$-Laplace differential operator and reaction given by a parametric $p$-convex term plus a $q$-concave one is investigated. A bifurcation-type result, describing changes in the set of positive solutions as the parameter $\lambda>0$ varies, is proven. Since for every admissible $\lambda$ the problem has a smallest positive solution $\bar u_{\lambda}$, both monotonicity and continuity of the map $ \lambda \mapsto \bar u_{\lambda}$ are studied.

Equations

$${\mathrm{P}}_{\lambda }$$

$${\Delta }_{r}u:=\mathrm{div}(|\nabla u{|}^{r-2}\nabla u)\forall u\in W_0^{1,r}(\Omega ).$$

$$\xi \mapsto \lambda {\xi }^{s-1}+{\xi }^{r-1},\xi \ge 0,$$

$$B_{\delta }(x):=\{z\in X:\Vert z-x\Vert <\delta \},B_{\delta }:=B_{\delta }(0).$$

$$x_n\rightharpoonup x\text{mbox}inX,\underset{n\to +\infty }{\mathrm{limsup}}⟨A(x_n),x_n-x⟩\le 0⟹x_n\to x.$$

$$x_n\rightharpoonup x\text{mbox}inX⟹\Phi (x)\le \underset{n\to \infty }{\mathrm{liminf}}\Phi (x_n).$$

$$K(\Phi ):=\{x\in X:{\Phi }^{\prime }(x)=0\}.$$

$$[u,v]:=\{w\in Y:u\le w\le v\},[u):=\{w\in Y:u\le w\}.$$

$$r^{\ast }:=\{\begin{array}{ll}\frac{Nr}{N-r}& \text{when r<N},\\ +\infty & \text{otherwise}.\end{array}$$

$$\Vert u{\Vert }_r:={\left({\int }_{\Omega }|u{|}^{r}dx\right)}^{1/r}\forall u\in L^r(\Omega ),\Vert u{\Vert }_{1,r}:={\left({\int }_{\Omega }|\nabla u{|}^{r}dx\right)}^{1/r}\forall u\in W_0^{1,r}(\Omega ),$$

$$C_{+}:=\{u\in C_0^1(\overline{\Omega }):u(x)\ge 0\text{ in Ω}\}$$

$$\mathrm{int}(C_{+})=\left\{u\in C_{+}:u(x)>0\forall x\in \Omega ,\frac{\partial u}{\partial n}(x)<0\forall x\in \partial \Omega \right\}.$$

$$⟨A_r(u),v⟩:={\int }_{\Omega }|\nabla u{|}^{r-2}\nabla u\cdot \nabla vdx,u,v\in W_0^{1,r}(\Omega ).$$

$$|g(x,t)|\le a(x)\left(1+|t{|}^{s-1}\right)\text{in}\Omega \times \mathbb{R},$$

$$\phi (u):=\frac{1}{p}\Vert \nabla u{\Vert }_p^p+\frac{1}{q}\Vert \nabla u{\Vert }_q^q-{\int }_{\Omega }G(x,u(x))dx,u\in W_0^{1,p}(\Omega ).$$

$$h_1\prec h_2⟺\underset{K}{\mathrm{ess}\text{leavevmode}\backslash nobreak\; inf}(h_2-h_1)>0\text{for any nonempty compact set}K\subseteq \Omega .$$

$$-{\Delta }_p{u}_i-{\Delta }_q{u}_i+a|u_i{|}^{p-2}{u}_i=h_i\text{in}\Omega ,i=1,2.$$

$$c_1{t}^{p-1}+c_2{t}^{q-1}\le f(x,t)\le c_0\left(t^{\theta -1}+t^{r-1}\right)\forall (x,t)\in \Omega \times {\mathbb{R}}_{+},$$

$$(r-p)\max \left\{N{p}^{-1},1\right\}<\beta <p^{\ast }.$$

$$0<\sigma F(x,\xi )\le f(x,\xi )\xi ,(x,\xi )\in \Omega \times [M,+\infty ).$$

$$f_1(t):=\{\begin{array}{ll}{t}^{p-1}+c_2{t}^{\tau -1}& \text{if }0\le t\le 1,\\ t^{r-1}+c_2{t}^{q-1}& \text{otherwise},\end{array}{f}_2(t):=t^{p-1}\log (1+t)+c_2{t}^{q-1},t\in {\mathbb{R}}_{+},$$

$$S_{\lambda }\subseteq \mathrm{int}(C_{+}).$$

$${\Psi }_{\lambda }(u):=\frac{1}{p}\Vert \nabla u{\Vert }_p^p+\frac{1}{q}\Vert \nabla u{\Vert }_q^q-{\int }_{\Omega }dx{\int }_0^{u(x)}{g}_{\lambda }(t)dt\forall u\in W_0^{1,p}(\Omega ),$$

$$g_{\lambda }(t):=(t^{+}{)}^{\tau -1}+\lambda c_0\left[(t^{+}{)}^{\theta -1}+(t^{+}{)}^{r-1}\right],t\in \mathbb{R}.$$

$$u\in \mathrm{int}(C_{+})⟹\underset{t\to +\infty }{\lim }{\Psi }_{\lambda }(tu)=-\infty$$

$$\Vert u{\Vert }_s\le c\Vert u{\Vert }_{p^{\ast }}\le C\Vert u\Vert \forall u\in X,$$

$$\begin{array}{rl}{\Psi }_{\lambda }(u)& \ge \frac{1}{p}\Vert u{\Vert }^p-c_4\Vert u{\Vert }^{\tau }-\lambda c_5\left[\Vert u{\Vert }^{\theta }+\Vert u{\Vert }^r\right]\\ & =\left[\frac{1}{p}-c_4\Vert u{\Vert }^{\tau -p}-\lambda c_5\left(\Vert u{\Vert }^{\theta -p}+\Vert u{\Vert }^{r-p}\right)\right]\Vert u{\Vert }^p,u\in X.\end{array}$$

$${\gamma }_{\lambda }(t):=c_4{t}^{\tau -p}+\lambda c_5(t^{\theta -p}+t^{r-p}),{\hat{\gamma }}_{\lambda }(t):=(c_4+\lambda c_5)t^{\tau -p}+2\lambda c_5{t}^{r-p}.$$

$$0<{\gamma }_{\lambda }(t)\le {\hat{\gamma }}_{\lambda }(t)\text{in}(0,+\infty ).$$

$$t_0:=t_0(\lambda ):={\left[\frac{(c_4+\lambda c_5)(p-\tau )}{2\lambda c_5(r-p)}\right]}^{\frac{1}{r-\tau }}$$

Full text

1. Introduction

Let $\Omega$ be a bounded domain in $\mathbb{R}^{N}$ with a $C^{2}$-boundary $\partial\Omega$, let $1<\tau<q<p<+\infty$, and let $f:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carathéodory function. Consider the Dirichlet problem

[TABLE]

where $\lambda>0$ is a parameter while $\Delta_{r}$, $r>1$, denotes the $r$-Laplacian, namely

[TABLE]

The nonhomogeneous differential operator $Au:=\Delta_{p}u+\Delta_{q}u$ that drives (${\rm P}_{\lambda}$) is usually called $(p,q)$-Laplacian. It stems from a wide range of important applications, including models of elementary particles [8], biophysics [9], plasma physics [26], reaction-diffusion equations [7], elasticity theory [27], etc. That’s why the relevant literature looks daily increasing and numerous meaningful works on this subject are by now available; see the survey paper [19] for a larger bibliography.

Since $\tau<q<p$, the function $\xi\mapsto\xi^{\tau-1}$ grows $(q-1)$-sublinearly at $+\infty$, whereas $\xi\mapsto f(x,\xi)$ is assumed to be $(p-1)$-superlinear near $+\infty$, although it need not satisfy the usual (in such cases) Ambrosetti-Rabinowitz condition. So, the reaction in (${\rm P}_{\lambda}$) exhibits the competing effects of concave and convex terms, with the latter multiplied by a positive parameter.

The aim of this paper is to investigate how the solution set of (${\rm P}_{\lambda}$) changes as $\lambda$ varies. In particular, we prove that there exists a critical parameter value $\lambda^{*}>0$ for which problem (${\rm P}_{\lambda}$) admits

• •

at least two solutions if $\lambda\in(0,\lambda^{*})$,

• •

at least one solution when $\lambda=\lambda^{*}$, and

• •

no solution provided $\lambda>\lambda^{*}$.

Moreover, we detect a smallest positive solution $\bar{u}_{\lambda}$ for each $\lambda\in(0,\lambda^{*}]$ and show that the map $\lambda\mapsto\bar{u}_{\lambda}$ turns out left-continuous, besides increasing.

The first bifurcation result for semilinear Dirichlet problems driven by the Laplace operator was established, more than twenty years ago, in the seminal paper[2] and then extended to the $p$-Laplacian in [11, 16]. These works treat the reaction

[TABLE]

where $1<s<p<r<p^{*}$, $\lambda>0$, and $p^{*}$ denotes the critical Sobolev exponent. A wider class of nonlinearities has recently been investigated in [22], while [24] deals with Robin boundary conditions. It should be noted that, unlike our case, $\lambda$ always multiplies the concave term, which changes the analysis of the problem. Finally, [4, 14, 23] contain analogous bifurcation theorems for problems of a different kind, whereas [20, 21] study $(p,q)$-Laplace equations having merely concave right-hand side.

Our approach is based on the critical point theory, combined with appropriate truncation and comparison techniques.

2. Mathematical background and hypotheses

Let $(X,\|\cdot\|)$ be a real Banach space. Given a set $V\subseteq X$, write $\overline{V}$ for the closure of $V$, $\partial V$ for the boundary of $V$, and ${\rm int}_{X}(V)$ or simply ${\rm int}(V)$, when no confusion can arise, for the interior of $V$. If $x\in X$ and $\delta>0$ then

[TABLE]

The symbol $(X^{*},\|\cdot\|_{X^{*}})$ denotes the dual space of $X$, $\langle\cdot\,,\cdot\rangle$ indicates the duality pairing between $X$ and $X^{*}$, while $x_{n}\to x$ (respectively, $x_{n}\rightharpoonup x$) in $X$ means ‘the sequence $\{x_{n}\}$ converges strongly (respectively, weakly) in $X$’. We say that $A:X\to X^{*}$ is of type $({\rm S})_{+}$ provided

[TABLE]

The function $\Phi:X\to\mathbb{R}$ is called coercive if $\displaystyle{\lim_{\|x\|\to+\infty}}\Phi(x)=+\infty$ and weakly sequentially lower semicontinuous when

[TABLE]

Suppose $\Phi\in C^{1}(X)$. We denote by $K(\Phi)$ the critical set of $\Phi$, i.e.,

[TABLE]

The classical Cerami compactness condition for $\Phi$ reads as follows:

• $({\rm C})$

Every $\{x_{n}\}\subseteq X$ such that $\{\Phi(x_{n})\}$ is bounded and $(1+\|x_{n}\|)\Phi^{\prime}(x_{n})\to 0$ in $X^{*}$ has a convergent subsequence.

From now on, $\Omega$ indicates a fixed bounded domain in $\mathbb{R}^{N}$ with a $C^{2}$-boundary $\partial\Omega$. Let $u,v:\Omega\to\mathbb{R}$ be measurable and let $t\in\mathbb{R}$. The symbol $u\leq v$ means $u(x)\leq v(x)$ for almost every $x\in\Omega$, $t^{\pm}:=\max\{\pm t,0\}$, $u^{\pm}(\cdot):=u(\cdot)^{\pm}$. If $u,v$ belong to a function space, say $Y$, then we set

[TABLE]

The conjugate exponent $r^{\prime}$ of a number $r\geq 1$ is defined by $r^{\prime}:=r/(r-1)$, while $r^{*}$ indicates its Sobolev conjugate, namely

[TABLE]

As usual,

[TABLE]

and $W^{-1,r^{\prime}}(\Omega)$ denotes the dual space of $W^{1,r}_{0}(\Omega)$. We will also employ the linear space $C^{1}_{0}(\overline{\Omega}):=\{u\in C^{1}(\overline{\Omega}):u\lfloor_{\partial\Omega}=0\}$, which is complete with respect to the standard $C^{1}(\overline{\Omega})$-norm. Its positive cone

[TABLE]

has a nonempty interior given by

[TABLE]

Here $n(x)$ denotes the outward unit normal to $\partial\Omega$ at $x$.

Let $A_{r}:W^{1,r}_{0}(\Omega)\to W^{-1,r^{\prime}}(\Omega)$ be the nonlinear operator stemming from the negative $r$-Laplacian, i.e.,

[TABLE]

We know [12, Section 6.2] that $A_{r}$ is bounded, continuous, strictly monotone, and of type $({\rm S})_{+}$. The Liusternik-Schnirelmann theory gives an increasing sequence $\{\lambda_{n,r}\}$ of eigenvalues for $A_{r}$. The following assertions can be found in [12, Section 6.2].

• $({\rm p}_{1})$

$\lambda_{1,r}$ is positive, isolated, and simple.

• $({\rm p}_{2})$

$\|u\|_{r}^{r}\leq\displaystyle{\frac{1}{\lambda_{1,r}}}\|u\|_{1,r}^{r}$ for all $u\in W^{1,r}_{0}(\Omega)$.

• $({\rm p}_{3})$

$\lambda_{1,r}$ admits an eigenfunction $\phi_{1,r}\in{\rm int}(C_{+})$ such that $\|\phi_{1,r}\|_{r}=1$.

Proposition 13 of [6] then ensures that

• $({\rm p}_{4})$

If $r\neq\hat{r}$ then $\phi_{1,r}$ and $\phi_{1,\hat{r}}$ are linearly independent.

Let $g:\Omega\times\mathbb{R}\to\mathbb{R}$ be a Carathéodory function satisfying the growth condition

[TABLE]

where $a\in L^{\infty}(\mathbb{R})$, $1<s\leq p^{*}$. Set $G(x,\xi):=\int_{0}^{\xi}g(x,t)\,dt$ and consider the $C^{1}$-functional $\varphi:W^{1,p}_{0}(\Omega)\to\mathbb{R}$ defined by

[TABLE]

Proposition 2.1** ([13], Proposition 2.6).**

If $u_{0}\in W^{1,p}_{0}(\Omega)$ is a local $C^{1}_{0}(\overline{\Omega})$-minimizer of $\varphi$ then $u_{0}\in C^{1,\alpha}(\overline{\Omega})$ for some $\alpha\in(0,1)$ and $u_{0}$ turns out to be a local $W^{1,p}_{0}(\Omega)$-minimizer of $\varphi$.

Combining this result with the strong comparison principle below, essentially due to Arcoya-Ruiz [3], shows that certain constrained minimizers actually are ‘global’ critical points. Recall that, given $h_{1},h_{2}\in L^{\infty}(\Omega),$

[TABLE]

Proposition 2.2**.**

Let $a\in\mathbb{R}_{+}$, $h_{1},h_{2}\in L^{\infty}(\Omega)$, $u_{1}\in C^{1}_{0}(\overline{\Omega})$, $u_{2}\in{\rm int}(C_{+})$. Suppose $h_{1}\prec h_{2}$ as well as

[TABLE]

Then, $u_{2}-u_{1}\in{\rm int}(C_{+})$.

Throughout the paper, ‘for every $x\in\Omega$’ will take the place of ‘for almost every $x\in\Omega$’, $c_{0},c_{1},\ldots$ indicate suitable positive constants, $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Carathéodory function such that $f(\cdot\,,t)=0$ provided $t\leq 0$, while $F(x,\xi):=\int_{0}^{\xi}f(x,t)\,dt$.

The following hypotheses will be posited.

• $({\rm h}_{1})$

There exist $\theta\in[\tau,q]$ and $r\in(p,p^{*})$ such that

[TABLE]

where $c_{2}>\lambda_{1,q}$.

• $({\rm h}_{2})$

$\lim\limits_{\xi\to+\infty}\frac{F(x,\xi)}{\xi^{p}}=+\infty$ uniformly with respect to $x\in\Omega$.

• $({\rm h}_{3})$

$\liminf\limits_{\xi\to+\infty}\frac{f(x,\xi)\xi-pF(x,\xi)}{\xi^{\beta}}\geq c_{3}$ uniformly in $x\in\Omega$. Here, $\beta>\tau$ and

[TABLE]

• $({\rm h}_{4})$

To every $\rho>0$ there corresponds $\mu_{\rho}>0$ such that $t\mapsto f(x,t)+\mu_{\rho}t^{p-1}$ is nondecreasing in $[0,\rho]$ for any $x\in\Omega$.

By $({\rm h}_{2})$–$({\rm h}_{3})$ the perturbation $f(x,\cdot)$ is $(p-1)$-superlinear at $+\infty$. In the literature, one usually treats this case via the well-known Ambrosetti-Rabinowitz condition, namely:

• (AR)

With appropriate $M>0$, $\sigma>p$ one has both $\operatorname*{ess\leavevmode\nobreak\ inf}\limits_{\Omega}F(\cdot\,,M)>0$ and

[TABLE]

It easily entails $c_{3}\xi^{\sigma}\leq F(x,\xi)$ in $\Omega\times[M,+\infty)$, which forces $({\rm h}_{2})$. However, nonlinearities having a growth rate ‘slower’ than $t^{\sigma-1}$ at $+\infty$ are excluded from (2.1). Thus, assumption $({\rm h}_{3})$ incorporates in our framework more situations.

Example 2.3**.**

Let $c_{2}>\lambda_{1,q}$. The functions $f_{1},f_{2}:\mathbb{R}_{+}\to\mathbb{R}$ defined by

[TABLE]

satisfy $({\rm h}_{1})$–$({\rm h}_{4})$. Nevertheless, $f_{1}$ alone complies with condition (AR).

3. A bifurcation-type theorem

Write $S_{\lambda}$ for the set of positive solutions to (${\rm P}_{\lambda}$). Lieberman’s nonlinear regularity theory [18, p. 320] and Pucci-Serrin’s maximum principle [25, pp. 111,120] yield

[TABLE]

Put $\mathcal{L}:=\{\lambda>0:S_{\lambda}\neq\emptyset\}$. Our first goal is to establish some basic properties of $\mathcal{L}$. From now on, $X:=W^{1,p}_{0}(\Omega)$ and $\|\cdot\|:=\|\cdot\|_{1,p}$.

Proposition 3.1**.**

Under $({\rm h}_{1})$ one has $\mathcal{L}\neq\emptyset$.

Proof.

Given $\lambda>0$, consider the $C^{1}$-functional $\Psi_{\lambda}:W^{1,p}_{0}(\Omega)\to\mathbb{R}$ defined by

[TABLE]

where

[TABLE]

Evidently, $g_{\lambda}$ fulfills (2.1) once $\sigma\in(p,r)$ and $M>0$ is big enough. So, condition (C) holds true for $\Psi_{\lambda}$. Moreover,

[TABLE]

because $r>p$. Observe next that if $s\in[1,p^{*}]$ then

[TABLE]

with $C:=C(s,\Omega)$. This easily leads to

[TABLE]

Let us set, for any $t>0$,

[TABLE]

From $\tau\leq\theta<p<r$ it follows $\lambda c_{5}t^{\theta-p}\leq\lambda c_{5}\left(t^{\tau-p}+t^{r-p}\right)$, which implies

[TABLE]

Since $\lim\limits_{t\to 0^{+}}\hat{\gamma}_{\lambda}(t)=\lim\limits_{t\to+\infty}\hat{\gamma}_{\lambda}(t)=+\infty$, there exists $t_{0}>0$ satisfying $\hat{\gamma}_{\lambda}^{\prime}(t_{0})=0$. One has

[TABLE]

and, via simple calculations, $\lim\limits_{\lambda\to 0^{+}}\hat{\gamma}_{\lambda}(t_{0})=0$. On account of (3.1)–(3.2) we can thus find $\lambda_{0}>0$ such that

[TABLE]

Pick $\lambda\in(0,\lambda_{0})$. The mountain pass theorem entails $\Psi_{\lambda}^{\prime}(\bar{u}_{\lambda})=0$ and $\Psi_{\lambda}(\bar{u}_{\lambda})\geq m_{\lambda}$ with appropriate $\bar{u}_{\lambda}\in X$. Hence,

[TABLE]

and $\bar{u}_{\lambda}\neq 0$. Choosing $v:=-\bar{u}_{\lambda}^{-}$ in (3.3) yields $\|\nabla\bar{u}_{\lambda}^{-}\|_{p}^{p}+\|\nabla\bar{u}_{\lambda}^{-}\|_{q}^{q}=0$, namely $\bar{u}_{\lambda}^{-}=0$. This forces $\bar{u}_{\lambda}\geq 0$ while, by (3.3) again,

[TABLE]

Lieberman’s nonlinear regularity theory and Pucci-Serrin’s maximum principle finally lead to $\bar{u}_{\lambda}\in{\rm int}(C_{+})$. Now define, provided $(x,\xi)\in\Omega\times\mathbb{R}$,

[TABLE]

An easy verification ensures that the associated $C^{1}$-functional

[TABLE]

is coercive and weakly sequentially lower semicontinuous. So, it attains its infimum at some point $u_{\lambda}\in X$. Assumption $({\rm h}_{1})$ produces

[TABLE]

i.e., $u_{\lambda}\neq 0$, because $\tau<q<p$. As before, from

[TABLE]

we infer $u_{\lambda}\geq 0$. Test (3.4) with $v:=(u_{\lambda}-\bar{u}_{\lambda})^{+}$, exploit $({\rm h}_{1})$ again, and recall (3.3) to arrive at

[TABLE]

which entails $u_{\lambda}\leq\bar{u}_{\lambda}$ by monotonicity. Summing up, $u_{\lambda}\in[0,\bar{u}_{\lambda}]\setminus\{0\}$. On account of (3.4), one thus has $u_{\lambda}\in S_{\lambda}$ for any $\lambda\in(0,\lambda_{0})$. This completes the proof. ∎

Our next result ensures that $\mathcal{L}$ is an interval.

Proposition 3.2**.**

Let $({\rm h}_{1})$ be satisfied. If $\hat{\lambda}\in\mathcal{L}$ then $(0,\hat{\lambda})\subseteq\mathcal{L}$.

Proof.

Pick $\hat{u}\in S_{\hat{\lambda}}$, $\lambda\in(0,\hat{\lambda})$, and define, provided $(x,\xi)\in\Omega\times\mathbb{R}$,

[TABLE]

The associated energy functional

[TABLE]

turns out coercive, weakly sequentially lower semicontinuous, besides $C^{1}$. Now, arguing exactly as above yields the conclusion. ∎

A careful reading of this proof allows one to state the next ‘monotonicity’ property.

Corollary 3.3**.**

Under hypothesis $({\rm h}_{1})$, for every $\hat{\lambda}\in\mathcal{L}$, $u_{\hat{\lambda}}\in S_{\hat{\lambda}}$, and $\lambda\in(0,\hat{\lambda})$ there exists $u_{\lambda}\in S_{\lambda}$ such that $u_{\lambda}\leq u_{\hat{\lambda}}$.

Actually, we can prove a more precise assertion.

Proposition 3.4**.**

Suppose $({\rm h}_{1})$ and $({\rm h}_{4})$ hold. Then to each $\hat{\lambda}\in\mathcal{L}$, $u_{\hat{\lambda}}\in S_{\hat{\lambda}}$, $\lambda\in(0,\hat{\lambda})$ there corresponds $u_{\lambda}\in S_{\lambda}$ fulfilling $u_{\hat{\lambda}}-u_{\lambda}\in{\rm int}(C_{+})$.

Proof.

Write $\rho:=\|u_{\hat{\lambda}}\|_{\infty}$. If $\mu_{\rho}$ is given by $({\rm h}_{4})$ while $u_{\lambda}$ comes from Corollary 3.3 then

[TABLE]

because $u_{\lambda}\leq u_{\hat{\lambda}}$ and $f(x,t)\geq 0$ once $t\geq 0$. The function $h(x):=(\hat{\lambda}-\lambda)f(x,u_{\hat{\lambda}}(x))$ lies in $L^{\infty}(\Omega)$. Indeed, on account of $({\rm h}_{1})$, we have

[TABLE]

Pick any compact set $K\subseteq\Omega$. Recalling that $u_{\hat{\lambda}}\in{\rm int}(C_{+})$ and using $({\rm h}_{1})$ again gives

[TABLE]

whence $0\prec h$. Now, (3.5) combined with Proposition 2.2 entail $u_{\hat{\lambda}}-u_{\lambda}\in{\rm int}(C_{+})$. ∎

The interval $\mathcal{L}$ turns out to be bounded.

Proposition 3.5**.**

Let $({\rm h}_{1})$ and $({\rm h}_{4})$ be satisfied. If $\lambda^{*}:=\sup\mathcal{L}$ then $\lambda^{*}<\infty$.

Proof.

Fix $\lambda\in\mathcal{L}$, $u_{\lambda}\in S_{\lambda}$. Note that we can suppose $\lambda>1$, otherwise $\mathcal{L}$ would be bounded, which of course entails $\lambda^{*}<\infty$. Define

[TABLE]

for every $(x,\xi)\in\Omega\times\mathbb{R}$, as well as

[TABLE]

The same arguments employed before yield here a global minimum point, say $\bar{u}_{\lambda}$, to $\Psi_{\lambda}$. So, in particular,

[TABLE]

Choosing $v:=-\bar{u}_{\lambda}^{-}$ first and then $v:=(\bar{u}_{\lambda}-u_{\lambda})^{+}$ we obtain $\bar{u}_{\lambda}\in[0,u_{\lambda}]$; cf. the proof of Proposition 3.1. Since, by $({\rm p}_{3})$ in Section 2, $u_{\lambda},\phi_{1,q}\in{\rm int}(C_{+})$, through [22, Proposition 1] one has $t\phi_{1,q}\leq u_{\lambda}$, with $t>0$ small enough. Thus, on account of $({\rm p}_{3})$ again,

[TABLE]

Now, recall that $q<p$ and decrease $t$ when necessary to achieve

[TABLE]

i.e., $\bar{u}_{\lambda}\neq 0$. Summing up, $\bar{u}_{\lambda}\in[0,u_{\lambda}]\setminus\{0\}$, whence, by (3.6), it turns out a positive solution of the equation

[TABLE]

Due to [5, Theorem 2.4], this prevents $\lambda$ from being arbitrary large, as desired. ∎

Le us finally prove that $\mathcal{L}=(0,\lambda^{*}]$. From now on, $\Phi_{\lambda}:X\to\mathbb{R}$ will denote the $C^{1}$-energy functional associated with problem (${\rm P}_{\lambda}$). Evidently,

[TABLE]

Proposition 3.6**.**

Under $({\rm h}_{1})$, $({\rm h}_{3})$, and $({\rm h}_{4})$ one has $\lambda^{*}\in\mathcal{L}$.

Proof.

Pick any $\{\lambda_{n}\}\subseteq(0,\lambda^{*})$ fulfilling $\lambda_{n}\uparrow\lambda^{*}$. Via Corollary 3.3, construct a sequence $\{u_{n}\}\subseteq X$ such that $u_{n}\in S_{\lambda_{n}}$, $u_{n}\leq u_{n+1}$. Then

[TABLE]

We can also assume $\Phi_{\lambda}(u_{n})<0$ (see the proof of Proposition 3.1), which means

[TABLE]

Testing (3.8) with $v:=u_{n}$ gives

[TABLE]

Since $q<p$ while $\lambda_{1}\leq\lambda_{n}$, from (3.9)–(3.10) it follows

[TABLE]

Observe next that, thanks to $({\rm h}_{1})$ and $({\rm h}_{3})$, one has

[TABLE]

Consequently, (3.11) becomes

[TABLE]

because $\tau<\beta$. This clearly forces

[TABLE]

If $r\leq\beta$ then $\{u_{n}\}$ turns out also bounded in $L^{r}(\Omega)$. Using (3.10) besides $({\rm h}_{1})$ entails

[TABLE]

whence $\{u_{n}\}\subseteq X$ is bounded. Suppose now $\beta<r<p^{*}$. Two cases may occur.

1. $p<N$. Let $t\in(0,1)$ satisfy

[TABLE]

The interpolation inequality [12, p. 905] yields $\|u_{n}\|_{r}\leq\|u_{n}\|_{\beta}^{1-t}\|u_{n}\|_{p^{*}}^{t}$. Via (3.12) we thus obtain

[TABLE]

Reasoning exactly as before and exploiting (3.15) produces

[TABLE]

Finally, note that $tr<p$. Indeed, $(r-p)\frac{N}{p}<\beta$ due to $({\rm h}_{3})$, while

[TABLE]

cf. (3.14). Now, the boundedness of $\{u_{n}\}\subseteq X$ directly stems from (3.16).

2. $p\geq N$, which implies $p^{*}=+\infty$. We will repeat the previous argument with $p^{*}$ replaced by any $\sigma>r$. Accordingly, if $t\in(0,1)$ fulfills $\frac{1}{r}=\frac{1-t}{\beta}+\frac{t}{\sigma}$ then $tr=\frac{\sigma(r-\beta)}{\sigma-\beta}$. Since, thanks to $({\rm h}_{3})$ again,

[TABLE]

one arrives at $tr<p$ for $\sigma$ large enough. This entails $\{u_{n}\}\subseteq X$ bounded once more.

Hence, in either case, we may assume

[TABLE]

where a subsequence is considered when necessary. Testing (3.8) with $v:=u_{n}-u^{*}$ thus yields, as $n\to+\infty$,

[TABLE]

whence, by monotonicity of $A_{q}$,

[TABLE]

On account of (3.17) it follows

[TABLE]

Recalling that $A_{p}$ enjoys the $({\rm S})_{+}$-property, we infer $u_{n}\to u^{*}$ in $X$, besides $0\leq u_{n}\leq u^{*}$ for all $n\in\mathbb{N}$. Finally, let $n\to+\infty$ in (3.8) to get

[TABLE]

i.e., $u^{*}\in S_{\lambda^{*}}$ and, a fortiori, $\lambda^{*}\in\mathcal{L}$. ∎

Some meaningful (bifurcation) properties of the set $S_{\lambda}$ will now be established.

Proposition 3.7**.**

Suppose $({\rm h}_{1})$–$({\rm h}_{4})$ hold true. Then, for every $\lambda\in(0,\lambda^{*})$, problem (${\rm P}_{\lambda}$) admits two solutions $u_{0},\hat{u}\in{\rm int}(C_{+})$ such that $u_{0}\leq\hat{u}$. Moreover, $u_{0}$ is a local minimizer of the associated energy functional $\Phi_{\lambda}$.

Proof.

Fix $\lambda\in(0,\lambda^{*})$ and choose $\eta\in(\lambda,\lambda^{*})$. By Proposition 3.2, there exists $u_{\eta}\in S_{\eta}$ while Proposition 3.4 provides $u_{0}\in S_{\lambda}$ satisfying

[TABLE]

The same reasoning adopted in the proof of Proposition 3.2 ensures here that $u_{0}$ is a global minimum point to the functional

[TABLE]

where $F_{\lambda,\eta}(x,\xi):=\int_{0}^{\xi}f_{\lambda,\eta}(x,t)\,dt$, with

[TABLE]

By (3.18), $u_{0}$ turns out a local $C^{1}_{0}(\overline{\Omega})$-minimizer of $\Phi_{\lambda}$, because $\Phi_{\lambda}\lfloor_{[0,u_{\eta}]}=\Phi_{\lambda,\eta}\lfloor_{[0,u_{\eta}]}$. Via Proposition 2.1 we then see that this remains valid with $C^{1}_{0}(\overline{\Omega})$ replaced by $X$. Set

[TABLE]

$(x,\xi)\in\Omega\times\mathbb{R}$, as well as

[TABLE]

From (3.19) and the nonlinear regularity theory it follows $u_{0}\in K(\Phi_{0})\subseteq[u_{0})\cap{\rm int}(C_{+})$. We may thus assume

[TABLE]

or else a second solution of (${\rm P}_{\lambda}$) bigger than $u_{0}$ would exist. Bearing in mind the proof of Proposition 3.6 and making small changes to accommodate the truncation at $u_{0}(x)$ shows that $\Phi_{0}$ satisfies condition (C). Let us next truncate $f_{0}(x,\cdot)$ at $u_{\eta}(x)$ to construct a new Carathéodory function $\tilde{f}$, with primitive $\tilde{F}$ and associated functional $\tilde{\Phi}$, defined like in (3.20) but replacing $F_{0}$ by $\tilde{F}$. Evidently,

[TABLE]

whence $K(\tilde{\Phi})=\{u_{0}\}$ because of (3.21). Since $\tilde{\Phi}$ is coercive and weakly sequentially lower semicontinuous, it possesses a global minimum point that must coincide with $u_{0}$. An easy verification gives $\Phi_{0}\lfloor_{[0,u_{\eta}]}=\tilde{\Phi}\lfloor_{[0,u_{\eta}]}$. So, thanks to (3.18), $u_{0}$ turns out a local $C_{0}^{1}(\overline{\Omega})$-minimizer of $\Phi_{0}$. This still holds when $X$ replaces $C^{1}_{0}(\overline{\Omega})$; cf. Proposition 2.1. We may suppose $K(\Phi_{0})$ finite, otherwise infinitely many solutions of (${\rm P}_{\lambda}$) bigger than $u_{0}$ do exist. Adapting the argument exploited in [1, Proposition 29] provides $\rho\in(0,1)$ such that

[TABLE]

Finally, if $u\in{\rm int}(C_{+})$ then simple calculations based on $({\rm h}_{2})$ entail $\Phi_{0}(tu)\to-\infty$ as $t\to+\infty$. Therefore, the mountain pass theorem can be applied, and there is $\hat{u}\in X$ fulfilling

[TABLE]

Via (3.22)–(3.23) one has $u_{0}\neq\hat{u}$ while the inclusion $K(\Phi_{0})\subseteq[u_{0})\cap{\rm int}(C_{+})$ forces $u_{0}\leq\hat{u}$, which ends the proof. ∎

Proposition 3.8**.**

Under $({\rm h}_{1})$–$({\rm h}_{4})$, the solution set $S_{\lambda}$ admits a smallest element $\bar{u}_{\lambda}$ for every $\lambda\in\mathcal{L}$.

Proof.

A standard procedure ensures that $S_{\lambda}$ turns out downward directed; see, e.g., [10, Section 4]. Lemma 3.10 at p. 178 of [17] yields

[TABLE]

for some decreasing sequence $\{u_{n}\}\subseteq S_{\lambda}$. Consequently, $0\leq u_{n}\leq u_{1}$ and

[TABLE]

Due to $({\rm h}_{1})$, testing (3.25) with $v:=u_{n}$ we thus obtain

[TABLE]

namely $\{u_{n}\}\subseteq X$ is bounded. Like before (cf. the proof of Proposition 3.6), this gives $u_{n}\to\bar{u}_{\lambda}$ in $X$, where a subsequence is considered if necessary. So, from (3.25) it easily follows

[TABLE]

Showing that $\bar{u}_{\lambda}\neq 0$ will entail $\bar{u}_{\lambda}\in S_{\lambda}$, whence the conclusion by (3.24). To the aim, consider the problem

[TABLE]

Its energy functional

[TABLE]

turns out coercive and weakly sequentially lower semicontinuous. Hence, there exists $\tilde{u}\in X$ satisfying $\Phi_{0}(\tilde{u})=\inf_{X}\Phi_{0}$. One has $u_{0}\neq 0$, because $\Phi_{0}(\tilde{u})<0=\Phi_{0}(0)$ (the argument is like in the proof of Proposition 3.5). Further, $\Phi_{0}^{\prime}(\tilde{u})=0$, i.e.,

[TABLE]

Choosing $v:=-\tilde{u}^{-}$ we see that $u$ is a positive solution to (3.26). Actually, $\tilde{u}\in{\rm int}(C_{+})$ and, through a standard procedure [15, Lemma 3.1], $\tilde{u}$ turns out unique.

Claim: $\tilde{u}\leq u$ for all $u\in S_{\lambda}$.

Indeed, fixed any $u\in S_{\lambda}$, define

[TABLE]

where

[TABLE]

The following assertions can be easily verified.

• •

$\Psi(u^{*})=\inf_{X}\Psi$, with appropriate $u^{*}\in X$.

• •

$\Psi(u^{*})<0=\Psi(0)$, whence $u^{*}\neq 0$.

• •

$u^{*}\in K(\Psi)\subseteq[0,u]\cap C_{+}$.

Therefore, $u^{*}$ is a positive solution of (3.26). By uniqueness, this implies $u^{*}=\tilde{u}$. Thus, a fortiori, $\tilde{u}\leq u$.

The claim brings $\tilde{u}\leq u_{n}$, $n\in\mathbb{N}$, which in turn provides $0<\tilde{u}\leq\bar{u}_{\lambda}$, as desired. ∎

Let us finally come to some meaningful properties of the map

[TABLE]

Proposition 3.9**.**

Suppose $({\rm h}_{1})$–$({\rm h}_{4})$ hold true. Then the function $k$ is both

• $({\rm i}_{1})$

strictly increasing, namely $\bar{u}_{\lambda_{2}}-\bar{u}_{\lambda_{1}}\in{\rm int}(C_{+})$ if $\lambda_{1}<\lambda_{2}$, and

• $({\rm i}_{2})$

left-continuous.

Proof.

Pick $\lambda_{1},\lambda_{2}\in\mathcal{L}$ such that $\lambda_{1}<\lambda_{2}$. Since $\bar{u}_{\lambda_{2}}\in S_{\lambda_{2}}$, Proposition 3.4 yields $u_{\lambda_{1}}\in S_{\lambda_{1}}$ fulfilling $\bar{u}_{\lambda_{2}}-u_{\lambda_{1}}\in{\rm int}(C_{+})$, while Proposition 3.8 entails $\bar{u}_{\lambda_{1}}\leq u_{\lambda_{1}}$. Hence, $\bar{u}_{\lambda_{2}}-\bar{u}_{\lambda_{1}}\in{\rm int}(C_{+})$. This shows $({\rm i}_{1})$.

If $\lambda_{n}\to\lambda^{-}$ in $\mathcal{L}$ then, by $({\rm i}_{1})$, the sequence $\{\bar{u}_{\lambda_{n}}\}$ turns out increasing. Its boundedness in $X$ immediately stems from $({\rm h}_{1})$; see the previous proof. Now, repeat the argument below (3.17) to arrive at

[TABLE]

whence $\tilde{u}_{\lambda}\in S_{\lambda}\subseteq{\rm int}(C_{+})$. We finally claim that $\tilde{u}_{\lambda}=\bar{u}_{\lambda}$. Assume on the contrary

[TABLE]

Lieberman’s nonlinear regularity theory gives $\{\bar{u}_{n}\}\subseteq C^{1,\alpha}_{0}(\overline{\Omega})$ as well as

[TABLE]

Since the embedding $C^{1,\alpha}_{0}(\overline{\Omega})\hookrightarrow C^{1}_{0}(\overline{\Omega})$ is compact, (3.27) becomes

[TABLE]

Because of (3.28), this implies $\bar{u}_{\lambda}(x_{0})<\bar{u}_{\lambda_{n}}(x_{0})$ for any $n$ large enough, against $({\rm i}_{1})$. Consequently, $\tilde{u}_{\lambda}=\bar{u}_{\lambda}$, and $({\rm i}_{2})$ follows from (3.27). ∎

Gathering Propositions 3.1–3.9 together we obtain the following

Theorem 3.10**.**

Let $({\rm h}_{1})$–$({\rm h}_{4})$ be satisfied. Then, there exists $\lambda^{*}>0$ such that problem (${\rm P}_{\lambda}$) admits

• $({\rm j}_{1})$

at least two solutions $u_{0},\hat{u}\in{\rm int}(C_{+})$, with $u_{0}\leq\hat{u}$, for every $\lambda\in(0,\lambda^{*})$,

• $({\rm j}_{2})$

at least one solution $u^{*}\in{\rm int}(C_{+})$ when $\lambda=\lambda^{*}$,

• $({\rm j}_{3})$

no positive solutions for all $\lambda>\lambda^{*}$,

• $({\rm j}_{4})$

a smallest positive solution $\bar{u}_{\lambda}\in{\rm int}(C_{+})$ provided $\lambda\in(0,\lambda^{*}]$.

Moreover, the map $\lambda\in(0,\lambda^{*}]\mapsto\bar{u}_{\lambda}\in C_{0}^{1}(\overline{\Omega})$ is strictly increasing and left-continuous.

Acknowledgment. This work is performed within the 2016–2018 Research Plan - Intervention Line 2: ‘Variational Methods and Differential Equations’, and partially supported by GNAMPA of INDAM.