In this paper, we deal with the existence, localization and multiplicity of positive radial solutions to the Dirichlet problem involving $\varphi$-Laplacian operators:

where $ℬ$ is the unit open ball in ${ℝ}^n$ $\left(n\ge 3\right)$ centered at the origin, the function $f:[0,1]\times {ℝ}_{+}\to {ℝ}_{+}$ is continuous and $\psi :(-a,a)\to ℝ$ is $C^1,$ such that $\varphi \left(s\right):=s\psi \left(s\right)$ is an increasing homeomorphism between two intervals $(-a,a)$ and $(-b,b)$

The following particular cases are of much interest due to their corresponding models arising from physics:

(a) $\varphi :ℝ\to ℝ,\varphi \left(s\right)={\left|s\right|}^{p-2}s,$ where $p>1$(here $a=b=+\mathrm{\infty }$), when the left side $Lv$ in (1.1) is

involved in a nonlinear Darcy law for flows through porous media [3];

(b) (singular homeomorphism) $\varphi :(-a,a)\to ℝ,\varphi \left(s\right)=\frac{s}{\sqrt{a^2-s^2}}$ (here and $b=+\mathrm{\infty }$), when

involved in the relativistic mechanics [2, 10];

(c) (bounded homeomorphism) $\varphi :ℝ\to (-b,b),\varphi \left(s\right)=\frac{bs}{\sqrt{1+s^2}}$ (here $a=+\mathrm{\infty }$ and ), when

associated to capillarity problems [1, 14].

Looking for radial solutions of (1.1), that is, functions of the form $v(x)=u(r)$ with $r=\left|x\right|$, the Dirichlet problem (1.1) reduces to the mixed boundary value problem

Radial and nonradial solutions for the Dirichlet problem involving $\varphi$-Laplace operators have been intensively investigated in the literature, both by means of topological and variational methods. We refer the interested reader to the papers [4, 5, 6, 7, 8, 9, 11, 12, 15] and the references therein.

Our approach here is based on fixed point index theory, namely on compression-expansion type homotopy arguments. The most known are those from Krasnosel’skiĭ’s compression-expansion theorem in a conical annulus defined by using the max-norm of the space. Applications to one-dimensional $\varphi$-Laplace equations are given in [13]. In the radial case considered in the present paper, the absence of a Harnack type inequality in terms of the max-norm, makes Krasnosel’skiĭ’s theorem inoperative and forces us to use instead, some other homotopy conditions and properties of the fixed point index.

The first paper in radial solutions that uses the compression-expansion technique, but in a variational form and only for $p$-Laplacian equations, is [17]. As explained there, the difficulty in applying the compression-expansion method consists in the necessity that, for the considered differential operator, a Harnack type inequality be available. In the present paper, such a key inequality is established for problem (1.2) with a general homeomorphism $\varphi$ satisfying some additional conditions. With its help, a precise localization of positive solutions is possible, allowing in a natural way to obtain multiple solutions. The results apply in particular for the $p$-Laplacian and the Minkowski mean curvature operator.

Our basic assumptions are as follow:

(H_ϕ)

$\varphi :(-a,a)\to (-b,b)$ is an odd increasing homeomorphism such that

$$\lambda \varphi \left(x\right)\ge \varphi \left(\lambda x\right)\text{ for all }\lambda \in [0,1],x\in [0,a)\text{(}\varphi \text{ is convex on }[0,a)\text{).}$$

(1.3)

(H_f)

$f:[0,1]\times {ℝ}_{+}\to [0,b)$ is continuous, with $f(\cdot ,s)$ nonincreasing in $[0,1]$ for every $s\in {ℝ}_{+}$ and $f(r,\cdot )$ nondecreasing in ${ℝ}_{+}$ for every $r\in [0,1]$.

Note that the homeomorphisms related to the $p$-Laplacian for $p\ge 2$ and the Minkowski mean curvature operator both satisfy condition (H_ϕ). Contrarily, the bounded homeomorphisms with $a=+\mathrm{\infty },$ for example the one involved by the Euclidian mean curvature operator, are not convex on $[0,+\mathrm{\infty })$ and thus they do not satisfy our assumption (H_ϕ).

In the space of functions $u\in C^1[0,1]$ satisfying $u^{\prime }\left(0\right)=u\left(1\right)=0$ we consider the following norms:

Let $\varphi$ satisfy (H_ϕ) and denote

First we prove a Harnack type inequality for problem (1.2), given in terms of the norm $\parallel \cdot {\parallel }_p$, with $1\le p\le +\mathrm{\infty }$.

A similar result has been established in [17] for the particular case of the $p$-Laplacian with $p>n$, for which $\varphi (s)={\left|s\right|}^{p-2}s$ and $a=b=+\mathrm{\infty }$. More exactly, it has been proved that

where ${\Vert u\Vert }_{1,p}={\left({\int }_0^1{r}^{n-1}{\left|u^{\prime }\left(r\right)\right|}^p𝑑r\right)}^{\frac{1}{p}}.$ In this case, since by using Hölder’s inequality one has

a Harnack type inequality in terms of the max-norm ${\left|u\right|}_{\mathrm{\infty }}={\max }_{r\in [0,1]}\left|u\left(r\right)\right|$ can be immediately derived from (2.5), namely

It is an open problem to obtain an analogue result for more general homeomorphisms $\varphi$ satisfying (H_ϕ).   At this moment we are only able to establish such an inequality in terms of a max-seminorm on $C[0,1].$ For example, taking $p=+\mathrm{\infty }$ in (2.4), we have the following Harnack type inequality related to a seminorm on $C[0,1].$

In the sequel, inequality (2.6) is a key ingredient for the localization and multiplicity of positive radial solutions. We will use the main ideas in [16] in order to localize the solutions in terms of a norm and a semi-norm.

Recall that by a (nonnegative) solution of (1.2) we mean a function $u\in C^1([0,1],{ℝ}_{+})$ with $u^{\prime }\left(0\right)=u\left(1\right)=0,$ for all $r\in [0,1]$, such that $r^{n-1}\varphi (u^{\prime })\in C^1[0,1]$ and (1.2) is satisfied. We will say that a nonnegative solution is positive if it is distinct from the identically zero function. Let $X$ be the Banach space of continuous functions $X=C[0,1]$ and $K_0$ its positive cone $K_0=\{u\in X:u\ge 0\text{ on }[0,1]\}.$ It is not difficult to see that a nonnegative function $u$ is a solution of problem (1.2) if and only if $u$ is a fixed point of the operator $T:K_0\to K_0$ given by

As proved in [4, 6], the operator $T$ is completely continuous.

Let us now consider a subcone of $K_0$ related to the Harnack inequality (2.6), namely

where $c:=\left(1-\nu \right){\eta }^{2n-2}\left(n-2\right).$

Now, for any number $\alpha >0$ consider the set

The operator $T$ being completely continuous, the set $T\left({\overline{U}}_{\alpha }\right)$ is bounded, so there is a number $\tilde{\alpha }\ge \alpha$ such that $T\left({\overline{U}}_{\alpha }\right)\subset {\overline{U}}_{\tilde{\alpha }}.$ Define the operator $\tilde{T}:{\overline{U}}_{\tilde{\alpha }}\to {\overline{U}}_{\tilde{\alpha }}$ by

Next, for a number $\beta >0$ consider the set

It is clear that $V_{\beta }$ is open in ${\overline{U}}_{\tilde{\alpha }}.$

By using the previous fixed point index computations, we deduce the following existence result.

Now we give sufficient conditions in order to guarantee the assumptions of the previous lemmas hold.

We will use the following notation. If , denote

If $b=+\mathrm{\infty }$, denote

Obviously, if $\varphi$ is a classical homeomorphism ($a=b=+\mathrm{\infty }$), conditions (3.5) and (3.6) can be rewritten as

Hence, if we assume in addition that $\varphi$ satisfies:

then the existence of both positive numbers $\alpha$ and $\beta$ is guaranteed under suitable asymptotic conditions about $f$ at zero and at infinity.

Note that assumption (3.7) holds in the case of the $p$-Laplacian operator and so it is commonly employed in the literature, see for instance [6, 11].

We show the applicability of our theory with an example involving radial solutions of $p$-Laplacian equations.