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The study of multiple solutions for quasilinear
elliptic problems under Dirichlet or nonlinear Neumann type
boundary conditions has received much attention over the last
decades. The main goal of this paper is to present multiple
solutions results for elliptic inclusions of Clarke's gradient
type under Dirichlet boundary condition involving the

Let

The second class of

The existence of multiple solutions for semilinear and quasilinear elliptic problems has been studied by a number of authors, for example, [

The aim of this section is to provide an existence result of multiple solutions for all values of the parameter

Let

For all

There are constants

For all

In

Let us provide an example where all the assumptions formulated in

For the sake of simplicity we drop the

The operator

Let the data

On the contrary there would exist a constant

We denote by

Assume

Fix a constant

For a fixed

On the other hand, by hypothesis we can find

The following result which asserts the existence of two solutions of problem (

Assume

Since the proof of the existence of the greatest negative solution follows the same lines, we only provide the arguments for the existence of the least positive solution.

Applying Lemma

We thus obtain that for every positive integer

By (

We claim that

In view of (

The main result of this section is as follows.

Under hypotheses

Let

We observe that if

Since the function

Since

As in the case of (

Thus, the proof reduces to consider the cases

The next step in the proof is to show that

At this point we apply the second deformation lemma (see, e.g., [

The goal of this section is to show that under hypotheses stronger than those in Theorem

The new hypotheses on the nonlinearity

For all

There are constants

For all

There exist constants

For every

We notice that hypotheses

We state now the main result of this section, which produces two sign-changing solutions for problem (

Assume that hypotheses

Since hypotheses

On the other hand, hypotheses

In fact, under hypotheses

The main goal of this section is to provide a detailed multiplicity analysis of the nonsmooth elliptic problem (

The set

Let

Fučik Spectrum.

We impose the following hypotheses on the nonlinearity

There exists

One has

One has

In view of assumptions (g1) and (g2) the function

Suppose

A function

Due to assumption (g3) we have

A function

Similarly, we define a supersolution as follows.

A function

Let

Let

In a similar way the following lemma on the existence of a negative subsolution can be proved.

Let

In the next lemma we demonstrate that small constant multiples of

Let

By (g3) there is a constant

Applying a recently obtained comparison result that holds for even more general elliptic inclusions (see [

Let hypotheses (g1)-(g2) be satisfied and assume the existence of a subsolution

Combining the results of Lemmas

Let hypotheses (g1)–(g4) be fulfilled. For every

Let

As

The proof of the existence of the greatest negative solution

Under hypotheses (g1)–(g4), Theorem

Let

A critical point

A critical point

A critical point

To prove (i) let

The following lemma provides a variational characterization of the extremal constant-sign solutions

Let

The functional

The functional

One easily verifies that

Theorem

Let hypotheses (g1)–(g4) and (H) be satisfied. Then problem (

Clearly the existence of the extremal positive and negative solution

Let us consider the case

Therefore, it remains to prove the existence of sign-changing solutions under the assumptions that the global minimizer

The multiplicity results and the existence of sign-changing solutions obtained in this section generalize very recent results due to the authors obtained in [

Theorem

Multiplicity results for a nonsmooth version of problem (

Multiplicity and sign-changing solutions results have been obtained recently in [