## Existence and multiplicity results for elliptic problems with critical growth and discontinuous nonlinearities.(English)Zbl 0877.35046

The paper provides existence and multiplicity results for the elliptic problem $-\Delta u=u^{p-1}+bh(u-a),\;u>0\quad\text{in }\Omega,\;u=0\quad\text{on} \partial\Omega\tag{1}$ where $$a,b>0$$, $$\Omega$$ is a bounded subset of $$\mathbb{R}^N$$ and $$h$$ is the Heaviside function $$h(s)=0$$, if $$s<0$$ and $$h(s)=1$$, if $$s\geq 0$$. The authors also study carefully the elliptic problem of the form (1) with nonmonotone nonlinearity, i.e. when the function $$h$$ is replaced by a lower jump function $$g$$ such that $$g(s)=1$$, if $$s\leq 0$$ and $$g(s)=0$$, if $$s>0$$. It is shown that for fixed positive $$b$$ and $$N\geq 5$$, problem (1) admits a weak solution $$u_a$$ in $$W^{2,q}(\Omega)\cap H^1_0(\Omega)$$ for every $$q\in[1,+\infty)$$ and every $$a>0$$, satisfying $$|u_a|_\infty>a$$. If $$N=3,4$$, the existence of such a solution to (1) is established only for $$a>0$$ small. As far as the multiplicity of solutions is considered, it is proved that for $$b\in(0,b^*]$$, with a suitable $$b^*>0$$, there exists $$a^*=a^*(b)>0$$ such that for every $$a\in(0,a^*)$$ problem (1) admits, in the same class, three ordered weak solutions in $$\Omega$$.
The approach used in this paper combines the sub- and super-solution method with the variational technique developed by Chang for a class of locally Lipschitz functionals.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations
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### References:

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