### On a class of semilinear elliptic equations with boundary conditions and potentials which change sign.

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Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ $div(x,\nabla u)+{a\left(x\right)\left|u\right|}^{p-2}u={g\left(x\right)\left|u\right|}^{p-2}u+h\left(x\right){\left|u\right|}^{s-1}u$ in ${\mathbb{R}}^{N}$ ⎨ ⎩ u > 0, $li{m}_{\left|x\right|\to \infty}u\left(x\right)=0$, where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

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